Discrete mathematics Books

Book SynopsisA practical guide simplifying discrete math for curious minds and demonstrating its application in solving problems related to software development, computer algorithms, and data scienceKey Features Apply the math of countable objects to practical problems in computer science Explore modern Python libraries such as scikit-learn, NumPy, and SciPy for performing mathematics Learn complex statistical and mathematical concepts with the help of hands-on examples and expert guidance Book DescriptionDiscrete mathematics deals with studying countable, distinct elements, and its principles are widely used in building algorithms for computer science and data science. The knowledge of discrete math concepts will help you understand the algorithms, binary, and general mathematics that sit at the core of data-driven tasks. Practical Discrete Mathematics is a comprehensive introduction for those who are new to the mathematics of countable objects. This book will help you get up to speed with using discrete math principles to take your computer science skills to a more advanced level. As you learn the language of discrete mathematics, you'll also cover methods crucial to studying and describing computer science and machine learning objects and algorithms. The chapters that follow will guide you through how memory and CPUs work. In addition to this, you'll understand how to analyze data for useful patterns, before finally exploring how to apply math concepts in network routing, web searching, and data science. By the end of this book, you'll have a deeper understanding of discrete math and its applications in computer science, and be ready to work on real-world algorithm development and machine learning.What you will learn Understand the terminology and methods in discrete math and their usage in algorithms and data problems Use Boolean algebra in formal logic and elementary control structures Implement combinatorics to measure computational complexity and manage memory allocation Use random variables, calculate descriptive statistics, and find average-case computational complexity Solve graph problems involved in routing, pathfinding, and graph searches, such as depth-first search Perform ML tasks such as data visualization, regression, and dimensionality reduction Who this book is forThis book is for computer scientists looking to expand their knowledge of discrete math, the core topic of their field. University students looking to get hands-on with computer science, mathematics, statistics, engineering, or related disciplines will also find this book useful. Basic Python programming skills and knowledge of elementary real-number algebra are required to get started with this book.Table of ContentsTable of Contents Key Concepts, Notation, Set Theory, Relations, and Functions Formal Logic and Constructing Mathematical Proofs Computing with Base-n Numbers Combinatorics Using SciPy Elements of Discrete Probability Computational Algorithms in Linear Algebra Computational Requirements for Algorithms Storage and Feature Extraction of Graphs, Trees, and Networks Searching Data Structures and Finding Shortest Paths Regression Analysis with NumPy and Scikit-Learn Web Searches with PageRank Principal Component Analysis with Scikit-Learn

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Book SynopsisDrawing on many years'experience of teaching discrete mathem atics to students of all levels, Anderson introduces such as pects as enumeration, graph theory and configurations or arr angements. Starting with an introduction to counting and rel ated problems, he moves on to the basic ideas of graph theor y with particular emphasis on trees and planar graphs. He de scribes the inclusion-exclusion principle followed by partit ions of sets which in turn leads to a study of Stirling and Bell numbers. Then follows a treatment of Hamiltonian cycles, Eulerian circuits in graphs, and Latin squares as well as proof of Hall's theorem. He concludes with the constructions of schedules and a brief introduction to block designs. Each chapter is backed by a number of examples, with straightforw ard applications of ideas and more challenging problems.Trade Review"Wie der Titel andeutet, handelt es sich bei diesem Buch um eine elementare Einführung in Denkweisen und Methoden der Diskreten Mathematik. Die fachlichen Voraussetzungen an den Leser sind minimal. Darauf aufbauend wird ein doch recht buntes Bild entwickelt, bestehend vor allem aus den wichtigsten Konzepten aus Kombinatorik und Graphentheorie sowie einigen spezielleren Themen wie Designs und Codes.... Der Vorteil besteht darin, dass auch dem mathematischen Laien auf knapp 200 Seiten ein durchaus einprägsames Bild von einem Zweig der Mathematik vermittelt wird, der in unserer Zeit u.a. durch die Allgegenwart der sogenannten Informationstechnologie extrem an Bedeutung gewonnen hat."Internationale Mathematische Nachrichten, Nr. 187, August 2001Table of Contents1. Counting and Binomial Coefficients.- 2. Recurrence.- 3. Introduction to Graphs.- 4. Travelling Round a Graph.- 5. Partitions and Colourings.- 6. The Inclusion Exclusion Principle.- 7. Latin Squares and Hall’s Theorem.- 8. Schedules and 1-Factorisations.- 9. Introduction to Designs.- Solutions.- Further Reading.

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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 SynopsisThe goal of this monograph is to give an accessible introduction to nonstandard methods and their applications, with an emphasis on combinatorics and Ramsey theory. It includes both new nonstandard proofs of classical results and recent developments initially obtained in the nonstandard setting. This makes it the first combinatorics-focused account of nonstandard methods to be aimed at a general (graduate-level) mathematical audience. This book will provide a natural starting point for researchers interested in approaching the rapidly growing literature on combinatorial results obtained via nonstandard methods. The primary audience consists of graduate students and specialists in logic and combinatorics who wish to pursue research at the interface between these areas.Table of Contents- Part I Preliminaries. - Ultrafilters. - Nonstandard Analysis. - Hyperfinite Generators of Ultrafilters. - Many Stars: Iterated Nonstandard Extensions. - LoebMeasure. - Part II Ramsey Theory. - Ramsey’s Theorem. - The Theorems of van der Waerden and Hales-Jewett. - From Hindman to Gowers. - Partition Regularity of Equations. - Part III Combinatorial Number Theory. - Densities and Structural Properties. - Working in the Remote Realm. - Jin’s Sumset Theorem. - Sumset Configurations in Sets of Positive Density. - Near Arithmetic Progressions in Sparse Sets. - The Interval Measure Property. - Part IV Other Topics. - Triangle Removal and Szemerédi Regularity. - Approximate Groups. - Foundations of Nonstandard Analysis.

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Book SynopsisThis easy-to-understand textbook introduces the mathematical language and problem-solving tools essential to anyone wishing to enter the world of computer and information sciences. Specifically designed for the student who is intimidated by mathematics, the book offers a concise treatment in an engaging style.The thoroughly revised third edition features a new chapter on relevance-sensitivity in logical reasoning and many additional explanations on points that students find puzzling, including the rationale for various shorthand ways of speaking and ‘abuses of language’ that are convenient but can give rise to misunderstandings. Solutions are now also provided for all exercises.Topics and features: presents an intuitive approach, emphasizing how finite mathematics supplies a valuable language for thinking about computation; discusses sets and the mathematical objects built with them, such as relations and functions, as well as recursion and induction; introduces core topics of mathematics, including combinatorics and finite probability, along with the structures known as trees; examines propositional and quantificational logic, how to build complex proofs from simple ones, and how to ensure relevance in logic; addresses questions that students find puzzling but may have difficulty articulating, through entertaining conversations between Alice and the Mad Hatter; provides an extensive set of solved exercises throughout the text.This clearly-written textbook offers invaluable guidance to students beginning an undergraduate degree in computer science. The coverage is also suitable for courses on formal methods offered to those studying mathematics, philosophy, linguistics, economics, and political science. Assuming only minimal mathematical background, it is ideal for both the classroom and independent study.Table of ContentsPart I: Sets Collecting Things Together: Sets Comparing Things: Relations Associating One Item with Another: Functions Recycling Outputs as Inputs: Induction and Recursion Part II: Math Counting Things: Combinatorics Weighing the Odds: Probability Squirrel Math: Trees Part III: Logic Yea and Nay: Propositional Logic Something about Everything: Quantificational Logic Just Supposing: Proof and Consequence Sticking to the Point: Relevance in Logic

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Book SynopsisThis book provides a self-contained introduction to convex geometry in Euclidean space. After covering the basic concepts and results, it develops Brunn–Minkowski theory, with an exposition of mixed volumes, the Brunn–Minkowski inequality, and some of its consequences, including the isoperimetric inequality. Further central topics are then treated, such as surface area measures, projection functions, zonoids, and geometric valuations. Finally, an introduction to integral-geometric formulas in Euclidean space is provided. The numerous exercises and the supplementary material at the end of each section form an essential part of the book. Convexity is an elementary and natural concept. It plays a key role in many mathematical fields, including functional analysis, optimization, probability theory, and stochastic geometry. Paving the way to the more advanced and specialized literature, the material will be accessible to students in the third year and can be covered in one semester.Trade Review“‘Lectures on convex geometry’ is an excellent graduate book about convex geometry. … The book is very well-organized. … The presentation is clear, well-composed and illustrated. The problems at the end of each subchapter are carefully selected and revised. The whole text is readable, interesting and easy to learn from. … the book is excellent and it can serve the studies of the future generation students in convex geometry.” (Gergely Kiss, zbMATH 1487.52001, 2022)“The book is informative, very interesting, and mathematically accessible, and the authors have achieved the purpose that they outline above. As the title suggests, Lectures on convex geometry is well suited to be used as the prescribed textbook for graduate courses in convex geometry; this is because of its pedagogical style and the quality of the exercises. It will also be useful to students intending to pursue a research career in the area … .” (Daniel John Fresen, Mathematical Reviews, June, 2022)Table of ContentsPreface.- Preliminaries and Notation.- 1. Convex Sets.- 2. Convex Functions.- 3. Brunn-Minkowski Theory.- 4. From Area Measures to Valuations.- 5. Integral Geometric Formulas.-6. Solutions of Selected Exercises.- References.- Index.

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Book SynopsisThis book provides a self-contained introduction to convex geometry in Euclidean space. After covering the basic concepts and results, it develops Brunn–Minkowski theory, with an exposition of mixed volumes, the Brunn–Minkowski inequality, and some of its consequences, including the isoperimetric inequality. Further central topics are then treated, such as surface area measures, projection functions, zonoids, and geometric valuations. Finally, an introduction to integral-geometric formulas in Euclidean space is provided. The numerous exercises and the supplementary material at the end of each section form an essential part of the book. Convexity is an elementary and natural concept. It plays a key role in many mathematical fields, including functional analysis, optimization, probability theory, and stochastic geometry. Paving the way to the more advanced and specialized literature, the material will be accessible to students in the third year and can be covered in one semester.Trade Review“‘Lectures on convex geometry’ is an excellent graduate book about convex geometry. … The book is very well-organized. … The presentation is clear, well-composed and illustrated. The problems at the end of each subchapter are carefully selected and revised. The whole text is readable, interesting and easy to learn from. … the book is excellent and it can serve the studies of the future generation students in convex geometry.” (Gergely Kiss, zbMATH 1487.52001, 2022)“The book is informative, very interesting, and mathematically accessible, and the authors have achieved the purpose that they outline above. As the title suggests, Lectures on convex geometry is well suited to be used as the prescribed textbook for graduate courses in convex geometry; this is because of its pedagogical style and the quality of the exercises. It will also be useful to students intending to pursue a research career in the area … .” (Daniel John Fresen, Mathematical Reviews, June, 2022)Table of ContentsPreface.- Preliminaries and Notation.- 1. Convex Sets.- 2. Convex Functions.- 3. Brunn-Minkowski Theory.- 4. From Area Measures to Valuations.- 5. Integral Geometric Formulas.-6. Solutions of Selected Exercises.- References.- Index.

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Book SynopsisThis textbook can serve as a comprehensive manual of discrete mathematics and graph theory for non-Computer Science majors; as a reference and study aid for professionals and researchers who have not taken any discrete math course before. It can also be used as a reference book for a course on Discrete Mathematics in Computer Science or Mathematics curricula. The study of discrete mathematics is one of the first courses on curricula in various disciplines such as Computer Science, Mathematics and Engineering education practices. Graphs are key data structures used to represent networks, chemical structures, games etc. and are increasingly used more in various applications such as bioinformatics and the Internet. Graph theory has gone through an unprecedented growth in the last few decades both in terms of theory and implementations; hence it deserves a thorough treatment which is not adequately found in any other contemporary books on discrete mathematics, whereas about 40% of this textbook is devoted to graph theory. The text follows an algorithmic approach for discrete mathematics and graph problems where applicable, to reinforce learning and to show how to implement the concepts in real-world applications.Trade Review“This accessible reference book should be well received by undergraduate-level CS, engineering, and mathematics students.” (Soubhik Chakraborty, Computing Reviews, July 12, 2022)“The book under review is an elementary introduction to mathematical logic, set theory, discrete mathematics, number theory, probability theory and graph theory. Its undoubted advantage is its good algorithmic support. … I would recommend this book to students studying computer science at the bachelor’s level.” (I. M. Erusalimskiy, zbMATH 1477.68004, 2022)Table of ContentsPreface.- Part I: Fundamentals of Discrete Mathematics.- Logic.- Proofs.- Algorithms.- Set Theory.- Relations and Functions.- Sequences, Induction and Recursion.- Introduction to Number Theory.- Counting and Probability.- Boolean Algebra and Combinational Circuits.- Introduction to the Theory of Computation.- Part II: Graph Theory.- Introduction to Graphs.- Trees and Traversals.- Subgraphs.- Connectivity, Network Flows and Shortest Paths.- Graph Applications.- A:.- Pseudocode Conventions.- Index.

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Book SynopsisThis textbook introduces enumerative combinatorics through the framework of formal languages and bijections. By starting with elementary operations on words and languages, the authors paint an insightful, unified picture for readers entering the field. Numerous concrete examples and illustrative metaphors motivate the theory throughout, while the overall approach illuminates the important connections between discrete mathematics and theoretical computer science. Beginning with the basics of formal languages, the first chapter quickly establishes a common setting for modeling and counting classical combinatorial objects and constructing bijective proofs. From here, topics are modular and offer substantial flexibility when designing a course. Chapters on generating functions and partitions build further fundamental tools for enumeration and include applications such as a combinatorial proof of the Lagrange inversion formula. Connections to linear algebra emerge in chapters studying Cayley trees, determinantal formulas, and the combinatorics that lie behind the classical Cayley–Hamilton theorem. The remaining chapters range across the Inclusion-Exclusion Principle, graph theory and coloring, exponential structures, matching and distinct representatives, with each topic opening many doors to further study. Generous exercise sets complement all chapters, and miscellaneous sections explore additional applications. Lessons in Enumerative Combinatorics captures the authors' distinctive style and flair for introducing newcomers to combinatorics. The conversational yet rigorous presentation suits students in mathematics and computer science at the graduate, or advanced undergraduate level. Knowledge of single-variable calculus and the basics of discrete mathematics is assumed; familiarity with linear algebra will enhance the study of certain chapters.Trade Review“The wide variety of slightly unusual topics makes the book an excellent resource for the instructor who wants to craft a combinatorics course that contains a diverse collection of attractive results … . The attentive student will certainly come away from a course based on this book with a solid understanding of the combinatorial way of thinking. … the book is an excellent resource for anyone teaching a class in combinatorics.” (Timothy Y. Chow, Mathematical Reviews, March, 2023)“A whole book whose backbone is enumeration by codifying the objects to be enumerated as words. … They do this in a skillfully structured fashion which makes the connections natural and unforced. … One of the remarkable features of this book is the care the authors have taken to make it reader-friendly and accessible to a wide range of students following a graduate mathematics course or an honours undergraduate course in mathematics and computer science.” (Josef Lauri, zbMATH 1478.05001, 2022)Table of Contents1. Basic Combinatorial Structures.- 2. Partitions and Generating Functions.- 3. Planar Trees and the Lagrange Inversion Formula.- 4. Cayley Trees.- 5. The Cayley–Hamilton Theorem.- 6. Exponential Structures and Polynomial Operators.- 7. The Inclusion-Exclusion Principle.- 8. Graphs, Chromatic Polynomials and Acyclic Orientations.- 9. Matching and Distinct Representatives.

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This stimulating textbook presents a broad and accessible guide to the fundamentals of discrete mathematics, highlighting how the techniques may be applied to various exciting areas in computing. The text is designed to motivate and inspire the reader, encouraging further study in this important skill. Features: This book provides an introduction to the building blocks of discrete mathematics, including sets, relations and functions; describes the basics of number theory, the techniques of induction and recursion, and the applications of mathematical sequences, series, permutations, and combinations; presents the essentials of algebra; explains the fundamentals of automata theory, matrices, graph theory, cryptography, coding theory, language theory, and the concepts of computability and decidability; reviews the history of logic, discussing propositional and predicate logic, as well as advanced topics such as the nature of theorem proving; examines the field of software engineering, including software reliability and dependability and describes formal methods; investigates probability and statistics and presents an overview of operations research and financial mathematics.

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Book SynopsisPreface.- Notation.- I. Discrete Structures.- 1. Introduction.-2. Mathematical Arguments.-3. Sets.- 4. Proof by Induction.- 5. Equivalence Relations.- 6. Partial Orders and Lattices.- 7. Floor and Ceiling Functions.- 8. Number Theory.- II. Summation and Asymptotics.- 10. Asymptotic Analysis.- III. Combinatorics.- 11. Counting.- 12. Generating Functions.- 13. Recurrence Relations.- 14. Graphs.- 15. Probability.- Bibliography.- Index.

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Book SynopsisEvery year there is at least one combinatorics problem in each of the major international mathematical olympiads. These problems can only be solved with a very high level of wit and creativity. This book explains all the problem-solving techniques necessary to tackle these problems, with clear examples from recent contests. It also includes a large problem section for each topic, including hints and full solutions so that the reader can practice the material covered in the book.​ The material will be useful not only to participants in the olympiads and their coaches but also in university courses on combinatorics.Trade ReviewFrom the reviews: “Soberón (Univ. College London, UK) presents tools, techniques, and some tricks to tackle problems of varying difficulty in combinatorial mathematics in this well-written book. … Salient features include the wealth of examples, exercises, and problems and two additional chapters with hints and solutions to the problems. Valuable for all readers interested in combinatorics and useful as a course resource on the subject. Summing Up: Highly recommended. Upper-division undergraduate through professional mathematics collections.” (D. V. Chopra, Choice, Vol. 51 (4), December, 2013)Table of ContentsIntroduction.- 1 First concepts.- 2 The pigeonhole principle.- 3 Invariants.- 4 Graph theory.- 5 Functions.- 6 Generating Functions.- 7 Partitions.- 8 Hints for the problems.- 9 Solutions to the problems.- Notation.- Further reading.- Index. ​

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Book SynopsisCryptography has become essential as bank transactions, credit card infor-mation, contracts, and sensitive medical information are sent through inse-cure channels. This book is concerned with the mathematical, especially algebraic, aspects of cryptography. It grew out of many courses presented by the authors over the past twenty years at various universities and covers a wide range of topics in mathematical cryptography. It is primarily geared towards graduate students and advanced undergraduates in mathematics and computer science, but may also be of interest to researchers in the area. Besides the classical methods of symmetric and private key encryption, the book treats the mathematics of cryptographic protocols and several unique topics such as Group-Based Cryptography Gröbner Basis Methods in Cryptography Lattice-Based Cryptography

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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 many fundamental results in Geometry and Discrete Mathematics along with their proofs and their history. In the second edition we include a new chapter on Topological Data Analysis and enhanced the chapter on Graph Theory for solving further classical problems such as the Traveling Salesman Problem.

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Book SynopsisThis book provides an introduction to hypergraphs, its aim being to overcome the lack of recent manuscripts on this theory. In the literature hypergraphs have many other names such as set systems and families of sets. This work presents the theory of hypergraphs in its most original aspects, while also introducing and assessing the latest concepts on hypergraphs. The variety of topics, their originality and novelty are intended to help readers better understand the hypergraphs in all their diversity in order to perceive their value and power as mathematical tools. This book will be a great asset to upper-level undergraduate and graduate students in computer science and mathematics. It has been the subject of an annual Master's course for many years, making it also ideally suited to Master's students in computer science, mathematics, bioinformatics, engineering, chemistry, and many other fields. It will also benefit scientists, engineers and anyone else who wants to understand hypergraphs theory.Trade ReviewFrom the reviews:“This book addresses the mathematics and theory of hypergraphs. The target audience includes graduate students and researchers with an interest in math and computer science (CS). … I expect readers of this book will be motivated to advance this field, which in turn can advance other sciences.” (Hsun-Hsien Chang, Computing Reviews, January, 2014)“The aim of this book is to introduce the basic concepts of hypergraphs, to present the knowledge of the theory and applications of hypergraphs in other fields. … This book is useful for anyone who wants to understand the basics of hypergraph theory. It is mainly for math and computer science majors, but it may also be useful for other fields which use the theory. … appropriate for both researchers and graduate students. It is very well-written and proofs are stated in a clear manner.” (Somayeh Moradi, zbMATH, Vol. 1269, 2013)Table of ContentsHypergraphs: basic concepts.- Hypergraphs: first properties.- Hypergraph coloring.- Some particular hypergraphs.- Reduction-contraction of Hypergraph.- Dirhypergraphs: basic concepts.- Applications of hypergraph theory : a brief overview.

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Book SynopsisThis book provides an introduction to hypergraphs, its aim being to overcome the lack of recent manuscripts on this theory. In the literature hypergraphs have many other names such as set systems and families of sets. This work presents the theory of hypergraphs in its most original aspects, while also introducing and assessing the latest concepts on hypergraphs. The variety of topics, their originality and novelty are intended to help readers better understand the hypergraphs in all their diversity in order to perceive their value and power as mathematical tools. This book will be a great asset to upper-level undergraduate and graduate students in computer science and mathematics. It has been the subject of an annual Master's course for many years, making it also ideally suited to Master's students in computer science, mathematics, bioinformatics, engineering, chemistry, and many other fields. It will also benefit scientists, engineers and anyone else who wants to understand hypergraphs theory.Trade ReviewFrom the reviews:“This book addresses the mathematics and theory of hypergraphs. The target audience includes graduate students and researchers with an interest in math and computer science (CS). … I expect readers of this book will be motivated to advance this field, which in turn can advance other sciences.” (Hsun-Hsien Chang, Computing Reviews, January, 2014)“The aim of this book is to introduce the basic concepts of hypergraphs, to present the knowledge of the theory and applications of hypergraphs in other fields. … This book is useful for anyone who wants to understand the basics of hypergraph theory. It is mainly for math and computer science majors, but it may also be useful for other fields which use the theory. … appropriate for both researchers and graduate students. It is very well-written and proofs are stated in a clear manner.” (Somayeh Moradi, zbMATH, Vol. 1269, 2013)Table of ContentsHypergraphs: basic concepts.- Hypergraphs: first properties.- Hypergraph coloring.- Some particular hypergraphs.- Reduction-contraction of Hypergraph.- Dirhypergraphs: basic concepts.- Applications of hypergraph theory : a brief overview.

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Book SynopsisTransactions are a concept related to the logical database as seen from the perspective of database application programmers: a transaction is a sequence of database actions that is to be executed as an atomic unit of work. The processing of transactions on databases is a well- established area with many of its foundations having already been laid in the late 1970s and early 1980s.The unique feature of this textbook is that it bridges the gap between the theory of transactions on the logical database and the implementation of the related actions on the underlying physical database. The authors relate the logical database, which is composed of a dynamically changing set of data items with unique keys, and the underlying physical database with a set of fixed-size data and index pages on disk. Their treatment of transaction processing builds on the “do-redo-undo” recovery paradigm, and all methods and algorithms presented are carefully designed to be compatible with this paradigm as well as with write-ahead logging, steal-and-no-force buffering, and fine-grained concurrency control.Chapters 1 to 6 address the basics needed to fully appreciate transaction processing on a centralized database system within the context of our transaction model, covering topics like ACID properties, database integrity, buffering, rollbacks, isolation, and the interplay of logical locks and physical latches. Chapters 7 and 8 present advanced features including deadlock-free algorithms for reading, inserting and deleting tuples, while the remaining chapters cover additional advanced topics extending on the preceding foundational chapters, including multi-granular locking, bulk actions, versioning, distributed updates, and write-intensive transactions.This book is primarily intended as a text for advanced undergraduate or graduate courses on database management in general or transaction processing in particular.Table of Contents1 Transactions on the Logical Database.- 2 Operations on the Physical Database.- 3 Logging and Buffering.- 4 Transaction Rollback and Restart Recovery.- 5 Transactional Isolation.- 6 Lock-Based Concurrency Control.- 7 B-Tree Traversals.- 8 B-Tree Structure Modifications.- 9 Advanced Locking Protocols.- 10 Bulk Operations on B-Trees.- 11 Online Index Construction and Maintenance.- 12 Concurrency Control by Versioning.- 13 Distributed Transactions.- 14 Transactions in Page-Server Systems.- 15 Processing of Write-Intensive Transactions.

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Book SynopsisThis textbook treats two important and related matters in convex geometry: the quantification of symmetry of a convex set—measures of symmetry—and the degree to which convex sets that nearly minimize such measures of symmetry are themselves nearly symmetric—the phenomenon of stability. By gathering the subject’s core ideas and highlights around Grünbaum’s general notion of measure of symmetry, it paints a coherent picture of the subject, and guides the reader from the basics to the state-of-the-art. The exposition takes various paths to results in order to develop the reader’s grasp of the unity of ideas, while interspersed remarks enrich the material with a behind-the-scenes view of corollaries and logical connections, alternative proofs, and allied results from the literature. Numerous illustrations elucidate definitions and key constructions, and over 70 exercises—with hints and references for the more difficult ones—test and sharpen the reader’s comprehension.The presentation includes: a basic course covering foundational notions in convex geometry, the three pillars of the combinatorial theory (the theorems of Carathéodory, Radon, and Helly), critical sets and Minkowski measure, the Minkowski–Radon inequality, and, to illustrate the general theory, a study of convex bodies of constant width; two proofs of F. John’s ellipsoid theorem; a treatment of the stability of Minkowski measure, the Banach–Mazur metric, and Groemer’s stability estimate for the Brunn–Minkowski inequality; important specializations of Grünbaum’s abstract measure of symmetry, such as Winternitz measure, the Rogers–Shepard volume ratio, and Guo’s Lp -Minkowski measure; a construction by the author of a new sequence of measures of symmetry, the kth mean Minkowski measure; and lastly, an intriguing application to the moduli space of certain distinguished maps from a Riemannian homogeneous space to spheres—illustrating the broad mathematical relevance of the book’s subject.Trade Review“The book under review is a graduate-level textbook on convexity, which presents the topic from a new and interesting point of view. … The book offers the reader a new approach to the study of convexity, focusing on the important topics of measures of symmetry and stability. It moves from the very beginning background to recent research, and therefore both students and researchers can benefit from it.” (María A. Hernández Cifre, Mathematical Reviews, December, 2016) “This is a graduate-level textbook on convex geometry in finite-dimensional Euclidean spaces, which has some interesting special features. … Each chapter has illustrating figures and concludes with exercises … . The book has a surprising appendix, where certain of the symmetry measures are applied to convex bodies … . This book is an unconventional introduction to convexity, full of appealing intuitive geometry; it may equally well serve the beginner and the experienced researcher in the field.” (Rolf Schneider, zbMATH 1335.52002, 2016)Table of ContentsFirst Things First on Convex Sets.- Affine Diameters and the Critical Set.- Measures of Stability and Symmetry.- Mean Minkowski Measures.

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Book SynopsisThis clearly written textbook presents an accessible introduction to discrete mathematics for computer science students, offering the reader an enjoyable and stimulating path to improve their programming competence. The text empowers students to think critically, to be effective problem solvers, to integrate theory and practice, and to recognize the importance of abstraction. Its motivational and interactive style provokes a conversation with the reader through a questioning commentary, and supplies detailed walkthroughs of several algorithms.This updated and enhanced new edition also includes new material on directed graphs, and on drawing and coloring graphs, in addition to more than 100 new exercises (with solutions to selected exercises).Topics and features: assumes no prior mathematical knowledge, and discusses concepts in programming as and when they are needed; designed for both classroom use and self-study, presenting modular and self-contained chapters that follow ACM curriculum recommendations; describes mathematical processes in an algorithmic manner, often supported by a walkthrough demonstrating how the algorithm performs the desired task; includes an extensive set of exercises throughout the text, together with numerous examples, and shaded boxes highlighting key concepts; selects examples that demonstrate a practical use for the concept in question.Students embarking on the start of their studies of computer science will find this book to be an easy-to-understand and fun-to-read primer, ideal for use in a mathematics course taken concurrently with their first programming course.Table of ContentsAlgorithms, Numbers and MachinesSets, Sequences and CountingBoolean Expressions, Logic and ProofSearching and SortingGraphs and TreesRelations: Especially on (Integer) SequencesSequences and SeriesGenerating Sequences and SubsetsDiscrete Probability and Average Case ComplexityTuring Machines

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Book SynopsisIt is gratifying that this textbook is still sufficiently popular to warrant a third edition. I have used the opportunity to improve and enlarge the book. When the second edition was prepared, only two pages on algebraic geometry codes were added. These have now been removed and replaced by a relatively long chapter on this subject. Although it is still only an introduction, the chapter requires more mathematical background of the reader than the remainder of this book. One of the very interesting recent developments concerns binary codes defined by using codes over the alphabet 7l.4• There is so much interest in this area that a chapter on the essentials was added. Knowledge of this chapter will allow the reader to study recent literature on 7l. -codes. 4 Furthermore, some material has been added that appeared in my Springer Lec­ ture Notes 201, but was not included in earlier editions of this book, e. g. Generalized Reed-Solomon Codes and Generalized Reed-Muller Codes. In Chapter 2, a section on "Coding Gain" ( the engineer's justification for using error-correcting codes) was added. For the author, preparing this third edition was a most welcome return to mathematics after seven years of administration. For valuable discussions on the new material, I thank C.P.l.M.Baggen, I. M.Duursma, H.D.L.Hollmann, H. C. A. van Tilborg, and R. M. Wilson. A special word of thanks to R. A. Pellikaan for his assistance with Chapter 10.Table of Contents1 Mathematical Background.- 1.1. Algebra.- 1.2. Krawtchouk Polynomials.- 1.3. Combinatorial Theory.- 1.4. Probability Theory.- 2 Shannon’s Theorem.- 2.1. Introduction.- 2.2. Shannon’s Theorem.- 2.3. On Coding Gain.- 2.4. Comments.- 2.5. Problems.- 3 Linear Codes.- 3.1. Block Codes.- 3.2. Linear Codes.- 3.3. Hamming Codes.- 3.4. Majority Logic Decoding.- 3.5. Weight Enumerators.- 3.6. The Lee Metric.- 3.7. Comments.- 3.8. Problems.- 4 Some Good Codes.- 4.1. Hadamard Codes and Generalizations.- 4.2. The Binary Golay Code.- 4.3. The Ternary Golay Code.- 4.4. Constructing Codes from Other Codes.- 4.5. Reed—Muller Codes.- 4.6. Kerdock Codes.- 4.7. Comments.- 4.8. Problems.- 5 Bounds on Codes.- 5.1. Introduction: The Gilbert Bound.- 5.2. Upper Bounds.- 5.3. The Linear Programming Bound.- 5.4. Comments.- 5.5. Problems.- 6 Cyclic Codes.- 6.1. Definitions.- 6.2. Generator Matrix and Check Polynomial.- 6.3. Zeros of a Cyclic Code.- 6.4. The Idempotent of a Cyclic Code.- 6.5. Other Representations of Cyclic Codes.- 6.6. BCH Codes.- 6.7. Decoding BCH Codes.- 6.8. Reed—Solomon Codes.- 6.9. Quadratic Residue Codes.- 6.10. Binary Cyclic Codes of Length 2n(n odd).- 6.11. Generalized Reed—Muller Codes.- 6.12. Comments.- 6.13. Problems.- 7 Perfect Codes and Uniformly Packed Codes.- 7.1. Lloyd’s Theorem.- 7.2. The Characteristic Polynomial of a Code.- 7.3. Uniformly Packed Codes.- 7.4. Examples of Uniformly Packed Codes.- 7.5. Nonexistence Theorems.- 7.6. Comments.- 7.7. Problems.- 8 Codes over ?4.- 8.1. Quaternary Codes.- 8.2. Binary Codes Derived from Codes over ?4.- 8.3. Galois Rings over ?4.- 8.4. Cyclic Codes over ?4.- 8.5. Problems.- 9 Goppa Codes.- 9.1. Motivation.- 9.2. Goppa Codes.- 9.3. The Minimum Distance of Goppa Codes.- 9.4. Asymptotic Behaviour of Goppa Codes.- 9.5. Decoding Goppa Codes.- 9.6. Generalized BCH Codes.- 9.7. Comments.- 9.8. Problems.- 10 Algebraic Geometry Codes.- 10.1. Introduction.- 10.2. Algebraic Curves.- 10.3. Divisors.- 10.4. Differentials on a Curve.- 10.5. The Riemann—Roch Theorem.- 10.6. Codes from Algebraic Curves.- 10.7. Some Geometric Codes.- 10.8. Improvement of the Gilbert—Varshamov Bound.- 10.9. Comments.- 10.10.Problems.- 11 Asymptotically Good Algebraic Codes.- 11.1. A Simple Nonconstructive Example.- 11.2. Justesen Codes.- 11.3. Comments.- 11.4. Problems.- 12 Arithmetic Codes.- 12.1. AN Codes.- 12.2. The Arithmetic and Modular Weight.- 12.3. Mandelbaum—Barrows Codes.- 12.4. Comments.- 12.5. Problems.- 13 Convolutional Codes.- 13.1. Introduction.- 13.2. Decoding of Convolutional Codes.- 13.3. An Analog of the Gilbert Bound for Some Convolutional Codes.- 13.4. Construction of Convolutional Codes from Cyclic Block Codes.- 13.5. Automorphisms of Convolutional Codes.- 13.6. Comments.- 13.7. Problems.- Hints and Solutions to Problems.- References.

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Book SynopsisBoth classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces, convex sets, etc., crucial ideas and above all abstract concepts needed for attaining the results are elucidated. These are conceptual notions, each built "above" the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs: the 'ladder' in the Old Testament, that angels ascended and descended... In all this, the aim of the book is to demonstrate to readers the unceasingly renewed spirit of geometry and that even so-called "elementary" geometry is very much alive and at the very heart of the work of numerous contemporary mathematicians. It is also shown that there are innumerable paths yet to be explored and concepts to be created. The book is visually rich and inviting, so that readers may open it at random places and find much pleasure throughout according their own intuitions and inclinations. Marcel Berger is the author of numerous successful books on geometry, this book once again is addressed to all students and teachers of mathematics with an affinity for geometry.Trade Review“It is a must own book for anyone serious about developing a conceptual understanding of the interconnected web of modern geometry and the ever-growing intertwining of geometry with practically all other branches of mathematics. … It is remarkable for a book to provide such a detailed glimpse of contemporary geometry via well developed discussions of so many questions of current interest. It provides the most extensive exposition of geometric thinking I’ve ever seen in a book at this level.” (William H. Barker, MAA Reviews, August, 2017)“Geometry Revealed is to give the reader a feel for the conceptual frameworks of modern geometry, attempting to reach as far as possible with a minimum of assumed knowledge and formal scaffolding. … Geometry Revealed being useful for research mathematicians as a still reasonably up-to-date survey. … Geometry Revealed offered an ascent into the wonders of a new world.” (Danny Yee, Danny Yee’s Book Reviews, dannyreviews.com, July, 2015)“By considering a hierarchy of ‘natural’ geometrical objects … it sets out to investigate significant geometrical problems which are either unsolved or were solved only recently. … it is undoubtedly a major tour de force, and if you really want to gain an idea of where geometry is going in the 21st century, you will find plenty of exquisite material here.” (Gerry Leversha, The Mathematical Gazette, Vol. 96 (356), July, 2012)“The book contains twelve chapters, each of them is a collection of such problems about geometric objects with more and more complexity … . The chapters are independent from each other, any of them can serve as a course. Researchers in geometry can use it as a source for further research. … the book is accessible to a wide audience of people who are interested in geometry.” (János Kincses, Acta Scientiarum Mathematicarum (Szeged), Vol. 78 (1-2), 2012)“‘Geometry Revealed’ is a massive text of 831 pages which is organized in twelve chapters and which additionally provides indices for names, subjects and symbols … throughout the author quite carefully lays out the historical perspective. … a typical chapter starts with an observation or a problem in elementary geometry. Large parts of the text are very accessible, and a reader who likes (mathematical) physics will often get something extra.” (Michael Joswig, Zentralblatt MATH, Vol. 1232, 2012)“The author provides the reader with an enormous amount of detailed information and thus yields deep insight into the various topics. … All in all an overwhelming book which is a must … for everyone having sufficient mathematical knowledge.” (G. Kowol, Monatshefte für Mathematik, Vol. 164 (2), October, 2011)“The book is a very readable account of several branches of geometry, classical and modern, elementary and advanced. … Every chapter is extremely interesting and alive. … The book is rich in ideas, written in an informal style, with no formulae and no unnecessary technical details. … Every part of this book is interesting and should be accessible to a wide audience of mathematicians. … Every mathematician will experience great pleasure in reading this book.” (Athanase Papadopoulos, Mathematical Reviews, Issue 2011 m)Table of ContentsPoints and lines in the plane.- Circles and spheres.- The sphere by itself: can we distribute points on it evenly?.- Conics and quadrics.- Plane curves.- Smooth surfaces.- Convexity and convex sets.- Polygons, polyhedra, polytopes.- Lattices, packings and tilings in the plane.- Lattices and packings in higher dimensions.- Geometry and dynamics I: billiards.- Geometry and dynamics II: geodesic flow on a surface.

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Book SynopsisA matroid is an abstract mathematical structure that captures combinatorial properties of matrices. This book offers a unique introduction to matroid theory, emphasizing motivations from matrix theory and applications to systems analysis. This book serves also as a comprehensive presentation of the theory and application of mixed matrices, developed primarily by the present author in the 1990's. A mixed matrix is a convenient mathematical tool for systems analysis, compatible with the physical observation that "fixed constants" and "system parameters" are to be distinguished in the description of engineering systems. This book will be extremely useful to graduate students and researchers in engineering, mathematics and computer science. From the reviews: "…The book has been prepared very carefully, contains a lot of interesting results and is highly recommended for graduate and postgraduate students." András Recski, Mathematical Reviews Clippings 2000m:93006Table of ContentsPreface I. Introduction to Structural Approach --- Overview of the Book 1 Structural Approach to Index of DAE 1.1 Index of differential-algebraic equations 1.2 Graph-theoretic structural approach 1.3 An embarrassing phenomenon 2 What Is Combinatorial Structure? 2.1 Two kinds of numbers 2.2 Descriptor form rather than standard form 2.3 Dimensional analysis 3 Mathematics on Mixed Polynomial Matrices 3.1 Formal definitions 3.2 Resolution of the index problem 3.3 Block-triangular decomposition II. Matrix, Graph and Matroid 4 Matrix 4.1 Polynomial and algebraic independence 4.2 Determinant 4.3 Rank, term-rank and generic-rank 4.4 Block-triangular forms 5 Graph 5.1 Directed graph and bipartite graph 5.2 Jordan-Holder-type theorem for submodular functions 5.3 Dulmage-Mendelsohn decomposition 5.4 Maximum flow and Menger-type linking 5.5 Minimum cost flow and weighted matching 6 Matroid 6.1 From matrix to matroid 6.2 Basic concepts 6.3 Examples 6.4 Basis exchange properties 6.5 Independent matching problem 6.6 Union 6.7 Bimatroid (linking system) III. Physical Observations for Mixed Matrix Formulation 7 Mixed Matrix for Modeling Two Kinds of Numbers 7.1 Two kinds of numbers 7.2 Mixed matrix and mixed polynomial matrix 8 Algebraic Implications of Dimensional Consistency 8.1 Introductory comments 8.2 Dimensioned matrix 8.3 Total unimodularity of dimensioned matrices 9 Physical Matrix 9.1 Physical matrix 9.2 Physical matrices in a dynamical system IV. Theory and Application of Mixed Matrices 10 Mixed Matrix and Layered Mixed Matrix 11 Rank of Mixed Matrices 11.1 Rank identities for LM-matrices 11.2 Rank identities for mixed matrices 11.3 Reduction to independent matching problems 11.4 Algorithms for the rank 11.4.1 Algorithm for LM-matrices 11.4.2 Algorithm for mixed matrices 12 Structural Solvability of Systems of Equations 12.1 Formulation of structural solvability 12.2 Graphical conditions for structural solvability 12.3 Matroidal conditions for structural solvability 13. Combinatorial Canonical Form of LM-matrices 13.1 LM-equivalence 13.2 Theorem of CCF 13.3 Construction of CCF 13.4 Algorithm for CCF 13.5 Decomposition of systems of equations by CCF 13.6 Application of CCF 13.7 CCF over rings 14 Irreducibility of LM-matrices 14.1 Theorems on LM-irreducibility 14.2 Proof of the irreducibility of determinant 15 Decomposition of Mixed Matrices 15.1 LU-decomposition of invertible mixed matrices 15.2 Block-triangularization of general mixed matrices 16 Related Decompositions 16.1 Partition as a matroid union 16.2 Multilayered matrix 16.3 Electrical network with admittance expression 17 Partitioned Matrix 17.1 Definitions 17.2 Existence of proper block-triangularization 17.3 Partial order among blocks 17.4 Generic partitioned matrix 18 Principal Structures of LM-matrices 18.1 Motivations 18.2 Principal structure of submodular systems 18.3 Principal structure of generic matrices 18.4 Vertical principal structure of LM-matrices 18.5 Horizontal principal structure of LM-matrices V. Polynomial Matrix and Valuated Matroid 19 Polynomial/Rational Matrix 19.1 Polynomial matrix and Smith form 19.2 Rational matrix and Smith-McMillan form at infinity 19.3 Matrix pencil and Kronecker form 20 Valuated Matroid 20.1 Introduction 20.2 Examples 20.3 Basic operations 20.4 Greedy algorithms 20.5 Valuated bimatroid 20.6 Induction through bipartite graphs 20.7 Characterizations 20.8 Further exchange properties 20.9 Valuated independent assignment problem 20.10 Optimality criteria 20.10.1 Potential criterion 20.10.2 Negative-cycle criterion 20.10.3 Proof of the optimality criteria 20.10.4 Extension to VIAP(k) 20.11 Application to triple matrix product 20.12 Cycle-canceling algorithms 20.12.1 Algorithms 20.12.2 Validity of the minimum-ratio cycle algorithm 20.13 Augmenting algorithms 20.13.1 Algorithms 20.13.2 Validity of the augmenting algorithm VI. Theory and Application of Mixed Polynomial Matrices 21 Descriptions of Dynamical Systems 21.1 Mixed polynomial mat

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Book SynopsisThe algorithmic solution of problems has always been one of the major concerns of mathematics. For a long time such solutions were based on an intuitive notion of algorithm. It is only in this century that metamathematical problems have led to the intensive search for a precise and sufficiently general formalization of the notions of computability and algorithm. In the 1930s, a number of quite different concepts for this purpose were pro­ posed, such as Turing machines, WHILE-programs, recursive functions, Markov algorithms, and Thue systems. All these concepts turned out to be equivalent, a fact summarized in Church's thesis, which says that the resulting definitions form an adequate formalization of the intuitive notion of computability. This had and continues to have an enormous effect. First of all, with these notions it has been possible to prove that various problems are algorithmically unsolvable. Among of group these undecidable problems are the halting problem, the word problem theory, the Post correspondence problem, and Hilbert's tenth problem. Secondly, concepts like Turing machines and WHILE-programs had a strong influence on the development of the first computers and programming languages. In the era of digital computers, the question of finding efficient solutions to algorithmically solvable problems has become increasingly important. In addition, the fact that some problems can be solved very efficiently, while others seem to defy all attempts to find an efficient solution, has called for a deeper under­ standing of the intrinsic computational difficulty of problems.Trade ReviewP. Bürgisser, M. Clausen, M.A. Shokrollahi, and T. Lickteig Algebraic Complexity Theory "The book contains interesting exercises and useful bibliographical notes. In short, this is a nice book."—MATHEMATICAL REVIEWS From the reviews: "This book is certainly the most complete reference on algebraic complexity theory that is available hitherto. … superb bibliographical and historical notes are given at the end of each chapter. … this book would most certainly make a great textbook for a graduate course on algebraic complexity theory. … In conclusion, any researchers already working in the area should own a copy of this book. … beginners at the graduate level who have been exposed to undergraduate pure mathematics would find this book accessible." (Anthony Widjaja, SIGACT News, Vol. 37 (2), 2006)Table of Contents1. Introduction.- I. Fundamental Algorithms.- 2. Efficient Polynomial Arithmetic.- 3. Efficient Algorithms with Branching.- II. Elementary Lower Bounds.- 4. Models of Computation.- 5. Preconditioning and Transcendence Degree.- 6. The Substitution Method.- 7. Differential Methods.- III. High Degree.- 8. The Degree Bound.- 9. Specific Polynomials which Are Hard to Compute.- 10. Branching and Degree.- 11. Branching and Connectivity.- 12. Additive Complexity.- IV. Low Degree.- 13. Linear Complexity.- 14. Multiplicative and Bilinear Complexity.- 15. Asymptotic Complexity of Matrix Multiplication.- 16. Problems Related to Matrix Multiplication.- 17. Lower Bounds for the Complexity of Algebras.- 18. Rank over Finite Fields and Codes.- 19. Rank of 2-Slice and 3-Slice Tensors.- 20. Typical Tensorial Rank.- V. Complete Problems.- 21. P Versus NP: A Nonuniform Algebraic Analogue.- List of Notation.

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Book SynopsisThis volume deals with the theory of finite topological spaces and its relationship with the homotopy and simple homotopy theory of polyhedra. The interaction between their intrinsic combinatorial and topological structures makes finite spaces a useful tool for studying problems in Topology, Algebra and Geometry from a new perspective. In particular, the methods developed in this manuscript are used to study Quillen's conjecture on the poset of p-subgroups of a finite group and the Andrews-Curtis conjecture on the 3-deformability of contractible two-dimensional complexes. This self-contained work constitutes the first detailed exposition on the algebraic topology of finite spaces. It is intended for topologists and combinatorialists, but it is also recommended for advanced undergraduate students and graduate students with a modest knowledge of Algebraic Topology.Trade ReviewFrom the reviews:“This book deals with the algebraic topology of finite topological spaces and its applications, and includes well-known results on finite spaces and original results developed by the author. The book is self-contained and well written. It is understandable and enjoyable to read. It contains a lot of examples and figures which help the readers to understand the theory.” (Fumihiro Ushitaki, Mathematical Reviews, March, 2014)“This book illustrates convincingly the idea that the study of finite non-Hausdorff spaces from a homotopical point of view is useful in many areas and can even be used to study well-known problems in classical algebraic topology. … This book is a revised version of the PhD Thesis of the author. … All the concepts introduced with the chapters are usefully illustrated by examples and the recollection of all these results gives a very nice introduction to a domain of growing interest.” (Etienne Fieux, Zentralblatt MATH, Vol. 1235, 2012)Table of Contents1 Preliminaries.- 2 Basic topological properties of finite spaces.- 3 Minimal finite models.- 4 Simple homotopy types and finite spaces.- 5 Strong homotopy types.- 6 Methods of reduction.- 7 h-regular complexes and quotients.- 8 Group actions and a conjecture of Quillen.- 9 Reduced lattices.- 10 Fixed points and the Lefschetz number.- 11 The Andrews-Curtis conjecture.

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Book SynopsisSince the publication of the first edition of our book, geometric algorithms and combinatorial optimization have kept growing at the same fast pace as before. Nevertheless, we do not feel that the ongoing research has made this book outdated. Rather, it seems that many of the new results build on the models, algorithms, and theorems presented here. For instance, the celebrated Dyer-Frieze-Kannan algorithm for approximating the volume of a convex body is based on the oracle model of convex bodies and uses the ellipsoid method as a preprocessing technique. The polynomial time equivalence of optimization, separation, and membership has become a commonly employed tool in the study of the complexity of combinatorial optimization problems and in the newly developing field of computational convexity. Implementations of the basis reduction algorithm can be found in various computer algebra software systems. On the other hand, several of the open problems discussed in the first edition are still unsolved. For example, there are still no combinatorial polynomial time algorithms known for minimizing a submodular function or finding a maximum clique in a perfect graph. Moreover, despite the success of the interior point methods for the solution of explicitly given linear programs there is still no method known that solves implicitly given linear programs, such as those described in this book, and that is both practically and theoretically efficient. In particular, it is not known how to adapt interior point methods to such linear programs.Table of Contents0. Mathematical Preliminaries.- 0.1 Linear Algebra and Linear Programming.- Basic Notation.- Hulls, Independence, Dimension.- Eigenvalues, Positive Definite Matrices.- Vector Norms, Balls.- Matrix Norms.- Some Inequalities.- Polyhedra, Inequality Systems.- Linear (Diophantine) Equations and Inequalities.- Linear Programming and Duality.- 0.2 Graph Theory.- Graphs.- Digraphs.- Walks, Paths, Circuits, Trees.- 1. Complexity, Oracles, and Numerical Computation.- 1.1 Complexity Theory: P and NP.- Problems.- Algorithms and Turing Machines.- Encoding.- Time and Space Complexity.- Decision Problems: The Classes P and NP.- 1.2 Oracles.- The Running Time of Oracle Algorithms.- Transformation and Reduction.- NP-Completeness and Related Notion.- 1.3 Approximation and Computation of Numbers.- Encoding Length of Numbers.- Polynomial and Strongly Polynomial Computations.- Polynomial Time Approximation of Real Numbers.- 1.4 Pivoting and Related Procedures.- Gaussian Elimination.- Gram-Schmidt Orthogonalization.- The Simplex Method.- Computation of the Hermite Normal Form.- 2. Algorithmic Aspects of Convex Sets: Formulation of the Problems.- 2.1 Basic Algorithmic Problems for Convex Sets.- 2.2 Nondeterministic Decision Problems for Convex Sets.- 3. The Ellipsoid Method.- 3.1 Geometric Background and an Informal Description.- Properties of Ellipsoids.- Description of the Basic Ellipsoid Method.- Proofs of Some Lemmas.- Implementation Problems and Polynomiality.- Some Examples.- 3.2 The Central-Cut Ellipsoid Method.- 3.3 The Shallow-Cut Ellipsoid Method.- 4. Algorithms for Convex Bodies.- 4.1 Summary of Results.- 4.2 Optimization from Separation.- 4.3 Optimization from Membership.- 4.4 Equivalence of the Basic Problems.- 4.5 Some Negative Results.- 4.6 Further Algorithmic Problems for Convex Bodies.- 4.7 Operations on Convex Bodies.- The Sum.- The Convex Hull of the Union.- The Intersection.- Polars, Blockers, Antiblockers.- 5. Diophantine Approximation and Basis Reduction.- 5.1 Continued Fractions.- 5.2 Simultaneous Diophantine Approximation: Formulation of the Problems.- 5.3 Basis Reduction in Lattices.- 5.4 More on Lattice Algorithms.- 6. Rational Polyhedra.- 6.1 Optimization over Polyhedra: A Preview.- 6.2 Complexity of Rational Polyhedra.- 6.3 Weak and Strong Problems.- 6.4 Equivalence of Strong Optimization and Separation.- 6.5 Further Problems for Polyhedra.- 6.6 Strongly Polynomial Algorithms.- 6.7 Integer Programming in Bounded Dimension.- 7. Combinatorial Optimization: Some Basic Examples.- 7.1 Flows and Cuts.- 7.2 Arborescences.- 7.3 Matching.- 7.4 Edge Coloring.- 7.5 Matroids.- 7.6 Subset Sums.- 7.7 Concluding Remarks.- 8. Combinatorial Optimization: A Tour d’Horizon.- 8.1 Blocking Hypergraphs and Polyhedra.- 8.2 Problems on Bipartite Graphs.- 8.3 Flows, Paths, Chains, and Cuts.- 8.4 Trees, Branchings, and Rooted and Directed Cuts.- Arborescences and Rooted Cuts.- Trees and Cuts in Undirected Graphs.- Dicuts and Dijoins.- 8.5 Matchings, Odd Cuts, and Generalizations.- Matching.- b-Matching.- T-Joins and T-Cuts.- Chinese Postmen and Traveling Salesmen.- 8.6 Multicommodity Flows.- 9. Stable Sets in Graphs.- 9.1 Odd Circuit Constraints and t-Perfect Graphs.- 9.2 Clique Constraints and Perfect Graphs.- Antiblockers of Hypergraphs.- 9.3 Orthonormal Representations.- 9.4 Coloring Perfect Graphs.- 9.5 More Algorithmic Results on Stable Sets.- 10. Submodular Functions.- 10.1 Submodular Functions and Polymatroids.- 10.2 Algorithms for Polymatroids and Submodular Functions.- Packing Bases of a Matroid.- 10.3 Submodular Functions on Lattice, Intersecting, and Crossing Families.- 10.4 Odd Submodular Function Minimization and Extensions.- References.- Notation Index.- Author Index.

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Book SynopsisThese lecture notes treat polynomial identity rings from both the combinatorial and structural points of view. The greater part of recent research in polynomial identity rings is about combinatorial questions, and the combinatorial part of the lecture notes gives an up-to-date account of recent research. On the other hand, the main structural results have been known for some time, and the emphasis there is on a presentation accessible to newcomers to the subject.Trade ReviewFrom the reviews: “The book under review consists of two excellent monographs on the PI-theory by two leading researchers, V. Drensky and E. Formanek … In summary, both expositions are very well written, and the book is recommended both for graduate students and researchers.” (MATHEMATICAL REVIEWS)Table of ContentsA Combinatorial Aspects in PI-Rings.- Vesselin Drensky.- 1 Basic Properties of PI-algebras.- 2 Quantitative Approach to PI-algebras.- 3 The Amitsur-Levitzki Theorem.- 4 Central Polynomials for Matrices.- 5 Invariant Theory of Matrices.- 6 The Nagata-Higman Theorem.- 7 The Shirshov Theorem for Finitely Generated PI-algebras.- 8 Growth of Codimensions of PI-algebras.- B Polynomial Identity Rings.- Edward Formanek.- 1 Polynomial Identities.- 2 The Amitsur-Levitzki Theorem.- 3 Central Polynomials.- 4 Kaplansky’s Theorem.- 5 Theorems of Amitsur and Levitzki on Radicals.- 6 Posner’s Theorem.- 7 Every PI-ring Satisfies a Power of the Standard Identity.- 8 Azumaya Algebras.- 9 Artin’s Theorem.- 10 Chain Conditions.- 11 Hilbert and Jacobson PI-Rings.- 12 The Ring of Generic Matrices.- 13 The Generic Division Ring of Two 2 x 2 Generic Matrices.- 14 The Center of the Generic Division Ring.- 15 Is the Center of the Generic Division Ring a Rational Function Field?.

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Book SynopsisIn July 2004, a conference on graph theory was held in Paris in memory of Claude Berge, one of the pioneers of the field. The event brought together many prominent specialists on topics such as perfect graphs and matching theory, upon which Claude Berge's work has had a major impact. This volume includes contributions to these and other topics from many of the participants.Table of ContentsClaude Berge — Sculptor of Graph Theory.- ?-path-connectivity and mk-generation: an Upper Bound on m.- Automated Results and Conjectures on Average Distance in Graphs.- Brambles, Prisms and Grids.- Dead Cell Analysis in Hex and the Shannon Game.- Ratios of Some Domination Parameters in Graphs and Claw-free Graphs.- Excessive Factorizations of Regular Graphs.- Odd Pairs of Cliques.- Recognition of Perfect Circular-arc Graphs.- On Edge-maps whose Inverse Preserves Flows or Tensions.- On the Extremal Number of Edges in 2-Factor Hamiltonian Graphs.- Generalized Colourings (Matrix Partitions) of Cographs.- A Note on [k, l]-sparse Graphs.- Even Pairs in Bull-reducible Graphs.- Kernels in Orientations of Pretransitive Orientable Graphs.- Nonrepetitive Graph Coloring.- A Characterization of the 1-well-covered Graphs with no 4-cycles.- A Graph-theoretical Generalization of Berge’s Analogue of the Erd?s-Ko-Rado Theorem.- Independence Polynomials and the Unimodality Conjecture for Very Well-covered, Quasi-regularizable, and Perfect Graphs.- Precoloring Extension on Chordal Graphs.- On the Enumeration of Bipartite Minimum Edge Colorings.- Kempe Equivalence of Colorings.- Acyclic 4-choosability of Planar Graphs with Girth at Least 5.- Automorphism Groups of Circulant Graphs — a Survey.- Hypo-matchings in Directed Graphs.- On Reed’s Conjecture about ?,? and ?.- On the Generalization of the Matroid Parity Problem.- Reconstruction of a Rank 3 Oriented Matroids from its Rank 2 Signed Circuits.- The Normal Graph Conjecture is True for Circulants.- Two-arc Transitive Near-polygonal Graphs.- Open Problems.

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Book SynopsisThe problem of enumerating maps (a map is a set of polygonal "countries" on a world of a certain topology, not necessarily the plane or the sphere) is an important problem in mathematics and physics, and it has many applications ranging from statistical physics, geometry, particle physics, telecommunications, biology, ... etc. This problem has been studied by many communities of researchers, mostly combinatorists, probabilists, and physicists. Since 1978, physicists have invented a method called "matrix models" to address that problem, and many results have been obtained.Besides, another important problem in mathematics and physics (in particular string theory), is to count Riemann surfaces. Riemann surfaces of a given topology are parametrized by a finite number of real parameters (called moduli), and the moduli space is a finite dimensional compact manifold or orbifold of complicated topology. The number of Riemann surfaces is the volume of that moduli space. More generally, an important problem in algebraic geometry is to characterize the moduli spaces, by computing not only their volumes, but also other characteristic numbers called intersection numbers.Witten's conjecture (which was first proved by Kontsevich), was the assertion that Riemann surfaces can be obtained as limits of polygonal surfaces (maps), made of a very large number of very small polygons. In other words, the number of maps in a certain limit, should give the intersection numbers of moduli spaces.In this book, we show how that limit takes place. The goal of this book is to explain the "matrix model" method, to show the main results obtained with it, and to compare it with methods used in combinatorics (bijective proofs, Tutte's equations), or algebraic geometry (Mirzakhani's recursions).The book intends to be self-contained and accessible to graduate students, and provides comprehensive proofs, several examples, and gives the general formula for the enumeration of maps on surfaces of any topology. In the end, the link with more general topics such as algebraic geometry, string theory, is discussed, and in particular a proof of the Witten-Kontsevich conjecture is provided.Trade Review“This book brings together details of topological recursion from many different papers and organizes them in an accessible way. … this book will be an invaluable resource for mathematicians learning about topological recursion.” (Daniel D. Moskovich, Mathematical Reviews, February, 2017) “The author explains how matrix models and counting surfaces are related and aims at presenting to mathematicians and physicists the random matrix approach to quantum gravity. … The book is an outstanding monograph of a recent research trend in surface theory.” (Gert Roepstorff, zbMATH 1338.81005, 2016)Table of ContentsI Maps and discrete surfaces.- II Formal matrix integrals.- III Solution of Tutte-loop equations.- IV Multicut case.- V Counting large maps.- VI Counting Riemann surfaces.- VII Topological recursion and symplectic invariants.- VIII Ising model.- Index.- Bibliography.

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Book SynopsisTrade Review"Discrete Mathematics is adequately written and well-documented.... This book presents the material on the topic in a cogently coherent manner thereby serving and justifying the purpose of writing books such as this one. The classroom-tested pedagogy and its 400 examples speak a lot about the kind and amount of sweat that must have gone into it." --zbMATH OpenTable of ContentsPart I: Logic 1. Propositional Logic 2. Predicate Logic Part II: Set Theory and Related Topics 3. Sets 4. Matrices 5. Relations 6. Functions 7. Boolean Algebra Part III: Proof Methods 8. Sequences 9. Recursion 10. Induction 11. General Proof Methods Part IV: Number Theory and Applications 12. Elementary Number Theory 13. Cryptography Part V: Probability 14. Counting Methods 15. Discrete Probability 16. Discrete Random Variables Part VI: Graph Theory 17. Graphs 18. Trees 19. Network Models Part VII: Algorithms and Finite State Machines 20. Algorithms

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Book SynopsisRandomized algorithms have become a central part of the algorithms curriculum, based on their increasingly widespread use in modern applications. This book presents a coherent and unified treatment of probabilistic techniques for obtaining high probability estimates on the performance of randomized algorithms. It covers the basic toolkit from the ChernoffâHoeffding bounds to more sophisticated techniques like martingales and isoperimetric inequalities, as well as some recent developments like Talagrand's inequality, transportation cost inequalities and log-Sobolev inequalities. Along the way, variations on the basic theme are examined, such as ChernoffâHoeffding bounds in dependent settings. The authors emphasise comparative study of the different methods, highlighting respective strengths and weaknesses in concrete example applications. The exposition is tailored to discrete settings sufficient for the analysis of algorithms, avoiding unnecessary measure-theoretic details, thus makingTrade ReviewReview of the hardback: 'It is beautifully written, contains all the major concentration results, and is a must to have on your desk.' Richard LiptonReview of the hardback: 'Concentration bounds are at the core of probabilistic analysis of algorithms. This excellent text provides a comprehensive treatment of this important subject, ranging from the very basic to the more advanced tools, including some recent developments in this area. The presentation is clear and includes numerous examples, demonstrating applications of the bounds in analysis of algorithms. This book is a valuable resource for both researchers and students in the field.' Eli Upfal, Brown UniversityReview of the hardback: 'Concentration inequalities are an essential tool for the analysis of algorithms in any probabilistic setting. There have been many recent developments on this subject, and this excellent text brings them together in a highly accessible form.' Alan Frieze, Carnegie Mellon UniversityReview of the hardback: 'The book does a superb job of describing a collection of powerful methodologies in a unified manner; what is even more striking is that basic combinatorial and probabilistic language is used in bringing out the power of such approaches. To summarize, the book has done a great job of synthesizing diverse and important material in a very accessible manner. Any student, researcher, or practitioner of computer science, electrical engineering, mathematics, operations research, and related fields, could benefit from this wonderful book. The book would also make for fruitful classes at the undergraduate and graduate levels. I highly recommend it.' Aravind Srinivasan, SIGACT NewsReview of the hardback: '… the strength of this book is that it is appropriate for both the beginner as well as the experienced researcher in the field of randomized algorithms … The exposition style […] combines informal discussion with formal definitions and proofs, giving first the intuition and motivation for the probabalistic technique at hand. … I highly recommend this book both as an advanced as well as an introductory textbook, which can also serve the needs of an experienced researcher in algorithmics.' Yannis C. Stamatiou, Mathematical ReviewsReviews of the hardback: 'This timely book brings together in a comprehensive and accessible form a sophisticated toolkit of powerful techniques for the analysis of randomized algorithms, illustrating their use with a wide array of insightful examples. This book is an invaluable resource for people venturing into this exciting field of contemporary computer science research.' Prabhakar Ragahavan, Yahoo ResearchTable of Contents1. Chernoff–Hoeffding bounds; 2. Applying the CH-bounds; 3. CH-bounds with dependencies; 4. Interlude: probabilistic recurrences; 5. Martingales and the MOBD; 6. The MOBD in action; 7. Averaged bounded difference; 8. The method of bounded variances; 9. Interlude: the infamous upper tail; 10. Isoperimetric inequalities and concentration; 11. Talagrand inequality; 12. Transportation cost and concentration; 13. Transportation cost and Talagrand's inequality; 14. Log–Sobolev inequalities; Appendix A. Summary of the most useful bounds.

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Book SynopsisEmphasizing the search for patterns within and between biological sequences, trees, and graphs, Combinatorial Pattern Matching Algorithms in Computational Biology Using Perl and R shows how combinatorial pattern matching algorithms can solve computational biology problems that arise in the analysis of genomic, transcriptomic, proteomic, metabolomic, and interactomic data. It implements the algorithms in Perl and R, two widely used scripting languages in computational biology. The book provides a well-rounded explanation of traditional issues as well as an up-to-date account of more recent developments, such as graph similarity and search. It is organized around the specific algorithmic problems that arise when dealing with structures that are commonly found in computational biology, including biological sequences, trees, and graphs. For each of these structures, the author makes a clear distinction between problems that arise in the analysis of one strTrade ReviewI like the hands-on approach this book offers, and the very pedagogical structure it follows … . The book also has tons of examples, thoughtfully chosen and beautifully laid out … the book is very well-written and accessible, undoubtedly written by an author who takes great care in preparing his manuscripts and teaching about an area he enjoys working on.—Anthony Labarre, SIGACT News, July 2012This text provides a solid foundation to the field. It will work as a practical handbook for pattern matching applications in computational biology. —Michael Goldberg, Computing Reviews, February 2010… the book makes a clear distinction between problems that emerge in the analysis of the structure and in the comparative analysis of two or more structures. … Well-known computational biology tools that allow searching nucleotide and protein databases for local sequence alignment are based on CPM algorithms only. The techniques presented in this book go beyond that. … detailed algorithm solutions in pseudocode, full Perl and R implementation, and pointers to software and implementation are presented. This … is what makes Valiente’s effort unique. …—Ernesto D’Avanzo, Computing Reviews, February 2010… It is a well-sorted collection of pattern matching algorithms that are used to work with problems in computational biology. … You can find all of the sources on the author’s website, which come in handy when you actually want to use them, since you do not have to retype them. And there is an introduction to Perl as well as to R, showing how to decode DNA/RNA-triplets to amino acids and giving some basic overview over standard functions. … I certainly recommend this as an introduction and reference to some algorithms of pattern matching in computational biology. You actually learn how algorithms over the most important data types are designed in a straightforward, logical way. …—Jannik Pewny, IACR Book Reviews, 2009…after a few minutes of random browsing, I was left with a feeling of total appreciation of the book, admiration for Prof. Gabriel Valiente, and a realization that this book will be part of my fundamental library for me and my group from the moment it is published. There are so many good things to say that I do not know where to start. The organization is straightforward with major sections that extend from simple sequences to trees to graphs. … This parallel structure makes it easy to apply lessons used on the simplest object (sequences) to objects of medium (trees) and significant (graphs) difficulty. …a wonderful way to learn leveraging … The Perl is beautifully clear and the examples have already taught me how to improve my own code.—Michael Levitt, Professor and Chair, Department of Structural Biology, Stanford University, California, USA…Balancing a careful mixture of formal methods, programming, and examples, Gabriel Valiente has managed to harmoniously bridge languages and contents into a self-contained source of lasting influence. It is not difficult to predict that this book will be studied indifferently by the specialist of biology and computer science, helping each to walk a few steps toward the other. It will entice new generations of scholars to engage in its beautiful subject.—From the Foreword, Alberto Apostolico, Professor, College of Computing, Georgia Tech, Atlanta, USAUnlocks the power for R for Perl programmers, and vice versa. Reveals R to be a powerful and accessible tool for bioinformatics. The title is a mouthful, but the use of both R and Perl for bioinformatics is revealing.—Steven Skiena, Professor, Department of Computer Science, Stony Brook University, New York, USAI like the hands-on approach this book offers, and the very pedagogical structure it follows … . The book also has tons of examples, thoughtfully chosen and beautifully laid out … the book is very well-written and accessible, undoubtedly written by an author who takes great care in preparing his manuscripts and teaching about an area he enjoys working on.—Anthony Labarre, SIGACT News, July 2012This text provides a solid foundation to the field. It will work as a practical handbook for pattern matching applications in computational biology. —Michael Goldberg, Computing Reviews, February 2010… the book makes a clear distinction between problems that emerge in the analysis of the structure and in the comparative analysis of two or more structures. … Well-known computational biology tools that allow searching nucleotide and protein databases for local sequence alignment are based on CPM algorithms only. The techniques presented in this book go beyond that. … detailed algorithm solutions in pseudocode, full Perl and R implementation, and pointers to software and implementation are presented. This … is what makes Valiente’s effort unique. …—Ernesto D’Avanzo, Computing Reviews, February 2010… It is a well-sorted collection of pattern matching algorithms that are used to work with problems in computational biology. … You can find all of the sources on the author’s website, which come in handy when you actually want to use them, since you do not have to retype them. And there is an introduction to Perl as well as to R, showing how to decode DNA/RNA-triplets to amino acids and giving some basic overview over standard functions. … I certainly recommend this as an introduction and reference to some algorithms of pattern matching in computational biology. You actually learn how algorithms over the most important data types are designed in a straightforward, logical way. …—Jannik Pewny, IACR Book Reviews, 2009…after a few minutes of random browsing, I was left with a feeling of total appreciation of the book, admiration for Prof. Gabriel Valiente, and a realization that this book will be part of my fundamental library for me and my group from the moment it is published. There are so many good things to say that I do not know where to start. The organization is straightforward with major sections that extend from simple sequences to trees to graphs. … This parallel structure makes it easy to apply lessons used on the simplest object (sequences) to objects of medium (trees) and significant (graphs) difficulty. …a wonderful way to learn leveraging … The Perl is beautifully clear and the examples have already taught me how to improve my own code.—Michael Levitt, Professor and Chair, Department of Structural Biology, Stanford University, California, USA…Balancing a careful mixture of formal methods, programming, and examples, Gabriel Valiente has managed to harmoniously bridge languages and contents into a self-contained source of lasting influence. It is not difficult to predict that this book will be studied indifferently by the specialist of biology and computer science, helping each to walk a few steps toward the other. It will entice new generations of scholars to engage in its beautiful subject.—From the Foreword, Alberto Apostolico, Professor, College of Computing, Georgia Tech, Atlanta, USAUnlocks the power for R for Perl programmers, and vice versa. Reveals R to be a powerful and accessible tool for bioinformatics. The title is a mouthful, but the use of both R and Perl for bioinformatics is revealing.—Steven Skiena, Professor, Department of Computer Science, Stony Brook University, New York, USATable of ContentsIntroduction. SEQUENCE PATTERN MATCHING: Sequences. Simple Pattern Matching in Sequences. General Pattern Matching in Sequences. TREE PATTERN MATCHING: Trees. Simple Pattern Matching in Trees. General Pattern Matching in Trees. GRAPH PATTERN MATCHING: Graphs. Simple Pattern Matching in Graphs. General Pattern Matching in Graphs. Appendices. References. Index.

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Book SynopsisSet Theoretical Aspects of Real Analysis is built around a number of questions in real analysis and classical measure theory, which are of a set theoretic flavor. Accessible to graduate students, and researchers the beginning of the book presents introductory topics on real analysis and Lebesgue measure theory. These topics highlight the boundary between fundamental concepts of measurability and nonmeasurability for point sets and functions. The remainder of the book deals with more specialized material on set theoretical real analysis. The book focuses on certain logical and set theoretical aspects of real analysis. It is expected that the first eleven chapters can be used in a course on Lebesque measure theory that highlights the fundamental concepts of measurability and non-measurability for point sets and functions. Provided in the book are problems of varying difficulty that range from simple observations to advanced results. Relatively difficult Table of ContentsZF theory and some point sets on the real line. Countable versions of AC and real analysis. Uncountable versions of AC and Lebesgue nonmeasurable sets. The Continuum Hypothesis and Lebesgue nonmeasurable sets. Measurability properties of sets and functions. Radon measures and nonmeasurable sets. Real-valued step functions with strange measurability properties. Relationships between certain classical constructions of Lebesgue nonmeasurable sets. Measurability properties of Vitali sets. A relationship between the measurability and continuity of real-valued functions. A relationship between absolutely nonmeasurable functions and Sierpinski-Zygmund functions. Sums of absolutely nonmeasurable injective functions. A large group of absolutely nonmeasurable additive functions. Additive properties of certain classes of pathological functions. Absolutely nonmeasurable homomorphisms of commutative groups. Measurable and nonmeasurable sets with homogeneous sections. A combinatorial problem on translation invariant extensions of the Lebesgue measure. Countable almost invariant partitions of G-spaces. Nonmeasurable unions of measure zero sections of plane sets. Measurability properties of well-orderings. Appendices. Bibliography. Subject Index.

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