Combinatorics and graph theory Books

Book SynopsisThis is a complete reworking of the out-of-print first volume of a three-volume treatise on finite projective spaces. There are numerous articles in journals, but this is the only extended work in the area. It also includes a comprehensive bibliography of more than 3000 items.Trade ReviewThe first edition of this work appeared in 1979, and was immediately recognized as an outstanding contribution to the field of finite geometry; the present volume is a complete revision of the earlier work...The book can serve as the text for a basic course...or as a detailed reference work for all topics in spaces of dimension two...The work is an indispensable aid to all workers in finite geometry, from beginning students to advanced researchers. * Short Book Reviews *...A complete reworking [of the first edition]. As before, the volume is concisely but clearly written, and contains a wealth of interesting material...the new trilogy, comprising the 1998, 1986 and 1991 volumes, looks set to be the standard reference work on projective spaces over finite fields for many years to come. * Bulletin of the London Mathematical Society *'Snap it up !' Bulletin London mathematical SocietyTable of Contents1. Finite fields ; 2. Projective spaces and algebraic varieties ; 3. Subspaces ; 4. Partitions ; 5. Canonical forms for varieties and polarities ; 6. The line ; 7. First properties of the plane ; 8. Ovals ; 9. Arithmetic of arcs of degree two ; 10. Arcs in ovals ; 11. Cubic curves ; 12. Arcs of higher degree ; 13. Blocking sets ; 14. Small planes ; Appendix ; Notation ; References

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Book SynopsisThis monograph sets out a body of mathematical theory for finite graphs with nodes placed randomly in Euclidean space and edges added to connect points that are close to each other. As an alternative to classical random graph models, these geometric graphs are relevant to the modelling of real-world networks having spatial content, arising in numerous applications such as wireless communications, parallel processing, classification, epidemiology, astronomy, and the internet. Aimed at graduate students and researchers in probability, combinatorics, statistics, and theoretical computer science, it covers topics such as edge and component counts, vertex degrees, cliques, colourings, connectivity, giant component phenomena, vertex ordering and partitioning problems. It also illustrates and extends the application to geometric probability of modern techniques including Stein''s method, martingale methods and continuum percolation.Trade ReviewThe book is suitable to design a graduate course in random geometric graphs. Its scope stretches far beyond geometric probability and includes exciting material from Poisson approximation, percolation and statistical physics. This elegantly written monograph belongs to the collection of important books vital for every probabilist. * Zentralblatt MATH *Table of Contents1. Introduction ; 2. Probabilistic ingredients ; 3. Subgraph and component counts ; 4. Typical vertex degrees ; 5. Geometrical ingredients ; 6. Maximum degree, cliques and colourings ; 7. Minimum degree: laws of large numbers ; 8. Minimum degree: convergence in distribution ; 9. Percolative ingredients ; 10. Percolation and the largest component ; 11. The largest component for a binomial process ; 12. Ordering and partitioning problems ; 13. Connectivity and the number of components ; References ; Index

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Book SynopsisThis concise introduction to the subject emphasizes the various classes of regular semigroups. More than 150 exercises, accompanied by references to the relevant research literature, direct readers to areas not explicitly covered in the text.Trade ReviewThe author succeeds admirably in his goal of providing an introduction to semigroup theory suitable for graduate students and non-specialists. At the same time, specialists will also find much to attract them. The book is highly readable, and the author takes pain to lead the reader gently through the longer proofs. I am sure that all specialists will want to own a copy of the book, that non-specialists will find it a most attractive and useful book for consulting, and that for many years to come research students will learn their subject from this book. * Bulletin of the London Mathematical Society *The book provides a useful survey of a rapidly developing topic and is suitable for specialists and as an introduction to the subject for non-specialists and graduate students. * Aslib Book Guide, vol.61, no.5, May 1996. *This book will still have its outstanding place as a general introduction to semigroup theory offering both an updated overview of the subject and a suitable entree for the graduate student * Monatshefte fur Mathematik Vol. 124 1997 *With this well-written and well-organised book I think the author has ensured that "Howie" will continue to be a byword for semigroup books * Edinburgh Mathematical Society 1997 *Table of Contents1. Introductory ideas ; 2. Green's equivalences; regular semigroups ; 3. 0-simple semigroups ; 4. Completely regular semigroups ; 5. Inverse semigroups ; 6. Other classes of regular semigroups ; 7. Free semigroups ; 8. Semigroup amalgams ; References ; List of symbols

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Book SynopsisAn introduction to recent developments in algebraic combinatorics and an illustration of how research in mathematics actually progresses.Trade Review'Proofs and Confirmations is one of the most brilliant examples of mathematical exposition that I have encountered, in many years of reading the same. This is not for the faint-hearted, nor is Proofs and Confirmations a book that can be read in an easy chair, like a novel; it demands active participation by the reader. But Bressoud rewards such readers with a panorama of combinatorics today and with renewed awe at the human ability to penetrate the deeply hidden mysteries of pure mathematics.' Herbert S. Wilf, Science'The unexpected twists and turns will hardly be matched in any novel - this book allows us all to share in the excitement … a brilliant book.' Alun O. Morris'I strongly recommend the book as an account of a remarkable mathematical development.' P. J. Cameron, Proceedings of the Edinburgh Mathematical Society'This is an excellent book which can be recommended without hesitation, not only to specialists in the field, but to any mathematician with time to read something interesting and nicely written.' EMSTable of Contents1. The conjecture; 2. Fundamental structures; 3. Lattice paths and plane partitions; 4. Symmetric functions; 5. Hypergeometric series; 6. Explorations; 7. Square ice.

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Book SynopsisEvery four years leading researchers gather to survey the latest developments in all aspects of group theory. Since 1981, the proceedings of these meetings have provided a regular snapshot of the state of the art in group theory and helped to shape the direction of research in the field. This volume contains selected papers from the 2022 meeting held in Newcastle. It includes substantial survey articles from the invited speakers, namely the mini course presenters Michel Brion, Fanny Kassel and Pham Huu Tiep; and the invited one-hour speakers Bettina Eick, Scott Harper and Simon Smith. It features these alongside contributed survey articles, including some new results, to provide an outstanding resource for graduate students and researchers.

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Book SynopsisDesign robust graph neural networks with PyTorch Geometric by combining graph theory and neural networks with the latest developments and apps Purchase of the print or Kindle book includes a free PDF eBook Key Features Implement -of-the-art graph neural architectures in Python Create your own graph datasets from tabular data Build powerful traffic forecasting, recommender systems, and anomaly detection applications Book DescriptionGraph neural networks are a highly effective tool for analyzing data that can be represented as a graph, such as networks, chemical compounds, or transportation networks. The past few years have seen an explosion in the use of graph neural networks, with their application ranging from natural language processing and computer vision to recommendation systems and drug discovery. Hands-On Graph Neural Networks Using Python begins with the fundamentals of graph theory and shows you how to create graph datasets from tabular data. As you advance, you’ll explore major graph neural network architectures and learn essential concepts such as graph convolution, self-attention, link prediction, and heterogeneous graphs. Finally, the book proposes applications to solve real-life problems, enabling you to build a professional portfolio. The code is readily available online and can be easily adapted to other datasets and apps. By the end of this book, you’ll have learned to create graph datasets, implement graph neural networks using Python and PyTorch Geometric, and apply them to solve real-world problems, along with building and training graph neural network models for node and graph classification, link prediction, and much more.What you will learn Understand the fundamental concepts of graph neural networks Implement graph neural networks using Python and PyTorch Geometric Classify nodes, graphs, and edges using millions of samples Predict and generate realistic graph topologies Combine heterogeneous sources to improve performance Forecast future events using topological information Apply graph neural networks to solve real-world problems Who this book is forThis book is for machine learning practitioners and data scientists interested in learning about graph neural networks and their applications, as well as students looking for a comprehensive reference on this rapidly growing field. Whether you’re new to graph neural networks or looking to take your knowledge to the next level, this book has something for you. Basic knowledge of machine learning and Python programming will help you get the most out of this book.Table of ContentsTable of Contents Getting Started with Graph Learning Graph Theory for Graph Neural Networks Creating Node Representations with DeepWalk Improving Embeddings with Biased Random Walks in Node2Vec Including Node Features with Vanilla Neural Networks Introducing Graph Convolutional Networks Graph Attention Networks Scaling Graph Neural Networks with GraphSAGE Defining Expressiveness for Graph Classification Predicting Links with Graph Neural Networks Generating Graphs Using Graph Neural Networks Learning from Heterogeneous Graphs Temporal Graph Neural Networks Explaining Graph Neural Networks Forecasting Traffic Using A3T-GCN Detecting Anomalies Using Heterogeneous Graph Neural Networks Building a Recommender System Using LightGCN Unlocking the Potential of Graph Neural Networks for Real-Word Applications

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Book SynopsisThis book explores topics in Gallai-Ramsey theory, which looks into whether rainbow colored subgraphs or monochromatic subgraphs exist in a sufficiently large edge-colored complete graphs. A comprehensive survey of all known results with complete references is provided for common proof methods. Fundamental definitions and preliminary results with illustrations guide readers to comprehend recent innovations. Complete proofs and influential results are discussed with numerous open problems and conjectures. Researchers and students with an interest in edge-coloring, Ramsey Theory, and colored subgraphs will find this book a valuable guide for entering Gallai-Ramsey Theory.Trade Review“In the opinion of the reviewer, Topics in Gallai-Ramsey theory is a well-organized, well-written, valuable compendium of results in the foundation and halfway up to the second story of the newly christened area of Gallai-Ramsey theory. … this book and efforts in the area will have helped to shape whatever comes next.” (Peter D. Johnson, Jr., Mathematical Reviews, April, 2022)“This book is an excellent and much sought after material addressing Gallai-Ramsey theory. … this book is meant for very serious researchers in this topic.” (V. Yegnanarayanan, zbMATH 1452.05001, 2021)Table of Contents1. Introduction and Basic Definitions.- 2. General Structure Under Forbidden Rainbow Subgraphs.- 3. Gallai-Ramsey Results for Rainbow Triangles.- 4. Gallai-Ramsey Results for Other Rainbow Subgraphs.- 5. Conclusion and Open Problems.

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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 concise monograph present the complete history of the domination game and its variants up to the most recent developments and will stimulate research on closely related topics, establishing a key reference for future developments. The crux of the discussion surrounds new methods and ideas that were developed within the theory, led by the imagination strategy, the Continuation Principle, and the discharging method of Bujtás, to prove results about domination game invariants. A toolbox of proof techniques is provided for the reader to obtain results on the domination game and its variants. Powerful proof methods such as the imagination strategy are presented. The Continuation Principle is developed, which provides a much-used monotonicity property of the game domination number. In addition, the reader is exposed to the discharging method of Bujtás. The power of this method was shown by improving the known upper bound, in terms of a graph's order, on the (ordinary) domination number of graphs with minimum degree between 5 and 50. The book is intended primarily for students in graph theory as well as established graph theorists and it can be enjoyed by anyone with a modicum of mathematical maturity.The authors include exact results for several families of graphs, present what is known about the domination game played on subgraphs and trees, and provide the reader with the computational complexity aspects of domination games. Versions of the games which involve only the “slow” player yield the Grundy domination numbers, which connect the topic of the book with some concepts from linear algebra such as zero-forcing sets and minimum rank. More than a dozen other related games on graphs and hypergraphs are presented in the book. In all these games there are problems waiting to be solved, so the area is rich for further research. The domination game belongs to the growing family of competitive optimization graph games. The game is played by two competitors who take turns adding a vertex to a set of chosen vertices. They collaboratively produce a special structure in the underlying host graph, namely a dominating set. The two players have complementary goals: one seeks to minimize the size of the chosen set while the other player tries to make it as large as possible. The game is not one that is either won or lost. Instead, if both players employ an optimal strategy that is consistent with their goals, the cardinality of the chosen set is a graphical invariant, called the game domination number of the graph. To demonstrate that this is indeed a graphical invariant, the game tree of a domination game played on a graph is presented for the first time in the literature. Table of Contents1. Introduction.- 2. Domination Game.-3. Total Domination Game.- 4. Games for Staller.- 5. Related Games on Graphs and Hypergraphs.-References.-Symbol Index.

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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 SynopsisAs the first book of a three-part series, this book is offered as a tribute to pioneers in vision, such as Béla Julesz, David Marr, King-Sun Fu, Ulf Grenander, and David Mumford. The authors hope to provide foundation and, perhaps more importantly, further inspiration for continued research in vision. This book covers David Marr's paradigm and various underlying statistical models for vision. The mathematical framework herein integrates three regimes of models (low-, mid-, and high-entropy regimes) and provides foundation for research in visual coding, recognition, and cognition. Concepts are first explained for understanding and then supported by findings in psychology and neuroscience, after which they are established by statistical models and associated learning and inference algorithms. A reader will gain a unified, cross-disciplinary view of research in vision and will accrue knowledge spanning from psychology to neuroscience to statistics. Table of ContentsPreface.- About the Authors.- 1 Introduction.- 2 Statistics of Natural Images.- 3 Textures.- 4 Textons.- 5 Gestalt Laws and Perceptual Organizations.- 6 Primal Sketch: Integrating Textures and Textons.- 7 2.1D Sketch and Layered Representation.- 8 2.5D Sketch and Depth Maps.- 9 Learning about information Projection.- 10 Informing Scaling and Regimes of Models.- 11 Deep Images and Models.- 12 A Tale of Three Families: Discriminative, Generative and Descriptive Models.- Bibliography

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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 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 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 SynopsisToday, with physician and hospital reimbursement being cut and tied to quality incentives, physicians and health plans are revisiting the concept of integration. Payers are demanding that the industry do more with less without sacrificing quality of care. As a result, physicians again find themselves integrating and aligning with hospitals that have the resources they lack or must develop together.Written by an acknowledged expert in the field of physician integration and managed care contracting, Physician Integration & Alignment: IPA, PHO, ACOs, and Beyond examines physician integration and alignment in the current healthcare market. It outlines the common characteristics of integrated groups and various organizational structures, and also explains how you can avoid making the same mistakes of the past. Filled with suggestions and ideas from successfully integrated practices, the book:Identifies industry drivers for the resurgence of intTable of Contents1994: The Initial Wave. Why Do It Again: The Drivers of the New Wave. Common Characteristics of Integrated Groups. Elements of Design for the New Integrated Networks. Avoiding the Mistakes of the Past. Antitrust and Other Regulatory Concerns. Options and More Options. A Checklist for the Design Process. Index

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Book Synopsis1 Groups.- 2 Graphs.- 3 Chemical Compounds.- 4 Asymptotic Evaluation of the Number of Combinations.- The Legacy of Pólya's Paper: Fifty Years of Pólya Theory.- References.Table of Contents1 Groups.- 2 Graphs.- 3 Chemical Compounds.- 4 Asymptotic Evaluation of the Number of Combinations.- The Legacy of Pólya’s Paper: Fifty Years of Pólya Theory.- References.

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Book SynopsisThe first part of this book introduces the Schubert Cells and varieties of the general linear group Gl (k^(r+1)) over a field k according to Ehresmann geometric way. Smooth resolutions for these varieties are constructed in terms of Flag Configurations in k^(r+1) given by linear graphs called Minimal Galleries. In the second part, Schubert Schemes, the Universal Schubert Scheme and their Canonical Smooth Resolution, in terms of the incidence relation in a Tits relative building are constructed for a Reductive Group Scheme as in Grothendieck''s SGAIII. This is a topic where algebra and algebraic geometry, combinatorics, and group theory interact in unusual and deep ways.Table of ContentsGrassmannians and Flag Varieties. Schubert Cell Decomposition of Grassmannians and Flag Varieties. Resolution of Singularities of a Schubert Variety. The Singular Locus of a Schubert Variety. The Flag Complex. Configurations and Galleries Varieties. Configurations Varieties as Galleries Varieties. The Coxeter Complex. Minimal Generalized Galleries in a Coxeter Complex. Minimal Generalized Galleries in a Reductive Group Building. Parabolic Subgroups in a Reductive Group Scheme. Associated Schemes to the Relative Building. Incidence Type Schemes of the Relative Building. Smooth Resolutions of Schubert Schemes. Contracted Products and Galleries Configurations Schemes. Functoriality of Schubert Schemes Smooth Resolutions and Base Changes. About the Coxeter Complex. Generators and Relations and the Building of a Reductive Group.

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Book SynopsisBasics of Counting.- Induction and Pigeon Hole Principle.- Binomial Theorem and Binomial Identities Partitions.- Permutations.- Combinations and Cycles.- Generating Functions.- Recurrence Relations.- Inclusion Exclusion Principle.- Partial Order and Lattices.- Polya's Theory.- More on Counting.- Discrete Probability.- Basic Concepts.- Paths Connectedness.- Trees.- Connectivity.- Eulerian and Hamiltonian Graphs.- Planar Graphs.- Independent Sets.- Coverings and Matchings.- Graph Coloring.- Ramsey Numbers and Ramsey Graphs.- Spectral Properties of Graphs.- Directed Graphs and Graph Algorithms.

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Book SynopsisThe new 6th edition of Applied Combinatorics builds on the previous editions with more in depth analysis of computer systems in order to help develop proficiency in basic discrete math problem solving. As one of the most widely used book in combinatorial problems, this edition explains how to reason and model combinatorically while stressing the systematic analysis of different possibilities, exploration of the logical structure of a problem, and ingenuity. Although important uses of combinatorics in computer science, operations research, and finite probability are mentioned, these applications are often used solely for motivation. Numerical examples involving the same concepts use more interesting settings such as poker probabilities or logical games. This book is designed for use by students with a wide range of ability and maturity (sophomores through beginning graduate students). The stronger the students, the harder the exercises that can be assigned. The booTable of ContentsPrelude xi Part One Graph Theory 1 Chapter 1 Elements of Graph Theory 3 1.1 Graph Models 3 1.2 Isomorphism 14 1.3 Edge Counting 24 1.4 Planar Graphs 31 1.5 Summary and References 44 Supplementary Exercises 45 Chapter 2 Covering Circuits and Graph Coloring 49 2.1 Euler Cycles 49 2.2 Hamilton Circuits 56 2.3 Graph Coloring 68 2.4 Coloring Theorems 77 2.5 Summary and References 86 Supplement: Graph Model for Instant Insanity 87 Supplement Exercises 92 Chapter 3 Trees and Searching 93 3.1 Properties of Trees 93 3.2 Search Trees and Spanning Trees 103 3.3 The Traveling Salesperson Problem 113 3.4 Tree Analysis of Sorting Algorithms 121 3.5 Summary and References 125 Chapter 4 Network Algorithms 127 4.1 Shortest Paths 127 4.2 Minimum Spanning Trees 131 4.3 Network Flows 135 4.4 Algorithmic Matching 153 4.5 The Transportation Problem 164 4.6 Summary and References 174 Part Two Enumeration 177 Chapter 5 General Counting Methods for Arrangements and Selections 179 5.1 Two Basic Counting Principles 179 5.2 Simple Arrangements and Selections 189 5.3 Arrangements and Selections with Repetitions 206 5.4 Distributions 214 5.5 Binomial Identities 226 5.6 Summary and References 236 Supplement: Selected Solutions to Problems in Chapter 5 237 Chapter 6 Generating Functions 249 6.1 Generating Function Models 249 6.2 Calculating Coefficients of Generating Functions 256 6.3 Partitions 266 6.4 Exponential Generating Functions 271 6.5 A Summation Method 277 6.6 Summary and References 281 Chapter 7 Recurrence Relations 283 7.1 Recurrence Relation Models 283 7.2 Divide-and-Conquer Relations 296 7.3 Solution of Linear Recurrence Relations 300 7.4 Solution of Inhomogeneous Recurrence Relations 304 7.5 Solutions with Generating Functions 308 7.6 Summary and References 316 Chapter 8 Inclusion–Exclusion 319 8.1 Counting with Venn Diagrams 319 8.2 Inclusion–Exclusion Formula 328 8.3 Restricted Positions and Rook Polynomials 340 8.4 Summary and Reference 351 Part Three Additional Topics 353 Chapter 9 Polya’s Enumeration Formula 355 9.1 Equivalence and Symmetry Groups 355 9.2 Burnside’s Theorem 363 9.3 The Cycle Index 369 9.4 Polya’s Formula 375 9.5 Summary and References 382 Chapter 10 Games with Graphs 385 10.1 Progressively Finite Games 385 10.2 Nim-Type Games 393 10.3 Summary and References 400 Postlude 401 Appendix 415 A.1 Set Theory 415 A.2 Mathematical Induction 420 A.3 A Little Probability 423 A.4 The Pigeonhole Principle 427 A.5 Computational Complexity and NP-Completeness 430 Glossary of Counting and Graph Theory Terms 435 Bibliography 439 Solutions To Odd-Numbered Problems 441 Index 475

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Book SynopsisThe material has been extensively class-tested for over ten years at both the author's own university and other institutions. The book is uniquely organized into two main sections, one on Fibonacci Numbers and one on Catalan Numbers, each containing subsections that explore related topics in intricate detail.Table of ContentsPreface xi Part One The Fibonacci Numbers 1. Historical Background 3 2. The Problem of the Rabbits 5 3. The Recursive Definition 7 4. Properties of the Fibonacci Numbers 8 5. Some Introductory Examples 13 6. Compositions and Palindromes 23 7. Tilings: Divisibility Properties of the Fibonacci Numbers 33 8. Chess Pieces on Chessboards 40 9. Optics, Botany, and the Fibonacci Numbers 46 10. Solving Linear Recurrence Relations: The Binet Form for Fn 51 11. More on α and β: Applications in Trigonometry, Physics, Continued Fractions, Probability, the Associative Law, and Computer Science 65 12. Examples from Graph Theory: An Introduction to the Lucas Numbers 79 13. The Lucas Numbers: Further Properties and Examples 100 14. Matrices, The Inverse Tangent Function, and an Infinite Sum 113 15. The gcd Property for the Fibonacci Numbers 121 16. Alternate Fibonacci Numbers 126 17. One Final Example? 140 Part Two The Catalan Numbers 18. Historical Background 147 19. A First Example: A Formula for the Catalan Numbers 150 20. Some Further Initial Examples 159 21. Dyck Paths, Peaks, and Valleys 169 22. Young Tableaux, Compositions, and Vertices and Arcs 183 23. Triangulating the Interior of a Convex Polygon 192 24. Some Examples from Graph Theory 195 25. Partial Orders, Total Orders, and Topological Sorting 205 26. Sequences and a Generating Tree 211 27. Maximal Cliques, a Computer Science Example, and the Tennis Ball Problem 219 28. The Catalan Numbers at Sporting Events 226 29. A Recurrence Relation for the Catalan Numbers 231 30. Triangulating the Interior of a Convex Polygon for the Second Time 236 31. Rooted Ordered Binary Trees, Pattern Avoidance, and Data Structures 238 32. Staircases, Arrangements of Coins, The Handshaking Problem, and Noncrossing Partitions 250 33. The Narayana Numbers 268 34. Related Number Sequences: The Motzkin Numbers, The Fine Numbers, and The Schröder Numbers 282 35. Generalized Catalan Numbers 290 36. One Final Example? 296 Solutions for the Odd-Numbered Exercises 301 Index 355

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Book SynopsisContains a wealth of information previously scattered in research journals, conference proceedings and technical reports. Identifies more than 200 unsolved problems. Every problem is stated in a self-contained, extremely accessible format, followed by comments on its history, related results and literature.Table of ContentsPlanar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.

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Book SynopsisDescribes various analytical techniques and systems for the development, validation, quality control, purification, and physicochemical testing of combinatorial libraries. This book provides coverage of applications of Nuclear Magnetic Resonance (NMR), liquid chromatography/mass spectrometry (LC/MS), and Fourier Transform Infrared (FTIR).Trade Review"…a timely and valuable volume that would be an excellent addition to university libraries and the collections of individuals…" (E-STREAMS, February 2005) "...a useful book for chemists entering the field from either analytical or synthetic organic chemistry backgrounds.” (Angewandte Chemie International Edition, September 6, 2004) "This useful volume is a worthwhile addition to institute libraries as well as to the libraries students and researchers who are working in analytical chemistry, medicinal chemistry, organic chemistry, biotechnology…" (Energy Sources, August 2004)Table of ContentsPreface. Contributors. PART I: ANALYSIS FOR FEASIBILITY AND OPTIMIZATION OF LIBRARY SYNTHESIS. Chapter 1. Quantitative Analysis in Organic Synthesis with NMR (L. Lucas & C. Larive). Chapter 2. 19F Gel-phase NMR Spectroscopy for Reaction Monitoring and Quantification of Resin Loading (J. Salvino). Chapter 3. The Application of Single-Bead FTIR and Color Test for Reaction Monitoring and Building Block Validation in Combinatorial Library Sysnthesis(J. Cournoyer, et al.). Chapter 4. HR-MAS NMR Analysis of Compounds Attached to Polymer Supports (M. Guinó & Y. de Miguel). Chapter 5. Multivariate Tools for Real-Time Monitoring and Optimization of Combinatorial Materials and Process Conditions (R. Potyrailo, et al.). Chapter 6. Mass Spectrometry and Soluble Polymeric Supports (C. Enjalbal, et al.). PART II: HIGH-THROUGHPUT ANALYSIS FOR LIBRARY QUALITY CONTROL. Chapter 7. High-Throughput NMR Techniques for Combinatorial Chemical Library Analysis (T. Hou & D. Raftery). Chapter 8. Micellar Electrokinetic Chromatography as a Tool for Combinatorial Chemistry Analysis: Theory and Applications (P. Simms). Chapter 9. Characterization of Split-Pool Encoded Combinatorial Libraries (J. Zhang & W. Fitch). PART III: HIGH-THROUGHPUT PURIFICATION TO IMPROVE LIBRARY QUALITY. Cha pter 10. Strategies and Methods for Purifying Organic Compounds and Combinatorial Libraries (J. Zhao, et al.). Chapter 11. HTP of Combinatorial Chemistry Libraries (J. Hochlowski). Chapter 12. Practical HPLC in High Throughput Analysis and Purification (H. Gumm & R. God). PART IV: ANALYSIS FOR COMPOUND STABILITY AND DRUGABILITY. Chapter 13. Organic Compound Stability in Large, Diverse Phatmaceutical Screening Collection (K. Morand & X. Cheng). Chapter 14. Quartz Crystal Microbalance in Biomolecular Recognition (M. Tseng, et al.). Chapter 15. High-Throughput Physicochemical Profiling: Potential and Limitations (B. Faller). Chapter 16. Solubility in the Design of Combinatorial Libraries (C. Lipinski). Chapter 17. High-Throughput Determination of Log D Values by LC/MS Method (J. Villena, et al.). Index.

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Book SynopsisComplete coverage of the prediction approach to survey sampling in a single resource Prediction theory has been extremely influential in survey sampling for nearly three decades, yet research findings on this model-based approach are scattered in disparate areas of the statistical literature.Trade Review"Valliant...is joined...to dispel the perception of dichotomy between mainstream statistics...and survey sampling..." (SciTech Book News, Vol. 24, No. 4, December 2000) "The vast majority of the book is devoted to prediction of a population mean or total, and as such it forms a cohesive and comprehensive treatment of the subject." (Mathematical Reviews, Issue 2001j) "A highly recommended book which is an essential read for all research workers in this area." (Short Book Reviews - Publication of the Int. Statistical Institute, December 2001) "This book is a welcome addition to the subject of survey sampling." (Zentralblatt MATH, Vol. 964, 2001/14)Table of ContentsIntroduction to Prediction Theory. Prediction Theory Under the General Linear Model. Bias-Robustness. Robustness and Efficiency. Variance Estimation. Stratified Populations. Models with Qualitative Auxiliaries. Clustered Populations. Robust Variance Estimation in Two-Stage Cluster Sampling. Alternative Variance Estimation Methods. Special Topics and Open Questions. Appendices. Bibliography. Answers to Select Exercises. Indexes.

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Book SynopsisDiscrete optimization models are used to tackle a wide variety of problems in many fields, including operations research, management science, engineering, and mathematics. Written by two internationally recognized integer programming experts, this book presents the mathematical foundations, theory, and algorithms of discrete optimization methods.Table of ContentsFOUNDATIONS. The Scope of Integer and Combinatorial Optimization. Linear Programming. Graphs and Networks. Polyhedral Theory. Computational Complexity. Polynomial-Time Algorithms for Linear Programming. Integer Lattices. GENERAL INTEGER PROGRAMMING. The Theory of Valid Inequalities. Strong Valid Inequalities and Facets for Structured Integer Programs. Duality and Relaxation. General Algorithms. Special-Purpose Algorithms. Applications of Special- Purpose Algorithms. COMBINATORIAL OPTIMIZATION. Integral Polyhedra. Matching. Matroid and Submodular Function Optimization. References. Indexes.

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Book SynopsisTable of Contents*Frontmatter, pg. i*CONTENTS, pg. v*PREFACE, pg. vii*PROBLEMS IN COMBINATORIAL GROUP THEORY, pg. 3*POINCARE DUALITY GROUPS OF DIMENSION TWO ARE SURFACE GROUPS, pg. 35*HOW TO GENERALIZE ONE-RELATOR GROUP THEORY, pg. 53*GRAPHICAL THEORY OF AUTOMORPHISMS OF FREE GROUPS, pg. 79*PEAK REDUCTION AND AUTOMORPHISMS OF FREE GROUPS AND FREE PRODUCTS, pg. 107*NONSINGULAR EQUATIONS OF SMALL WEIGHT OVER GROUPS, pg. 121*GRAPH-THEORETIC LEMMA AND GROUP-EMBEDDINGS, pg. 145*THE TODD-COXETER PROCESS, USING GRAPHS, pg. 157*A SUBGROUP THEOREM FOR PREGROUPS, pg. 163*GROUPS WITH A RATIONAL CROSS-SECTION, pg. 175*ON THE RATIONAL GROWTH OF VIRTUALLY NILPOTENT GROUPS, pg. 185*SJOGREN'S THEOREM FOR DIMENSION SUBGROUPS - THE METABELIAN CASE, pg. 197*ON GROUP PRESENTATIONS, COPRODUCTS AND INVERSES, pg. 213*ON COMPLEXES DOMINATED BY A TWO-COMPLEX, pg. 221*SUBCOMPLEXES OF TWO-COMPLEXES AND PROJECTIVE CROSSED MODULES, pg. 255*LENGTH FUNCTIONS OF GROUP ACTIONS ON A-TREES, pg. 265*RESIDUAL FINITENESS FOR 3-MANIFOLDS, pg. 379*THE NIELSEN-THURSTON THEORY OF SURFACE AUTOMORPHISMS, pg. 397*WHITEHEAD GROUPS OF CERTAIN HYPERBOLIC MANIFOLDS, II, pg. 415*CHARACTERIZATION OF FINITE SUBGROUPS OF THE MAPPING-CLASS GROUP, pg. 433*A SEQUENCE OF PSEUDO-ANOSOV DIFFEOMORPHISMS, pg. 443*DEHN'S ALGORITHM REVISITED, WITH APPLICATIONS TO SIMPLE CURVES ON SURFACES, pg. 451*PATHS OF GEODESICS AND GEOMETRIC INTERSECTION NUMBERS: I, pg. 479*PATHS OF GEODESICS AND GEOMETRIC INTERSECTION NUMBERS: II, pg. 501*SELECTED PROBLEMS, pg. 545*Backmatter, pg. 552

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Book SynopsisApplies model theoretic methods to the study of certain finite permutation groups, the automorphism groups of structures for a fixed finite language with a bounded number of orbits on 4-tuples. This book treats the general case by developing analogs of the model theoretic methods of geometric stability theory.

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Book SynopsisUses the phenomenon called 'six degrees of separation' as a prelude to a more general exploration: under what conditions can a small world arise in any kind of network? This book is intended for a variety of fields, including physics and mathematics, as well as sociology, economics, and biology.Trade Review"An engaging and informative introduction."--Science "Playfully and clearly written... [Watts] uses examples adroitly, and mixes abstract theory with real-world anecdotes with superb skill... I have not enjoyed reading a book this much in a long time."--Peter Kareiva, Quarterly Review of Biology "[Small Worlds] will be seized on by those seeking a first rough map of this fascinating new mathematical land. Those entering can expect to find some amazing connections between areas of research with apparently nothing in common, such as neurology to business studies. But then, it's a small world."--Robert Matthews, New Scientist "Informally written and aimed at a wide audience, this book shows how mathematics yields new vistas on ubiquitous and seemingly familiar aspects of our world."--ChoiceTable of ContentsPREFACE xiii 1 Kevin Bacon, the Small World, and Why It All Matters 3 PART I STRUCTURE 9 2 An Overview of the Small-World Phenomenon 11 2.1 Social Networks and the Small World 11 2.1.1 A Brief History of the Small World 12 2.1.2 Difficulties with the Real World 20 2.1.3 Reframing the Question to Consider All Worlds 24 2.2 Background on the Theory of Graphs 25 2.2.1 Basic Definitions 25 2.2.2 Length and Length Scaling 27 2.2.3 Neighbourhoods and Distribution Sequences 31 2.2.4 Clustering 32 2.2.5 "Lattice Graphs" and Random Graphs 33 2.2.6 Dimension and Embedding of Graphs 39 3 Big Worlds and Small Worlds: Models of Graphs 41 3.1 Relational Graphs 42 3.1.1 a-Graphs 42 3.1.2 A Stripped-Down Model: B-Graphs 66 3.1.3 Shortcuts and Contractions: Model Invariance 70 3.1.4 Lies, Damned Lies, and (More) Statistics 87 3.2 Spatial Graphs 91 3.2.1 Uniform Spatial Graphs 93 3.2.2 Gaussian Spatial Graphs 98 3.3 Main Points in Review 100 4 Explanations and Ruminations 101 4.1 Going to Extremes 101 4.1.1 The Connected-Caveman World 102 4.1.2 Moore Graphs as Approximate Random Graphs 109 4.2 Transitions in Relational Graphs 114 4.2.1 Local and Global Length Scales 114 4.2.2 Length and Length Scaling 116 4.2.3 Clustering Coefficient 117 4.2.4 Contractions 118 4.2.5 Results and Comparisons with B-Model 120 4.3 Transitions in Spatial Graphs 127 4.3.1 Spatial Length versus Graph Length 127 4.3.2 Length and Length Scaling 128 4.3.3 Clustering 130 4.3.4 Results and Comparisons 132 4.4 Variations on Spatial and Relational Graphs 133 4.5 Main Points in Review 136 5 "It's a Small World after All": Three Real Graphs 138 5.1 Making Bacon 140 5.1.1 Examining the Graph 141 5.1.2 Comparisons 143 5.2 The Power of Networks 147 5.2.1 Examining the System 147 5.2.2 Comparisons 150 5.3 A Worm's Eye View 153 5.3.1 Examining the System 154 5.3.2 Comparisons 156 5.4 Other Systems 159 5.5 Main Points in Review 161 PART II DYNAMICS 163 6 The Spread of Infectious Disease in Structured Populations 165 6.1 A Brief Review of Disease Spreading 166 6.2 Analysis and Results 168 6.2.1 Introduction of the Problem 168 6.2.2 Permanent-Removal Dynamics 169 6.2.3 Temporary-Removal Dynamics 176 6.3 Main Points in Review 180 7 Global Computation in Cellular Automata 181 7.1 Background 181 7.1.1 Global Computation 184 7.2 Cellular Automata on Graphs 187 7.2.1 Density Classification 187 7.2.2 Synchronisation 195 7.3 Main Points in Review 198 8 Cooperation in a Small World: Games on Graphs 199 8.1 Background 199 8.1.1 The Prisoner's Dilemma 200 8.1.2 Spatial Prisoner's Dilemma 204 8.1.3 N-Player Prisoner's Dilemma 206 8.1.4 Evolution of Strategies 207 8.2 Emergence of Cooperation in a Homogeneous Population 208 8.2.1 Generalised Tit-for-Tat 209 8.2.2 Win-Stay, Lose-Shift 214 8.3 Evolution of Cooperation in a Heterogeneous Population 219 8.4 Main Points in Review 221 9 Global Synchrony in Populations of Coupled Phase Oscillators 223 9.1 Background 223 9.2 Kuramoto Oscillators on Graphs 228 9.3 Main Points in Review 238 10 Conclusions 240 NOTES 243 BIBLIOGRAPHY 249 INDEX 257

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Book SynopsisContains a collection of mathematical applications of linear algebra, mainly in combinatorics, geometry, and algorithms. Each chapter covers a single main result with motivation and full proof, and assumes only a modest background in linear algebra. The topics include Hamming codes, the matrix-tree theorem, the Lovász bound on the Shannon capacity, and a counterexample to Borsuk's conjecture.Trade ReviewFinding examples of "linear algebra in action" that are both accessible and convincing is difficult. Thirty-three Miniatures is an attempt to present some usable examples. . . . For me, the biggest impact of the book came from noticing the tools that are used. Many linear algebra textbooks, including the one I use, delay discussion of inner products and transpose matrices till later in the course, which sometimes means they don't get discussed at all. Seeing how often the transpose matrix shows up in Matousek's miniatures made me realize space must be made for it. Similarly, the theorem relating the rank of the product of two matrices to the ranks of the factors plays a big role here. Most linear algebra instructors would benefit from this kind of insight. . . . Thirty-three Miniatures would be an excellent book for an informal seminar offered to students after their first linear algebra course. It may also be the germ of many interesting undergraduate talks. And it's fun as well." - Fernando Q. Gouvêa, MAA Reviews"[This book] is an excellent collection of clever applications of linear algebra to various areas of (primarily) discrete/combinatiorial mathematics. ... The style of exposition is very lively, with fairly standard usage of terminologies and notations. ... Highly recommended." - ChoiceTable of Contents Preface Notation Fibonacci numbers, quickly Fibonacci numbers, the formula The clubs of Oddtown Same-size intersections Error-correcting codes Odd distances Are these distances Euclidean? Packing complete bipartite graphs Equiangular lines Where is the triangle? Checking matrix multiplication Tiling a rectangle by squares Three Petersens are not enough Petersen, Hoffman–Singleton, and maybe 57 Only two distances Covering a cube minus one vertex Medium-size intersection is hard to avoid On the difficulty of reducing the diameter The end of the small coins Walking in the yard Counting spanning trees In how many ways can a man tile a board? More bricks—more walls? Perfect matchings and determinants Turning a ladder over a finite field Counting compositions Is it associative? The secret agent and umbrella Shannon capacity of the union: a tale of two fields Equilateral sets Cutting cheaply using eigenvectors Rotating the cube Set pairs and exterior products Index

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