Combinatorics and graph theory Books

Book SynopsisThis title provides a comprehensive survey over the subject of probabilistic combinatorial optimization, discussing probabilistic versions of some of the most paradigmatic combinatorial problems on graphs, such as the maximum independent set, the minimum vertex covering, the longest path and the minimum coloring. Those who possess a sound knowledge of the subject mater will find the title of great interest, but those who have only some mathematical familiarity and knowledge about complexity and approximation theory will also find it an accessible and informative read.Table of ContentsPreface 11 Chapter 1. A Short Insight into Probabilistic Combinatorial Optimization 15 1.1. Motivations and applications 15 1.2. A formalism for probabilistic combinatorial optimization 19 1.3. The main methodological issues dealing with probabilistic combinatorial optimization 24 1.3.1. Complexity issues 24 1.3.1.1. Membership in NPO is not always obvious 24 1.3.1.2. Complexity of deterministic vs. complexity of probabilistic optimization problems 24 1.3.2. Solution issues 26 1.3.2.1. Characterization of optimal a priori solutions 26 1.3.2.2. Polynomial subcases 28 1.3.2.3. Exact solutions and polynomial approximation issues 29 1.4. Miscellaneous and bibliographic notes 31 FIRST PART. PROBABILISTIC GRAPH-PROBLEMS 35 Chapter 2. The Probabilistic Maximum Independent Set 37 2.1. The modification strategies and a preliminary result 39 2.1.1. Strategy M1 39 2.1.2. Strategies M2 and M3 39 2.1.3. Strategy M4 41 2.1.4. Strategy M5 41 2.1.5. A general mathematical formulation for the five functionals 42 2.2. PROBABILISTIC MAX INDEPENDENT SET1 44 2.2.1. Computing optimal a priori solutions 44 2.2.2. Approximating optimal solutions 45 2.2.3. Dealing with bipartite graphs 46 2.3. PROBABILISTIC MAX INDEPENDENT SET2 and 3 47 2.3.1. Expressions for E(G, S, M2) and E(G, S, M3) 47 2.3.2. An upper bound for the complexity of E(G, S, M2) 48 2.3.3. Bounds for E(G, S, M2) 49 2.3.4. Approximating optimal solutions 51 2.3.4.1. Using argmax{_vi∈S pi} as an a priori solution 51 2.3.4.2. Using approximations of MAX INDEPENDENT SET 53 2.3.5. Dealing with bipartite graphs 53 2.4. PROBABILISTIC MAX INDEPENDENT SET4 55 2.4.1. An expression for E(G, S, M4) 55 2.4.2. Using S∗ or argmax{_vi∈S pi} as an a priori solution 56 2.4.3. Dealing with bipartite graphs 57 2.5. PROBABILISTIC MAX INDEPENDENT SET5 58 2.5.1. In general graphs 58 2.5.2. In bipartite graphs 60 2.6. Summary of the results 61 2.7. Methodological questions 63 2.7.1. Maximizing a criterion associated with gain 65 2.7.1.1. The minimum gain criterion 65 2.7.1.2. The maximum gain criterion 66 2.7.2. Minimizing a criterion associated with regret 68 2.7.2.1. The maximum regret criterion 68 2.7.3. Optimizing expectation 70 2.8. Proofs of the results 71 2.8.1. Proof of Proposition 2.1 71 2.8.2. Proof of Theorem 2.6 74 2.8.3. Proof of Proposition 2.3 77 2.8.4. Proof of Theorem 2.13 78 Chapter 3. The Probabilistic Minimum Vertex Cover 81 3.1. The strategies M1, M2 and M3 and a general preliminary result 82 3.1.1. Specification of M1, M2 and M3 82 3.1.1.1. Strategy M1 82 3.1.1.2. Strategy M2 83 3.1.1.3. Strategy M3 83 3.1.2. A first expression for the functionals 84 3.2. PROBABILISTIC MIN VERTEX COVER1 84 3.3. PROBABILISTIC MIN VERTEX COVER2 86 3.4. PROBABILISTIC MIN VERTEX COVER3 87 3.4.1. Building E(G, C, M3) 87 3.4.2. Bounds for E(G, C, M3) 88 3.5. Some methodological questions 89 3.6. Proofs of the results 91 3.6.1. Proof of Theorem 3.3 91 3.6.2. On the the bounds obtained in Theorem 3.3 93 Chapter 4. The Probabilistic Longest Path 99 4.1. Probabilistic longest path in terms of vertices 100 4.2. Probabilistic longest path in terms of arcs 102 4.2.1. An interesting algebraic expression 104 4.2.2. Metric PROBABILISTIC ARC WEIGHTED LONGEST PATH 105 4.3. Why the strategies used are pertinent 109 4.4. Proofs of the results 110 4.4.1. Proof of Theorem 4.1 110 4.4.2. Proof of Theorem 4.2 112 4.4.3. An algebraic proof for Theorem 4.3 114 4.4.4. Proof of Lemma 4.1 116 4.4.5. Proof of Lemma 4.2 117 4.4.6. Proof of Theorem 4.4 117 Chapter 5. Probabilistic Minimum Coloring 125 5.1. The functional E(G,C) 127 5.2. Basic properties of probabilistic coloring 131 5.2.1. Properties under non-identical vertex-probabilities 131 5.2.2. Properties under identical vertex-probabilities 131 5.3. PROBABILISTIC MIN COLORING in general graphs 132 5.3.1. The complexity of probabilistic coloring 132 5.3.2. Approximation 132 5.3.2.1. The main result 132 5.3.2.2. Further approximation results 137 5.4. PROBABILISTIC MIN COLORING in bipartite graphs 139 5.4.1. A basic property 139 5.4.2. General bipartite graphs 141 5.4.3. Bipartite complements of bipartite matchings 147 5.4.4. Trees 151 5.4.5. Cycles 154 5.5. Complements of bipartite graphs 155 5.6. Split graphs 156 5.6.1. The complexity of PROBABILISTIC MIN COLORING 156 5.6.2. Approximation results 159 5.7. Determining the best k-coloring in k-colorable graphs 164 5.7.1. Bipartite graphs 164 5.7.1.1. PROBABILISTIC MIN 3-COLORING 164 5.7.1.2. PROBABILISTIC MIN k-COLORING fork > 3 168 5.7.1.3. Bipartite complements of bipartite matchings 171 5.7.2. The complements of bipartite graphs 171 5.7.3. Approximation in particular classes of graphs 174 5.8. Comments and open problems 175 5.9. Proofs of the different results 178 5.9.1. Proof of [5.5] 178 5.9.2. Proof of [5.4] 179 5.9.3. Proof of Property 5.1 180 5.9.4. Proof of Proposition 5.2 181 5.9.5. Proof of Lemma 5.11 183 SECOND PART. STRUCTURAL RESULTS 185 Chapter 6. Classification of Probabilistic Graph-problems 187 6.1. When MS is feasible 187 6.1.1. The a priori solution is a subset of the initial vertex-set 188 6.1.2. The a priori solution is a collection of subsets of the initial vertex-set 191 6.1.3. The a priori solution is a subset of the initial edge-set 193 6.2. When application of MS itself does not lead to feasible solutions 198 6.2.1. The functional associated with MSC 198 6.2.2. Applications 199 6.2.2.1. The a priori solution is a cycle 200 6.2.2.2. The a priori solution is a tree 201 6.3. Some comments 205 6.4. Proof of Theorem 6.4 206 Chapter 7. A Compendium of Probabilistic NPO Problems on Graphs 211 7.1. Covering and partitioning 214 7.1.1. MIN VERTEX COVER 214 7.1.2. MIN COLORING 214 7.1.3. MAX ACHROMATIC NUMBER 215 7.1.4. MIN DOMINATING SET 215 7.1.5. MAX DOMATIC PARTITION 216 7.1.6. MIN EDGE-DOMINATING SET 216 7.1.7. MIN INDEPENDENT DOMINATING SET 217 7.1.8. MIN CHROMATIC SUM 217 7.1.9. MIN EDGE COLORING 218 7.1.10. MIN FEEDBACK VERTEX-SET 219 7.1.11. MIN FEEDBACK ARC-SET 220 7.1.12. MAX MATCHING 220 7.1.13. MIN MAXIMAL MATCHING 220 7.1.14. MAX TRIANGLE PACKING 220 7.1.15. MAX H-MATCHING 221 7.1.16. MIN PARTITION INTO CLIQUES 222 7.1.17. MIN CLIQUE COVER 222 7.1.18. MIN k-CAPACITED TREE PARTITION 222 7.1.19. MAX BALANCED CONNECTED PARTITION 223 7.1.20. MIN COMPLETE BIPARTITE SUBGRAPH COVER 223 7.1.21. MIN VERTEX-DISJOINT CYCLE COVER 223 7.1.22. MIN CUT COVER 224 7.2. Subgraphs and supergraphs 224 7.2.1. MAX INDEPENDENT SET 224 7.2.2. MAX CLIQUE 224 7.2.3. MAX INDEPENDENT SEQUENCE 225 7.2.4. MAX INDUCED SUBGRAPH WITH PROPERTY π 225 7.2.5. MIN VERTEX DELETION TO OBTAIN SUBGRAPH WITH PROPERTY π 225 7.2.6. MIN EDGE DELETION TO OBTAIN SUBGRAPH WITH PROPERTY π 226 7.2.7. MAX CONNECTED SUBGRAPH WITH PROPERTY π 226 7.2.8. MIN VERTEX DELETION TO OBTAIN CONNECTED SUBGRAPH WITH PROPERTY π 226 7.2.9. MAX DEGREE-BOUNDED CONNECTED SUBGRAPH 226 7.2.10. MAX PLANAR SUBGRAPH 227 7.2.11. MIN EDGE DELETION k-PARTITION 227 7.2.12. MAX k-COLORABLE SUBGRAPH 227 7.2.13. MAX SUBFOREST 228 7.2.14. MAX EDGE SUBGRAPH or DENSE k-SUBGRAPH 228 7.2.15. MIN EDGE K-SPANNER 228 7.2.16. MAX k-COLORABLE INDUCED SUBGRAPH 229 7.2.17. MIN EQUIVALENT DIGRAPH 229 7.2.18. MIN CHORDAL GRAPH COMPLETION 229 7.3. Iso- and other morphisms 229 7.3.1. MAX COMMON SUBGRAPH 229 7.3.2. MAX COMMON INDUCED SUBGRAPH 230 7.3.3. MAX COMMON EMBEDDED SUBTREE 230 7.3.4. MIN GRAPH TRANSFORMATION 230 7.4. Cuts and connectivity 231 7.4.1. MAX CUT 231 7.4.2. MAX DIRECTED CUT 231 7.4.3. MIN CROSSING NUMBER 231 7.4.4. MAX k-CUT 232 7.4.5. MIN k-CUT 233 7.4.6. MIN NETWORK INHIBITION ON PLANAR GRAPHS 233 7.4.7. MIN VERTEX k-CUT 234 7.4.8. MIN MULTI-WAY CUT 234 7.4.9. MIN MULTI-CUT 234 7.4.10. MIN RATIO-CUT 235 7.4.11. MIN b-BALANCED CUT 236 7.4.12. MIN b-VERTEX SEPARATOR 236 7.4.13. MIN QUOTIENT CUT 236 7.4.14. MIN k-VERTEX CONNECTED SUBGRAPH 236 7.4.15. MIN k-EDGE CONNECTED SUBGRAPH 237 7.4.16. MIN BICONNECTIVITY AUGMENTATION 237 7.4.17. MIN STRONG CONNECTIVITY AUGMENTATION 237 7.4.18. MIN BOUNDED DIAMETER AUGMENTATION 237 Appendix A. Mathematical Preliminaries 239 A.1. Sets, relations and functions 239 A.2. Basic concepts from graph-theory 242 A.3. Elements from discrete probabilities 246 Appendix B. Elements of the Complexity and the Approximation Theory 249 B.1. Problem, algorithm, complexity 249 B.2. Some notorious complexity classes 250 B.3. Reductions and NP-completeness 251 B.4. Approximation of NP-hard problems 252 Bibliography 255 Index 261

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Book SynopsisThe 2017 PIMS-CRM Summer School in Probability was held at the Pacific Institute for the Mathematical Sciences (PIMS) at the University of British Columbia in Vancouver, Canada, during June 5-30, 2017. It had 125 participants from 20 different countries, and featured two main courses, three mini-courses, and twenty-nine lectures. The lecture notes contained in this volume provide introductory accounts of three of the most active and fascinating areas of research in modern probability theory, especially designed for graduate students entering research: Scaling limits of random trees and random graphs (Christina Goldschmidt) Lectures on the Ising and Potts models on the hypercubic lattice (Hugo Duminil-Copin) Extrema of the two-dimensional discrete Gaussian free field (Marek Biskup) Each of these contributions provides a thorough introduction that will be of value to beginners and experts alike.Table of ContentsScaling Limits of Random Trees and Random Graphs (C. Goldschmidt).- Lectures on the Ising and Potts Models on the Hypercubic Lattice (H. Duminil-Copin).- Extrema of the Two-Dimensional Discrete Gaussian Free Field (M. Biskup).

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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 volume comprises 17 contributions that present advanced topics in graph domination, featuring open problems, modern techniques, and recent results. The book is divided into 3 parts. The first part focuses on several domination-related concepts: broadcast domination, alliances, domatic numbers, dominator colorings, irredundance in graphs, private neighbor concepts, game domination, varieties of Roman domination and spectral graph theory. The second part covers domination in hypergraphs, chessboards, and digraphs and tournaments. The third part focuses on the development of algorithms and complexity of signed, minus and majority domination, power domination, and alliances in graphs. The third part also includes a chapter on self-stabilizing algorithms. Of extra benefit to the reader, the first chapter includes a glossary of commonly used terms.The book is intended to provide a reference for established researchers in the fields of domination and graph theory and graduate students who wish to gain knowledge of the topics covered as well as an overview of the major accomplishments and proof techniques used in the field. Table of Contents1. Glossary of Common Terms (Haynes).- Part 1. Related Parameters: 2. Broadcast Domination in Graphs (MacGillivray).- 3. Alliances and Related Domination Parameters (Haynes).- 4. Fractional Domatic, Idomatic and Total Domatic Numbers of a Graph (Goddard).- 5. Dominator and Total Dominator Colorings in Graphs (Henning).- 6. Irredundance (Mynhardt).- 7. The Private Neighbor Concept (McRae).- 8. An Introduction to Game Domination in Graphs (Henning).- 9. Domination and Spectral Graph Theory (Hoppen).- 10. Varieties of Roman Domination (Chellali).- Part 2. Domination in Selected Graph Families: 11. Domination and Total Domination in Hypergraphs (Yeo).- 12. Domination in Chessboards (Hedetniemi).- 13. Domination in Digraphs (Haynes).- Part 3. Algorithms and Complexity: 14. Algorithms and Complexity of Signed, Minus and Majority Domination (McRae).- 15. Algorithms and Complexity of Power Domination in Graphs (Mohan).- 16. Self-Stabilizing Domination Algorithms (Hedetniemi).- 17. Algorithms and Complexity of Alliances in Graphs (Hedetniemi)

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Book SynopsisThis volume comprises 17 contributions that present advanced topics in graph domination, featuring open problems, modern techniques, and recent results. The book is divided into 3 parts. The first part focuses on several domination-related concepts: broadcast domination, alliances, domatic numbers, dominator colorings, irredundance in graphs, private neighbor concepts, game domination, varieties of Roman domination and spectral graph theory. The second part covers domination in hypergraphs, chessboards, and digraphs and tournaments. The third part focuses on the development of algorithms and complexity of signed, minus and majority domination, power domination, and alliances in graphs. The third part also includes a chapter on self-stabilizing algorithms. Of extra benefit to the reader, the first chapter includes a glossary of commonly used terms.The book is intended to provide a reference for established researchers in the fields of domination and graph theory and graduate students who wish to gain knowledge of the topics covered as well as an overview of the major accomplishments and proof techniques used in the field. Table of Contents1. Glossary of Common Terms (Haynes).- Part 1. Related Parameters: 2. Broadcast Domination in Graphs (MacGillivray).- 3. Alliances and Related Domination Parameters (Haynes).- 4. Fractional Domatic, Idomatic and Total Domatic Numbers of a Graph (Goddard).- 5. Dominator and Total Dominator Colorings in Graphs (Henning).- 6. Irredundance (Mynhardt).- 7. The Private Neighbor Concept (McRae).- 8. An Introduction to Game Domination in Graphs (Henning).- 9. Domination and Spectral Graph Theory (Hoppen).- 10. Varieties of Roman Domination (Chellali).- Part 2. Domination in Selected Graph Families: 11. Domination and Total Domination in Hypergraphs (Yeo).- 12. Domination in Chessboards (Hedetniemi).- 13. Domination in Digraphs (Haynes).- Part 3. Algorithms and Complexity: 14. Algorithms and Complexity of Signed, Minus and Majority Domination (McRae).- 15. Algorithms and Complexity of Power Domination in Graphs (Mohan).- 16. Self-Stabilizing Domination Algorithms (Hedetniemi).- 17. Algorithms and Complexity of Alliances in Graphs (Hedetniemi)

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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 SynopsisIn the present era dominated by computers, graph theory has come into its own as an area of mathematics, prominent for both its theory and its applications. One of the richest and most studied types of graph structures is that of the line graph, where the focus is more on the edges of a graph than on the vertices. A subject worthy of exploration in itself, line graphs are closely connected to other areas of mathematics and computer science. This book is unique in its extensive coverage of many areas of graph theory applicable to line graphs. The book has three parts. Part I covers line graphs and their properties, while Part II looks at features that apply specifically to directed graphs, and Part III presents generalizations and variations of both line graphs and line digraphs.Line Graphs and Line Digraphs is the first comprehensive monograph on the topic. With minimal prerequisites, the book is accessible to most mathematicians and computer scientists who have had an introduction graph theory, and will be a valuable reference for researchers working in graph theory and related fields.Table of ContentsPart I Line Graphs.- 1 Fundamentals of Line Graphs.- 2 Line Graph Isomorphisms.- 3 Characterization of Line Graphs.- 4 Spectral Properties of Line Graphs.- 5 Planarity of Line Graphs.- 6 Connectivity of Line Graphs.- 7 Tranversability in Line Graphs.- 8 Colorability in Line Graphs.- 9 Distance and Transitivity in Line Graphs.- Part II Line Digraphs.- 10 Fundamentals of Line Digraphs.- 11 Characterizations of Line Digraphs.- 12 Iterated Line Digraphs.- Part III Generalizations.- 13 Total Graphs and Total Digraphs.- 14 Path Graphs and Path Digraphs.- 15 Super Line Graphs and Super Line Digraphs.- 16 Line Graphs of Signed Graphs.- 17 The Krausz Dimension of Graph.- Reference. Index of Names.- Index of Definitions.

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Book SynopsisThis book provides an overview of many interesting properties of natural numbers, demonstrating their applications in areas such as cryptography, geometry, astronomy, mechanics, computer science, and recreational mathematics. In particular, it presents the main ideas of error-detecting and error-correcting codes, digital signatures, hashing functions, generators of pseudorandom numbers, and the RSA method based on large prime numbers. A diverse array of topics is covered, from the properties and applications of prime numbers, some surprising connections between number theory and graph theory, pseudoprimes, Fibonacci and Lucas numbers, and the construction of Magic and Latin squares, to the mathematics behind Prague’s astronomical clock. Introducing a general mathematical audience to some of the basic ideas and algebraic methods connected with various types of natural numbers, the book will provide invaluable reading for amateurs and professionals alike.Trade Review“This is a nicely written book that can be read with profit by undergraduates with a background in elementary number theory, and it may serve as bedtime reading for the experts.” (Franz Lemmermeyer, zbMATH 1486.11001, 2022)“It also has more applications than is usual in either kind of book. Apart from that it is very conventional and has the theorems and proofs that you would expect. … The book does cover a number of newer discoveries … .” (Allen Stenger, MAA Reviews, December 27, 2021)Table of ContentsForeword.- 1. Divisibility and Congruence.- 2. Prime and Composite Numbers.- 3. Properties of Prime Numbers.- 4. Special Types of Primes.- 5. On a Connection of Number Theory with Graph Theory.- 6. Pseudoprimes.- 7. Fibonacci and Lucas Numbers.- 8. Further Special Types of Integers.- 9. Magic and Latin Squares.- 10. The Mathematics Behind Prague's Horologe.- 11. Applications of Primes.- 12. Further Applications of Number Theory.- Tables.- References.

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Book SynopsisScientific visualization has always been an integral part of discovery, starting first with simplified drawings of the pre-Enlightenment and progressing to present day. Mathematical formalism often supersedes visual methods, but their use is at the core of the mental process. As historical examples, a spatial description of flow led to electromagnetic theory, and without visualization of crystals, structural chemistry would not exist. With the advent of computer graphics technology, visualization has become a driving force in modern computing. A Concise Introduction to Scientific Visualization – Past, Present, and Future serves as a primer to visualization without assuming prior knowledge. It discusses both the history of visualization in scientific endeavour, and how scientific visualization is currently shaping the progress of science as a multi-disciplinary domain. Table of ContentsPreface.- Early Visual Models.- Illustration and Analysis.- Scientific Visualization in the 19th Century.- A Convergence with Computer Science.- Recent Developments.- The Future.- Bibliography

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Book SynopsisThis book describes Python3 programming resources for implementing decision aiding algorithms in the context of a bipolar-valued outranking approach. These computing resources, made available under the name Digraph3, are useful in the field of Algorithmic Decision Theory and more specifically in outranking-based Multiple-Criteria Decision Aiding (MCDA). The first part of the book presents a set of tutorials introducing the Digraph3 collection of Python3 modules and its main objects, such as bipolar-valued digraphs and outranking digraphs. In eight methodological chapters, the second part illustrates multiple-criteria evaluation models and decision algorithms. These chapters are largely problem-oriented and demonstrate how to edit a new multiple-criteria performance tableau, how to build a best choice recommendation, how to compute the winner of an election and how to make rankings or ratings using incommensurable criteria. The book’s third part presents three real-world decision case studies, while the fourth part addresses more advanced topics, such as computing ordinal correlations between bipolar-valued outranking digraphs, computing kernels in bipolar-valued digraphs, testing for confidence or stability of outranking statements when facing uncertain or solely ordinal criteria significance weights, and tempering plurality tyranny effects in social choice problems. The fifth and last part is more specifically focused on working with undirected graphs, tree graphs and forests. The closing chapter explores comparability, split, interval and permutation graphs. The book is primarily intended for graduate students in management sciences, computational statistics and operations research. The chapters presenting algorithms for ranking multicriteria performance records will be of computational interest for designers of web recommender systems. Similarly, the relative and absolute quantile-rating algorithms, discussed and illustrated in several chapters, will be of practical interest to public and private performance auditors. Trade Review“The book … guides reader through each topic through sub-chapters and using links, even leading to the python.org interface for related content. From the moment reader download the Diagraph3 software, it helps reader understand with copyable code snippets and separate warning Note content. It guides reader through the operation of the algorithm through concrete, solved example.” (Rózsa Horváth–Bokor, zbMATH 1497.91004, 2022)Table of ContentsPart I: Introduction to the DIGRAPH3 Python Resources.- 1. Working with the DIGRAPH3 Python Resources.- 2. Working with Bipolar-Valued Digraphs.- 3. Working with Outranking Digraphs.- Part II: Evaluation Models and Decision Algorithms.- 4. Building a Best Choice Recommendation.- 5. How to Create a New Multiple-Criteria Performance Tableau.- 6. Generating Random Performance Tableaux.- 7. Who Wins the Election?.- 8. Ranking with Multiple Incommensurable Criteria.- 9. Rating by Sorting into Relative Performance Quantiles.- 10. Rating-by-Ranking with Learned Performance Quantile Norms.- 11. HPC Ranking of Big Performance Tableaux.- Part III: Evaluation and Decision Case Studies.- 12. Alice’s Best Choice: A Selection Case Study.- 13. The Best Academic Computer Science Depts: A Ranking Case Study.- 14. The Best Students, Where Do They Study? A Rating Case Study.- 15. Exercises.- Part IV: Advanced Topics.- 16. On Measuring the Fitness of a Multiple-Criteria Ranking.- 17. On Computing Digraph Kernels.- 18. On Confident Outrankings with Uncertain Criteria Significance Weights.- 19. Robustness Analysis of Outranking Digraphs.- 20. Tempering Plurality Tyranny Effects in Social Choice.- Part V: Working with Undirected Graphs.- 21. Bipolar-Valued Undirected Graphs.- 22. On Tree Graphs and Graph Forests.- 23. About Split, Comparability, Interval, and Permutation Graphs.

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Book SynopsisThis book provides a complete introduction into spatial networks. It offers the mathematical tools needed to characterize these structures and how they evolve in time and presents the most important models of spatial networks.The book puts a special emphasis on analyzing complex systems which are organized under the form of networks where nodes and edges are embedded in space. In these networks, space is relevant, and topology alone does not contain all the information. Characterizing and understanding the structure and the evolution of spatial networks is thus crucial for many different fields, ranging from urbanism to epidemiology.This subject is therefore at the crossroad of many fields and is of potential interest to a broad audience comprising physicists, mathematicians, engineers, geographers or urbanists. In this book, the author has expanded his previous book ("Morphogenesis of Spatial Networks") to serve as a textbook and reference on this topic for a wide range of students and professional researchers.Trade Review“This book, written by a statistical physicist, has the style of a survey rather than a mathematics textbook. It outlines numerous results (around 500 papers are cited) via descriptions of statistics and models and back-of-envelope calculations and simulation results together with real-world data examples. It is fairly technically undemanding, meaning mostly accessible to an advanced undergraduate mathematics student. … This book succeeds admirably in its stated ‘complete Introduction’ goal … .” (David J. Aldous, Mathematical Reviews, October, 2022)Table of Contents0. IntroductionI. Characterization 1. Planar graphs 2. Simple measures 3. Betweenness centrality 4. Simplicity and Entropy 5. The shape of shortest paths 6. Spatial dominance 7. Typology of spatial networks 8. Time evolution of spatial networksII. Models 1. Spatial random graphs 2. Tesselations of the plane 3. Random geometric graphs 4. beta-skeletons 5. Loops and branches 6. Optimal networks 7. Growing networks 8. Greedy models 9. Transitions in spatial networks 10. Multilayer networksIII. Discussion and perspectives

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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 SynopsisThis textbook covers a diversity of topics in graph and network theory, both from a theoretical standpoint, and from an applied modelling point of view. Mathematica® is used to demonstrate much of the modelling aspects. Graph theory and model building tools are developed in tandem with effective techniques for solving practical problems via computer implementation. The book is designed with three primary readerships in mind. Individual syllabi or suggested sequences for study are provided for each of three student audiences: mathematics, applied mathematics/operations research, and computer science. In addition to the visual appeal of each page, the text contains an abundance of gems. Most chapters open with real-life problem descriptions which serve as motivation for the theoretical development of the subject matter. Each chapter concludes with three different sets of exercises. The first set of exercises are standard and geared toward the more mathematically inclined reader. Many of these are routine exercises, designed to test understanding of the material in the text, but some are more challenging. The second set of exercises is earmarked for the computer technologically savvy reader and offer computer exercises using Mathematica. The final set consists of larger projects aimed at equipping those readers with backgrounds in the applied sciences to apply the necessary skills learned in the chapter in the context of real-world problem solving. Additionally, each chapter offers biographical notes as well as pictures of graph theorists and mathematicians who have contributed significantly to the development of the results documented in the chapter. These notes are meant to bring the topics covered to life, allowing the reader to associate faces with some of the important discoveries and results presented. In total, approximately 100 biographical notes are presented throughout the book. The material in this book has been organized into three distinct parts, each with a different focus. The first part is devoted to topics in network optimization, with a focus on basic notions in algorithmic complexity and the computation of optimal paths, shortest spanning trees, maximum flows and minimum-cost flows in networks, as well as the solution of network location problems. The second part is devoted to a variety of classical problems in graph theory, including problems related to matchings, edge and vertex traversal, connectivity, planarity, edge and vertex coloring, and orientations of graphs. Finally, the focus in the third part is on modern areas of study in graph theory, covering graph domination, Ramsey theory, extremal graph theory, graph enumeration, and application of the probabilistic method.Table of ContentsPreface.- List of Algorithms.- List of Bibliographical Notes.- Part 1. Topics in network optimisation.- 1. An introduction to graphs.- 2. Graph connectedness.- 3. Algorithmic complexity.- 4. Optimal paths.- 5. Trees.- 6. Location problems.- 7. Maximum flow networks.- 8. Minimum-cost network flows.- Part 2. Topics in classical graph theory.- 9. Matchings.- 10. Eulerian graphs.- 11. Hamiltonian graphs.- 12. Graph connectivity.- 13. Planarity.- 14. Graph colouring.- 15. Oriented graphs. Part 3. Topics in modern graph theory.- 16. Domination in graphs.- 17. Ramsey Theory.- 18. Extremal graph theory.- 19. Graph enumeration.- 20. The probabilistic method.- Index.

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Book SynopsisThis proceedings volume gathers selected, revised papers presented at the 51st Southeastern International Conference on Combinatorics, Graph Theory and Computing (SEICCGTC 2020), held at Florida Atlantic University in Boca Raton, USA, on March 9-13, 2020. The SEICCGTC is broadly considered to be a trendsetter for other conferences around the world – many of the ideas and themes first discussed at it have subsequently been explored at other conferences and symposia.The conference has been held annually since 1970, in Baton Rouge, Louisiana and Boca Raton, Florida. Over the years, it has grown to become the major annual conference in its fields, and plays a major role in disseminating results and in fostering collaborative work.This volume is intended for the community of pure and applied mathematicians, in academia, industry and government, working in combinatorics and graph theory, as well as related areas of computer science and the interactions among these fields.Table of ContentsRatio Balancing Numbers(Bartz et al).- An Unexpected Digit Permutation from Multiplying in any Number Base(Qu et al).- A & Z Sequences for Double Riordan Arrays (Branch et al).- Constructing Clifford Algebras for Windmill and Dutch Windmill Graphs; A New Proof of The Friendship Theorem(Myers).- Finding Exact Values of a Character Sum (Peart et al).- On Minimum Index Stanton 4-cycle Designs (Bunge et al).- k-Plane Matroids and Whiteley’s Flattening Conjectures (Servatius et al).- Bounding the edge cover of a hypergraph (Shahrokhi).- A Generalization on Neighborhood Representatives (Holliday).- Harmonious Labelings of Disconnected Graphs involving Cycles and Multiple Components Consisting of Starlike Trees(Abueida et al).- On Rainbow Mean Colorings of Trees (Hallas et al).- Examples of Edge Critical Graphs in Peg Solitaire (Beeler et al).- Regular Tournaments with Minimum Split Domination Number and Cycle Extendability (Factor et al).- Independence and Domination of Chess Pieces on Triangular Boards and on the Surface of a Tetrahedron(Munger et al).- Efficient and Non-efficient Domination of Z-stacked Archimedean Lattices (Paskowitz et al).- On subdivision graphs which are 2-steps Hamiltonian graphs and hereditary non 2-steps Hamiltonian graphs (Lee et al).- On the Erd}os-S_os Conjecture for graphs with circumference at most k + 1 (Heissan et al).- Regular graph and some vertex-deleted subgraph (Egawa et al).- Connectivity and Extendability in Digraphs (Beasle).-On the extraconnectivity of arrangement graphs (Cheng et al).- k-Paths of k-Trees(Bickle).-Rearrangement of the Simple Random Walk(Skyers et al).- On the Energy of Transposition Graphs(DeDeo).- A Smaller Upper Bound for the (4; 82) Lattice Site Percolation Threshold(Wierman).

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Book SynopsisThis monograph is designed to be an in-depth introduction to domination in graphs. It focuses on three core concepts: domination, total domination, and independent domination. It contains major results on these foundational domination numbers, including a wide variety of in-depth proofs of selected results providing the reader with a toolbox of proof techniques used in domination theory. Additionally, the book is intended as an invaluable reference resource for a variety of readerships, namely, established researchers in the field of domination who want an updated, comprehensive coverage of domination theory; next, researchers in graph theory who wish to become acquainted with newer topics in domination, along with major developments in the field and some of the proof techniques used; and, graduate students with interests in graph theory, who might find the theory and many real-world applications of domination of interest for masters and doctoral thesis topics. The focused coverage also provides a good basis for seminars in domination theory or domination algorithms and complexity. The authors set out to provide the community with an updated and comprehensive treatment on the major topics in domination in graphs. And by Jove, they’ve done it! In recent years, the authors have curated and published two contributed volumes: Topics in Domination in Graphs, © 2020 and Structures of Domination in Graphs, © 2021. This book rounds out the coverage entirely. The reader is assumed to be acquainted with the basic concepts of graph theory and has had some exposure to graph theory at an introductory level. As graph theory terminology sometimes varies, a glossary of terms and notation is provided at the end of the book.Table of Contents1. Introduction.- 2. Historic background.- 3. Domination Fundamentals.- 4. Bounds in terms of order and size, and probability.- 5. Bounds in terms of degree.- 6. Bounds with girth and diameter conditions.- 7. Bounds in terms of forbidden subgraphs.- 8. Domination in graph families : Trees.- 9. Domination in graph families: Claw-free graphs.- 10. Domination in regular graphs including Cubic graphs.- 11. Domination in graph families: Planar graph.- 12. Domination in graph families: Chordal, bipartite, interval, etc.- 13. Domination in grid graphs and graph products.- 14. Progress on Vizing's Conjecture.- 15. Sums and Products (Nordhaus-Gaddum).- 16. Domination Games.- 17. Criticality.- 18. Complexity and Algorithms.- 19. The Upper Domination Number.- 20. Domatic Numbers (for lower and upper gamma) and other dominating partitions, including the newly introduced Upper Domatic Number.- 21. Concluding Remarks, Conjectures, and Open Problems.

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Book SynopsisThis book details recent developments in the emerging area of plug-and-play (PnP) visual subgraph query interfaces (VQI). These PnP interfaces are grounded in the principles of human-computer interaction (HCI) and cognitive psychology to address long-standing limitations to bottom-up search capabilities in graph databases using traditional graph query languages, which often require domain experts and specialist programmers. This book explains how PnP interfaces go against the traditional mantra of VQI construction by taking a data-driven approach and giving end users the freedom to easily and quickly construct and maintain a VQI for any data sources without resorting to coding. The book walks readers through the intuitive PnP interface that uses templates where the underlying graph repository represents the socket and user-specified requirements represent the plug. Hence, a PnP interface enables an end user to change the socket (i.e., graph repository) or the plug (i.e., requirements) as necessary to automatically and effortlessly generate VQIs. The book argues that such a data-driven paradigm creates several benefits, including superior support for visual subgraph query construction, significant reduction in the manual cost of constructing and maintaining a VQI for any graph data source, and portability of the interface across diverse sources and querying applications. This book provides a comprehensive introduction to the notion of PnP interfaces, compares it to its classical manual counterpart, and reviews techniques for automatic construction and maintenance of these new interfaces. In synthesizing current research on plug-and-play visual subgraph query interface management, this book gives readers a snapshot of the state of the art in this topic as well as future research directions.Table of ContentsChapter 1 - The Future is Democratized Graphs.- Chapter 2 - Background.- Chapter 3 - The World of Visual Graph Query Interfaces: An Overview.- Chapter 4 - Plug-and-Play Visual Subgraph Query Interfaces.- Chapter 5 - The Building Block of PnP Interfaces: Canned Patterns.- Chapter 6 - Pattern Selection for Graph Databases.- Chapter 7 - Pattern Selection for Large Networks.- Chapter 8 - Maintenance of Patterns.- Chapter 9 - The Road Ahead.

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Book SynopsisThis book highlights cutting-edge research in the field of network science, offering scientists, researchers, students, and practitioners a unique update on the latest advances in theory and a multitude of applications. It presents the peer-reviewed proceedings of the XI International Conference on Complex Networks and their Applications (COMPLEX NETWORKS 2022). The carefully selected papers cover a wide range of theoretical topics such as network models and measures; community structure, network dynamics; diffusion, epidemics, and spreading processes; resilience and control as well as all the main network applications, including social and political networks; networks in finance and economics; biological and neuroscience networks and technological networks.Table of ContentsPart I: Information Spreading in Social Media.- Cognitive Cascades within Media Ecosystems: Simulating Fragmentation, Selective Exposure and Media Tactics to Investigate Polarization.- Properties of Reddit News Topical Interactions.- Will You Take the Knee? Italian Twitter Echo Chambers’ Genesis during EURO 2020.- A simple model of knowledge scaffolding.- Using knowledge graphs to detect partisanship in online political discourse.- The wisdom_of_crowds: an efficient, philosophically-validated, social epistemological network profiling toolkit.- Opening up echo chambers via optimal content recommendation.- Change my Mind: Data Driven Estimate of Open-Mindedness from Political Discussions.- The effects of message sorting in the diffusion of information in online social media.- Gradual Network Sparsification and Georeferencing for Location-Aware Event Detection in Microblogging Services.- Manipulation during the French presidential campaign : Coordinated inauthentic behaviors and astroturfing analysis on text and images.- Part II: Modeling Human Behavior.- Lexical networks constructed to correspond students’ short written responses: A quantum semantic approach.- Attributed Stream-Hypernetwork analysis: Homophilic Behaviors in Pairwise and Group Political Discussions on Reddit.- Individual Fairness for Social Media Influencers.- Multidimensional online American politics: Mining emergent social cleavages in social graphs.- Classical and quantum random walks to identify leaders in criminal networks.- Random walk for generalization in goal-directed human navigation on Wikipedia.- Sometimes Less is More: When Aggregating Networks Masks Effects.- An Adaptive Network Model Simulating the Effects of Different Culture Types and Leader Qualities on Mistake Handling and Organisational Learning.- Part III: Biological Networks.- Modeling of Hardy-Weinberg Equilibrium using dynamic random networks in an ABM framework.- IntegrOmics: A computational Framework to analyze RNA-Seq and Methylation data through heterogeneous multi-layer networks.- A Network-based Approach for Inferring Thresholds in Co-expression Networks.- Building Differential Co-expression Networks with Variable Selection and Regularization.- Inferring probabilistic Boolean networks from steady-state gene data samples.- Quantifying High-Order Interactions in Complex Physiological Networks: a frequency-specific approach.- A Novel Reverse Engineering Approach for Gene Regulatory Networks.- Using the Duplication-Divergence Network Model to Predict Protein-Protein Interactions.- Part IV: Machine Learning and Networks.- SignedS2V: structural embedding method for signed networks.- HM-LDM: A Hybrid-Membership Latent Distance Model.- The Structure of Interdisciplinary Science: Uncovering and Explaining Roles in Citation Graphs.- Inferring Parsimonious Coupling Statistics in Nonlinear Dynamics with Variational Gaussian Processes.- Detection of Sparsity in Multidimensional Data Using Network Degree Distribution and Improved Supervised Learning with Correction of Data Weighting.- Network Structure vs Chemical Information in Drug-Drug Interaction Prediction.- Geometric Deep Learning graph pruning to speed-up the run-time of Maximum Clique Enumerarion algorithms.- Graph Mining and Machine Learning for Shader Codes Analysis to Accelerate GPU Tuning.- Part V: Networks in Finance and Economics.- Pattern Analysis of Money Flows in the Bitcoin Blockchain.- On the Empirical Association between Trade Network Complexity and Global Gross Domestic Product.- Measuring the Stability of Technical Cooperation Network Based on the Nested Structure Theory.- Dynamic transition graph for estimating the predictability of financial and economical processes.- A network analysis of world trade structural changes (1996-2019).- Green Sector Space: The evolution and capabilities spillover of economic green sectors in the United States.- Statistical inference of lead-lag between asynchronous time series from p-values of transfer entropy at various timescales.- Part VI: Networks and Mobility.- Extracting Metro Passenger Flow Predictors from Network’s Complex Characteristics.- Estimating Peak-Hour Urban Traffic Congestion.- Adaptive Routing Potential in Road Networks.- Part VII: Diffusion and Epidemics.- Detecting Global Community Structure in a COVID-19 Activity Correlation Network.- Overcoming vaccine hesitancy by multiplex social network targeting.- Analyzing Community-aware Centrality Measures Using The Independent Cascade Model.- Paths for emergence of superspreaders in dengue fever spreading network.- Part VIII: Multilayer Networks.- Structural Cores and Problems of Vulnerability of Partially Overlapped Multilayer Networks.- Multilayer Block Models for Exploratory Analysis of Computer Event Logs.- On the Effectiveness of Using Link Weights and Link Direction for Community Detection in Multilayer Networks.

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Book SynopsisThis book is designed to meet the requirement of undergraduate and postgraduate students pursuing computer science, information technology, mathematical science, and physical science course. No formal prerequisites are needed to understand the text matter except a very reasonable background in college algebra. The text contains in-depth coverage of all major topics proposed by professional institutions and universities for a discrete mathematics course. It emphasizes on problem-solving techniques, pattern recognition, conjecturing, induction, applications of varying nature, proof technique, algorithmic development, algorithm correctness, and numeric computations. A sufficient amount of theory is included for those who enjoy the beauty in development of the subject and a wealth of applications as well as for those who enjoy the power of problem-solving techniques. Biographical sketches of nearly 25 mathematicians and computer scientists who have played a significant role in the development of the field are threaded into the text to provide a human dimension and attach a human face to major discoveries. Each section of the book contains a generous selection of carefully tailored examples to classify and illuminate various concepts and facts. Theorems are backbone of mathematics. Consequently, this book contains the various proof techniques, explained and illustrated in details. Most of the concepts, definitions, and theorems in the book are illustrated with appropriate examples. Proofs shed additional light on the topic and enable students to sharpen thin problem-solving skills. Each chapter ends with a summary of important vocabulary, formulae, properties developed in the chapter, and list of selected references for further exploration and enrichment.Table of Contents0. PRELIMINARIES 1–140.1 Numbers 10.2 Euclid’s Algorithm 30.3 Fundamental Theorem of Arithmetic 40.4 Euclid’s Theorem 60.5 Congruence Modulo m 60.6 Chinese Remainder Theorem 70.7 Fermat’s and Euler’s Theorems 90.8 Exponents and Logarithms 100.9 Sums 110.10 Mapping 12Suggested Readings 141. THE LANGUAGE OF SETS 15–661.1 Introduction 151.2 Elements and Notations of Sets 161.3 Construction of Sets 171.4 Types of Sets 191.5 Set Operations 251.5.1 Intersection of Sets 251.5.2 Union of Sets 261.5.3 Disjoint Set (Mutually Exclusive) 271.5.4 Ordinary Difference of Sets (A – B) 271.5.5 Complementation of Sets 27Contentsxii Contents1.5.6 Universal Set and its Complement 271.5.7 Symmetric Difference (Boolean Sum) 281.6 Venn Diagrams 281.7 Some Basic Results 321.8 Properties of Set Operations 341.8.1 Properties of Intersection on Sets 341.8.2 Properties of Union of Sets 351.8.3 Number of Elements in a Union of two or more Sets 391.9 De-Morgan’s Laws 401.10 General form of Principle of Inclusion and Exclusion 441.11 Laws of Sets 63Summary 63Suggested Readings 652. BASIC COMBINATORICS 67–1142.1 Introduction 672.2 Basic Counting Principles 682.2.1 The Principle of Disjunctive Counting (Sum Rule) 682.2.2 The Principle of Sequential Counting (Product Rule) 692.3 Factorial 712.4 Permutation and Combination 732.4.1 Cyclic Permutation 762.4.2 Pascal’s Identity 762.4.3 Vandermonde’s Identity 772.4.4 Pigeonhole Principle 782.4.5 Inclusion–Exclusion Principle 792.5 The Binomial Theorem 932.6 nth Catalan Number 952.7 Principle of Mathematical Induction (P.M.I) 962.8 Recurrence Relations 99Summary 110Suggested Readings 113Contents xiii3. MATHEMATICAL LOGIC 115–1803.1 Introduction 1153.2 Propositions (Statements) 1173.3 Connectives 1173.3.1 Negation 1183.3.2 Conjunction 1193.3.3 Disjunction 1193.3.4 Conditional 1203.3.5 Biconditional 1203.4 Equivalence of Formulae 1213.5 Well-Formed Formulae (WFF) 1223.6 Tautologies 1223.7 Principle of Duality 1233.8 Two State Devices 1283.9 The Relay-Switching Devices 1293.10 Logic Gates and Modules 1303.10.1 OR, AND and NOT Gates 1303.10.2 Two-Level Networks 1323.10.3 NOR and NAND Gates 1323.11 Normal Forms (Decision Problems) 1413.11.1 Disjunctive Normal Form (DNF) 1413.11.2 Conjunctive Normal Form (CNF) 1453.11.3 Principal Disjunctive Normal Form (PDNF) 1463.11.4 Principal Conjuctive Normal Forms (PCNF) 1483.12 Rules of Inference 1513.13 Automatic Proving System (Theorems) 1523.14 The Predicate Calculus 1643.14.1 Statement Functions, Variables and Quantifiers 1663.14.2 Free and Bound Variables 1663.14.3 Special Valid Formulae using Quantifiers 1673.14.4 Theory of Inference for the Predicate Calculus 1683.14.5 Formulae Involving More than one Quantifier 169Summary 175Suggested Readings 179xiv Contents4. RELATIONS 181–2364.1 Introduction 1814.2 Product Sets 1824.3 Partitions 1824.4 Relations 1834.5 Binary Relations in a Set 1834.6 Domain and Range of a Relation 1844.6.1 Number of Distinct Relation From set A to B 1854.6.2 Solution sets and Graph of Relations 1894.6.3 Relation as Sets of Ordered Pairs 1904.7 The Matrix of a Relation and Digraphs 1904.8 Paths in Relations and Digraphs 1914.9 Boolean Matrices 1944.9.1 Boolean Operations AND and OR 1954.9.2 Joint and Meet 1954.9.3 Boolean Product 1954.9.4 Boolean Power of a Boolean Matrix 1954.10 Adjacency Matrix of a Relation 1984.11 Gray Code 1984.12 Properties of Relations 2004.12.1 Reflexive and Irreflexive Relations 2014.12.2 Symmetric, Asymmetric and AntisymmetricRelations 2014.12.3 Transitive Relation 2024.13 Equivalence Relations 2054.14 Closure of Relations 2074.15 Manipulation and Composition of Relations 2084.16 Warshall’s Algorithm 2164.17 Partial Order Relation 2254.17.1 Totally Ordered Set 2264.17.2 Lexicographic Order 2264.17.3 Hasse Diagrams 228Summary 230Suggested Readings 235Contents xv5. FUNCTIONS 237–2705.1 Introduction 2385.1.1 Sum and Product of Functions 2395.2 Special Types of Functions 2425.2.1 Polynomial Function 2445.2.2 Exponential and Logarithmic Function 2445.2.3 Floor and Ceiling Functions 2455.2.4 Transcedental Function 2475.2.5 Identity Function 2475.2.6 Integer Value and Absolute Value Functions 2475.2.7 Remainder Function 2485.3 Composition of Functions 2485.4 Inverse of a Function 2505.5 Hashing Functions 2565.6 Countable and Uncountable Sets 2575.7 Characteristic Function of a Set 2595.8 Permutation Function 2615.9 Growth of Functions 2625.10 The Relation Θ 262Summary 267Suggested Readings 2696. LATTICE THEORY 271–3046.1 Introduction 2716.2 Partial Ordered Sets 2726.2.1 Some Important Terms 2736.2.2 Diagramatical Representation of a Poset(Hasse Diagram) 2756.2.3 Isomorphism 2766.2.4 Duality 2786.2.5 Product of two Posets 2806.3 Lattices as Posets 2826.3.1 Some Properties of Lattices 2836.3.2 Lattices as Algebraic Systems 284xvi Contents6.3.3 Complete Lattice 2906.3.4 Bounded Lattice 2906.3.5 Sublattices 2916.3.6 Ideals of Lattices 2916.4 Modular and Distributive Lattices 292Summary 302Suggested Readings 3047. BOOLEAN ALGEBRAS AND APPLICATIONS 305–3547.1 Introduction 3057.2 Boolean Algebra (Analytic Approach) 3067.2.1 Sub-Boolean Algebra 3087.2.2 Boolean Homomorphism 3097.3 Boolean Functions 3187.3.1 Equality of Boolean Expressions 3197.3.2 Minterms and Maxterms 3197.3.3 Functional Completeness 3207.3.4 NAND and NOR 3207.4 Combinatorial Circuits (Synthesis of Circuits) 3267.4.1 Half-Adder and Full-Adder 3267.4.2 Equivalent Combinatorial Circuits 3287.5 Karnaugh Map 3317.5.1 Don’t Care Conditions 3347.5.2 Minimization Process 3357.6 Finite State Machines 344Summary 347Suggested Readings 3528. FUZZY ALGEBRA 355–3928.1 Introduction 3558.2 Crisp Sets and Fuzzy Sets 3578.3 Some Useful Definitions 3608.4 Operations of Fuzzy Sets 3628.5 Interval-Valued Fuzzy Sets (I-V Fuzzy Sets) 3678.5.1 Union and Intersection of two I–V Fuzzy Sets 368Contents xvii8.6 Fuzzy Relations 3698.6 Fuzzy Measures 3738.7.1 Belief and Plausibility Measures 3738.7.2 Probability Measure 3748.7.3 Uncertainty and Measures of Fuzziness 3748.7.4 Uncertainty and Information 3758.8 Applications of Fuzzy Algebras 3768.8.1 Natural, Life and Social Sciences 3768.8.2 Engineering 3788.8.3 Medical Sciences 3818.8.4 Management Sciences and Decision MakingProcess 3828.8.5 Computer Science 3838.9 Uniqueness of Uncertainty Measures 3848.9.1 Shannon’s Entropy 3848.9.2 U-uncertainty 3868.9.3 Uniqueness of the U-uncertainty forTwo-Value Possibility Distributions 388Summary 389Suggested Readings 3909. FORMAL LANGUAGES AND AUTOMATATHEORY 393–4289.1 Introduction 3939.2 Formal Languages 3969.2.1 Equality of Words 3979.2.2 Concatenation of Languages 3989.2.3 Kleene Closure 3999.3 Grammars 4039.3.1 Phase-structure Grammar 4069.3.2 Derivations of Grammar 4069.3.3 Backus-Normal Form (BNF) or BackusNaur Form 4079.3.4 Chomsky Grammar 4109.3.5 Ambiguous Grammar 411xviii Contents9.4 Finite-State Automation (FSA) 4139.4.1 Counting to Five 4149.4.2 Process of Getting up in the Morning (Alarm) 4149.4.3 Traffic Light 4159.4.4 Vending Machine 4169.5 Finite-State Machine (FSM) 4169.6 Finite-State Automata 4189.6.1 Deterministic Finite-State Automata (DFSA) 4189.6.2 Nondeterministic Finite-State Automata 4199.6.3 Equivalent Nondeterministic Finite StateAutomata 420Summary 424Suggested Readings 42810. THE BASICS OF GRAPH THEORY 429–48010.1 Introduction 42910.2 Graph. What is it? 43010.2.1 Simple Graph 43010.2.2 Graph 43310.2.3 Loops 43610.2.4 Degree of Vertices 43610.2.5 Equivalence Relation 44110.2.6 Random Graph Model 44210.2.7 Isolated Vertex, Pendent Vertex and Null Graph 44210.3 Digraphs 44310.4 Path, Trail, Walk and Vertex Sequence 44610.5 Subgraph 44710.6 Circuit and Cycle 44710.7 Cycles and Multiple Paths 44910.8 Connected Graph 44910.9 Spanning Subgraph and Induced Subgraph 45010.10 Eulerian Graph (Eulerian Trail and Circuit) 45010.11 Hamiltonian Graph 45110.12 Biconnected Graph 452Contents xix10.13 Algebraic terms and operations used in Graph Theory 45310.13.1 Graphs Isomorphism 45310.13.2 Union of two Graphs 45510.13.3 Intersection of two Graphs 45510.13.4 Addition of two Graphs 45610.13.5 Direct Sum or Ring Sum of two Graphs 45610.13.6 Product of two Graphs 45710.13.7 Composition of two Graphs 45710.13.8 Complement of a Graph 45710.13.9 Fusion of a Graph 45810.13.10 Rank and Nullity 45910.13.11 Adjacency Matrix 45910.13.12 Some Important Theorems 46010.14 Some Popular Problems in Graph Theory 46510.14.1 Tournament Ranking Problem 46510.14.2 The Königsberg Bridge Problem 46710.14.3 Four Colour Problem 46710.14.4 Three Utilities Problem 46810.14.5 Traveling - Salesman Problem 46810.14.6 MTNL’S Networking Problem 47010.14.7 Electrical Network Problems 47010.14.8 Satellite Channel Problem 47110.15 Applications of Graphs 471Summary 475Suggested Readings 48011. TREES 481–52011.1 Introduction 48111.2 Definitions of a Tree 48211.3 Forest 48311.4 Rooted Graph 48411.5 Parent, Child, Sibling and Leaf 48511.6 Rooted Plane Tree 48511.7 Binary Trees 492xx Contents11.8 Spanning Trees 49411.9 Breadth – First Search and Depth – FirstSearch (BFS and DFS) 49611.10 Minimal Spanning Trees 50411.10.1 Kruskal’s Algorithm (for Finding a MinimalSpanning Tree) 50411.10.2 Prim’s Algorithm 50911.11 Directed Trees 511Summary 516Suggested Readings 51812. PLANAR GRAPHS 521–54412.1 Introduction 52112.2 Geometrical Representation of Graphs 52212.3 Bipertite Graph 52412.4 Homeomorphic Graph 52512.5 Kuratowski’s Graphs 52612.6 Dual Graphs 53012.7 Euler’s Formula 53212.8 Outerplanar Graphs 53512.8.1 k-outerplanar Graphs 536Summary 542Suggested Readings 54313. DIRECTED GRAPHS 545–57413.1 Introduction 54513.2 Directed Paths 54713.3 Tournament 54913.4 Directed Cycles 55013.5 Acyclic Graph 55413.6 Di-Orientable Graph 55513.7 Applications of Directed Graphs 55813.7.1 Job Sequencing Problem 55813.7.2 To Design an Efficient Computer Drum 56013.7.3 Ranking of the Participants in a Tournament 562Contents xxi13.8 Network Flows 56413.9 Improvable Flows 56513.10 Max-Flow Min-Cut Theorem 56713.11 k-flow 56813.12 Tutte’s Problem 569Summary 571Suggested Readings 57414. MATCHING AND COVERING 575–60814.1 Introduction 57514.2 Matching and Covering in Bipertite Graphs 57714.2.1 Covering 58214.3 Perfect Matching 58414.4 Factor-critical Graph 58814.5 Complete Matching 59014.6 Matrix Method to Find Matching of a Bipertite Graph 59214.7 Path Covers 59514.8 Applications 59614.8.1 The Personnel Assignment Problem 59614.8.2 The Optimal Assignment Problem 60114.8.3 Covering to Switching Functions 602Summary 604Suggested Readings 60715. COLOURING OF GRAPHS 609–64015.1 Introduction 60915.2 Vertex Colouring 61215.3 Chromatic Polynomial 61315.3.1 Bounds of the Chromatic Number 61415.4 Exams Scheduling Problem 61715.5 Edge Colouring 62515.6 List Colouring 63015.7 Greedy Colouring 63115.8 Applications 63515.8.1 The Time Table Problem 635xxii Contents15.8.2 Scheduling of Jobs 63615.8.3 Ramsey Theory 63715.8.4 Storage Problem 637Summary 638Suggested Readings 639References 641–642Index 643–648​

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Book SynopsisChapter(s) “Chapter Name or No.” is/are available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.Table of ContentsA stochastic algorithm for non-monotone DR-submodular maximziation over a convex set.- Flow shop scheduling problems with transportation constraints revisited.- LotterySampling: A Randomized Algorithm for the Heavy Hitters and Top-k Problems in Data Streams.- Approximation Algorithms for the Min-Max Mixed Rural Postmen Cover Problem and Its Variants.- Large k-gons in a 1.5D Terrain.- Nondeterministic Auxiliary Depth-Bounded Storage Automata and Semi-Unbounded Fan-in Cascading Circuits (Extended Abstract).-Analysis of Approximate sorting in I/O model.-Two Generalizations of Proper Coloring: Hardness and Approximability.-Approximation Algorithms for Capacitated Assignment with Budget Constraints and Applications in Transportation Systems.-On the Complexity of Minimum Maximal Acyclic Matchings.-Online non-monotone DR-submodular maximization: 1/4 approximation ratio and sublinear regret.-Fair Division with Minimal Withheld Information in Social Networks.- Facility Location Games with Ordinal Preferences.-Fully Dynamic $k$-Center Clustering with Outliers.-Refutation of Spectral Graph Theory Conjectures with Monte Carlo Search.-Online one-sided smooth function maximization.-Revisiting Maximum Satisfiability and Related Problems in Data Streams.-Turing Machines with Two-level Memory: A Deep Look into the Input/Output Complexity.- A quantum version of Pollard's Rho of which Shor's Algorithm is a particular case.-Single machine scheduling with rejection to minimize the $k$-th power of the makespan.- Escape from the Room.- Algorithms for hard-constraint point processes via discretization.-Space Limited Graph Algorithms on Big Data Counting Cycles on Planar Graphs in Subexponential Time.-Semi-strict chordal digraphs.- Reallocation Problems with Minimum Completion Time.-The bound coverage problem by aligned disks in L1 metric.-Facility Location Games with Group Externalities.- Some New Results on Gallai Theorem and Perfect Matching for k-Uniform Hypergraphs.- Refined Computational Complexities of Hospitals/Residents Problem with Regional Caps.- Customizable Hub Labeling: Properties and Algorithms Linear-Time Algorithm for Paired-Domination on Distance-Hereditary Graphs.-Bounding the Number of Eulerian Tours in Undirected Graphs.- A Probabilistic Model Revealing Shortcomings in Lua's Hybrid Tables.- A 4-Space Bounded.- Approximation Algorithm \\for Online Bin Packing Problem.-Generalized Sweeping Line Spanners.- Rooting Gene Trees via Phylogenetic Networks.- An evolving network model from clique extension.- Online semi-matching problem with two heterogeneous sensors in a metric space.- Two-Stage BP Maximization under $p$-matroid Constraint.- The Hamiltonian Path Graph is Connected for Simple $s,t$ Paths in Rectangular Grid Graphs.- An $O(n^3)$-Time Algorithm for the Min-Gap.- Unit-Length Job Scheduling Problem.- Approximation Schemes for k-Facility Location.- Improved Deterministic Algorithms for Non-monotone Submodular Maximization.- Distributed Dominating Sets in Interval Graphs.- Optimal Window Queries on Line Segments using the Trapezoidal Search DAG.- On Rotation Distance, Transpositions and Rank Bounded Trees.- Hitting Geometric Objects Online via Points in $\mathbb{Z}^d$.- Capacitated Facility Location with Outliers/Penalties Improved Separated Red Blue Center Clustering.- Proper colorability of segment intersection graphs.

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Book SynopsisThese Lecture Notes provide an introduction to the study of those discrete surfaces which are obtained by randomly gluing polygons along their sides in a plane. The focus is on the geometry of such random planar maps (diameter, volume growth, scaling and local limits...) as well as the behavior of statistical mechanics models on them (percolation, simple random walks, self-avoiding random walks...).A “Markovian” approach is adopted to explore these random discrete surfaces, which is then related to the analogous one-dimensional random walk processes. This technique, known as "peeling exploration" in the literature, can be seen as a generalization of the well-known coding processes for random trees (e.g. breadth first or depth first search). It is revealed that different types of Markovian explorations can yield different types of information about a surface. Based on an École d'Été de Probabilités de Saint-Flour course delivered by the author in 2019, the book is aimed at PhD students and researchers interested in graph theory, combinatorial probability and geometry. Featuring open problems and a wealth of interesting figures, it is the first book to be published on the theory of random planar maps.Table of Contents- Part I (Planar) Maps. - 1. Discrete Random Surfaces in High Genus. - 2. Why Are Planar Maps Exceptional?. - 3. The Miraculous Enumeration of Bipartite Maps. - Part II Peeling Explorations. - 4. Peeling of Finite Boltzmann Maps. - 5. Classification of Weight Sequences. - Part III Infinite Boltzmann Maps. - 6. Infinite Boltzmann Maps of the Half-Plane. - 7. Infinite Boltzmann Maps of the Plane. - 8. Hyperbolic Random Maps. - 9. Simple Boundary, Yet a Bit More Complicated. - 10. Scaling Limit for the Peeling Process. - Part IV Percolation(s). - 11. Percolation Thresholds in the Half-Plane. - 12. More on Bond Percolation. - Part V Geometry. - 13. Metric Growths. - 14. A Taste of Scaling Limit. - Part VI Simple Random Walk. - 15. Recurrence, Transience, Liouville and Speed. - 16. Subdiffusivity and Pioneer Points.

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Book SynopsisThis book provides a comprehensive algebraic treatment of hypergroups, as defined by F. Marty in 1934. It starts with structural results, which are developed along the lines of the structure theory of groups. The focus then turns to a number of concrete classes of hypergroups with small parameters, and continues with a closer look at the role of involutions (modeled after the definition of group-theoretic involutions) within the theory of hypergroups. Hypergroups generated by involutions lead to the exchange condition (a genuine generalization of the group-theoretic exchange condition), and this condition defines the so-called Coxeter hypergroups. Coxeter hypergroups can be treated in a similar way to Coxeter groups. On the other hand, their regular actions are mathematically equivalent to buildings (in the sense of Jacques Tits). A similar equivalence is discussed for twin buildings. The primary audience for the monograph will be researchers working in Algebra and/or Algebraic Combinatorics, in particular on association schemes.Table of Contents1 Basic Facts : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11.1 Neutral Elements and Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Complex Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Thin Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Groups and Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Actions of Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.7 Hypergroups Admitting Regular Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.8 Association Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Closed Subsets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 272.1 Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Dedekind Modularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3 Generating Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.5 Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.6 The Thin Radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.7 Foldings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Elementary Structure Theory: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 473.1 Centralizers and Normalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Su cient Conditions for Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3 Strong Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.4 Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.5 Computations in Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.6 Homomorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.7 The Homomorphism Theorem and the Isomorphism Theorems . . . . . . . . . . 714 Subnormality and Thin Residues : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 794.1 Subnormal Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2 Composition Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.3 The Thin Residue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.4 Thin Residues of Thin Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.5 Residually Thin Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.6 Finite Residually Thin Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.7 Solvable Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045 Tight Hypergroups : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1075.1 Tight Hypergroup Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.2 The Set S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.3 The Sets a b \ Fc and Sa;b(Fc) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.4 The Sets bf1b  \ Fa and Sb;(f1;:::;fn)(Fa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.5 Structure Constants of Finite Tight Hypergroups . . . . . . . . . . . . . . . . . . . . . 1225.6 Rings Arising from Certain Finite Tight Hypergroups . . . . . . . . . . . . . . . . . 1265.7 Finite Metathin Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.8 Finite Metathin Hypergroups with Restricted Thin Residue . . . . . . . . . . . . 1326 Involutions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1376.1 Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386.2 Cosets of Closed Subsets Generated by an Involution, I . . . . . . . . . . . . . . . . 1426.3 Cosets of Closed Subsets Generated by an Involution, II . . . . . . . . . . . . . . . 1456.4 Cosets of Closed Subsets Generated by an Involution, III . . . . . . . . . . . . . . . 1476.5 Length Functions De ned by Sets of Involutions . . . . . . . . . . . . . . . . . . . . . . 1526.6 Hypergroups Generated by Two Distinct Involutions . . . . . . . . . . . . . . . . . . 1566.7 Dichotomy and the Exchange Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1616.8 Projective Hypergroups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1647 Hypergroups with a Small Number of Elements : : : : : : : : : : : : : : : : : : : : : : 1717.1 Hypergroups of Cardinality at Most 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1727.2 Non-Symmetric Hypergroups of Cardinality 4 . . . . . . . . . . . . . . . . . . . . . . . . 1797.3 Hypergroups of Cardinality 6 with a Non-Normal Closed Subset, I . . . . . . 1907.4 Hypergroups of Cardinality 6 with a Non-Normal Closed Subset, II . . . . . . 2027.5 Non-Normal Closed Subsets Missing Four Elements . . . . . . . . . . . . . . . . . . . 2157.6 Non-Normal Closed Subsets Missing Four Elements and Thin Elements . . 2218 Constrained Sets of Involutions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2238.1 Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2248.2 Constrained Sets of Involutions and Cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . 2288.3 Constrained Sets of Involutions and the Thin Radical . . . . . . . . . . . . . . . . . . 2308.4 Constrained Sets of Involutions and Dichotomy . . . . . . . . . . . . . . . . . . . . . . . 2338.5 Constrained Sets of Non-Thin Involutions and Dichotomy . . . . . . . . . . . . . . 2398.6 Constrained Sets of Involutions and Foldings . . . . . . . . . . . . . . . . . . . . . . . . . 2448.7 Dichotomic Constrained Sets of Involutions and Foldings . . . . . . . . . . . . . . . 2489 Coxeter Sets of Involutions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2519.1 General Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2529.2 The Sets V1(U) for Subsets U of Coxeter Sets V of Involutions . . . . . . . . . . 2569.3 The Sets V����1(U) for Subsets U of Coxeter Sets V of Involutions . . . . . . . . . 2639.4 Sets of Subsets of Coxeter Sets of Involutions . . . . . . . . . . . . . . . . . . . . . . . . . 2659.5 Spherical Coxeter Sets of Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2689.6 Subsets of Spherical Coxeter Sets of Involutions . . . . . . . . . . . . . . . . . . . . . . . 2739.7 Coxeter Sets of Involutions and Foldings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2779.8 Coxeter Sets of Involutions and Their Coxeter Numbers . . . . . . . . . . . . . . . . 2809.9 Coxeter Sets of Involutions and Type Preserving Bijections . . . . . . . . . . . . . 28610 Regular Actions of (Twin) Coxeter Hypergroups: : : : : : : : : : : : : : : : : : : : : 29310.1 Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29310.2 Twin Buildings, I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29810.3 Twin Buildings, II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30110.4 Regular Actions of Coxeter Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30510.5 Regular Actions of Twin Coxeter Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . 315References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 333

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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 SynopsisThis textbook provides an introduction to the Catalan numbers and their remarkable properties, along with their various applications in combinatorics. Intended to be accessible to students new to the subject, the book begins with more elementary topics before progressing to more mathematically sophisticated topics. Each chapter focuses on a specific combinatorial object counted by these numbers, including paths, trees, tilings of a staircase, null sums in Zn+1, interval structures, partitions, permutations, semiorders, and more. Exercises are included at the end of book, along with hints and solutions, to help students obtain a better grasp of the material. The text is ideal for undergraduate students studying combinatorics, but will also appeal to anyone with a mathematical background who has an interest in learning about the Catalan numbers.“Roman does an admirable job of providing an introduction to Catalan numbers of a different nature from the previous ones. He has made an excellent choice of topics in order to convey the flavor of Catalan combinatorics. [Readers] will acquire a good feeling for why so many mathematicians are enthralled by the remarkable ubiquity and elegance of Catalan numbers.” - From the foreword by Richard StanleyTrade Review“The pace of this book is of an introductory nature, the coverage of Catalan numbers is rigorous and will provide the reader with a firm grasp of many of the properties of these numbers. … a scholarly work and one that number theorists will find well worth reading.” (James Van Speybroeck, Computing Reviews, April, 2016)“The book is supplemented by a set of exercises for those who want to go further. The reader is helped by 70 carefully designed figures throughout the book. The 24 enumeration problems are selected carefully to show Catalan numbers from very different viewpoints. Several of these problems likely have their most readable write-up in this book. A must for anyone, who wants to understand the significance of Catalan numbers!” (László Székely, zbMATH 1342.05002, 2016)Table of ContentsIntroduction.- Dyck Words.- The Catalan Numbers.- Catalan Numbers and Paths.- Catalan Numbers and Trees.- Catalan Numbers and Geometric Widgits.- Catalan Numbers and Algebraic Widgits.- Catalan Numbers and Interval Structures.- Catalan Numbers and Partitions.- Catalan Numbers and Permutations.- Catalan Numbers and Semiorders.- Exercises.- Solutions and Hints.- Appendix A: A Brief Introduction to Partially Ordered Sets.- Appendix B: A Brief Introduction to Graphs and Trees.- Index.

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Book SynopsisThis textbook provides an exciting new addition to the area of network science featuring a stronger and more methodical link of models to their mathematical origin and explains how these relate to each other with special focus on epidemic spread on networks. The content of the book is at the interface of graph theory, stochastic processes and dynamical systems. The authors set out to make a significant contribution to closing the gap between model development and the supporting mathematics. This is done by: Summarising and presenting the state-of-the-art in modeling epidemics on networks with results and readily usable models signposted throughout the book; Presenting different mathematical approaches to formulate exact and solvable models; Identifying the concrete links between approximate models and their rigorous mathematical representation; Presenting a model hierarchy and clearly highlighting the links between model assumptions and model complexity; Providing a reference source for advanced undergraduate students, as well as doctoral students, postdoctoral researchers and academic experts who are engaged in modeling stochastic processes on networks; Providing software that can solve differential equation models or directly simulate epidemics on networks. Replete with numerous diagrams, examples, instructive exercises, and online access to simulation algorithms and readily usable code, this book will appeal to a wide spectrum of readers from different backgrounds and academic levels. Appropriate for students with or without a strong background in mathematics, this textbook can form the basis of an advanced undergraduate or graduate course in both mathematics and other departments alike. Trade Review“The book adds to the knowledge of epidemic modeling on networks by providing a number of rigorous mathematical arguments and confirming the validity and optimal range of applicability of the epidemic models. It serves as a good reference guide for researchers and a comprehensive textbook for graduate students.” (Yilun Shang, Mathematical Reviews, November, 2017)“This is one of the first books to appear on modeling epidemics on networks. … This is a comprehensive and well-written text aimed at students with a serious interest in mathematical epidemiology. It is most appropriate for strong advanced undergraduates or graduate students with some background in differential equations, dynamical systems, probability and stochastic processes.” (William J. Satzer, MAA Reviews, September, 2017)Table of ContentsPreface.- ​Introduction to Networks and Diseases.- Exact Propagation Models: Top Down.- Exact Propagation Models: Bottom-Up.- Mean-Field Approximations for Heterogeneous Networks.- Percolation-Based Approaches for Disease Modelling.- Hierarchies of SIR Models.- Dynamic and Adaptive Networks.- Non-Markovian Epidemics.- PDE Limits for Large Networks.- Disease Spread in Networks with Large-scale structure.- Appendix: Stochastic Simulation.- Index.

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Book SynopsisThis text presents six mini-courses, all devoted to interactions between representation theory of algebras, homological algebra, and the new ever-expanding theory of cluster algebras. The interplay between the topics discussed in this text will continue to grow and this collection of courses stands as a partial testimony to this new development. The courses are useful for any mathematician who would like to learn more about this rapidly developing field; the primary aim is to engage graduate students and young researchers. Prerequisites include knowledge of some noncommutative algebra or homological algebra. Homological algebra has always been considered as one of the main tools in the study of finite-dimensional algebras. The strong relationship with cluster algebras is more recent and has quickly established itself as one of the important highlights of today’s mathematical landscape. This connection has been fruitful to both areas—representation theory provides a categorification of cluster algebras, while the study of cluster algebras provides representation theory with new objects of study.The six mini-courses comprising this text were delivered March 7–18, 2016 at a CIMPA (Centre International de Mathématiques Pures et Appliquées) research school held at the Universidad Nacional de Mar del Plata, Argentina. This research school was dedicated to the founder of the Argentinian research group in representation theory, M.I. Platzeck.The courses held were: Advanced homological algebra Introduction to the representation theory of algebras Auslander-Reiten theory for algebras of infinite representation type Cluster algebras arising from surfaces Cluster tilted algebras Cluster characters Introduction to K-theory Brauer graph algebras and applications to cluster algebras Table of ContentsIntroduction to the Representation Theory of Finite-Dimensional Algebras: The Functorial Approach (M. I. Platzeck).- Auslander–Reiten Theory for Finite-Dimensional Algebras (P. Malicki).- Cluster Algebras From Surfaces (R. Schiffler).- Cluster Characters (P.-G. Plamondon).- A Course on Cluster Tilted Algebras (I. Assem).- Brauer Graph Algebras (S. Schroll).

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Book SynopsisTable of Contents1. Das Zeigerdiagramm.- 1.1. Darstellung einer zeitlich sinusförmigen Größe durch einen Zeiger.- 1.2. Zeigerdiagramm bei einfachen Schaltelementen.- 1.2.1. Ohmwiderstand.- 1.2.2. Kondensator.- 1.2.3. Drosselspule.- 1.2.4. Beispiele.- 1.3. Zeigerdiagramm bei zusammengesetzten Schaltungen.- 1.3.1. Verknüpfungsgesetze.- 1.3.2. Parallelschaltung G-C.- 1.3.3. Reihenschaltung R-L.- 1.3.4. Andere Schaltungen.- 2. Leistung bei Wechselstrom.- 2.1. Scheinleistung, Wirkleistung.- 2.1.1. Definition.- 2.1.2. Ohmwiderstand.- 2.1.3. Kondensator.- 2.1.4. Spule.- 2.2. Wirkstrom und Blindstrom.- 3. Beschreibung von Wechselstrom mit Hilfe der komplexen Rechnung.- 3.1. Komplexe Zahlen.- 3.2. Anwendung der komplexen Rechnung auf Wechsel strom Schaltung.- 3.3. Komplexer Widerstand (Impedanz) und Leitwert (Admittanz).- 3.4. Leistung in komplexer Schreibweise.- 3.5. Berechnung einfacher Schaltungen.- 3.5.1. Parallelschaltung G-L.- 3.5.2. Reihenschaltung R-C.- 3.5.3. Abgleichbedingung der Maxwell-Brücke.- 3.6. Zusammenfassung.- 4. Resonanzschaltungen.- 4.1. Parallel- und Reihenschwingkreis.- 4.2. Blindstromkompensation.- 5. Der Transformator.- 5.1. Magnetische Kopplung zweier Stromkreise.- 5.2. Ersatzschaltbild und Zeigerdiagramm.- 5.3. Vereinfachtes Ersatzschaltbild.- 5.4. Einige Sonderfälle.- 5.4.1. Leerlaufender Transformator.- 5.4.2. Sekundär kurzgeschlossener Transformator.- 6. Allgemeine Verfahren zur Berechnung linearer Netzwerke.- 6.1. Aufgabenstellung und Lösungsweg.- 6.2. Berechnung des Netzwerkes durch Ansatz von Kreisströmen.- 6.2.1. Begründung.- 6.2.2. Beispiel und Verallgemeinerung.- 6.2.3. Beispiel: Berechnung der Vierpol-Eigenschaften einer Brückenschaltung.- 6.2.4. Erweiterung des Kreisstromverfahrens auf Wechselstrom.- 6.3. Berechnung der Zweigströme mit Hilfe der Knotenpunktsspannungen.- 6.3.1. Begründung.- 6.3.2. Beispiel: Messung der induzierten Spannung einer Gleich Strommaschine.- 6.4. Das Überlagerungsverfahren.- 6.4.1. Begründung.- 6.4.2. Beispiele.- 7. Spezielle Verfahren zur Berechnung linearer Netzwerke.- 7.1. Ersatz-Spannungsquelle und Ersatz-Stromquelle.- 7.1.1. Aufgabenstellung und Lösung.- 7.1.2. Beispiele.- 7.2. Netzwerksumwandlung.- 7.2.1. Allgemeines.- 7.2.2. Stern-Dreieck-Umwandlung.- 7.2.3. Verallgemeinerung.- 7.2.4. Beispiele.- 8. Vierpole.- 8.1. Vierpolgleichungen.- 8.2. Darstellung eines Vierpols in T- oder ?-Schaltung.- 8.3. Reziproke Vierpoleigenschaften.- 9. Drehstromsystem mit sinusförmigen Spannungen und Strömen.- 9.1. Symmetrisches Dreh strom system 9.- 9.1.1. Allgemeines, Erzeugung von Dreh strom.- 9.1.2. Dreh Strombelastung.- 9.1.3. Leistung bei Dreh strom.- 9.2. Unsymmetrisches Dreh Stromsystem.- 9.3. Beispiel: Erdschluß-Löschung in einem Hochspannungsnetz.- 10. Nicht sinusförmige periodische Vorgänge.- 10.1. Allgemeines.- 10.2. Darstellung periodischer Vorgänge durch Fouriersche Reihen.- 10.3. Anregung einer linearen Schaltung durch nicht sinusförmige Spannungen und Ströme.- 10.4. Nachrichtenübertragung.- 10.5. Leistung und Effektivwert bei nicht sinusförmigen periodischen Vorgängen.- 10.5.1. Erweiterte Definition des Effektivwertes.- 10.5.2. Berechnung des Effektivwertes aus dem Frequenzspektrum.- 10.5.3. Klirrfaktor.- 10.6. Symmetrisches Drehstromsystem mit Oberschwingungen.- 10.6.1. Ableitung.- 10.6.2. Anwendung.- 11. Darstellung komplexer Funktionen durch Ortskurven.- 11.1. Komplexe Funktion einer reellen Veränderlichen.- 11.2. Komplexe Funktion einer komplexen Veränderlichen.- 11.3. Die Abbildung durch die Funktion F = 1/w.- 11.4. Abbildung durch eine allgemeine lineare Funktion.- 11.5. Anwendung zur Berechnung von Ortskurven.- 11.5.1. Reihenschaltung R-L.- 11.5.2. Parallelschwingkreis.- 11.5.3. Frequenzgang eines RC-Vierpols im Leerlauf.- 11.5.4. Frequenzgang eines LC-Tiefpasses.- 12. Berechnung nichtstationärer Vorgänge in linearen Netzwerken mit Hilfe der Differentialgleichung.- 12.1. Energiespeicher.- 12.2. Ansatz der Differentialgleichung.- 12.3. Vorgänge beim Einschalten einer Gleichspannung.- 12.3.1. RC-Tiefpaß.- 12.3.2. Induktiver Stromkreis.- 12.3.3. Einschaltvorgang eines Reihenschwingkreises.- 12.3.4. Einschaltvorgang eines Impulsübertragers.- 12.3.5. Speisung eines Netzwerkes durch eine periodische Rechteckspannung.- 12.4. Vorgänge beim Einschalten einer Wechselspannung.- 13. Zeitbereich und Frequenzbereich.- 13.1. Allgemeine stationäre Lösung der Differentialgleichung.- 13.2. Komplexe Frequenz.- 13.3. Kontinuierliches Spektrum, Fourier- und Laplace-Transformation.- 13.3.1. Diskretes Frequenz Spektrum, Fourier-Reihe.- 13.3.2. Kontinuierliches Frequenzspektrum (Fourier-Transformation).- 13.3.3. Laplace-Transformation.- 13.4. Berechnung einiger Korrespondenzen der Laplace-Transformation 177 1.3.4.1. Exponentialfunktion.- 13.4.2. Schaltfunktion, Sprungfunktion.- 13.4.3. Dirac-Impuls.- 13.4.4. Anstiegsfunktion.- 13.4.5. Linearität.- 13.5. Laplace-Transformation und Übertragungsfunktion.- 14. Berechnung von Einschaltvorgängen mit der Laplace-Transformation.- 14.1. Sprungantwort und Impulsantwort.- 14.2. Partialbruchzerlegung.- 14.3. Rücktransformation durch komplexe Integration.- 14.4. Beispiele zur Anwendung der Laplace-Transformation auf die Berechnung von Einschaltvorgängen.- 14.4.1. Einschaltvorgang bei einem RC-Vierpol.- 14.4.2. Einschalten eines Gleichstromes auf einen Parallelschwingkreis.- 14.4.3. Impulsanregung eines kritisch gedämpften Schwingkreises.- 14.4.4. Einschaltvorgang eines Transformators.- 14.5. Heavisidesche Formel.- 15. Berechnung von Einschwingvorgängen durch Transformation der Differentialgleichung.- 15.1. Transformation der Differential-und Integraloperation.- 15.1.1. Differentiation.- 15.1.2. Integration.- 15.2. Lösung durch Transformation der Differentialgleichung.- 15.3. Schwingkreis mit Anfangsenergie.- Anhang: Formeln zur Laplace-Transformation.- Literatur.- Sachwortverzeichnis.

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Book SynopsisWie sieht eine Kurve aus, die die ganze Ebene oder den Raum vollständig ausfüllt? Kann man einen Polyeder flexibel bewegen, ja sogar umstülpen? Was ist die projektive Ebene oder der vierdimensionale Raum? Gibt es Seifenblasen, die keine runden Kugel sind? Wie kann man die komplizierte Struktur von Strömungen besser verstehen?In diesem Buch erleben Sie die Mathematik von ihrer anschaulichen Seite und finden faszinierende und bisher nie gesehene Bilder, die Ihnen illustrative Antworten zu all diesen Fragestellungen geben. Zu allen Bildern gibt es kurze Erklärungstexte, viele Literaturhinweise und jede Menge Web-Links. Das Buch ist für alle Freunde der Mathematik, die nicht nur trockenen Text und endlose Formeln sehen wollen. Vom Schüler zum Lehrer, vom Studenten zum Professor. Es soll sie alle inspirieren und anregen, sich mit diesem oder jenem vermeintlich nur Insidern vorbehaltenem Thema zu beschäftigen. Lernen Sie die Mathematik von einer ganz neuen und bunten Seite kennen. Die Neuauflage ist vollständig durchgesehen und um acht Doppelseiten mit neuen und spektakulären Bildern ergänzt. Stimmen zur 1. Auflage: „Die durchweg exzellenten grafischen Veranschaulichungen geben gute Beispiele, wie man elegant und sauber argumentiert. Möge dieses Buch viele Leserinnen und Leser zur Mathematik verführen." c't 17/09„In den ‚Bildern der Mathematik‘ kann man nach Herzenslust schmökern. Denn die einzelnen Mathematik-Häppchen und kleinen Geschichten sind zwar thematisch geordnet, bauen aber nicht aufeinander auf. So ist dieses Buch – für ein mathematisches Sachbuch sicher erstaunlich – sogar für den Nachttisch geeignet." Deutschlandradio KulturTrade Review“... Hier liegt ein Buch für alle jene vor, denen die Begeisterung für und das Verständnis von Mathematik am Herzen liegen. Wie das Buch eindrucksvoll zeigt, können dabei zielgerichtet erstellte Illustrationen wie eine Art Katalysator wirken. ... es leistet so einen wichtigen Beitrag zur Propagierung unseres Faches. ... An vielen Stellen finden sich auch nützliche Querverweise auf Ressourcen im Internet. Insgesamt untermauert dieses schöne Buch die Gültigkeit einer uns allen vertrauten Redensart: Ein Bild sagt mehr als tausendWorte!” (Hans Havlicek, in: Mathematische Semesterberichte, Jg. 62, 2015, S. 118)Table of ContentsEinleitung.- Polyedrische Modelle.- Geometrie in der Ebene.- Alte und neue Probleme.- Formeln und Zahlen.- Funktionen und Grenzwerte.- Kurven und Knoten.- Geometrie und Topologie von Flächen.- Minimalflächen und Seifenblasen.- Parkette und Packungen.- Raumformen und Dimensionen.- Graphen und Inzidenzen.- Bewegliche Formen.- Fraktale Mengen.- Landkarten und Abbildungen.- Formen und Verfahren in Natur und Technik.- Bildnachweis.- Index.

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Book SynopsisDie Graphentheorie gehört zu den Gebieten der Mathematik, die sich heute am stärksten entwickeln, zum Teil angestoßen durch Erfordernisse der Praxis, aber auch aus rein mathematischem Interesse. Dieses Kapitel der diskreten Mathematik auch Nicht-Fachleuten zugänglich zu machen, ist der Sinn dieses Buches. Es ist deshalb so geschrieben, dass es im Wesentlichen mathematisch exakt, aber auch ohne mathematische Vorkenntnisse verständlich und vor allem leicht lesbar ist. In Beispielen wird die Denkweise der modernen Mathematik nachvollziehbar und es werden auch Probleme dargestellt, die heute noch ungelöst sind. Der Autor hat wiederholt große Teile aus seinem Buch in verschiedenen Jahrgangsstufen erprobt: den Schülerinnen und Schülern hat Graphentheorie mehr Spaß gemacht als die sonstige Mathematik!Trade Review"Ein recht unterhaltsames Buch rund um die Graphentheorie." Die Wurzel, 02/2006 "Der Autor war Fachleiter für Mathematik an einem Berliner Gymnasium. Er hat sein Buch für Kollegen und Schüler mit besonderem Interesse geschrieben. Es eignet sich aber auch bestens für Studierende der Mathematik (insbesondere des Lehramts) für einen ersten Einblick." PM Praxis der Mathematik in der Schule, 03/2005 "Der Gymnasiallehrer Nitzsche gibt neun ansprechende, lebendig und anschaulich gestaltete Kapitel über eulersche, hamiltonsche und bipartite Graphen, Digraphen, Farben, Körper und Flächen." ekz-Informationsdienst, 50/04Table of ContentsErste Graphen - Über alle Brücken: Eulersche Graphen - Durch alle Städte: Hamiltonsche Graphen - Mehr über Grade von Ecken - Bäume - Bipartite Graphen - Graphen mit Richtungen - Körper und Flächen - Farben

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Book SynopsisDas Buch enthält eine Einführung in graphentheoretische Grundbegriffe und Basissätze. Graphen werden als Modellierungswerkzeuge für verschiedene Anwendungen aus dem Bereich der Standortplanung, Logistik, Verkehrsplanung, des Scheduling und der Planung von Kommunikationsnetzen vorgestellt. Für die entstehenden graphentheoretischen Probleme werden effiziente Verfahren vorgestellt und rigoros analysiert. Für komplexitätstheoretisch "schwierige" Probleme enthält das Buch effiziente Näherungsverfahren, die schnell Lösungen mit beweisbarer Güte liefern.Table of ContentsEinleitung - Graphentheoretische Grundbegriffe - Wege, Kreise, Zusammenhang - Färbungen und Überdeckungen - Transitive Hülle und irreduzible Kerne - Bäume, Wälder, Matroide - Suchstrategien - Kürzeste Wege - Flüsse und Strömungen - Matchings - Netzwerkdesign und Routing - Planare Graphen - Graphtransformationen - Baumweite

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Book SynopsisIn the tradition of EuroComb'01 (Barcelona), Eurocomb'03 (Prague), EuroComb'05 (Berlin), Eurocomb'07 (Seville), Eurocomb'09 (Bordeaux), and Eurocomb'11 (Budapest), this volume covers recent advances in combinatorics and graph theory including applications in other areas of mathematics, computer science and engineering. Topics include, but are not limited to: Algebraic combinatorics, combinatorial geometry, combinatorial number theory, combinatorial optimization, designs and configurations, enumerative combinatorics, extremal combinatorics, ordered sets, random methods, topological combinatorics.Table of Contents90 to 100 extended abstracts accepted by the Program Committee of Eurocomb 2013, formed by 24 experts in the field, for presentation at the forthcoming Conference Eurocomb 2013, Pisa, September 3-9, 2013.

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Book SynopsisThis book focuses on combinatorial problems in mathematical competitions. It provides basic knowledge on how to solve combinatorial problems in mathematical competitions, and also introduces important solutions to combinatorial problems and some typical problems with often-used solutions. Some enlightening and novel examples and exercises are well chosen in this book.With this book, readers can explore, analyze and summarize the ideas and methods of solving combinatorial problems. Their mathematical culture and ability will be improved remarkably after reading this book.Table of ContentsCounting Principles and Counting Formulas; Pigeonhole Principles and Mean Value Principles; Generating Functions; Recurrence Sequence of Numbers; Classification and Method of Fractional Steps; Corresponding Method; Counting in Two Ways; Recurrence Method; Coloring Method and Evaluation Method; Proof by Contradiction and Extreme Principle; Locally Adjusted Method; Constructive Method; Combinatorial Counting Problems; Existence Problems and the Proof of Inequalities in Combinatorial Problems; Combinatorial Extremum Problems.

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Book SynopsisThis is a textbook for an introductory combinatorics course lasting one or two semesters. An extensive list of problems, ranging from routine exercises to research questions, is included. In each section, there are also exercises that contain material not explicitly discussed in the preceding text, so as to provide instructors with extra choices if they want to shift the emphasis of their course.Just as with the first three editions, the new edition walks the reader through the classic parts of combinatorial enumeration and graph theory, while also discussing some recent progress in the area: on the one hand, providing material that will help students learn the basic techniques, and on the other hand, showing that some questions at the forefront of research are comprehensible and accessible to the talented and hardworking undergraduate. The basic topics discussed are: the twelvefold way, cycles in permutations, the formula of inclusion and exclusion, the notion of graphs and trees, matchings, Eulerian and Hamiltonian cycles, and planar graphs.New to this edition are the Quick Check exercises at the end of each section. In all, the new edition contains about 240 new exercises. Extra examples were added to some sections where readers asked for them.The selected advanced topics are: Ramsey theory, pattern avoidance, the probabilistic method, partially ordered sets, the theory of designs, enumeration under group action, generating functions of labeled and unlabeled structures and algorithms and complexity.The book encourages students to learn more combinatorics, provides them with a not only useful but also enjoyable and engaging reading.The Solution Manual is available upon request for all instructors who adopt this book as a course text. Please send your request to sales@wspc.com.The previous edition of this textbook has been adopted at various schools including UCLA, MIT, University of Michigan, and Swarthmore College. It was also translated into Korean.

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Book SynopsisThe solutions to each problem are written from a first principles approach, which would further augment the understanding of the important and recurring concepts in each chapter. Moreover, the solutions are written in a relatively self-contained manner, with very little knowledge of undergraduate mathematics assumed. In that regard, the solutions manual appeals to a wide range of readers, from secondary school and junior college students, undergraduates, to teachers and professors.

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Book SynopsisThis is a textbook for an introductory combinatorics course lasting one or two semesters. An extensive list of problems, ranging from routine exercises to research questions, is included. In each section, there are also exercises that contain material not explicitly discussed in the preceding text, so as to provide instructors with extra choices if they want to shift the emphasis of their course.Just as with the first two editions, the new edition walks the reader through the classic parts of combinatorial enumeration and graph theory, while also discussing some recent progress in the area: on the one hand, providing material that will help students learn the basic techniques, and on the other hand, showing that some questions at the forefront of research are comprehensible and accessible to the talented and hardworking undergraduate. The basic topics discussed are: the twelvefold way, cycles in permutations, the formula of inclusion and exclusion, the notion of graphs and trees, matchings, Eulerian and Hamiltonian cycles, and planar graphs.The selected advanced topics are: Ramsey theory, pattern avoidance, the probabilistic method, partially ordered sets, the theory of designs (new to this edition), enumeration under group action (new to this edition), generating functions of labeled and unlabeled structures and algorithms and complexity.As the goal of the book is to encourage students to learn more combinatorics, every effort has been made to provide them with a not only useful, but also enjoyable and engaging reading.The Solution Manual is available upon request for all instructors who adopt this book as a course text. Please send your request to sales@wspc.com.Table of ContentsBasic Methods Seven is More Than Six. The Pigeon-Hole Principle; One Step at a Time. The Method of Mathematical Induction; Enumerative Combinatorics There are a Lot of Them. Elementary Counting Problems; No Matter How You Slice It. The Binomial Theorem and Related Identities; Divide and Conquer. Partitions; Not So Vicious Cycles. Cycles in Permutations; You Shall Not Overcount. The Sieve; A Function is Worth Many Numbers. Generating Functions; Graph Theory Dots and Lines. The Origins of Graph Theory; Staying Connected. Trees; Finding a Good Match. Coloring and Matching; Do Not Cross. Planar Graphs; Horizons Does It clique? Ramsey Theory; So Hard to Avoid. Subsequence Conditions on Permutations; Who Knows What It Looks Like, But It Exists. The Probabilistic Method; At Least Some Order. Partial Orders and Lattices; As Evenly as Possible. Block Designs and Error Correcting Codes; Are They Really Different? Counting Unlabeled Structures; The Sooner the Better. Combinatorial Algorithms; Does Many Mean More Than One? Computational Complexity.

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Book SynopsisIn China, lots of excellent students who are good at maths takes an active part in various maths contests and the best six senior high school students will be selected to form the IMO National Team to compete in the International Mathematical Olympiad. In the past ten years China's IMO Team has achieved outstanding results — they have won the first place almost every year.The author is one of the coaches of China's IMO National Team, whose students have won many gold medals many times in IMO.This book is part of the Mathematical Olympiad Series which discusses several aspects related to maths contests, such as algebra, number theory, combinatorics, graph theory and geometry. The book elaborates on methods of discrete extremization, such as inequality control, repeated extremum, partial adjustment, exploiting symmetry, polishing transform, space estimates, etc.Table of ContentsInequality Control; Repeated Extremum; Partial Adjustment; Exploiting Symmetry; Polishing Transform; Space Estimates; Block Estimates; Guesses and Contradiction; Global Estimates; Parameter Estimates; Counting in Two Ways; Shrinking the Encirclement; Considering Special Cases; Solutions to Exercises;

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Book SynopsisSolving Fuzzy Linear Equations.- Graphs, Fuzzy Graphs and Intuitionistic Fuzzy Graphs.- Fuzzy Measures, Possibility and Necessity.- Compositional Rule of Inference.- Fuzzy Topological Spaces.- Fuzzy Subgroups and Fuzzy Normal Subgroups.- Application of Fuzzy Mathematics in Other Disciplines.

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