Combinatorics and graph theory Books

Book SynopsisCombinatorics is mathematics of enumeration, existence, construction, and optimization questions concerning finite sets. This text focuses on the first three types of questions and covers basic counting and existence principles, distributions, generating functions, recurrence relations, Polya theory, combinatorial designs, error correcting codes, partially ordered sets, and selected applications to graph theory including the enumeration of trees, the chromatic polynomial, and introductory Ramsey theory. The only prerequisites are single-variable calculus and familiarity with sets and basic proof techniques. The text emphasizes the brands of thinking that are characteristic of combinatorics: bijective and combinatorial proofs, recursive analysis, and counting problem classification. It is flexible enough to be used for undergraduate courses in combinatorics, second courses in discrete mathematics, introductory graduate courses in applied mathematics programs, as well as for independent study or reading courses. What makes this text a guided tour are the approximately 350 reading questions spread throughout its eight chapters. These questions provide checkpoints for learning and prepare the reader for the end-of-section exercises of which there are over 470. Most sections conclude with Travel Notes that add color to the material of the section via anecdotes, open problems, suggestions for further reading, and biographical information about mathematicians involved in the discoveries.Trade ReviewThis is a well-written, reader-friendly, and self-contained undergraduate course on combinatorics, focusing on enumeration. The book includes plenty of exercises, and about half of them come with hints."" - M. Bona, Choice Magazine""The delineation of the topics is first rate-better than I have ever seen in any other book. ... CAGT has both good breadth and great presentation; it is in fact a new standard in presentation for combinatorics, essential as a resource for any instructor, including those teaching out of a different text. For the student: If you are just starting to build a library in combinatorics, this should be your first book."" - The UMAP Journal""... [This book] is an excellent candidate for a special topics course for mathematics majors; with the broad spectrum of applications that course can simultaneously be an advanced and a capstone course. This book would be an excellent selection for the textbook of such a course. ... This book is the best candidate for a textbook in combinatorics that I have encountered."" - Charles Ashbacher

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Book SynopsisPresents an introductory and up-to-date course on game theory addressed to mathematicians and economists, and to other scientists having a basic mathematical background. The book is self-contained, providing a formal description of the classic game-theoretic concepts together with rigorous proofs of the main results in the field.Table of Contents Introduction to decision theory Strategic games Extensive games Games with incomplete information Fundamentals of cooperative games Applications of cooperative games Bibliography Notations Index Index of solution concepts Subject index.

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Book SynopsisCompressed sensing is a relatively recent area of research that refers to the recovery of high-dimensional but low-complexity objects from a limited number of measurements. The topic has applications to signal/image processing and computer algorithms, and it draws from a variety of mathematical techniques such as graph theory, probability theory, linear algebra, and optimization. The author presents significant concepts never before discussed as well as new advances in the theory, providing an in-depth initiation to the field of compressed sensing.An Introduction to Compressed Sensing contains substantial material on graph theory and the design of binary measurement matrices, which is missing in recent texts despite being poised to play a key role in the future of compressed sensing theory. It also covers several new developments in the field and is the only book to thoroughly study the problem of matrix recovery. The book supplies relevant results alongside their proofs in a compact and streamlined presentation that is easy to navigate.The core audience for this book is engineers, computer scientists, and statisticians who are interested in compressed sensing. Professionals working in image processing, speech processing, or seismic signal processing will also find the book of interest.

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Book SynopsisAdvanced Graph Theory focuses on some of the main notions arising in graph theory with an emphasis from the very start of the book on the possible applications of the theory and the fruitful links existing with linear algebra. The second part of the book covers basic material related to linear recurrence relations with application to counting and the asymptotic estimate of the rate of growth of a sequence satisfying a recurrence relation.Table of ContentsForeword ix Introduction xi Chapter 1. A First Encounter with Graphs 1 1.1. A few definitions 1 1.1.1. Directed graphs 1 1.1.2. Unoriented graphs 9 1.2. Paths and connected components 14 1.2.1. Connected components 16 1.2.2. Stronger notions of connectivity 18 1.3. Eulerian graphs 23 1.4. Defining Hamiltonian graphs 25 1.5. Distance and shortest path 27 1.6. A few applications 30 1.7. Comments 35 1.8. Exercises 37 Chapter 2. A Glimpse at Complexity Theory 43 2.1. Some complexity classes 43 2.2. Polynomial reductions 46 2.3. More hard problems in graph theory 49 Chapter 3. Hamiltonian Graphs 53 3.1. A necessary condition 53 3.2. A theorem of Dirac 55 3.3. A theorem of Ore and the closure of a graph 56 3.4. Chvátal’s condition on degrees 59 3.5. Partition of Kn into Hamiltonian circuits 62 3.6. De Bruijn graphs and magic tricks 65 3.7. Exercises 68 Chapter 4. Topological Sort and Graph Traversals 69 4.1. Trees 69 4.2. Acyclic graphs 79 4.3. Exercises 82 Chapter 5. Building New Graphs from Old Ones 85 5.1. Some natural transformations 85 5.2. Products 90 5.3. Quotients 92 5.4. Counting spanning trees 93 5.5. Unraveling 94 5.6. Exercises 96 Chapter 6. Planar Graphs 99 6.1. Formal definitions 99 6.2. Euler’s formula 104 6.3. Steinitz’ theorem 109 6.4. About the four-color theorem 113 6.5. The five-color theorem 115 6.6. From Kuratowski’s theorem to minors 120 6.7. Exercises 123 Chapter 7. Colorings 127 7.1. Homomorphisms of graphs 127 7.2. A digression: isomorphisms and labeled vertices 131 7.3. Link with colorings 134 7.4. Chromatic number and chromatic polynomial 136 7.5. Ramsey numbers 140 7.6. Exercises 147 Chapter 8. Algebraic Graph Theory 151 8.1. Prerequisites 151 8.2. Adjacency matrix 154 8.3. Playing with linear recurrences 160 8.4. Interpretation of the coefficients 168 8.5. A theorem of Hoffman 169 8.6. Counting directed spanning trees 172 8.7. Comments 177 8.8. Exercises 178 Chapter 9. Perron–Frobenius Theory 183 9.1. Primitive graphs and Perron’s theorem 183 9.2. Irreducible graphs 188 9.3. Applications 190 9.4. Asymptotic properties 195 9.4.1. Canonical form 196 9.4.2. Graphs with primitive components 197 9.4.3. Structure of connected graphs 206 9.4.4. Period and the Perron–Frobenius theorem 214 9.4.5. Concluding examples 218 9.5. The case of polynomial growth 224 9.6. Exercises 231 Chapter 10. Google’s Page Rank 233 10.1. Defining the Google matrix 238 10.2. Harvesting the primitivity of the Google matrix 241 10.3. Computation 246 10.4. Probabilistic interpretation 246 10.5. Dependence on the parameter α 247 10.6. Comments 248 Bibliography 249 Index 263

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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 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 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 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 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 text is a comprehensive survey of the literature surrounding star-critical Ramsey numbers. First defined by Jonelle Hook in her 2010 dissertation, these numbers aim to measure the sharpness of the corresponding Ramsey numbers by determining the minimum number of edges needed to be added to a critical graph for the Ramsey property to hold. Despite being in its infancy, the topic has gained significant attention among Ramsey theorists.This work provides researchers and students with a resource for studying known results and their complete proofs. It covers typical results, including multicolor star-critical Ramsey numbers for complete graphs, trees, cycles, wheels, and n-good graphs, among others. The proofs are streamlined and, in some cases, simplified, with a few new results included. The book also explores the connection between star-critical Ramsey numbers and deleted edge numbers, which focus on destroying the Ramsey property by removing edges.The book concludes with open problems and conjectures for researchers to consider, making it a valuable resource for those studying the field of star-critical Ramsey numbers.Table of Contents1. Multi Star-Critical Ramsey Numbers.- 2. Non-Complete Graphs.- 3. Generalizations of Star-Critical Ramsey Numbers.- 4. Open Problems.

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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 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 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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Book SynopsisTriple systems are among the simplest combinatorial designs. They have applications in coding theory, cryptography, computer science, statistcs, and many other areas. This book provides the first systematic and comprehensive treatment of triple systems. It gives an accurate picture of an incredibly rich and vibrant area of combinatorial mathematics.Table of ContentsHistorical introduction ; 1. Design-theoretic fundamentals ; 2. Existence: direct methods ; 3. Existence:recursive methods ; 4. Isomorphism and invariants ; 5. Enumeration ; 6. Subsystems and holes ; 7. Automorphisms I: small groups ; 8. Automorphisms II: large groups ; 9. Leaves and partial tripls systems ; 10. Excesses and coverings ; 11. Embedding and its variants ; 12. Neighbourhoods ; 13. Configurations ; 14. Intersections ; 15. Large sets and partitions ; 16. Support sizes ; 17. Independent sets ; 18. Chromatic number ; 19. Chromatic index and resolvability ; 20. Orthogonal resolutions ; 21. Nested and derived triple systems ; 22. Decomposability ; 23. Directed triple systems ; 24. Mendelsohn triple systems ; Bibliographies ; Index

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Book SynopsisGraph connectivities and submodular functions are two widely applied and fast developing fields of combinatorial optimization. This book not only includes the most recent results, but also highlights several surprising connections between diverse topics within combinatorial optimization. It offers a unified treatment of developments in the concepts and algorithmic methods of the area, starting from basic results on graphs, matroids and polyhedral combinatorics, through the advanced topics of connectivity issues of graphs and networks, to the abstract theory and applications of submodular optimization. Difficult theorems and algorithms are made accessible to graduate students in mathematics, computer science, operations research, informatics and communication. The book is not only a rich source of elegant material for an advanced course in combinatorial optimization, but it also serves as a reference for established researchers by providing efficient tools for applied areas like infocomTrade ReviewThe title of the book is wisely chosen: it deals, among other subjects, with graph connectivity, and it provides connections between graph theoretical results and underlying combinatorial structures...The book is readable for students, researchers, possibly also practitioners. * Mathematical Reviews *Table of ContentsPART I - BASIC COMBINATORIAL OPTIMIZATION; PART II - HIGHER-ORDER CONNECTIONS; PART III - SEMIMODULAR OPTIMIZATION

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Book SynopsisThis book includes introduction of several algorithms which are exclusively for graph based problems, namely combinatorial optimization problems, path formation problems, etc. Each chapter includes the introduction of the basic traditional nature inspired algorithm and discussion of the modified version for discrete algorithms including problems pertaining to discussed algorithms. Trade Review"Each chapter includes detailed problem formulation, practical examples, flowcharts illustrating special algorithms, questions and solved exercises which reinforce important topics. Besides being very useful to those who are interested in discrete optimizations problems and applying various metaheuristics to them, involved reader can also benefit from the easy way it presents various ideas and approaches to problem solutions. It is written in a clean and easily understandable, but still highly scientific language and it is a beneficial reading for post-docs and researchers interested in metaheuristic approaches to graph-based discrete optimization problems."—Zentralblatt MATHTable of Contents1. Introduction to Optimization Problems 2. Particle Swarm Optimization 3. Genetic Algorithms 4. Ant Colony Optimization 5. Bat Algorithm 6. Cuckoo Search Algorithm 7. Artificial Bee Colony 8. Shuffled Frog Leap Algorithm 9. Brain Storm Swarm Optimization Algorithm 10. Intelligent Water Drop Algorithm 11. Egyptian Vulture Algorithm 12. Biogeography-Based Optimization 13. Invasive Weed Optimization 14. Glowworm Swarm Optimization 15. Bacteria Foraging Optimization Algorithm 16. Flower Pollination Algorithm

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Book Synopsis Analytic Combinatorics: A Multidimensional Approach is written in a reader-friendly fashion to better facilitate the understanding of the subject. Naturally, it is a firm introduction to the concept of analytic combinatorics and is a valuable tool to help readers better understand the structure and large-scale behavior of discrete objects. Primarily, the textbook is a gateway to the interactions between complex analysis and combinatorics. The study will lead readers through connections to number theory, algebraic geometry, probability and formal language theory. The textbook starts by discussing objects that can be enumerated using generating functions, such as tree classes and lattice walks. It also introduces multivariate generating functions including the topics of the kernel method, and diagonal constructions. The second part explains methods of counting these objects, which involves deep mathematics coming from outside combinatorics, such aTable of ContentsA Primer on Combinatorical CalculusCombinatorical ParametersDerived and Transcendental ClassesGenerating Functions as Analytic ObjectsParallel TaxonomiesSingularities of Multvariable Rational FunctionsIntegration and Multivariable Coefficient AsymptoticsMultiple PointsPartitionsBibliographyGlossaryIndex

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Book SynopsisIntroduction to Number Theory is a classroom-tested, student-friendly text that covers a diverse array of number theory topics, from the ancient Euclidean algorithm for finding the greatest common divisor of two integers to recent developments such as cryptography, the theory of elliptic curves, and the negative solution of Hilbertâs tenth problem. The authors illustrate the connections between number theory and other areas of mathematics, including algebra, analysis, and combinatorics. They also describe applications of number theory to real-world problems, such as congruences in the ISBN system, modular arithmetic and Eulerâs theorem in RSA encryption, and quadratic residues in the construction of tournaments. Ideal for a one- or two-semester undergraduate-level course, this Second Edition: Features a more flexible structure that offers a greater range of options for course design Adds new sections on the representations of integTrade ReviewPraise for the Previous Edition "The authors succeed in presenting the topics of number theory in a very easy and natural way, and the presence of interesting anecdotes, applications, and recent problems alongside the obvious mathematical rigor makes the book even more appealing. … a valid and flexible textbook for any undergraduate number theory course."—International Association for Cryptologic Research Book Reviews, May 2011 "… a welcome addition to the stable of elementary number theory works for all good undergraduate libraries."—J. McCleary, Vassar College, Poughkeepsie, New York, USA, from CHOICE, Vol. 46, No. 1, August 2009 "… a reader-friendly text. … provides all of the tools to achieve a solid foundation in number theory."—L’Enseignement Mathématique, Vol. 54, No. 2, 2008 The theory of numbers is a core subject of mathematics. The authors have written a solid update to the first edition (CH, Aug'09, 46-6857) of this classic topic. There is no shortage of introductions to number theory, and this book does not offer significantly different information. Nonetheless, the authors manage to give the subject a fresh, new feel. The writing style is simple, clear, and easy to follow for standard readers. The book contains all the essential topics of a first-semester course and enough advanced topics to fill a second. In particular, it includes several modern aspects of number theory, which are often ignored in other texts, such as the use of factoring in computer security, searching for large prime numbers, and connections to other branches of mathematics. Each section contains supplementary homework exercises of various difficulties, a crucial ingredient of any good textbook. Finally, much emphasis is placed on calculating with computers, a staple of modern number theory. Overall, this title should be considered by any student or professor seeking an excellent text on the subject. --A. Misseldine, Southern Utah University, Choice magazine 2016 Praise for the Previous Edition "The authors succeed in presenting the topics of number theory in a very easy and natural way, and the presence of interesting anecdotes, applications, and recent problems alongside the obvious mathematical rigor makes the book even more appealing. … a valid and flexible textbook for any undergraduate number theory course."—International Association for Cryptologic Research Book Reviews, May 2011 "… a welcome addition to the stable of elementary number theory works for all good undergraduate libraries."—J. McCleary, Vassar College, Poughkeepsie, New York, USA, from CHOICE, Vol. 46, No. 1, August 2009 "… a reader-friendly text. … provides all of the tools to achieve a solid foundation in number theory."—L’Enseignement Mathématique, Vol. 54, No. 2, 2008 The theory of numbers is a core subject of mathematics. The authors have written a solid update to the first edition (CH, Aug'09, 46-6857) of this classic topic. There is no shortage of introductions to number theory, and this book does not offer significantly different information. Nonetheless, the authors manage to give the subject a fresh, new feel. The writing style is simple, clear, and easy to follow for standard readers. The book contains all the essential topics of a first-semester course and enough advanced topics to fill a second. In particular, it includes several modern aspects of number theory, which are often ignored in other texts, such as the use of factoring in computer security, searching for large prime numbers, and connections to other branches of mathematics. Each section contains supplementary homework exercises of various difficulties, a crucial ingredient of any good textbook. Finally, much emphasis is placed on calculating with computers, a staple of modern number theory. Overall, this title should be considered by any student or professor seeking an excellent text on the subject. --A. Misseldine, Southern Utah University, Choice magazine 2016 Table of ContentsIntroduction. Divisibility. Greatest Common Divisor. Primes. Congruences. Special Congruences. Primitive Roots. Cryptography. Quadratic Residues. Applications of Quadratic Residues. Sums of Squares. Further Topics in Diophantine Equations. Continued Fractions. Continued Fraction Expansions of Quadratic Irrationals. Arithmetic Functions. Large Primes. Analytic Number Theory. Elliptic Curves.

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Book SynopsisCombinatorics deals with ways of arranging and distributing mathematical objects, and involves ideas from geometry, algebra and analysis. The theory has broad applications, including codes, circuit design and algorithm complexity. Graph theory, enumeration, external problems, projective geometry, designs, colourings and codes, amongst others, are dealt with in a unified way.Trade Review'Both for the professional with a passing interest in combinatorics and for the students for whom it is primarily intended, this is a valuable book.' The Times Higher Education Supplement'… it will no doubt become a standard choice among the many texts on combinatorics … fascinating … it is highly recommended reading.' Dieter Jungnichel, Zentralblatt MATH'This well written textbook can be highly recommended to any student of combinatorics and, because of its breadth, has many new things to tell researchers in the field also.' EMS'This is a fascinating introduction to almost all aspects of combinatorics. Plenty of interesting problems, concrete examples, useful notes and references complement the main text. This book can be highly recommended to everyone interested in combinatorics.' Monatshefe für Mathematik'… becoming a modern classic … every good student should progress to this book at some stage: it is a wonderful source of elegant proofs and tantalising examples. No-one will find it easy, but every budding or established combinatorialist will be enriched by it … This text is unashamedly and impressively mathematical; it will challenge and inform every reader and is a very significant achievement.' The Mathematical GazetteTable of ContentsPreface; 1. Graphs; 2. Trees; 3. Colorings of graphs and Ramsey's theorem; 4. Turán's theorem and extremal graphs; 5. Systems of distinct representatives; 6. Dilworth's theorem and extremal set theory; 7. Flows in networks; 8. De Bruijn sequences; 9. The addressing problem for graphs; 10. The principle of inclusion and exclusion: inversion formulae; 11. Permanents; 12. The Van der Waerden conjecture; 13. Elementary counting: Stirling numbers; 14. Recursions and generating functions; 15. Partitions; 16. (0,1)-matrices; 17. Latin squares; 18. Hadamard matrices, Reed-Muller codes; 19. Designs; 20. Codes and designs; 21. Strongly regular graphs and partial geometries; 22. Orthogonal Latin squares; 23. Projective and combinatorial geometries; 24. Gaussian numbers and q-analogues; 25. Lattices and Möbius inversion; 26. Combinatorial designs and projective geometries; 27. Difference sets and automorphisms; 28. Difference sets and the group ring; 29. Codes and symmetric designs; 30. Association schemes; 31. Algebraic graph theory: eigenvalue techniques; 32. Graphs: planarity and duality; 33. Graphs: colorings and embeddings; 34. Electrical networks and squared squares; 35. Pólya theory of counting; 36. Baranyai's theorem; Appendices; Name index; Subject index.

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