Discrete mathematics Books

Book SynopsisMany believe mathematics is only about calculations, formulas, numbers, and strange letters. But mathematics is much more than just crunching numbers or manipulating symbols. Mathematics is about discovering patterns, uncovering hidden structures, finding counterexamples, and thinking logically. Mathematics is a way of thinking. It is an activity that is both highly creative and challenging. This book offers an introduction to mathematical reasoning for beginning university or college students, providing a solid foundation for further study in mathematics, computer science, and related disciplines. Written in a manner that directly conveys the sense of excitement and discovery at the heart of doing science, its 25 short and visually appealing chapters cover the basics of set theory, logic, proof methods, combinatorics, graph theory, and much more. In the book you will, among other things, find answers to: What is a proof? What is a counterexample? What does it mean to say that something follows logically from a set of premises? What does it mean to abstract over something? How can knowledge and information be represented and used in calculations? What is the connection between Morse code and Fibonacci numbers? Why could it take billions of years to solve Hanoi's Tower? Logical Methods is especially appropriate for students encountering such concepts for the very first time. Designed to ease the transition to a university or college level study of mathematics or computer science, it also provides an accessible and fascinating gateway to logical thinking for students of all disciplines.Trade Review"The definitions are followed by examples to help explain their meaning, along with counterexamples ... . Therefore, very little basic knowledge is required for this introduction to logical methods ... which is written in an accessible style ... . contained in the book are several hundred small figures; arrow, Venn, and Hasse diagrams; and simplifies visual representations ... . The author has also elected to use color to draw the reader's attention ... ." “From personal teaching experience, knowledge of these mathematical areas is necessary for disparate fields of CS and informatics. These foundations are needed for many fields, from database theory to various domains of information systems applications. The book’s presentation of topics and incentives for problem-solving, along with its exercises, is very useful for university-level instructors and students. The compact chapters contain clear explanations, diagrams, and brief descriptions of interesting facts.” (Bálint Molnár, Computing Reviews, July 27, 2021)Table of ContentsPreface.- 0 The Art of Thinking Abstractly and Mathematically.- 1 Basic Set Theory.- 2 Propositional Logic.- 3 Semantics from Propositional Logic.- 4 Concepts in Propositional Logic.- 5 Proofs, Conjectures, and Counterexamples.- 6 Relations.- 7 Functions.- 8 A Little More Set Theory.- 9 Closures and Inductively Defined Sets.- 10 Recursively Defined Functions.- 11 Mathematical Induction.- 12 Structural Induction.- 13 First-Order Languages.- 14 Representation of Quantified Statements.- 15 Interpretation in Models.- 16 Reasoning About Models.- 17 Abstraction with Equivalences and Partitions.- 18 Combinatorics.- 19 A Little More Combinatorics.- 20 A Bit of Abstract Algebra.- 21 Graph Theory.- 22 Walks in Graphs.- 23 Formal Languages and Grammars.- 24 Natural Deduction.- The Road Ahead.- Index. Symbols.

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

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

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Book SynopsisGraph algorithms is a well-established subject in mathematics and computer science. Beyond classical application fields, such as approximation, combinatorial optimization, graphics, and operations research, graph algorithms have recently attracted increased attention from computational molecular biology and computational chemistry. Centered around the fundamental issue of graph isomorphism, this text goes beyond classical graph problems of shortest paths, spanning trees, flows in networks, and matchings in bipartite graphs. Advanced algorithmic results and techniques of practical relevance are presented in a coherent and consolidated way. This book introduces graph algorithms on an intuitive basis followed by a detailed exposition in a literate programming style, with correctness proofs as well as worst-case analyses. Furthermore, full C++ implementations of all algorithms presented are given using the LEDA library of efficient data structures and algorithms.Table of Contents1. Introduction.- 2. Algorithmic Techniques.- 3. Tree Traversal.- 4. Tree Isomorphism.- 5. Graph Traversal.- 6. Clique, Independent Set, and Vertex Cover.- 7. Graph Isomorphism.

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This textbook discusses the design and implementation of basic algebraic graph algorithms, and algebraic graph algorithms for complex networks, employing matroids whenever possible. The text describes the design of a simple parallel matrix algorithm kernel that can be used for parallel processing of algebraic graph algorithms. Example code is presented in pseudocode, together with case studies in Python and MPI. The text assumes readers have a background in graph theory and/or graph algorithms.

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Book SynopsisThis book constitutes the refereed proceedings of the First Workshop on Monte Carlo Search, MCS 2020, organized in conjunction with IJCAI 2020. The event was supposed to take place in Yokohama, Japan, in July 2020, but due to the Covid-19 pandemic was held virtually on January 7, 2021. The 9 full papers of the specialized project were carefully reviewed and selected from 15 submissions. The following topics are covered in the contributions: discrete mathematics in computer science, games, optimization, search algorithms, Monte Carlo methods, neural networks, reinforcement learning, machine learning.Table of ContentsThe αµ Search Algorithm for the Game of Bridge.- Stabilized Nested Rollout Policy Adaptation.- zoNNscan: A Boundary-Entropy Index for Zone Inspection of Neural Models.- Ordinal Monte Carlo Tree Search.- Monte Carlo Game Solver.- Generalized Nested Rollout Policy Adaptation.- Monte Carlo Inverse Folding.- Monte Carlo Graph Coloring.- Enhancing Playout Policy Adaptation for General Game Playing.

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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 book constitutes the refereed proceedings of the 15th International Conference on Graph Transformation, ICGT 2022, which took place Nantes, France in July 2022.The 10 full papers and 1 tool paper presented in this book were carefully reviewed and selected from 19 submissions. The conference focuses on describing new unpublished contributions in the theory and applications of graph transformation as well as tool presentation papers that demonstrate main new features and functionalities of graph-based tools.Table of ContentsTheoretical Advances.- Application Domains.- Tool Presentation.

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Book SynopsisThis book constitutes the proceedings of the 16th International Conference on Algorithmic Aspects in Information and Management, AAIM 2022, which was held online during August 13-14, 2022. The conference was originally planned to take place in Guangzhou, China, but changed to a virtual event due to the COVID-19 pandemic.The 41 regular papers included in this book were carefully reviewed and selected from 59 submissions. Table of ContentsAn improvement of the bound on the odd chromatic number of 1-planar graphs.- AoI Minimizing of Wireless Rechargeable Sensor Network based on Trajectory Optimization of Laser-Charged UAV.- Monotone k-Submodular Knapsack Maximization: An Analysis of the Greedy+Singleton Algorithm.- The constrained parallel-machine scheduling problem with divisible processing times and penalties.- Energy-constrained Geometric Covering Problem.- Fast searching on $k$-combinable graphs.- Three Algorithms for Converting Control Flow Statements from Python to XD-M.- Class Ramsey numbers involving induced graphs.- An Approximation Algorithm for the Clustered Path Travelling Salesman Problem.- Hyperspectral Image Reconstruction for SD-CASSI systems based on Residual Attention Network.- Improved Approximation Algorithm for the Asymmetric Prize-Collecting TSP.- Injective edge coloring of power graphs and necklaces.- Guarantees for Maximization of $k$-Submodular Functions with a Knapsack and a Matroid Constraint.- Incremental SDN Deployment to Achieve Load Balance in ISP Networks.- Approximation scheme for single-machine rescheduling with job delay and rejection.- Defense of Scapegoating Attack in Network Tomography.- A Binary Search Double Greedy Algorithm for Non-monotone DR-submodular Maximization.- Streaming Adaptive Submodular Maximization.- Constrained Stochastic Submodular Maximization with State-Dependent Costs.- Online early work maximization problem on two hierarchical machines with buffer or rearrangements.- Polynomial time algorithm for k-vertex-edge dominating problem in interval graphs.- Adaptive Competition-based Diversified-profit Maximization with Online Seed Allocation.- Collaborative Service Caching in Mobile Edge Nodes.- A Decentralized Auction Framework with Privacy Protection in Mobile Crowdsourcing.- On-line single machine scheduling with release dates and submodular rejection penalties.- Obnoxious Facility Location Games with Candidate Locations.- Profit Maximization for Multiple Products in Community-based Social Networks.- MCM: A Robust Map Matching Method by Tracking Multiple Road Candidates.- Security on Ethereum: Ponzi Scheme Detection in Smart Contract.- Cyclically orderable generalized Petersen graphs.- The r-dynamic chromatic number of planar graphs without special short cycles.- Distance Labeling of the Halved Folded $n$-Cube.- Signed network embedding based on muti-attention mechanism.- Balanced Graph Partitioning based on Mixed 0-1 Linear Programming and Iteration Vertex Relocation Algorithm.- Partial inverse min-max spanning tree problem under the weighted bottleneck Hamming distance.- Mixed Metric Dimension of Some Plane Graphs.- The Optimal Dynamic Rationing Policy in the Stock-Rationing Queue.- Pilot Pattern Design with Branch and Bound in PSA-OFDM System.- Bicriteria Algorithms for Maximizing the Difference Between Submodular Function and Linear Function under Noise.- On the Transversal Number of k-Uniform Connected Hypergraphs.- Total coloring of planar graphs without some adjacent cycles.

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Book SynopsisThis textbook aims to point out the most important principles of data analysis from the mathematical point of view. Specifically, it selected these questions for exploring: Which are the principles necessary to understand the implications of an application, and which are necessary to understand the conditions for the success of methods used? Theory is presented only to the degree necessary to apply it properly, striving for the balance between excessive complexity and oversimplification. Its primary focus is on principles crucial for application success. Topics and features: Focuses on approaches supported by mathematical arguments, rather than sole computing experiences Investigates conditions under which numerical algorithms used in data science operate, and what performance can be expected from them Considers key data science problems: problem formulation including optimality measure; learning and generalization in relationships to training set size and number of free parameters; and convergence of numerical algorithms Examines original mathematical disciplines (statistics, numerical mathematics, system theory) as they are specifically relevant to a given problem Addresses the trade-off between model size and volume of data available for its identification and its consequences for model parametrization Investigates the mathematical principles involves with natural language processing and computer vision Keeps subject coverage intentionally compact, focusing on key issues of each topic to encourage full comprehension of the entire book Although this core textbook aims directly at students of computer science and/or data science, it will be of real appeal, too, to researchers in the field who want to gain a proper understanding of the mathematical foundations “beyond” the sole computing experience.Table of Contents1. Data Science and its Tasks.- 2. Application Specific Mappings and Measuring the Fit to Data.- 3. Data Processing by Neural Networks.- 4. Learning and Generalization.- 5. Numerical Algorithms for Network Learning.- 6. Specific Problems of Natural Language Processing.- 7. Specific Problems of Computer Vision.

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Book SynopsisGenerating function (GF) is a mathematical technique to concisely represent a known ordered sequence into a simple continuous algebraic function in dummy variable(s). This Second Edition introduces commonly encountered generating functions (GFs) in engineering and applied sciences, such as ordinary GF (OGF), exponential GF (EGF), as also Dirichlet GF (DGF), Lambert GF (LGF), Logarithmic GF (LogGF), Hurwitz GF (HGF), Mittag-Lefler GF (MLGF), etc. This book is intended mainly for beginners in applied science and engineering fields to help them understand single-variable GFs and illustrate how to apply them in various practical problems. Specifically, the book discusses probability GFs (PGF), moment and cumulant GFs (MGF, CGF), mean deviation GFs (MDGF), survival function GFs (SFGF), rising and falling factorial GFs, factorial moment, and inverse factorial moment GFs. Applications of GFs in algebra, analysis of algorithms, bioinformatics, combinatorics, economics, finance, genomics, geometry, graph theory, management, number theory, polymer chemistry, reliability, statistics and structural engineering have been added to this new edition. This book is written in such a way that readers who do not have prior knowledge of the topic can easily follow through the chapters and apply the lessons learned in their respective disciplines.Table of ContentsTypes of Generating Functions.- Operations on Generating Functions.- Generating Functions in Statistics.- Applications of Generating Functions.- Bibliography.

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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 SynopsisThis book constitutes the proceedings of the 30th International Symposium on Graph Drawing and Network Visualization, GD 2022, held in Tokyo, Japan, during September 13-16, 2022. The 25 full papers, 7 short papers, presented together with 2 invited talks, one report on graph drawing contest, and one obituary in these proceedings were carefully reviewed and selected from 70 submissions. The abstracts of 5 posters presented at the conference can be found in the back matter of the volume. The contributions were organized in topical sections as follows: properties of drawings of complete graphs; stress-based visualizations of graphs; planar and orthogonal drawings; drawings and properties of directed graphs; beyond planarity; dynamic graph visualization; linear layouts; and contact and visibility graph representations. Table of ContentsProperties of Drawings of Complete Graphs.- Stress-based Visualizations of Graphs.- Planar and Orthogonal Drawings.- Drawings and Properties of Directed Graphs.- Beyond Planarity.- Dynamic Graph Visualization.- Linear Layouts.- Contact and Visibility Graph Representations.

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Book Synopsis

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Book SynopsisThis book constitutes the refereed proceedings of the 24th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2023, held in Madison, WI, USA, during June 21–23, 2023. The 33 full papers presented were carefully reviewed and selected from 119 submissions. IPCO is under the auspices of the Mathematical Optimization Society, and it is an important forum for presenting present recent developments in theory, computation, and applications. The scope of IPCO is viewed in a broad sense, to include algorithmic and structural results in integer programming and combinatorial optimization as well as revealing computational studies and novel applications of discrete optimization to practical problems.

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Book SynopsisThis volume constitutes the proceedings of the 9th International Conference on Variable Neighborhood Search, ICVNS 2023, held in Abu Dhabi, United Arab Emirates, in October 2022.The 11 full papers presented in this volume were carefully reviewed and selected from 29 submissions. The papers describe recent advances in methods and applications of variable neighborhood search.Table of ContentsA metaheuristic approach for solving Monitor Placement Problem.- A VNS-based heuristic for the minimum number of resources under a perfect schedule.- BVNS for Overlapping Community Detection.- A Simulation-Based Variable Neighborhood Search Approach for Optimizing Cross-Training Policies.- Multi-Objective Variable Neighborhood Search for improving software modularity.- An Effective VNS for Delivery Districting.- BVNS for the Minimum Sitting Arrangement problem in a cycle.- Assigning Multi-Skill Confgurations to Multiple Servers with a Reduced VNS.- Multi-Round Infuence Maximization: A Variable Neighborhood Search Approach.- A VNS based heuristic for a 2D Open Dimension Problem.- BVNS for the bi-objective multi row equal facility layout problem.

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Book SynopsisThis book constitutes the refereed proceedings of the 17th International Joint Conference on Theoretical Computer Science-Frontier of Algorithmic Wisdom (IJTCS-FAW 2023), consisting of the 17th International Conference on Frontier of Algorithmic Wisdom (FAW) and the 4th International Joint Conference on Theoretical Computer Science (IJTCS), held in Macau, China, during August 14–18, 2023.FAW started as the Frontiers of Algorithmic Workshop in 2007 at Lanzhou, China, and was held annually from 2007 to 2021 and published archival proceedings. IJTCS, the International joint theoretical Computer Science Conference, started in 2020, aimed to bring in presentations covering active topics in selected tracks in theoretical computer science. To accommodate the diversified new research directions in theoretical computer science, FAW and IJTCS joined their forces together to organize an event for information exchange of new findings and work of enduring value in the field. The 21 full papers included in this book were carefully reviewed and selected from 34 submissions. They were organized in topical sections as follows: algorithmic game theory; algorithms and data structures; combinatorial optimization; and computational economics.Table of ContentsUnderstanding the Relationship Between Core Constraints and Core-Selecting Payment Rules in Combinatorial Auctions.- An Improved Analysis of the Greedy+Singleton Algorithm for k-Submodular Knapsack Maximization.- Generalized Sorting with Predictions Revisited.- Eliciting Truthful Reports with Partial Signals in Repeated Games.- On the NP-hardness of two scheduling problems under linear constraints.- On the Matching Number of k-Uniform Connected Hypergraphs with Maximum Degree.- Max-Min Greedy Matching Problem: Hardness for the Adversary and Fractional Variant.- Approximate Core Allocations for Edge Cover Games.- Random Approximation Algorithms for Monotone k-Submodular Function Maximization with Size Constraints.- Additive Approximation Algorithms for Sliding Puzzle.- Differential Game Analysis for Cooperation Models in Automotive Supply Chain under Low-Carbon Emission Reduction Policies.- Adaptivity Gap for Influence Maximization with Linear Threshold Model on Trees.- Physically Verifying the First Nonzero Term in a Sequence: Physical ZKPs for ABC End View and Goishi Hiroi.- Mechanism Design in Fair Sequencing.- Red-Blue Rectangular Annulus Cover Problem.- Applying Johnson's Rule in Scheduling Multiple Parallel Two-Stage Flowshops.- The Fair k-Center with Outliers Problem: FPT and Polynomial Approximations.- Constrained Graph Searching on Trees.- EFX Allocations Exist for Binary Valuations.- Maximize Egalitarian Welfare for Cake Cutting.- Stackelberg Strategies on Epidemic Containment Games.

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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 SynopsisThis volume constitutes the thoroughly refereed proceedings of the 49th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2023. The 33 full papers presented in this volume were carefully reviewed and selected from a total of 116 submissions. The WG 2022 workshop aims to merge theory and practice by demonstrating how concepts from graph theory can be applied to various areas in computer science, or by extracting new graph theoretic problems from applications.Table of ContentsProportionally Fair Matching with Multiple Groups.- Reconstructing Graphs from Connected Triples.- Parameterized Complexity of Vertex Splitting to Pathwidth at most 1.- Odd Chromatic Number of Graph Classes.- Deciding the Erdos-P osa property in 3-connected digraphs.- New Width Parameters for Independent Set: One-sided-mim-width and Neighbor-depth.- Computational Complexity of Covering Colored Mixed Multigraphswith Degree Partition Equivalence Classes of Size at Most Two.- Cutting Barnette graphs perfectly is hard.- Metric dimension parameterized by treewidth in chordal graphs.- Efficient Constructions for the Gyori-Lovasz Theorem on Almost Chordal Graphs.- Generating faster algorithms for d-Path Vertex Cover.- A new width parameter of graphs based on edge cuts: -edge-crossing width.- Snakes and Ladders: a Treewidth Story.- Parameterized Results on Acyclic Matchings with Implications for Related Problems.- P-matchings Parameterized by Treewidth.- Algorithms and hardness for Metric Dimension on digraphs.- Degreewidth : a New Parameter for Solving Problems on Tournaments.- Approximating Bin Packing with Con ict Graphs via Maximization Techniques.- i-Metric Graphs: Radius, Diameter and all Eccentricities.- Maximum edge colouring problem on graphs that exclude a xed minor.- Bounds on Functionality and Symmetric Di erence { Two Intriguing Graph Parameters.- Cops and Robbers on Multi-layer Graphs.- Parameterized Complexity of Broadcasting in Graphs.- Turan's Theorem Through Algorithmic Lens.- On the Frank number and nowhere-zero ows on graphs.- On the minimum number of arcs in 4-dicritical oriented graphs.- Tight Algorithms for Connectivity Problems Parameterized byModular-Treewidth.

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Book Synopsis^ the= study= of= convex= polyhedra= in= ordinary= space= is= a= central= piece= classical= and= modern= geometry= that= has= had= significant= impact= on= many= areas= mathematics= also= computer= science.= present= book= project= by= joseph= o'rourke= costin= vîlcu= brings= together= two= important= strands= subject= = combinatorics= polyhedra,= intrinsic= underlying= surface.= this= leads= to= remarkable= interplay= concepts= come= life= wide= range= very= attractive= topics= concerning= polyhedra.= gets= message= across= thetheory= although= with= roots,= still= much= alive= today= continues= be= inspiration= basis= lot= current= research= activity.= work= presented= manuscript= interesting= applications= discrete= computational= geometry,= as= well= other= mathematics.= treated= detail= include= unfolding= onto= surfaces,= continuous= flattening= convexity= theory= minimal= length= enclosing= polygons.= along= way,= open= problems= suitable= for= graduate= students= are= raised,=

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Book SynopsisThis book constitutes the proceedings of the 49th International Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2024, held in Cochem, Germany, in February 2024. The 33 full papers presented in this book were carefully reviewed and selected from 81 submissions. The book also contains one invited talk in full paper length. They focus on original research and challenges in foundations of computer science including algorithms, AI-based methods, computational complexity, and formal models.Table of ContentsThe Information Extraction Framework of Document Spanners - A Very Informal Survey.- Generalized Distance Polymatrix Games.- Relaxed agreement forests.- On the Computational Complexity of Generalized Common Shape Puzzles.- Fractional Bamboo Trimming and Distributed Windows Scheduling.- New support size bounds and proximity bounds for integer linear programming.- On the Parameterized Complexity of Minus Domination.- Exact and Parameterized Algorithms for Choosability.- Parameterized Algorithms for Covering by Arithmetic Progressions.- Row-column combination of Dyck words.- Group Testing in Arbitrary Hypergraphs and Related Combinatorial Structures.- On the parameterized complexity of the Perfect Phylogeny problem.- Data reduction for directed feedback vertex set on graphs without long induced cycles.- Visualization of Bipartite Graphs in Limited Window Size.- Outerplanar and Forest Storyplans.- The Complexity of Cluster Vertex Splitting and Company.- Morphing Graph Drawings in the Presence of Point Obstacles.- Word-Representable Graphs from a Word’s Perspective.- Removable Online Knapsack with Bounded Size Items.- The Complexity of Online Graph Games.- Faster Winner Determination Algorithms for (Colored) Arc Kayles.- Automata Classes Accepting Languages Whose Commutative Closure is Regular.- Shortest Characteristic Factors of a Deterministic Finite Automaton and Computing Its Positive Position Run by Pattern Set Matching.- Query Learning of Minimal Deterministic Symbolic Finite Automata Separating Regular Languages.- Apportionment with Thresholds: Strategic Campaigns Are Easy in the Top-Choice But Hard in the Second-Chance Mode.- Local Certification of Majority Dynamics.- Complexity of Spherical Equations in Finite Groups.- Positive Characteristic Sets for Relational Pattern Languages.- Algorithms and Turing Kernels for Detecting and Counting Small Patterns in Unit Disk Graphs.- The Weighted HOM-Problem over Fields.- Combinatorics of block-parallel automata networks.- On the piecewise complexity of words and periodic words.- Distance Labeling for Families of Cycles.- On the induced problem for fixed-template CSPs.

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Book Synopsis- Mathematics and computer science in the information revolution.- NLP and some research results in Senegal.- On absolute valued algebras with nonzero central element.- On algebraic algebras without divisors of zero satisfying (xp, xq, xr) = 0.- Computing minimal free resolutions over monomial semirings with coefficients in D-A rings.- Schur complement and inequalities of eigenvalues on block Hadamard product.- A perturbed Mann-type algorithm for zeros of maximal monotone mappings.- On Rickart and Baer semimodules.- Completion fractions modules of filtered modules over non-necessarily commutative filtered rings.- On S-lifting semimodules over semirings.- A contribution to the study of a class of noncommutative ideals admitting finite Gröbner bases.- Construction of numbers with the same normality properties as a given number. - Robustness of imputation methods with backpropagation algorithm in nonlinear multiple regression.- A better random forest classifier: Labels guided Mondrian forest.- Remote sensing of artisanal mines buried in the ground by infrared thermography using UAV.- Implementation of EdDSA in the Ethereum blockchain.- Vulnerability prediction of web applications from source code based on machine learning and deep learning: Where are we at?.- Business process management and process mining on the large: Overview, challenges and research directions.

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

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

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Book SynopsisThe starting point for this monograph is the previously unknown connection between the Continuum Hypothesis and the saturation of the non-stationary ideal on ω1; and the principle result of this monograph is the identification of a canonical model in which the Continuum Hypothesis is false. This is the first example of such a model and moreover the model can be characterized in terms of maximality principles concerning the universal-existential theory of all sets of countable ordinals. This model is arguably the long sought goal of the study of forcing axioms and iterated forcing but is obtained by completely different methods, for example no theory of iterated forcing whatsoever is required. The construction of the model reveals a powerful technique for obtaining independence results regarding the combinatorics of the continuum, yielding a number of results which have yet to be obtained by any other method. This monograph is directed to researchers and advanced graduate students in Set Theory. The second edition is updated to take into account some of the developments in the decade since the first edition appeared, this includes a revised discussion of Ω-logic and related matters.

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

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Book SynopsisIn the two-volume set ‘A Selection of Highlights’ we present basics of mathematics in an exciting and pedagogically sound way. This volume examines many fundamental results in Geometry and Discrete Mathematics along with their proofs and their history. In the second edition we include a new chapter on Topological Data Analysis and enhanced the chapter on Graph Theory for solving further classical problems such as the Traveling Salesman Problem.

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

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Book SynopsisAdapted from a modular undergraduate course on computational mathematics, Concise Computer Mathematics delivers an easily accessible, self-contained introduction to the basic notions of mathematics necessary for a computer science degree. The text reflects the need to quickly introduce students from a variety of educational backgrounds to a number of essential mathematical concepts. The material is divided into four units: discrete mathematics (sets, relations, functions), logic (Boolean types, truth tables, proofs), linear algebra (vectors, matrices and graphics), and special topics (graph theory, number theory, basic elements of calculus). The chapters contain a brief theoretical presentation of the topic, followed by a selection of problems (which are direct applications of the theory) and additional supplementary problems (which may require a bit more work). Each chapter ends with answers or worked solutions for all of the problems.Trade ReviewFrom the reviews:“The book is ideally suited as an adjunct to a course in computer mathematics or as a refresher for someone with some background in computer mathematics. … The book fulfills its purpose of providing a distilled treatment of the mathematics most commonly used in computer science. It is of most value to computer science students who need a place to find a succinct treatment of the topics covered.” (Marlin Thomas, Computing Reviews, April, 2014)“Each of the chapters opens with a short summary followed by a set of essential problems and then a set of supplementary problems. … it would be very useful for someone that needs a quick and effective review that includes problems.” (Charles Ashbacher, MAA Reviews, January, 2014)Table of ContentsSets and NumbersRelations and DatabasesFunctionsBoolean Algebra, Logic and QuantifiersNormal Forms, Proof and ArgumentVectors and Complex NumbersMatrices and ApplicationsMatrix Transformations for Computer GraphicsElements of Graph TheoryElements of Number Theory and CryptographyElements of CalculusElementary Numerical Methods

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Book SynopsisThis book takes the reader on a mathematical journey, from a number-theoretic point of view, to the realm of Markov’s theorem and the uniqueness conjecture, gradually unfolding many beautiful connections until everything falls into place in the proof of Markov’s theorem. What makes the Markov theme so attractive is that it appears in an astounding variety of different fields, from number theory to combinatorics, from classical groups and geometry to the world of graphs and words.On the way, there are also introductory forays into some fascinating topics that do not belong to the standard curriculum, such as Farey fractions, modular and free groups, hyperbolic planes, and algebraic words. The book closes with a discussion of the current state of knowledge about the uniqueness conjecture, which remains an open challenge to this day.All the material should be accessible to upper-level undergraduates with some background in number theory, and anything beyond this level is fully explained in the text.This is not a monograph in the usual sense concentrating on a specific topic. Instead, it narrates in five parts – Numbers, Trees, Groups, Words, Finale – the story of a discovery in one field and its many manifestations in others, as a tribute to a great mathematical achievement and as an intellectual pleasure, contemplating the marvellous unity of all mathematics.Trade ReviewFrom the book reviews:“The topic, and its presentation, does make it a fine source for seminar usage. … this is a fine text for students who are ready to see material connecting various areas of mathematics. It reveals the beauty and hints at the excitement of ‘live’ mathematics.” (Thomas A. Schmidt, Mathematical Reviews, September, 2014)“In number theory, Markov’s theorem (1879) reveals surprising structure within a set of real numbers, called the Lagrange spectrum, which collects precise information about approximability of irrational numbers. … Summing Up: Highly recommended. Upper-division undergraduates and above.” (D. V. Feldman, Choice, Vol. 51 (8), April, 2014)“Book tells the story of a celebrated theorem and an intriguing conjecture: Markov’s theorem from 1879 and the uniqueness conjecture formulated by Frobenius … . author takes the opportunity to look at this theorem and this conjecture from many different viewpoints … . He offers a journey through the mathematical world around Markov’s theorem in a leisurely and relaxed style, making his book very pleasant to read. … An undergraduate student will certainly enjoy this reading and learn a lot.” (Yann Bugeaud, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 116, 2014)“This beautiful book gives readers a chance to familiarize themselves with a very simple and yet very difficult problem in number theory, and teaches them that it pays to look at a problem from many different angles. I recommend it to all students who are already hooked to number theory, and perhaps even more to those who are not.” (Franz Lemmermeyer, zbMATH, Vol. 1276, 2014)Table of ContentsApproximation of Irrational Numbers.- Markov's Theorem and the Uniqueness Conjecture.- The Markov Tree.- The Cohn Tree.- The Modular Group SL(2,Z).- The Free Group F2.- Christoffel Words.- Sturmian Words.- Proof of Markov's Theorem.- The Uniqueness Conjecture. ​

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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 is an elegant and rigorous presentation of integer programming, exposing the subject’s mathematical depth and broad applicability. Special attention is given to the theory behind the algorithms used in state-of-the-art solvers. An abundance of concrete examples and exercises of both theoretical and real-world interest explore the wide range of applications and ramifications of the theory. Each chapter is accompanied by an expertly informed guide to the literature and special topics, rounding out the reader’s understanding and serving as a gateway to deeper study.Key topics include: formulations polyhedral theory cutting planes decomposition enumeration semidefinite relaxations Written by renowned experts in integer programming and combinatorial optimization, Integer Programming is destined to become an essential text in the field.Trade Review“Integer Programming begins by introducing the subject and giving several examples of integer programming problems. … This book would be suitable for a graduate level course on the mathematics of cutting plane methods. … This book might also be of interest as a reference for researchers working in this area. … This book offers a more focused presentation that makes it better suited for use as a textbook.” (Brian Borchers, MAA Reviews, maa.org, December, 2015)“The book is written in a very clear and didactic style. … very useful for mathematically mature undergraduates, graduate students, postdocs, and established researchers who are interested in the techniques. …This is an excellent and impressive book. We wholeheartedly recommend it as a textbook for advanced undergraduate and introductory graduate courses on integer programming.” (Jakub Marecek, Interfaces, Vol. 45 (5), September-October, 2015)“The authors deliver a comprehensive presentation of integer programming. … Everything is presented in a rigorous way, but on the other hand, the form makes it easy to understand for everyone. Each chapter is followed by the exercises, that allow to recall the contents. … the book is an essential text in the field of integer programing, that should be recommended as a very useful textbook for students, but also a valuable introduction for the researchers in this area.” (Marcin Anholcer, zbMATH 1307.90001, 2015)Table of ContentsPreface.- 1 Getting Started.- 2 Integer Programming Models.- 3 Linear Inequalities and Polyhedra.- 4 Perfect Formulations.- 5 Split and Gomory Inequalities.- 6 Intersection Cuts and Corner Polyhedra.- 7 Valid Inequalities for Structured Integer Programs.- 8 Reformulations and Relaxations.- 9 Enumeration.- 10 Semidefinite Bounds.- Bibliography.- Index.

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

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Book SynopsisProviding a self-contained resource for upper undergraduate courses in combinatorics, this text emphasizes computation, problem solving, and proof technique. In particular, the book places special emphasis the Principle of Inclusion and Exclusion and the Multiplication Principle. To this end, exercise sets are included at the end of every section, ranging from simple computations (evaluate a formula for a given set of values) to more advanced proofs. The exercises are designed to test students' understanding of new material, while reinforcing a working mastery of the key concepts previously developed in the book. Intuitive descriptions for many abstract techniques are included. Students often struggle with certain topics, such as generating functions, and this intuitive approach to the problem is helpful in their understanding. When possible, the book introduces concepts using combinatorial methods (as opposed to induction or algebra) to prove identities. Students are also asked to prove identities using combinatorial methods as part of their exercises. These methods have several advantages over induction or algebra.Trade Review“The book is an excellent introduction to combinatorics. … The author uses a clear language and often provides an easy intuitive access to abstract subjects. The presentation is well motivated, the explanations are transparent and illustrated by carefully selected examples. Each section ends with a list of well formulated exercises which make the book ideally suited for self-instruction.” (Astrid Reifegerste, zbMATH 1328.05001, 2016)“This book by Beeler … is an excellent introductory text on combinatorics. The author gives the right balance of theory, computation, and applications, and he presents introductory-level topics, such as the multiplication principle, binomial theorem, and distribution problems in a clear manner. … Summing Up: Highly recommended. Upper-division undergraduates through researchers and faculty.” (S. L. Sullivan, Choice, Vol. 53 (1), September, 2015)Table of ContentsPreliminaries.- Basic Counting.- The Binomial Coefficient.- Distribution Problems.- Generating Functions.- Recurrence Relations.- Advanced Counting - Inclusion and Exclusion.- Advanced Counting - Polya Theory.- Application: Probability.- Application: Combinatorial Designs.- Application: Graph Theory.- Appendices.

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

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Book SynopsisThis 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 highly readable book aims to ease the many challenges of starting undergraduate research. It accomplishes this by presenting a diverse series of self-contained, accessible articles which include specific open problems and prepare the reader to tackle them with ample background material and references. Each article also contains a carefully selected bibliography for further reading.The content spans the breadth of mathematics, including many topics that are not normally addressed by the undergraduate curriculum (such as matroid theory, mathematical biology, and operations research), yet have few enough prerequisites that the interested student can start exploring them under the guidance of a faculty member. Whether trying to start an undergraduate thesis, embarking on a summer REU, or preparing for graduate school, this book is appropriate for a variety of students and the faculty who guide them. Trade Review“This book is a superb resource for students and faculty mentors embarking on undergraduate research in mathematics. Its focus is on topics and applications rarely covered in the traditional undergraduate math curriculum, offering novice researchers a sturdy jumping-off point to a broad array of research problems. … A valuable resource for students and faculty mentors interested in undergraduate research.” (V. K. Chellamuthu, Choice, Vol. 56 (2), October, 2018)Table of ContentsCoxeter Groups and the Davis Complex (T.A. Schroeder).- A Tale of Two Symmetries: Embeddable and Non-Embeddable Group Actions on Surfaces (V. Peterson, A. Wootton).- Tile Invariants for Tackling Tiling Questions (M.P. Hitchman).- Forbidden Minors: Finding the Finite Few (T.W. Mattman).- Introduction to competitive graph coloring (C. Dunn, V. Larsen, J.F. Nordstrom).- Matrioids (E. McNicholas, N.A. Neudauer, C. Starr).- Finite Frame Theory (S. Datta, J. Oldroyd).- Mathematical decision-making with linear and convex programming (J. Kotas).- Computing weight multiplicities (P. E. Harris).- Vaccination strategies for small worlds. (W. Just, H. C. Highlander).- Steady and Stable: Numerical Investigations of Nonlinear Partial Differential Equations (R. C. Harwood).

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

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Book SynopsisDas Buch stellt wesentliche Ansätze, Ergebnisse und Methoden der linearen und ganzzahligen Optimierung dar. Ziel ist es, eine solide mathematische Grundlage des Gebietes und seiner wichtigsten algorithmischen Ansätze zu entwickeln. Methodisch zentral ist der geometrische Zugang.Table of ContentsEinleitung. - Einstiege: Ungleichungssysteme und diskrete Strukturen. - Einstiege: Algorithmen und Komplexität. - Konvexitätstheorie - Der Simplex-Algorithmus. - LP-Dualität.

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Book SynopsisThis introduction to the recent theory of abstract tubes describes the framework for establishing improved inclusion-exclusion identities and Bonferroni inequalities, which are provably at least as sharp as their classical counterparts while involving fewer terms. All necessary definitions from graph theory, lattice theory and topology are provided. The role of closure and kernel operators is emphasized, and examples are provided throughout to demonstrate the applicability of this new theory. Applications are given to system and network reliability, reliability covering problems and chromatic graph theory. Topics also covered include Zeilberger's abstract lace expansion, matroid polynomials and Möbius functions.Table of Contents1. Introduction and Overview.- 2. Preliminaries.- 3.Bonferroni Inequalities via Abstract Tubes.- 4. Abstract Tubes via Closure and Kernel Operators.- 5. Recursive Schemes.- 6. Reliability Applications.- 7. Combinatorial Applications and Related Topics.- Bibliography.- Index.

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

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

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

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

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