The emergence of multilayer networks as a concept from the field of complex systems provides many new opportunities for the visualization of network complexity, and has also raised many new exciting challenges. The multilayer network model recognizes that the complexity of relationships between entities in real-world systems is better embraced as several interdependent subsystems (or layers) rather than a simple graph approach. Despite only recently being formalized and defined, this model can be applied to problems in the domains of life sciences, sociology, digital humanities, and more. Within the domain of network visualization there already are many existing systems, which visualize data sets having many characteristics of multilayer networks, and many techniques, which are applicable to their visualization. In this Synthesis Lecture, we provide an overview and structured analysis of contemporary multilayer network visualization. This is not only for researchers in visualization, but also for those who aim to visualize multilayer networks in the domain of complex systems, as well as those solving problems within application domains. We have explored the visualization literature to survey visualization techniques suitable for multilayer network visualization, as well as tools, tasks, and analytic techniques from within application domains. We also identify the research opportunities and examine outstanding challenges for multilayer network visualization along with potential solutions and future research directions for addressing them.

This textbook teaches readers how to turn geometry into an image on a computer screen. This exciting journey begins in the schools of the ancient Greek philosophers, and describes the major events that changed people's perception of geometry. The readers will learn how to see geometry and colors beyond simple mathematical formulas and how to represent geometric shapes, transformations and motions by digital sampling of various mathematical functions.Special multiplatform visualization software developed by the author will allow readers to explore the exciting world of visual immersive mathematics, and the book software repository will provide a starting point for their own sophisticated visualization applications. Making Images with Mathematics serves as a self-contained text for a one-semester computer graphics and visualization course for computer science and engineering students, as well as a reference manual for researchers and developers.

This book bridges the gaps between logic, mathematics and computer science by delving into the theory of well-quasi orders, also known as wqos. This highly active branch of combinatorics is deeply rooted in and between many fields of mathematics and logic, including proof theory, commutative algebra, braid groups, graph theory, analytic combinatorics, theory of relations, reverse mathematics and subrecursive hierarchies. As a unifying concept for slick finiteness or termination proofs, wqos have been rediscovered in diverse contexts, and proven to be extremely useful in computer science. The book introduces readers to the many facets of, and recent developments in, wqos through chapters contributed by scholars from various fields. As such, it offers a valuable asset for logicians, mathematicians and computer scientists, as well as scholars and students.

Die Theorie der regularen Graphen (The Theory of Regular Graphs), written by the Danish Mathematician Julius Petersen in 1891, is often considered the first strictly theoretical paper dealing with graphs. In the 130 years since then, regular graphs have been a common and popular area of study. While regular graphs are typically considered to be graphs whose vertices all have the same degree, a more general interpretation is that of graphs possessing some common characteristic throughout their structure. During the past several decades, however, there has been some increased interest in investigating graphs possessing a property that is, in a sense, opposite to regularity. It is this topic with which this book deals, giving rise to a study of what might be called irregularity in graphs. Here, various irregularity concepts dealing with several topics in graph theory are described, such as degrees of vertices, graph labelings, weightings, colorings, graph structures, Eulerian and Hamiltonian properties, graph decompositions, and Ramsey-type problems.

This open access book offers an innovative account of how relief organizations' visual depiction of Syrian displacement contributes to reproduce and reinforce a securitized account of refugees. Through visual analysis, the book demonstrates how the securitization process takes place in three different ways. First of all, even if marginally, it occurs through the reproduction of mainstream media and political accounts that have depicted refugees in terms of threats. Secondly, and more consistently, through a representation of Syrian displaced people that, despite the undeniable innovative aesthetic patterns focusing on dignity and empowerment, continue to reinforce a visual narrative around refugees in terms of victimhood and passivity. The reproduction of a securitized account takes also place through the dialectic between what is made visible in the pictures and what is not. At the same time the book identifies visual glimmers and minor displacements in the humanitarian discourse that have the potentiality to produce alternative discourses on refugees and displacement beyond the mainstream securitized ones. By showing how relief organizations' visual representation contributes to the securitization of the refugee issue, this book provides a great resource to students and academics in migration, visuality, humanitarianism and securitization, as well as social scientists and policy-makers.

This is the first book that focuses entirely on the fundamental questions in visualization. Unlike other existing books in the field, it contains discussions that go far beyond individual visual representations and individual visualization algorithms. It offers a collection of investigative discourses that probe these questions from different perspectives, including concepts that help frame these questions and their potential answers, mathematical methods that underpin the scientific reasoning of these questions, empirical methods that facilitate the validation and falsification of potential answers, and case studies that stimulate hypotheses about potential answers while providing practical evidence for such hypotheses. Readers are not instructed to follow a specific theory, but their attention is brought to a broad range of schools of thoughts and different ways of investigating fundamental questions. As such, the book represents the by now most significant collective effort for gathering a large collection of discourses on the foundation of data visualization. Data visualization is a relatively young scientific discipline. Over the last three decades, a large collection of computer-supported visualization techniques have been developed, and the merits and benefits of using these techniques have been evidenced by numerous applications in practice. These technical advancements have given rise to the scientific curiosity about some fundamental questions such as why and how visualization works, when it is useful or effective and when it is not, what are the primary factors affecting its usefulness and effectiveness, and so on. This book signifies timely and exciting opportunities to answer such fundamental questions by building on the wealth of knowledge and experience accumulated in developing and deploying visualization technology in practice.

This volume provides accessible and self-contained research problems designed for undergraduate student projects, and simultaneously promotes the development of sustainable undergraduate research programs. The chapters in this work span a variety of topical areas of pure and applied mathematics and mathematics education. Each chapter gives a self-contained introduction on a research topic with an emphasis on the specific tools and knowledge needed to create and maintain fruitful research programs for undergraduates. Some of the topics discussed include:* Disease modeling* Tropical curves and surfaces* Numerical semigroups* Mathematics EducationThis volume will primarily appeal to undergraduate students interested in pursuing research projects and faculty members seeking to mentor them. It may also aid students and faculty participating in independent studies and capstone projects.

Oriented matroids are a very natural mathematical concept which presents itself in many different guises and which has connections and applications to many different areas. These include discrete and computational geometry, combinatorics, convexity, topology, algebraic geometry, operations research, computer science and theoretical chemistry. This is the second edition of the first comprehensive, accessible account of the subject. It is intended for a diverse audience: graduate students who wish to learn the subject from scratch; researchers in the various fields of application who want to concentrate on certain aspects of the theory; specialists who need a thorough reference work; and others at academic points in between. A list of exercises and open problems ends each chapter. For the second edition, the authors have expanded the bibliography greatly to ensure that it remains comprehensive and up-to-date, and they have also added an appendix surveying research since the work was first published.

Graph-Theoretical Matrices in Chemistry presents a systematic survey of graph-theoretical matrices and highlights their potential uses. This comprehensive volume is an updated, extended version of a former bestseller featuring a series of mathematical chemistry monographs. In this edition, nearly 200 graph-theoretical matrices are included. This second edition is organized like the previous one-after an introduction, graph-theoretical matrices are presented in five chapters: The Adjacency Matrix and Related Matrices, Incidence Matrices, The Distance Matrix and Related Matrices, Special Matrices, and Graphical Matrices. Each of these chapters is followed by a list of references. Among the matrices presented several are novel and some are known only to a few. The properties and potential usefulness of many of the presented graph-theoretical matrices in chemistry have yet to be investigated. Most of the graph-theoretical matrices presented have been used as sources of molecular descriptors usually referred to as topological indices. They are particularly concerned with a special class of graphs that represents chemical structures involving molecules. Due to its multidisciplinary scope, this book will appeal to a broad audience ranging from chemistry and mathematics to pharmacology.

This book focuses on the latest developments in behaviormetrics and data science, covering a wide range of topics in data analysis and related areas of data science, including analysis of complex data, analysis of qualitative data, methods for high-dimensional data, dimensionality reduction, visualization of such data, multivariate statistical methods, analysis of asymmetric relational data, and various applications to real data. In addition to theoretical and methodological results, it also shows how to apply the proposed methods to a variety of problems, for example in consumer behavior, decision making, marketing data, and social network structures. Moreover, it discuses methodological aspects and applications in a wide range of areas, such as behaviormetrics; behavioral science; psychology; and marketing, management and social sciences. Combining methodological advances with real-world applications collected from a variety of research fields, the book is a valuable resource for researchers and practitioners, as well as for applied statisticians and data analysts.

This proceedings presents the result of the 8th International Conference in Network Analysis, held at the Higher School of Economics, Moscow, in May 2018. The conference brought together scientists, engineers, and researchers from academia, industry, and government. Contributions in this book focus on the development of network algorithms for data mining and its applications. Researchers and students in mathematics, economics, statistics, computer science, and engineering find this collection a valuable resource filled with the latest research in network analysis. Computational aspects and applications of large-scale networks in market models, neural networks, social networks, power transmission grids, maximum clique problem, telecommunication networks, and complexity graphs are included with new tools for efficient network analysis of large-scale networks. Machine learning techniques in network settings including community detection, clustering, and biclustering algorithms are presented with applications to social network analysis.

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

The contributions included in the volume are drawn from presentations at ODS2019 - International Conference on Optimization and Decision Science, which was the 49th annual meeting of the Italian Operations Research Society (AIRO) held at Genoa, Italy, on 4-7 September 2019. This book presents very recent results in the field of Optimization and Decision Science. While the book is addressed primarily to the Operations Research (OR) community, the interdisciplinary contents ensure that it will also be of very high interest for scholars and researchers from many scientific disciplines, including computer sciences, economics, mathematics, and engineering. Operations Research is known as the discipline of optimization applied to real-world problems and to complex decision-making fields. The focus is on mathematical and quantitative methods aimed at determining optimal or near-optimal solutions in acceptable computation times. This volume not only presents theoretical results but also covers real industrial applications, making it interesting for practitioners facing decision problems in logistics, manufacturing production, and services. Readers will accordingly find innovative ideas from both a methodological and an applied perspective.

This book covers pattern recognition techniques applied to various areas of biomedicine, including disease diagnosis and prognosis, and several problems of classification, with a special focus on-but not limited to-pattern recognition modeling of biomedical signals and images. Multidisciplinary by definition, the book's topic blends computing, mathematics and other technical sciences towards the development of computational tools and methodologies that can be applied to pattern recognition processes. In this work, the efficacy of such methods and techniques for processing medical information is analyzed and compared, and auxiliary criteria for determining the correct diagnosis and treatment strategies are recommended and applied. Researchers in applied mathematics, the computer sciences, engineering and related fields with a focus on medical applications will benefit from this book, as well as professionals with a special interest in state-of-the-art pattern recognition techniques as applied to biomedicine.

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

Discrete mathematics has been rising in prominence in the past fifty years, both as a tool with practical applications and as a source of new and interesting mathematics. The topics in discrete mathematics have become so well developed that it is easy to forget that common threads connect the different areas, and it is through discovering and using these connections that progress is often made. For over fifty years, Ron Graham has been able to illuminate some of these connections and has helped to bring the field of discrete mathematics to where it is today. To celebrate his contribution, this volume brings together many of the best researchers working in discrete mathematics, including Fan Chung, Erik D. Demaine, Persi Diaconis, Peter Frankl, Alfred W. Hales, Jeffrey C. Lagarias, Allen Knutson, Janos Pach, Carl Pomerance, N. J. A. Sloane, and of course, Ron Graham himself.

This textbook explores probability and stochastic processes at a level that does not require any prior knowledge except basic calculus. It presents the fundamental concepts in a step-by-step manner, and offers remarks and warnings for deeper insights. The chapters include basic examples, which are revisited as the new concepts are introduced. To aid learning, figures and diagrams are used to help readers grasp the concepts, and the solutions to the exercises and problems. Further, a table format is also used where relevant for better comparison of the ideas and formulae. The first part of the book introduces readers to the essentials of probability, including combinatorial analysis, conditional probability, and discrete and continuous random variable. The second part then covers fundamental stochastic processes, including point, counting, renewal and regenerative processes, the Poisson process, Markov chains, queuing models and reliability theory. Primarily intended for undergraduate engineering students, it is also useful for graduate-level students wanting to refresh their knowledge of the basics of probability and stochastic processes.

The inverse scattering problem is central to many areas of science and technology such as radar, sonar, medical imaging, geophysical exploration and nondestructive testing. This book is devoted to the mathematical and numerical analysis of the inverse scattering problem for acoustic and electromagnetic waves. In this fourth edition, a number of significant additions have been made including a new chapter on transmission eigenvalues and a new section on the impedance boundary condition where particular attention has been made to the generalized impedance boundary condition and to nonlocal impedance boundary conditions. Brief discussions on the generalized linear sampling method, the method of recursive linearization, anisotropic media and the use of target signatures in inverse scattering theory have also been added.

This book provides a detailed description of network science concepts applied to power systems and electricity markets, offering an appropriate blend of theoretical background and practical applications for operation and power system planning. It discusses an approach to understanding power systems from a network science perspective using the direct recognition of the interconnectivity provided by the transmission system. Further, it explores the network properties in detail and characterizes them as a tool for online and offline applications for power system operation. The book includes an in-depth explanation of electricity markets problems that can be addressed from a graph theory perspective. It is intended for advanced undergraduate and graduate students in the fields of electric energy systems, operations research, management science and economics. Practitioners in the electric energy sector also benefit from the concepts and techniques presented here.

This textbook can serve as a comprehensive manual of discrete mathematics and graph theory for non-Computer Science majors; as a reference and study aid for professionals and researchers who have not taken any discrete math course before. It can also be used as a reference book for a course on Discrete Mathematics in Computer Science or Mathematics curricula. The study of discrete mathematics is one of the first courses on curricula in various disciplines such as Computer Science, Mathematics and Engineering education practices. Graphs are key data structures used to represent networks, chemical structures, games etc. and are increasingly used more in various applications such as bioinformatics and the Internet. Graph theory has gone through an unprecedented growth in the last few decades both in terms of theory and implementations; hence it deserves a thorough treatment which is not adequately found in any other contemporary books on discrete mathematics, whereas about 40% of this textbook is devoted to graph theory. The text follows an algorithmic approach for discrete mathematics and graph problems where applicable, to reinforce learning and to show how to implement the concepts in real-world applications.

The canonical way to establish the central limit theorem for i.i.d. random variables is to use characteristic functions and Levy's continuity theorem. This monograph focuses on this characteristic function approach and presents a renormalization theory called mod- convergence. This type of convergence is a relatively new concept with many deep ramifications, and has not previously been published in a single accessible volume. The authors construct an extremely flexible framework using this concept in order to study limit theorems and large deviations for a number of probabilistic models related to classical probability, combinatorics, non-commutative random variables, as well as geometric and number-theoretical objects. Intended for researchers in probability theory, the text is carefully well-written and well-structured, containing a great amount of detail and interesting examples.

Magic and antimagic labelings are among the oldest labeling schemes in graph theory. This book takes readers on a journey through these labelings, from early beginnings with magic squares up to the latest results and beyond. Starting from the very basics, the book offers a detailed account of all magic and antimagic type labelings of undirected graphs. Long-standing problems are surveyed and presented along with recent results in classical labelings. In addition, the book covers an assortment of variations on the labeling theme, all in one self-contained monograph. Assuming only basic familiarity with graphs, this book, complete with carefully written proofs of most results, is an ideal introduction to graph labeling for students learning the subject. More than 150 open problems and conjectures make it an invaluable guide for postgraduate and early career researchers, as well as an excellent reference for established graph theorists.

This book comprehensively covers the important efforts in improving the quality of images in visual cryptography (VC), with a focus on cases with gray scale images. It not only covers schemes in traditional VC and extended VC for binary secret images, but also the latest development in the analysis-by-synthesis approach. This book distinguishes itself from the existing literature in three ways. First, it not only reviews traditional VC for binary secret images, but also covers recent efforts in improving visual quality for gray scale secret images. Second, not only traditional quality measures are reviewed, but also measures that were not used for measuring perceptual quality of decrypted secret images, such as Radially Averaged Power Spectrum Density (RAPSD) and residual variance, are employed for evaluating and guiding the design of VC algorithms. Third, unlike most VC books following a mathematical formal style, this book tries to make a balance between engineering intuition and mathematical reasoning. All the targeted problems and corresponding solutions are fully motivated by practical applications and evaluated by experimental tests, while important security issues are presented as mathematical proof. Furthermore, important algorithms are summarized as pseudocodes, thus enabling the readers to reproduce the results in the book. Therefore, this book serves as a tutorial for readers with an engineering background as well as for experts in related areas to understand the basics and research frontiers in visual cryptography.

Graduate students and researchers in applied mathematics, optimization, engineering, computer science, and management science will find this book a useful reference which provides an introduction to applications and fundamental theories in nonlinear combinatorial optimization. Nonlinear combinatorial optimization is a new research area within combinatorial optimization and includes numerous applications to technological developments, such as wireless communication, cloud computing, data science, and social networks. Theoretical developments including discrete Newton methods, primal-dual methods with convex relaxation, submodular optimization, discrete DC program, along with several applications are discussed and explored in this book through articles by leading experts.

The primary purpose of this textbook is to introduce the reader to a wide variety of elementary permutation statistical methods. Permutation methods are optimal for small data sets and non-random samples, and are free of distributional assumptions. The book follows the conventional structure of most introductory books on statistical methods, and features chapters on central tendency and variability, one-sample tests, two-sample tests, matched-pairs tests, one-way fully-randomized analysis of variance, one-way randomized-blocks analysis of variance, simple regression and correlation, and the analysis of contingency tables. In addition, it introduces and describes a comparatively new permutation-based, chance-corrected measure of effect size. Because permutation tests and measures are distribution-free, do not assume normality, and do not rely on squared deviations among sample values, they are currently being applied in a wide variety of disciplines. This book presents permutation alternatives to existing classical statistics, and is intended as a textbook for undergraduate statistics courses or graduate courses in the natural, social, and physical sciences, while assuming only an elementary grasp of statistics.