This book is a result of a workshop, the 8th of the successful TopoInVis workshop series, held in 2019 in Nykoeping, Sweden. The workshop regularly gathers some of the world's leading experts in this field. Thereby, it provides a forum for discussions on the latest advances in the field with a focus on finding practical solutions to open problems in topological data analysis for visualization. The contributions provide introductory and novel research articles including new concepts for the analysis of multivariate and time-dependent data, robust computational approaches for the extraction and approximations of topological structures with theoretical guarantees, and applications of topological scalar and vector field analysis for visualization. The applications span a wide range of scientific areas comprising climate science, material sciences, fluid dynamics, and astronomy. In addition, community efforts with respect to joint software development are reported and discussed.

Filling a gap in literature, this self-contained book presents theoretical and application-oriented results that allow for a structural exploration of complex networks. The work focuses not only on classical graph-theoretic methods, but also demonstrates the usefulness of structural graph theory as a tool for solving interdisciplinary problems. Applications to biology, chemistry, linguistics, and data analysis are emphasized.

The book is suitable for a broad, interdisciplinary readership of researchers, practitioners, and graduate students in discrete mathematics, statistics, computer science, machine learning, artificial intelligence, computational and systems biology, cognitive science, computational linguistics, and mathematical chemistry. It may also be used as a supplementary textbook in graduate-level seminars on structural graph analysis, complex networks, or network-based machine learning methods.

Unique in its approach, Models of Network Reliability: Analysis, Combinatorics, and Monte Carlo provides a brief introduction to Monte Carlo methods along with a concise exposition of reliability theory ideas. From there, the text investigates a collection of principal network reliability models, such as terminal connectivity for networks with unreliable edges and/or nodes, network lifetime distribution in the process of its destruction, network stationary behavior for renewable components, importance measures of network elements, reliability gradient, and network optimal reliability synthesis.

Solutions to most principal network reliability problems including medium-sized computer networks are presented in the form of efficient Monte Carlo algorithms and illustrated with numerical examples and tables. Written by reliability experts with significant teaching experience, this reader-friendly text is an excellent resource for software engineering, operations research, industrial engineering, and reliability engineering students, researchers, and engineers.

Stressing intuitive explanations and providing detailed proofs of difficult statements, this self-contained resource includes a wealth of end-of-chapter exercises, numerical examples, tables, and offers a solutions manual making it ideal for self-study and practical use.

The most recent methods in various branches of lattice path and enumerative combinatorics along with relevant applications are nicely grouped together and represented in this research contributed volume. Contributions to this edited volume will be mainly research articles however it will also include several captivating, expository articles (along with pictures) on the life and mathematical work of leading researchers in lattice path combinatorics and beyond. There will be four or five expository articles in memory of Shreeram Shankar Abhyankar and Philippe Flajolet and honoring George Andrews and Lajos Takacs. There may be another brief article in memory of Professors Jagdish Narayan Srivastava and Joti Lal Jain. New research results include the kernel method developed by Flajolet and others for counting different classes of lattice paths continues to produce new results in counting lattice paths. The recent investigation of Fishburn numbers has led to interesting counting interpretations and a family of fascinating congruences. Formulas for new methods to obtain the number of Fq-rational points of Schubert varieties in Grassmannians continues to have research interest and will be presented here. Topics to be included are far reaching and will include lattice path enumeration, tilings, bijections between paths and other combinatoric structures, non-intersecting lattice paths, varieties, Young tableaux, partitions, enumerative combinatorics, discrete distributions, applications to queueing theory and other continuous time models, graph theory and applications. Many leading mathematicians who spoke at the conference from which this volume derives, are expected to send contributions including. This volume also presents the stimulating ideas of some exciting newcomers to the Lattice Path Combinatorics Conference series; "The 8th Conference on Lattice Path Combinatorics and Applications" provided opportunities for new collaborations; some of the products of these collaborations will also appear in this book. This book will have interest for researchers in lattice path combinatorics and enumerative combinatorics. This will include subsets of researchers in mathematics, statistics, operations research and computer science. The applications of the material covered in this edited volume extends beyond the primary audience to scholars interested queuing theory, graph theory, tiling, partitions, distributions, etc. An attractive bonus within our book is the collection of special articles describing the top recent researchers in this area of study and documenting the interesting history of who, when and how these beautiful combinatorial results were originally discovered.

Current language technology is dominated by approaches that either enumerate a large set of rules, or are focused on a large amount of manually labelled data. The creation of both is time-consuming and expensive, which is commonly thought to be the reason why automated natural language understanding has still not made its way into "real-life" applications yet.

This book sets an ambitious goal: to shift the development of language processing systems to a much more automated setting than previous works. A new approach is defined: what if computers analysed large samples of language data on their own, identifying structural regularities that perform the necessary abstractions and generalisations in order to better understand language in the process?
After defining the framework of Structure Discovery and shedding light on the nature and the graphic structure of natural language data, several procedures are described that do exactly this: let the computer discover structures without supervision in order to boost the performance of language technology applications. Here, multilingual documents are sorted by language, word classes are identified, and semantic ambiguities are discovered and resolved without using a dictionary or other explicit human input. The book concludes with an outlook on the possibilities implied by this paradigm and sets the methods in perspective to human computer interaction.

The target audience are academics on all levels (undergraduate and graduate students, lecturers and professors) working in the fields of natural language processing and computational linguistics, as well as natural language engineers who are seeking to improve their systems.

Features Written in an accessible style for non-experts yet extensive enough for experts Serves as a comprehensive and accessible introduction to the theory of graph polynomials for researchers in mathematics, physics, and computer science Provides an extensive reference volume for the evaluations, theorems, and properties of the Tutte polynomial and related graph, matroid, and knot invariants Offers broad coverage, touching on the wide range of applications of the Tutte polynomial and its various specializations

This concise, self-contained textbook gives an in-depth look at problem-solving from a mathematician's point-of-view. Each chapter builds off the previous one, while introducing a variety of methods that could be used when approaching any given problem. Creative thinking is the key to solving mathematical problems, and this book outlines the tools necessary to improve the reader's technique. The text is divided into twelve chapters, each providing corresponding hints, explanations, and finalization of solutions for the problems in the given chapter. For the reader's convenience, each exercise is marked with the required background level. This book implements a variety of strategies that can be used to solve mathematical problems in fields such as analysis, calculus, linear and multilinear algebra and combinatorics. It includes applications to mathematical physics, geometry, and other branches of mathematics. Also provided within the text are real-life problems in engineering and technology. Thinking in Problems is intended for advanced undergraduate and graduate students in the classroom or as a self-study guide. Prerequisites include linear algebra and analysis.

Networked computers are ubiquitous, and are subject to attack, misuse, and abuse. One method to counteracting this cyber threat is to provide security analysts with better tools to discover patterns, detect anomalies, identify correlations, and communicate their findings. Visualization for computer security (VizSec) researchers and developers are doing just that. VizSec is about putting robust information visualization tools into the hands of human analysts to take advantage of the power of the human perceptual and cognitive processes in solving computer security problems. This volume collects the papers presented at the 4th International Workshop on Computer Security - VizSec 2007.

A symbol of the Divine, a good luck charm, a cosmogram of the world order, a template for fengshui through the ages, the luoshu, or magic squre of order three, has fascinated people of many different cultures.

In this riveting account of cultural detective work, renowned mathematics educator, Frank J. Swetz relates how he uncovered the previously hidden history of the luoshu, from its Chinese origins, shrouded in legend, through its eventual association with Chinese fortunetelling, Daoism, and fengshui, to its incorporation into Islamic astrology and alchemy and its migration into Kabbalistic lore and other occult traditions of the West.

Combinatorial theory is one of the fastest growing areas of modern mathematics. Focusing on a major part of this subject, Introduction to Combinatorial Designs, Second Edition provides a solid foundation in the classical areas of design theory as well as in more contemporary designs based on applications in a variety of fields.

After an overview of basic concepts, the text introduces balanced designs and finite geometries. The author then delves into balanced incomplete block designs, covering difference methods, residual and derived designs, and resolvability. Following a chapter on the existence theorem of Bruck, Ryser, and Chowla, the book discusses Latin squares, one-factorizations, triple systems, Hadamard matrices, and Room squares. It concludes with a number of statistical applications of designs.

Reflecting recent results in design theory and outlining several applications, this new edition of a standard text presents a comprehensive look at the combinatorial theory of experimental design. Suitable for a one-semester course or for self-study, it will prepare readers for further exploration in the field.

To access supplemental materials for this volume, visit the author's website at http: //www.math.siu.edu/Wallis/designs

This book develops the combinatorics of Young tableaux and shows them in action in the algebra of symmetric functions, representations of the symmetric and general linear groups, and the geometry of flag varieties. The first part of the book is a self-contained presentation of the basic combinatorics of Young tableaux, including the remarkable constructions of "bumping" and "sliding", and several interesting correspondences. In Part II the author uses these results to study representations with geometry on Grassmannians and flag manifolds, including their Schubert subvarieties, and the related Schubert polynomials. Much of this material has never before appeared in book form. There are numerous exercises throughout, with hints and answers provided. Researchers in representation theory and algebraic geometry as well as in combinatorics will find this book interesting and useful, while students will find the intuitive presentation easy to follow.

Combinatorics plays a prominent role in contemporary mathematics, due to the vibrant development it has experienced in the last two decades and its many interactions with other subjects. This book arises from the INdAM conference "CoMeTA 2013 - Combinatorial Methods in Topology and Algebra,'' which was held in Cortona in September 2013. The event brought together emerging and leading researchers at the crossroads of Combinatorics, Topology and Algebra, with a particular focus on new trends in subjects such as: hyperplane arrangements; discrete geometry and combinatorial topology; polytope theory and triangulations of manifolds; combinatorial algebraic geometry and commutative algebra; algebraic combinatorics; and combinatorial representation theory. The book is divided into two parts. The first expands on the topics discussed at the conference by providing additional background and explanations, while the second presents original contributions on new trends in the topics addressed by the conference.

In the past two decades, convex analysis and optimization have been developed in Hadamard spaces. This book represents a first attempt to give a systematic account on the subject. Hadamard spaces are complete geodesic spaces of nonpositive curvature. They include Hilbert spaces, Hadamard manifolds, Euclidean buildings and many other important spaces. While the role of Hadamard spaces in geometry and geometric group theory has been studied for a long time, first analytical results appeared as late as in the 1990s. Remarkably, it turns out that Hadamard spaces are appropriate for the theory of convex sets and convex functions outside of linear spaces. Since convexity underpins a large number of results in the geometry of Hadamard spaces, we believe that its systematic study is of substantial interest. Optimization methods then address various computational issues and provide us with approximation algorithms which may be useful in sciences and engineering. We present a detailed description of such an application to computational phylogenetics. The book is primarily aimed at both graduate students and researchers in analysis and optimization, but it is accessible to advanced undergraduate students as well.

This book reports on the development and assessment of a novel framework for studying neural interactions (the connectome) and their dynamics (the chronnectome). Using EEG recordings taken during an auditory oddball task performed by 48 patients with schizophrenia and 87 healthy controls, and applying local and network measures, changes in brain activation from pre-stimulus to cognitive response were assessed, and significant differences were observed between the patients and controls. This book investigates the source of the network abnormalities and presents new evidence for the disconnection hypothesis and the aberrant salience hypothesis with regard to schizophrenia. Moreover, it puts forward a novel approach to combining local regularity measures and graph measures in order to characterize schizophrenia brain dynamics, and presents interesting findings on the regularity of brain patterns in healthy control subjects versus patients with schizophrenia. Besides providing new evidence for the disconnection hypothesis, it offers a source of inspiration for future research directions in the field.

The papers presented here describe research to improve the general understanding of the application of SAMR to practical problems, to identify issues critical to efficient and effective implementation on high performance computers and to stimulate the development of a community code repository for software including benchmarks to assist in the evaluation of software and compiler technologies. The ten chapters have been divided into two parts reflecting two major issues in the topic: programming complexity of SAMR algorithms and the applicability and numerical challenges of SAMR methods.

This book offers a detailed introduction to graph theoretic methods in profinite groups and applications to abstract groups. It is the first to provide a comprehensive treatment of the subject. The author begins by carefully developing relevant notions in topology, profinite groups and homology, including free products of profinite groups, cohomological methods in profinite groups, and fixed points of automorphisms of free pro-p groups. The final part of the book is dedicated to applications of the profinite theory to abstract groups, with sections on finitely generated subgroups of free groups, separability conditions in free and amalgamated products, and algorithms in free groups and finite monoids. Profinite Graphs and Groups will appeal to students and researchers interested in profinite groups, geometric group theory, graphs and connections with the theory of formal languages. A complete reference on the subject, the book includes historical and bibliographical notes as well as a discussion of open questions and suggestions for further reading.

Wie sieht eine Kurve aus, die die ganze Ebene oder den Raum vollstandig ausfullt? Kann man einen Polyeder flexibel bewegen, ja sogar umstulpen? Was ist die projektive Ebene oder der vierdimensionale Raum? Gibt es Seifenblasen, die keine runden Kugel sind? Wie kann man die komplizierte Struktur von Stroemungen besser verstehen? In diesem Buch erleben Sie die Mathematik von ihrer anschaulichen Seite und finden faszinierende und bisher nie gesehene Bilder, die Ihnen illustrative Antworten zu all diesen Fragestellungen geben. Zu allen Bildern gibt es kurze Erklarungstexte, viele Literaturhinweise und jede Menge Web-Links. Das Buch ist fur alle Freunde der Mathematik, die nicht nur trockenen Text und endlose Formeln sehen wollen. Vom Schuler zum Lehrer, vom Studenten zum Professor. Es soll sie alle inspirieren und anregen, sich mit diesem oder jenem vermeintlich nur Insidern vorbehaltenem Thema zu beschaftigen. Lernen Sie die Mathematik von einer ganz neuen und bunten Seite kennen. Die Neuauflage ist vollstandig durchgesehen und um acht Doppelseiten mit neuen und spektakularen Bildern erganzt.

Automatic sequences are sequences over a finite alphabet generated by a finite-state machine. This book presents a novel viewpoint on automatic sequences, and more generally on combinatorics on words, by introducing a decision method through which many new results in combinatorics and number theory can be automatically proved or disproved with little or no human intervention. This approach to proving theorems is extremely powerful, allowing long and error-prone case-based arguments to be replaced by simple computations. Readers will learn how to phrase their desired results in first-order logic, using free software to automate the computation process. Results that normally require multipage proofs can emerge in milliseconds, allowing users to engage with mathematical questions that would otherwise be difficult to solve. With more than 150 exercises included, this text is an ideal resource for researchers, graduate students, and advanced undergraduates studying combinatorics, sequences, and number theory.

"Addresses the key topic in combinatorial synthesis--how to optimize the quality of a combinatorial library--by determining the usefulness of synthesized compunds, the reliability of biological assay results, and analyzing acadmic and industrial applications, real-world examples, and case studies of successful and unsuccessful technologies."

This is the first full-length book on the major theme of symmetry in graphs. Forming part of algebraic graph theory, this fast-growing field is concerned with the study of highly symmetric graphs, particularly vertex-transitive graphs, and other combinatorial structures, primarily by group-theoretic techniques. In practice the street goes both ways and these investigations shed new light on permutation groups and related algebraic structures. The book assumes a first course in graph theory and group theory but no specialized knowledge of the theory of permutation groups or vertex-transitive graphs. It begins with the basic material before introducing the field's major problems and most active research themes in order to motivate the detailed discussion of individual topics that follows. Featuring many examples and over 450 exercises, it is an essential introduction to the field for graduate students and a valuable addition to any algebraic graph theorist's bookshelf.

Since the early eighteenth century, the theory of networks and graphs has matured into an indispensable tool for describing countless real-world phenomena. However, the study of large-scale features of a network often requires unrealistic limits, such as taking the network size to infinity or assuming a continuum. These asymptotic and analytic approaches can significantly diverge from real or simulated networks when applied at the finite scales of real-world applications. This book offers an approach to overcoming these limitations by introducing operator graph theory, an exact, non-asymptotic set of tools combining graph theory with operator calculus. The book is intended for mathematicians, physicists, and other scientists interested in discrete finite systems and their graph-theoretical description, and in delineating the abstract algebraic structures that characterise such systems. All the necessary background on graph theory and operator calculus is included for readers to understand the potential applications of operator graph theory.

The challenge of dividing an asset fairly, from cakes to more important properties, is of great practical importance in many situations. Since the famous Polish school of mathematicians (Steinhaus, Banach, and Knaster) introduced and described algorithms for the fair division problem in the 1940s, the concept has been widely popularized.

This book gathers into one readable and inclusive source a comprehensive discussion of the state of the art in cake-cutting problems for both the novice and the professional. It offers a complete treatment of all cake-cutting algorithms under all the considered definitions of "fair" and presents them in a coherent, reader-friendly manner. Robertson and Webb have brought this elegant problem to life for both the bright high school student and the professional researcher.

This book is a tribute to Paul Erd\H{o}s, the wandering mathematician once described as the "prince of problem solvers and the absolute monarch of problem posers." It examines -- within the context of his unique personality and lifestyle -- the legacy of open problems he left to the world after his death in 1996. Unwilling to succumb to the temptations of money and position, Erd\H{o}s never had a home and never held a job. His "home" was a bag or two containing all his belongings and a record of the collective activities of the mathematical community. His "job" was one at which he excelled: identifying a fundamental roadblock in some particular line of approach and capturing it in a well-chosen, often innocent-looking problem, whose solution would likewise provide insight into the underlying theory. By cataloguing the unsolved problems of Erd\H{o}s in a comprehensive and well-documented volume, the authors hope to continue the work of an unusual and special man who fundamentally influenced the field of mathematics.

Cryptology: Classical and Modern, Second Edition proficiently introduces readers to the fascinating field of cryptology. The book covers classical methods including substitution, transposition, Alberti, Vigenere, and Hill ciphers. It also includes coverage of the Enigma machine, Turing bombe, and Navajo code. Additionally, the book presents modern methods like RSA, ElGamal, and stream ciphers, as well as the Diffie-Hellman key exchange and Advanced Encryption Standard. When possible, the book details methods for breaking both classical and modern methods. The new edition expands upon the material from the first edition which was oriented for students in non-technical fields. At the same time, the second edition supplements this material with new content that serves students in more technical fields as well. Thus, the second edition can be fully utilized by both technical and non-technical students at all levels of study. The authors include a wealth of material for a one-semester cryptology course, and research exercises that can be used for supplemental projects. Hints and answers to selected exercises are found at the end of the book. Features: Requires no prior programming knowledge or background in college-level mathematics Illustrates the importance of cryptology in cultural and historical contexts, including the Enigma machine, Turing bombe, and Navajo code Gives straightforward explanations of the Advanced Encryption Standard, public-key ciphers, and message authentication Describes the implementation and cryptanalysis of classical ciphers, such as substitution, transposition, shift, affine, Alberti, Vigenere, and Hill