Graphs drawn on two-dimensional surfaces have always attracted researchers by their beauty and by the variety of difficult questions to which they give rise. The theory of such embedded graphs, which long seemed rather isolated, has witnessed the appearance of entirely unexpected new applications in recent decades, ranging from Galois theory to quantum gravity models, and has become a kind of a focus of a vast field of research. The book provides an accessible introduction to this new domain, including such topics as coverings of Riemann surfaces, the Galois group action on embedded graphs (Grothendieck's theory of "dessins d'enfants"), the matrix integral method, moduli spaces of curves, the topology of meromorphic functions, and combinatorial aspects of Vassiliev's knot invariants and, in an appendix by Don Zagier, the use of finite group representation theory. The presentation is concrete throughout, with numerous figures, examples (including computer calculations) and exercises, and should appeal to both graduate students and researchers.

Image analysis is a computational feat which humans show excellence in, in comp- ison with computers. Yet the list of applications that rely on automatic processing of images has been growing at a fast pace. Biometric authentication by face, ?ngerprint, and iris, online character recognition in cell phones as well as drug design tools are but a few of its benefactors appearing on the headlines. This is, of course, facilitated by the valuable output of the resarch community in the past 30 years. The pattern recognition and computer vision communities that study image analysis have large conferences, which regularly draw 1000 parti- pants. In a way this is not surprising, because much of the human-speci?c activities critically rely on intelligent use of vision. If routine parts of these activities can be automated, much is to be gained in comfort and sustainable development. The - search ?eld could equally be called visualintelligence because it concerns nearly all activities of awake humans. Humans use or rely on pictures or pictorial languages to represent, analyze, and develop abstract metaphors related to nearly every aspect of thinking and behaving, be it science, mathematics, philosopy, religion, music, or emotions. The present volume is an introductory textbook on signal analysis of visual c- putation for senior-level undergraduates or for graduate students in science and - gineering. My modest goal has been to present the frequently used techniques to analyze images in a common framework-directional image processing.

Leave nothing to chance. This cliche embodies the common belief that ran domness has no place in carefully planned methodologies, every step should be spelled out, each i dotted and each t crossed. In discrete mathematics at least, nothing could be further from the truth. Introducing random choices into algorithms can improve their performance. The application of proba bilistic tools has led to the resolution of combinatorial problems which had resisted attack for decades. The chapters in this volume explore and celebrate this fact. Our intention was to bring together, for the first time, accessible discus sions of the disparate ways in which probabilistic ideas are enriching discrete mathematics. These discussions are aimed at mathematicians with a good combinatorial background but require only a passing acquaintance with the basic definitions in probability (e.g. expected value, conditional probability). A reader who already has a firm grasp on the area will be interested in the original research, novel syntheses, and discussions of ongoing developments scattered throughout the book. Some of the most convincing demonstrations of the power of these tech niques are randomized algorithms for estimating quantities which are hard to compute exactly. One example is the randomized algorithm of Dyer, Frieze and Kannan for estimating the volume of a polyhedron. To illustrate these techniques, we consider a simple related problem. Suppose S is some region of the unit square defined by a system of polynomial inequalities: Pi (x. y) ~ o.

A 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."

Andras Recski, Mathematical Reviews Clippings 2000m:93006"

From the reviews: " A unique and fascinating blend, which is shown to be useful for a variety of applications, including robotics, geometrical optics, computer animation, and geometric design. The contents of the book are visualized by a wealth of carefully chosen illustrations, making the book a shear pleasure to read, or even to just browse in." Mathematical Reviews

What is the "most uniform" way of distributing n points in the unit square? How big is the "irregularity" necessarily present in any such distribution? This book is an accessible and lively introduction to the area of geometric discrepancy theory, with numerous exercises and illustrations. In separate, more specialized parts, it also provides a comprehensive guide to recent research.

In 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.

This book illustrates how modern mathematical wavelet transform techniques offer fresh insights into the complex behavior of neural systems at different levels: from the microscopic dynamics of individual cells to the macroscopic behavior of large neural networks. It also demonstrates how and where wavelet-based mathematical tools can provide an advantage over classical approaches used in neuroscience. The authors well describe single neuron and populational neural recordings. This 2nd edition discusses novel areas and significant advances resulting from experimental techniques and computational approaches developed since 2015, and includes three new topics: * Detection of fEPSPs in multielectrode LFPs recordings. * Analysis of Visual Sensory Processing in the Brain and BCI for Human Attention Control; * Analysis and Real-time Classification of Motor-related EEG Patterns; The book is a valuable resource for neurophysiologists and physicists familiar with nonlinear dynamical systems and data processing, as well as for graduate students specializing in these and related areas.

There has been dramatic growth in the development and application of Bayesian inference in statistics. Berger (2000) documents the increase in Bayesian activity by the number of published research articles, the number of books, andtheextensivenumberofapplicationsofBayesianarticlesinapplied disciplines such as science and engineering. One reason for the dramatic growth in Bayesian modeling is the availab- ity of computational algorithms to compute the range of integrals that are necessary in a Bayesian posterior analysis. Due to the speed of modern c- puters, it is now possible to use the Bayesian paradigm to 't very complex models that cannot be 't by alternative frequentist methods. To 't Bayesian models, one needs a statistical computing environment. This environment should be such that one can: write short scripts to de?ne a Bayesian model use or write functions to summarize a posterior distribution use functions to simulate from the posterior distribution construct graphs to illustrate the posterior inference An environment that meets these requirements is the R system. R provides a wide range of functions for data manipulation, calculation, and graphical d- plays. Moreover, it includes a well-developed, simple programming language that users can extend by adding new functions. Many such extensions of the language in the form of packages are easily downloadable from the Comp- hensive R Archive Network (CRAN)

The lectures concentrate on highlights in Combinatorial (ChaptersII and III) and Number Theoretical (ChapterIV) Extremal Theory, in particular on the solution of famous problems which were open for many decades. However, the organization of the lectures in six chapters does neither follow the historic developments nor the connections between ideas in several cases. With the speci?ed auxiliary results in ChapterI on Probability Theory, Graph Theory, etc., all chapters can be read and taught independently of one another. In addition to the 16 lectures organized in 6 chapters of the main part of the book, there is supplementary material for most of them in the Appendix. In parti- lar, there are applications and further exercises, research problems, conjectures, and even research programs. The following books and reports [B97], [ACDKPSWZ00], [A01], and [ABCABDM06], mostly of the authors, are frequently cited in this book, especially in the Appendix, and we therefore mark them by short labels as [B], [N], [E], and [G]. We emphasize that there are also "Exercises" in [B], a "Problem Section" with contributions by several authors on pages 1063-1105 of [G], which are often of a combinatorial nature, and "Problems and Conjectures" on pages 172-173 of [E].

This volume surveys the development of combinatorics since 1930 by presenting in chronological order the fundamental results of the subject proved in over five decades of original papers by: T. van Aardenne-Ehrenfest.- R.L. Brooks.- N.G. de Bruijn.- G.F. Clements.- H.H. Crapo.- R.P. Dilworth.- J. Edmonds.- P. Erd s.- L.R. Ford, Jr.- D.R. Fulkerson.- D. Gale.- L. Geissinger.- I.J. Good.- R.L. Graham.- A.W. Hales.- P. Hall.- P.R. Halmos.- R.I. Jewett.- I. Kaplansky.- P.W. Kasteleyn.- G. Katona.- D.J. Kleitman.- K. Leeb.- B. Lindstr m.- L. Lov sz.- D. Lubell.- C. St. J.A. Nash-Williams.- G. P lya.-R. Rado.- F.P. Ramsey.- G.-C. Rota.- B.L. Rothschild.- H.J. Ryser.- C. Schensted.- M.P. Sch tzenberger.- R.P. Stanley.- G. Szekeres.- W.T. Tutte.- H.E. Vaughan.- H. Whitney.

A brilliant treatment of a knotty problem in computing. This volume contains chapters written by reputable researchers and provides the state of the art in theory and algorithms for the traveling salesman problem (TSP). The book covers all important areas of study on TSP, including polyhedral theory for symmetric and asymmetric TSP, branch and bound, and branch and cut algorithms, probabilistic aspects of TSP, and includes a thorough computational analysis of heuristic and metaheuristic algorithms.

Marking 94 years since its first appearance, this book provides an annotated translation of Sainte-Lague's seminal monograph Les reseaux (ou graphes), drawing attention to its fundamental principles and ideas. Sainte-Lague's 1926 monograph appeared only in French, but in the 1990s H. Gropp published a number of English papers describing several aspects of the book. He expressed his hope that an English translation might sometime be available to the mathematics community. In the 10 years following the appearance of Les reseaux (ou graphes), the development of graph theory continued, culminating in the publication of the first full book on the theory of finite and infinite graphs in 1936 by Denes Koenig. This remained the only well-known text until Claude Berge's 1958 book on the theory and applications of graphs. By 1960, graph theory had emerged as a significant mathematical discipline of its own. This book will be of interest to graph theorists and mathematical historians.

How many possible sudoku puzzles are there? In the lottery, what is the chance that two winning balls have consecutive numbers? Who invented Pascal's triangle? (it was not Pascal) Combinatorics, the branch of mathematics concerned with selecting, arranging, and listing or counting collections of objects, works to answer all these questions. Dating back some 3000 years, and initially consisting mainly of the study of permutations and combinations, its scope has broadened to include topics such as graph theory, partitions of numbers, block designs, design of codes, and latin squares. In this Very Short Introduction Robin Wilson gives an overview of the field and its applications in mathematics and computer theory, considering problems from the shortest routes covering certain stops to the minimum number of colours needed to colour a map with different colours for neighbouring countries. ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.

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.

A Path to Combinatorics for Undergraduates is a lively introduction not only to combinatorics, but also to mathematical ingenuity, rigor, and the joy of solving puzzles. This unique approach to combinatorics is centered around unconventional, essay-type combinatorial examples, followed by a number of carefully selected, challenging problems and extensive discussions of their solutions. Topics encompass permutations and combinations, binomial coefficients and their applications, bijections, inclusions and exclusions, and generating functions. Each chapter features fully-worked problems, including many from Olympiads and other competitions, as well as a number of problems original to the authors; at the end of each chapter are further exercises to reinforce understanding, encourage creativity, and build a repertory of problem-solving techniques.

The authors' previous text, 102 Combinatorial Problems, makes a fine companion volume to the present work, which is ideal for Olympiad participants and coaches, advanced high school students, undergraduates, and college instructors. The book's unusual problems and examples will interest seasoned mathematicians as well.

"102 Combinatorial Problems" consists of carefully selected problems that have been used in the training and testing of the USA International Mathematical Olympiad (IMO) team. Key features: * Provides in-depth enrichment in the important areas of combinatorics by reorganizing and enhancing problem-solving tactics and strategies * Topics include: combinatorial arguments and identities, generating functions, graph theory, recursive relations, sums and products, probability, number theory, polynomials, theory of equations, complex numbers in geometry, algorithmic proofs, combinatorial and advanced geometry, functional equations and classical inequalities The book is systematically organized, gradually building combinatorial skills and techniques and broadening the student's view of mathematics. Aside from its practical use in training teachers and students engaged in mathematical competitions, it is a source of enrichment that is bound to stimulate interest in a variety of mathematical areas that are tangential to combinatorics.

The four-colour theorem is one of the famous problems of mathematics, that frustrated generations of mathematicians from its birth in 1852 to its solution (using substantial assistance from electronic computers) in 1976. The theorem asks whether four colours are sufficient to colour all conceivable maps, in such a way that countries with a common border are coloured with different colours.

The book discusses various attempts to solve this problem, and some of the mathematics which developed out of these attempts. Much of this mathematics has developed a life of its own, and forms a fascinating part of the subject now known as graph theory. The book is designed to be self-contained, and develops all the graph-theoretical tools needed as it goes along. It includes all the elementary graph theory that should be included in an introduction to the subject, before concentrating on specific topics relevant to the four-colour problem.

This book arose from the Euroconference "Algebraic Combinatorics and Applications" held in Gössweinstein, Germany, in September 1999, where both senior and young researchers in pure mathematics, applied mathematics, computer science, physics, and chemistry from different European countries met. The main theme of the conference was group actions in various areas, a wide spectrum of which is presented in these proceedings. This volume will be a useful tool for researchers and graduate students in discrete mathematics and theoretical computer science.

Among the simplest combinatorial designs, triple systems are a natural generalization of graphs and have connections with geometry, algebra, group theory, finite fields, and cyclotomy. Applications of triple systems are found in coding theory, cryptography, computer science, and statistics. In many cases, triple systems provide the prototype for deep results in combinatorial design theory, and a number of important results were first understood in the context of triple systems and then generalized. This book attempts to survey current knowledge on the subject, to gather together common themes, and to provide an accurate portrait of the huge variety of problems and results. It includes representative samples of the major styles of proof technique and a comprehensive bibliography.

This book, now in a thoroughly revised second edition, provides a comprehensive and accessible introduction to modern set theory. Following an overview of basic notions in combinatorics and first-order logic, the author outlines the main topics of classical set theory in the second part, including Ramsey theory and the axiom of choice. The revised edition contains new permutation models and recent results in set theory without the axiom of choice. The third part explains the sophisticated technique of forcing in great detail, now including a separate chapter on Suslin's problem. The technique is used to show that certain statements are neither provable nor disprovable from the axioms of set theory. In the final part, some topics of classical set theory are revisited and further developed in light of forcing, with new chapters on Sacks Forcing and Shelah's astonishing construction of a model with finitely many Ramsey ultrafilters. Written for graduate students in axiomatic set theory, Combinatorial Set Theory will appeal to all researchers interested in the foundations of mathematics. With extensive reference lists and historical remarks at the end of each chapter, this book is suitable for self-study.

This book is intended as a teacher's manual and as an independent-study handbook for students and mathematical competitors. Based on a traditional teaching philosophy and a non-traditional writing approach (the stair-step method), this book consists of new problems with solutions created by the authors. The main idea of this approach is to start from relatively easy problems and "step-by-step" increase the level of difficulty toward effectively maximizing students' learning potential. In addition to providing solutions, a separate table of answers is also given at the end of the book. A broad view of mathematics is covered, well beyond the typical elementary level, by providing more in depth treatment of Geometry and Trigonometry, Number Theory, Algebra, Calculus, and Combinatorics.

The second edition of this undergraduate textbook is now available in paperback. Covering up-to-date as well as established material, it is the only textbook which deals with all the main areas of number theory, taught in the third year of a mathematics course. Each chapter ends with a collection of problems, and hints and sketch solutions are provided at the end of the book, together with useful tables.

Diagram geometry provides a range of techniques that enable an interaction between group theory and geometry. These techniques allow the mathematician to get information on a multi-dimensional geometric object from some knowledge of its bi-dimensional properties. This book introduces these techniques and provides a survey of the development of the subject of diagram geometry. The first three chapters are descriptive; a number of examples are presented, basic concepts are explained, and the reader is introduced to the language of diagram geometries. The theory is developed in the next three chapters and in chapter 7 a number of characterizations are proved. This is continued in later chapters following a survey of more advanced concepts and techniques.

In China, lots of excellent students who are good at maths takes an active part in various maths contests and the best six senior high school students will be selected to form the IMO National Team to compete in the International Mathematical Olympiad. In the past ten years China's IMO Team has achieved outstanding results - they have won the first place almost every year.The author is one of the coaches of China's IMO National Team, whose students have won many gold medals many times in IMO.This book is part of the Mathematical Olympiad Series which discusses several aspects related to maths contests, such as algebra, number theory, combinatorics, graph theory and geometry. The book elaborates on methods of discrete extremization, such as inequality control, repeated extremum, partial adjustment, exploiting symmetry, polishing transform, space estimates, etc.