В настоящей книге, написанной известными отечественными математиками, большинство задач относится к математическим дисциплинам, изучаемым только в высшей школе, — к теории вероятностей, проективной геометрии, топологии, интегральному исчислению, теории чисел. В то же время ни одна из собранных здесь задач не требует для своего решения знаний, выходящих за пределы школьного курса математики (кроме кратких разъяснений, приведенных в отдельных местах книги перед условиями соответствующих задач), — и по формулировкам, и по методам решения все эти задачи вполне элементарны. Книга состоит из условий задач, решений и ответов с указаниями. Главная цель книги — познакомить читателя с рядом математических фактов, идей и методов; форма задачника выбрана для того, чтобы стимулировать активную, творческую работу над всем этим материалом. Книга рассчитана на увлекающихся математикой школьников старших классов и студентов младших курсов ВУЗов, на преподавателей математики и вообще на всех любителей этой науки; она может быть использована в работе школьных и студенческих математических кружков.

Introduction to Combinatorics; In Romanian

It has been known for some time that geometries over finite fields, their automorphism groups and certain counting formulae involving these geometries have interesting guises when one lets the size of the field go to 1. On the other hand, the nonexistent field with one element, F1 , presents itself as a ghost candidate for an absolute basis in Algebraic Geometry to perform the Deninger–Manin program, which aims at solving the classical Riemann Hypothesis. This book, which is the first of its kind in the F1 -world, covers several areas in F1 -theory, and is divided into four main parts – Combinatorial Theory, Homological Algebra, Algebraic Geometry and Absolute Arithmetic. Topics treated include the combinatorial theory and geometry behind F1 , categorical foundations, the blend of different scheme theories over F1 which are presently available, motives and zeta functions, the Habiro topology, Witt vectors and total positivity, moduli operads, and at the end, even some arithmetic. Each chapter is carefully written by experts, and besides elaborating on known results, brand new results, open problems and conjectures are also met along the way. The diversity of the contents, together with the mystery surrounding the field with one element, should attract any mathematician, regardless of speciality. Keywords: The field with one element, F1 -geometry, combinatorial F1-geometry, non-additive category, Deitmar scheme, graph, monoid, motive, zeta function, automorphism group, blueprint, Euler characteristic, K-theory, Grassmannian, Witt ring, noncommutative geometry, Witt vector, total positivity, moduli space of curves, operad, torificiation, Absolute Arithmetic, counting function, Weil conjectures, Riemann Hypothesis

Main subject categories: • Combinatorics • Graph TheoryThis compact textbook consists of lecture notes given as a fourth-year undergraduate course of the mathematics degree at the Universitat Politècnica de Catalunya, including topics in enumerative combinatorics, finite geometry, and graph theory. This text covers a single-semester course and is aimed at advanced undergraduates and masters-level students. Each chapter is intended to be covered in 6-8 hours of classes, which includes time to solve the exercises. The text is also ideally suited for independent study. Some hints are given to help solve the exercises and if the exercise has a numerical solution, then this is given. The material covered allows the reader with a rudimentary knowledge of discrete mathematics to acquire an advanced level on all aspects of combinatorics, from enumeration, through finite geometries to graph theory.The intended audience of this book assumes a mathematical background of third-year students in mathematics, allowing for a swifter use of mathematical tools in analysis, algebra, and other topics, as these tools are routinely incorporated in contemporary combinatorics. Some chapters take on more modern approaches such as Chapters 1, 2, and 9. The authors have also taken particular care in looking for clear concise proofs of well-known results matching the mathematical maturity of the intended audience.

Computer science seeks to provide a scientific basis for the study of inform a tion processing, the solution of problems by algorithms, and the design and programming of computers. The last forty years have seen increasing sophistication in the science, in the microelectronics which has made machines of staggering complexity economically feasible, in the advances in programming methodology which allow immense programs to be designed with increasing speed and reduced error, and in the development of mathematical techniques to allow the rigorous specification of program, process, and machine. The present volume is one of a series, The AKM Series in Theoretical Computer Science, designed to make key mathe matical developments in computer science readily accessible to under graduate and beginning graduate students. Specifically, this volume takes readers with little or no mathematical background beyond high school algebra, and gives them a taste of a number of topics in theoretical computer science while laying the mathematical foundation for the later, more detailed, study of such topics as formal language theory, computability theory, programming language semantics, and the study of program verification and correctness. Chapter 1 introduces the basic concepts of set theory, with special emphasis on functions and relations, using a simple algorithm to provide motivation. Chapter 2 presents the notion of inductive proof and gives the reader a good grasp on one of the most important notions of computer science: the recursive definition of functions and data structures. Erscheinungsdatum: 14.12.2011

The history of triangular norms started with the paper'Statistical metrics'[Menger 1942]. The main idea of Karl Menger was to construct metric spaces where probability distributions rather than numbers are used in order to de­ scribe the distance between two elements of the space in question. Triangular norms (t-norms for short) naturally came into the picture in the course of the generalization of the classical triangle inequality to this more general set­ ting. The original set of axioms for t-norms was considerably weaker, including among others also the functions which are known today as triangular conorms. Consequently, the first field where t-norms played a major role was the theory of probabilistic metric spaces ( as statistical metric spaces were called after 1964). Berthold Schweizer and Abe Sklar in [Schweizer & Sklar 1958, 1960, 1961] provided the axioms oft-norms, as they are used today, and a redefinition of statistical metric spaces given in [Serstnev 1962]led to a rapid development of the field. Many results concerning t-norms were obtained in the course of this development, most of which are summarized in the monograph [Schweizer & Sklar 1983]. Mathematically speaking, the theory of (continuous) t-norms has two rather independent roots, namely, the field of (specific) functional equations and the theory of (special topological) semigroups.

In this third edition of Foundation Mathematics for Computer Science, John Vince has reviewed and edited the second edition, and added chapters on systems of counting, area and volume. These subjects complement the existing chapters on visual mathematics, numbers, algebra, logic, combinatorics, probability, modular arithmetic, trigonometry, coordinate systems, determinants, vectors, complex numbers, matrices, geometric matrix transforms, differential and integral calculus. During this journey, the author touches upon more esoteric topics such as quaternions, octonions, Grassmann algebra, Barrycentric coordinates, transfinite sets and prime numbers. John Vince describes a range of mathematical topics that provide a solid foundation for an undergraduate course in computer science, starting with a review of number systems and their relevance to digital computers, and finishing with calculating area and volume using calculus. Readers will find that the author’s visual approach should greatly improve their understanding as to why certain mathematical structures exist, together with how they are used in real-world applications. This third edition includes new, full-colour illustrations to clarify the mathematical descriptions, and in some cases, equations are also coloured to reveal vital algebraic patterns. The numerous worked examples will help consolidate the understanding of abstract mathematical concepts. Whether you intend to pursue a career in programming, scientific visualisation, artificial intelligence, systems design, or real-time computing, you should find the author’s literary style refreshingly lucid and engaging, and prepare you for more advanced texts.

Combinatorics has come of age. It had its beginnings in a number of puzzles which have still not lost their charm. Among these are EULER'S problem of the 36 officers and the KONIGSBERG bridge problem, BACHET's problem of the weights, and the Reverend T.P. KIRKMAN'S problem of the schoolgirls. Many of the topics treated in ROUSE BALL'S Recreational Mathe­ matics belong to combinatorial theory. All of this has now changed. The solution of the puzzles has led to a large and sophisticated theory with many complex ramifications. And it seems probable that the four color problem will only be solved in terms of as yet undiscovered deep results in graph theory. Combinatorics and the theory of numbers have much in common. In both theories there are many prob­ lems which are easy to state in terms understandable by the layman, but whose solution depends on complicated and abstruse methods. And there are now interconnections between these theories in terms of which each enriches the other. Combinatorics includes a diversity of topics which do however have interrelations in superficially unexpected ways. The instructional lectures included in these proceedings have been divided into six major areas: 1. Theory of designs; 2. Graph theory; 3. Combinatorial group theory; 4. Finite geometry; 5. Foundations, partitions and combinatorial geometry; 6. Coding theory. They are designed to give an overview of the classical foundations of the subjects treated and also some indication of the present frontiers of research.

This two-volume set presents combinatorial functional equations using an algebraic approach, and illustrates their applications in combinatorial maps, graphs, networks, etc. The second volume mainly presents several kinds of meson functional equations which are divided into three types: outer, inner and surface. It is suited for a wide readership, including pure and applied mathematicians, and also computer scientists. * A first attempt to present combinatorial functional equations in an algebraic manner. * Presents a rigorous mathematical theory both qualitative and quantitative for functional equations * Multi-disciplinary topics suited for wide readership.

В книге собраны красивые и глубокие теоремы из различных областей теории чисел, геометрии, анализа, комбинаторики, теории графов. Доказательства этих теорем используют неожиданные сочетания разнородных идей. Изложение материала сопровождается большим числом иллюстраций. Книга предназначена всем, кто увлечен математикой: в первую очередь студентам, аспирантам, а также преподавателям, научным работникам и просто любителям изящных математических рассуждений. Многое в книге доступно школьникам старших классов.

__Random Generation of Trees__ is about a field on the crossroads between computer science, combinatorics and probability theory. Computer scientists need random generators for performance analysis, simulation, image synthesis, etc. In this context random generation of trees is of particular interest. The algorithms presented here are efficient and easy to code. Some aspects of Horton--Strahler numbers, programs written in C and pictures are presented in the appendices. The complexity analysis is done rigorously both in the worst and average cases. __Random Generation of Trees__ is intended for students in computer science and applied mathematics as well as researchers interested in random generation.

Die grundlegenden Konzepte der Wahrscheinlichkeitstheorie werden in diesem Buch ausführlich und verständlich diskutiert. Mit vielen exemplarisch durchgerechneten Aufgaben, einer Vielzahl weiterer Problemstellungen und ausführlichen Lösungen bietet es dem Leser die Möglichkeit, die eigenen Fähigkeiten ständig zu erweitern und kritisch zu überprüfen und bilden so die Grundlage für ein solides Verständnis der Materie. Die realitätsnahen Anwendungen ermöglichen einen Ausblick in die breite Verwendbarkeit dieser wichtigen Theorie. Auf die Entwicklung der Begriffsbildung und mathematischen Konzepte wird besonderer Wert gelegt, so dass man Ihre Bedeutung bei der Erzeugung wie auch ständige Verbesserung von Forschungsinstrumenten für die Untersuchung unserer Welt erleben kann. Gerichtet ist das Buch an Gymnasiasten, Studienanfänger an Hochschulen, Lehrer und Interessierte, die sich mit diesem Gebiet vertraut machen möchten.

Dieses Buch wird alle Liebhaber der Mathematik (und die es werden wollen) durch eine Fülle von reizvollen und unterhaltsamen Problemstellungen aus Algebra, Geometrie, Kombinatorik und Zahlentheorie begeistern.  Ausgewählte Aufgaben aus 45 Jahren Bundeswettbewerb Mathematik werden mit ausführlichen Lösungen, Hintergrundinformation und inhaltlichen Variationen reich illustriert präsentiert.   Von der Mathematik geht für Viele seit jeher eine besondere Faszination aus. Wer tiefer in sie eindringt entdeckt ihre Schönheit, ihre Eleganz und ihre Vielfalt und stößt immer wieder auf überraschende Resultate. Hiervon bereits Schülerinnen und Schülern etwas nahe zu bringen, ist eines der Anliegen des Bundeswettbewerbs Mathematik. Mit seinen außergewöhnlichen Aufgaben regt er seit 46 Jahren Jugendliche an, sich eine Zeit lang intensiv mit Mathematik zu beschäftigen und Erfahrungen im Problemlösen zu sammeln. Anhand von ausgewählten, in den Augen des Aufgabenausschusses besonders gelungenen Aufgaben dokumentiert dieses Buch die Vielfalt der Aufgabenstellungen und beleuchtet ihren jeweiligen mathematischen Hintergrund. Darüber hinaus sind hier zum ersten Mal alle Aufgaben, die seit dem ersten Lauf des Bundeswettbewerbs Mathematik im Schuljahr 1970/71 gestellt wurden, komplett versammelt.

One of the most effective ways to stimulate students to enjoy intellectual efforts is the scientific competition. In 1894 the Hungarian Mathematical and Physical Society introduced a mathematical competition for high school students. The success of high school competitions led the Mathematical Society to found a college level contest, named after Miklós Schweitzer. The problems of the Schweitzer Contests are proposed and selected by the most prominent Hungarian mathematicians. This book collects the problems posed in the contests between 1962 and 1991 which range from algebra, combinatorics, theory of functions, geometry, measure theory, number theory, operator theory, probability theory, topology, to set theory. The second part contains the solutions. The Schweitzer competition is one of the most unique in the world. The experience shows that this competition helps to identify research talents. This collection of problems and solutions in several fields in mathematics can serve as a guide for many undergraduates and young mathematicians. The large variety of research level problems might be of interest for more mature mathematicians and historians of mathematics as well. Erscheinungsdatum: 30.09.2012

Spectral graph theory starts by associating matrices to graphs – notably, the adjacency matrix and the Laplacian matrix. The general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenvalues to structural properties of graphs. As it turns out, the spectral perspective is a powerful tool. Some of its loveliest applications concern facts that are, in principle, purely graph theoretic or combinatorial. This text is an introduction to spectral graph theory, but it could also be seen as an invitation to algebraic graph theory. The first half is devoted to graphs, finite fields, and how they come together. This part provides an appealing motivation and context of the second, spectral, half. The text is enriched by many exercises and their solutions. The target audience are students from the upper undergraduate level onwards. We assume only a familiarity with linear algebra and basic group theory. Graph theory, finite fields, and character theory for abelian groups receive a concise overview and render the text essentially self-contained. Keywords: Adjacency eigenvalues of graphs, Laplacian eigenvalues of graphs, Cayley graphs, algebraic graphs over finite fields, character sums

This book develops a method to obtain limit theorems for various functionals of random graphs. The method is based on a certain orthogonal decomposition. Janson's results include limit theorems for the two standard random graph models, $G_{n,p}$ and $G_{n,m}$, as well as functional limit theorems for the evolution of a random graph and results on the maximum of a function during the evolution. Janson obtains both normal and nonnormal limits, and the method provides an explanation for the appearance of nonnormal limits. Applications to subgraph counts and to vertex degrees are presented as examples.

The goal of the research out of which this monograph grew, was to make annealing as much as possible a general purpose optimization routine. At first glance this may seem a straight-forward task, for the formulation of its concept suggests applicability to any combinatorial optimization problem. All that is needed to run annealing on such a problem is a unique representation for each configuration, a procedure for measuring its quality, and a neighbor relation. Much more is needed however for obtaining acceptable results consistently in a reasonably short time. It is even doubtful whether the problem can be formulated such that annealing becomes an adequate approach for all instances of an optimization problem. Questions such as what is the best formulation for a given instance, and how should the process be controlled, have to be answered. Although much progress has been made in the years after the introduction of the concept into the field of combinatorial optimization in 1981, some important questions still do not have a definitive answer. In this book the reader will find the foundations of annealing in a self-contained and consistent presentation. Although the physical analogue from which the con cept emanated is mentioned in the first chapter, all theory is developed within the framework of markov chains. To achieve a high degree of instance independence adaptive strategies are introduced. Erscheinungsdatum: 06.10.2011

This volume contains the accounts of papers delivered at the Nato Advanced Study Institute on Finite and Infinite Combinatorics in Sets and Logic held at the Banff Centre, Alberta, Canada from April 21 to May 4, 1991. As the title suggests the meeting brought together workers interested in the interplay between finite and infinite combinatorics, set theory, graph theory and logic. It used to be that infinite set theory, finite combinatorics and logic could be viewed as quite separate and independent subjects. But more and more those disciplines grow together and become interdependent of each other with ever more problems and results appearing which concern all of those disciplines. I appreciate the financial support which was provided by the N. A. T. O. Advanced Study Institute programme, the Natural Sciences and Engineering Research Council of Canada and the Department of Mathematics and Statistics of the University of Calgary. 11l'te meeting on Finite and Infinite Combinatorics in Sets and Logic followed two other meetings on discrete mathematics held in Banff, the Symposium on Ordered Sets in 1981 and the Symposium on Graphs and Order in 1984. The growing inter-relation between the different areas in discrete mathematics is maybe best illustrated by the fact that many of the participants who were present at the previous meetings also attended this meeting on Finite and Infinite Combinatorics in Sets and Logic.

The subject of the book could be described as fuzzy discrete structures. The book deals with fuzzy logic, fuzzy switching functions, fuzzy decision trees, fuzzy networks, fuzzy petri nets, fuzzy path problems, fuzzy automata, and fuzzy languages. The notion of a fuzzy decision tree and a branch-bound-backtrack algorithm is presented. In the Appendix, this algorithm is described in C++. Network flow problems with various capacities are examined. Fuzzy shortest path techniques are presented whereby there is an actual path which corresponds to the fuzzy shortest path. Recent results on fuzzy automata and fuzzy languages are also presented.

Our intention in this collection is to provide, largely through original writings, an ex­ tended account of pi from the dawn of mathematical time to the present. The story of pi reflects the most seminal, the most serious, and sometimes the most whimsical aspects of mathematics. A surprising amount of the most important mathematics and a signifi­ cant number of the most important mathematicians have contributed to its unfolding­ directly or otherwise. Pi is one of the few mathematical concepts whose mention evokes a response of recog­ nition and interest in those not concerned professionally with the subject. It has been a part of human culture and the educated imagination for more than twenty-five hundred years. The computation of pi is virtually the only topic from the most ancient stratum of mathematics that is still of serious interest to modern mathematical research. To pursue this topic as it developed throughout the millennia is to follow a thread through the history of mathematics that winds through geometry, analysis and special functions, numerical analysis, algebra, and number theory. It offers a subject that provides mathe­ maticians with examples of many current mathematical techniques as weIl as a palpable sense of their historical development. Why a Source Book? Few books serve wider potential audiences than does a source book. To our knowledge, there is at present no easy access to the bulk of the material we have collected.

Пер. с англ. А. А. Слинкин. Эта книга помогает познакомиться с олимпиадным программированием. Она подробно описывает, как проходят олимпиады, что требуется от участника, в чем их цель, как к ним готовиться. Подробно разобраны базовые темы, трюки и алгоритмы. Спортивное программирование – это самый перспективный интеллектуаль­ный вид спорта, который можно назвать шахматами будущего. Уже сейчас им увлекаются лучшие умы планеты, и число участников растет год от года. Рост популярности олимпиадного программирования положительно влияет на другие сферы жизнедеятельности человека. Навыки быстрого решения сложнейших задач помогают сегодняшним студентам в будущем эффективно справляться с реальными проблемами человечества. Издание будет полезно прежде всего студентам, начинающим принимать участие в соревнованиях по программированию.

This Volume Contains Two Types Of Papers—a Selection Of Contributions From The “second International Conference In Network Analysis” Held In Nizhny Novgorod On May 7–9, 2012, And Papers Submitted To An Open Call For Papers Reflecting The Activities Of Latna At The Higher School For Economics. This Volume Contains Many New Results In Modeling And Powerful Algorithmic Solutions Applied To Problems In •         Vehicle Routing •         Single Machine Scheduling •         Modern Financial Markets •         Cell Formation In Group Technology •         Brain Activities Of Left- And Right-handers •         Speeding Up Algorithms For The Maximum Clique Problem •         Analysis And Applications Of Different Measures In Clustering  the Broad Range Of Applications That Can Be Described And Analyzed By Means Of A Network Brings Together Researchers, Practitioners, And Other Scientific Communities From Numerous Fields Such As Operations Research, Computer Science, Bioinformatics, Medicine, Transportation, Energy, Social Sciences, And More. The Contributions Not Only Come From Different Fields, But Also Cover A Broad Range Of Topics Relevant To The Theory And Practice Of Network Analysis. Researchers, Students, And Engineers From Various Disciplines Will Benefit From The State-of-the-art In Models, Algorithms, Technologies, And Techniques Including New Research Directions And Open Questions. Models, Algorithms, And Technologies For Network Analysis (v.a. Kalyagin, B. Goldengorin, P.m. Pardalos) -- Tolerance-based Vs. Cost-based Branching For The Asymmetric Capacitated Vehicle Routing Problem (m. Batsyn, B. Goldengorin, A. Kocheturov, p.m. Pardalos) -- Lower And Upper Bounds For The Preemptive Single Machine Scheduling Problem With Equal Processing Times (m. Batsyn, B. Goldengorin, P. Sukhov, P.m. Pardalos) -- Comparative Analysis Of Two Similarity Measures For The Market Graph Construction (g. A. Bautin, V.a. Kalyagin, A.p. Koldanov) -- Heuristic Algorithm For The Cell Formation Problem (i. Bychkov, M. Batsyn, P. Sukhov, P.m. Pardalos) -- Efficiency Analysis Of Branch Network (p.a. Koldanov) -- Eeg Coherence In Right- And Left-handers In Passive Visual Perception Of Lines With Different Slope Angles (l.m. Viktorovich) -- Speeding Up Mcs Algorithm For The Maximum Clique Problem With Ils Heuristic And Other Enhancements (e. Maslov, M. Batsyn, P.m. Pardalos) -- Summary And Semi-average Similarity Criteria For Individual Clusters (b. Mirkin) -- Kernel Principal Component Analysis: Applications, Implementation, And Comparison (d. Olsson, P. Georgiev, V. Boginski, A. Sorokin, P.m. Pardalos) -- Distance-based Clique Relaxations In Networks: S-clique And S-club (s. Shahinpour, S. Butenko) -- Grasp With Path-relinking For Facility Layout (r.m.a. Silva, M.g.c. Resende, P.m. Pardalos, G.r. Mateus, G. De Tomi) -- Comparative Analysis Of The Bric Countries Stock Markets Using Network Approach (a. Vizgunov, A. Glotov, P.m. Pardalos) -- Sensor Cover And Double Partition (l. Wu, W. Wu, L. Lu, Y. Zhu, D.-z. Du). Edited By Boris I. Goldengorin, Valery A. Kalyagin, Panos M. Pardalos.

Because of its wide applicability, graph theory is one of the fast-growing areas of modern mathematics. Graphs arise as mathematical models in areas as diverse as management science, chemistry, resource planning, and computing. Moreover, the theory of graphs provides a spectrum of methods of proof and is a good train­ ing ground for pure mathematics. Thus, many colleges and universities provide a first course in graph theory that is intended primarily for mathematics majors but accessible to other students at the senior Ievel. This text is intended for such a course. I have presented this course many times. Over the years classes have included mainly mathematics and computer science majors, but there have been several engineers and occasional psychologists as weil. Often undergraduate and graduate students are in the same dass. Many instructors will no doubt find themselves with similar mixed groups. lt is to be expected that anyone enrolling in a senior Ievel mathematics course will be comfortable with mathematical ideas and notation. In particular, I assume the reader is familiar with the basic concepts of set theory, has seen mathematical induction, and has a passing acquaintance with matrices and algebra. However, one cannot assume that the students in a first graph theory course will have a good knowledge of any specific advanced area. My reaction to this is to avoid too many specific prerequisites. The main requirement, namely a little mathematical maturity, may have been acquired in a variety of ways.

Covering the major topics of evolutionary game theory, Game-Theoretical Models in Biology presents both abstract and practical mathematical models of real biological situations. It discusses the static aspects of game theory in a mathematically rigorous way that is appealing to mathematicians. In addition, the authors explore many applications of game theory to biology, making the text useful to biologists as well. The book describes a wide range of topics in evolutionary games, including matrix games, replicator dynamics, the hawk-dove game, and the prisoner’s dilemma. It covers the evolutionarily stable strategy, a key concept in biological games, and offers in-depth details of the mathematical models. Most chapters illustrate how to use MATLAB ® to solve various games. Important biological phenomena, such as the sex ratio of so many species being close to a half, the evolution of cooperative behavior, and the existence of adornments (for example, the peacock’s tail), have been explained using ideas underpinned by game theoretical modeling. Suitable for readers studying and working at the interface of mathematics and the life sciences, this book shows how evolutionary game theory is used in the modeling of these diverse biological phenomena.

Знакомы ли вы с технологиями, лежащими в основе вашей собственной программы? Почему «правильный» код не хочет работать? Истина проста и банальна — нужно сразу создавать код, который будет работать хорошо и не будет прятать в себе трудноуловимые ошибки. Для этого Джонатан Стейнхарт исследует фундаментальные концепции, лежащие в основе работы компьютеров. Он рассматривает аппаратное обеспечение, поведение программ на определенных устройствах, чтобы показать, как на самом деле должен работать ваш код. Узнайте, что на самом деле происходит, когда вы запускаете код на компьютере, и вы научитесь программировать лучше и эффективнее.

Sri Gopal Mohanty has made pioneering contributions to lattice path counting and its applications to probability and statistics. This is clearly evident from his lifetime publications list and the numerous citations his publications have received over the past three decades. My association with him began in 1982 when I came to McMaster Univer­ sity. Since then, I have been associated with him on many different issues at professional as well as cultural levels; I have benefited greatly from him on both these grounds. I have enjoyed very much being his colleague in the statistics group here at McMaster University and also as his friend. While I admire him for his honesty, sincerity and dedication, I appreciate very much his kindness, modesty and broad-mindedness. Aside from our common interest in mathematics and statistics, we both have great love for Indian classical music and dance. We have spent numerous many different subjects associated with the Indian music and hours discussing dance. I still remember fondly the long drive (to Amherst, Massachusetts) I had a few years ago with him and his wife, Shantimayee, and all the hearty discussions we had during that journey. Combinatorics and applications of combinatorial methods in probability and statistics has become a very active and fertile area of research in the recent past.

Пер. с англ. А.А. Слинкина. Сопоставление строк – одна из самых старых тем в теории алгоритмов, но по-прежнему занимает важное место в информатике. За прошедшие 20 лет мы видели технологические прорывы в таких разных приложениях, как информационный поиск и сжатие информации. Эта книга, представляющая собой богатое собрание задач и упражнений по важнейшим вопросам алгоритмов обработки текстов и комбинаторных свойств слов, предлагает студентам и исследователям приятный и прямой путь к изучению и практическому освоению концепций повышенного уровня. Задачи взяты из многочисленных научных публикаций – как уже ставших классическими, так и сравнительно новых. Начав с основ, авторы рассматривают все более сложные задачи по комбинаторным свойствам слов (включая слова Фибоначчи и Туэ–Морса), поиску строк в тексте (включая алгоритмы Кнута–Морриса–Пратта и Бойера–Мура), эффективным структурам данных для представления текстов (включая суффиксные деревья и суффиксные массивы) и сжатия текста (включая методы Хаффмана, Лемпеля–Зива и Барроуза–Уилера). Издание будет полезно в качестве пособия для подготовки к олимпиадам по информатике.

Includes overviews of evolving areas in convexity, discrete geometry and graph theory Presents easily understandable, but surprising, properties, obtained using topological, geometric and graph theoretic tools Written by experts from all over the word This volume presents easy-to-understand yet surprising properties obtained using topological, geometric and graph theoretic tools in the areas covered by the Geometry Conference that took place in Mulhouse, France from September 7–11, 2014 in honour of Tudor Zamfirescu on the occasion of his 70th anniversary. The contributions address subjects in convexity and discrete geometry, in distance geometry or with geometrical flavor in combinatorics, graph theory or non-linear analysis. Written by top experts, these papers highlight the close connections between these fields, as well as ties to other domains of geometry and their reciprocal influence. They offer an overview on recent developments in geometry and its border with discrete mathematics, and provide answers to several open questions. The volume addresses a large audience in mathematics, including researchers and graduate students interested in geometry and geometrical problems. Topics Convex and Discrete Geometry Graph Theory Combinatorics Global Analysis and Analysis on Manifolds

The Virasoro algebra is an infinite dimensional Lie algebra that plays an increasingly important role in mathematics and theoretical physics. This book describes some fundamental facts about the representation theory of the Virasoro algebra in a self-contained manner. Topics include the structure of Verma modules and Fock modules, the classification of (unitarizable) Harish-Chandra modules, tilting equivalence, and the rational vertex operator algebras associated to the so-called minimal series representations. Covering a wide range of material, this book has three appendices which provide background information required for some of the chapters. Fundamental results are organized in a unified way and existing proofs refined. For instance in chapter three, a generalization of Jantzen filtration is reformulated in an algebraic manner, and geometric interpretation is provided. Statements, widely believed to be true, are collated, and results which are known but not verified are proven, such as the corrected structure theorem of Fock modules in chapter eight. This book will be of interest to a wide range of mathematicians and physicists from the level of graduate students to researchers.

With the advent of computers, algorithmic principles play an ever increasing role in mathematics. Algorithms have to exploit the structure of the underlying mathematical object, and properties exploited by algorithms are often closely tied to classical structural analysis in mathematics. This connection between algorithms and structure is in particular apparent in discrete mathematics, where proofs are often constructive, and can be turned into algorithms more directly. The principle of greediness plays a fundamental role both in the design of continuous algorithms (where it is called the steepest descent or gradient method) and of discrete algorithms. The discrete structure most closely related to greediness is a matroid; in fact, matroids may be characterized axiomatically as those independence systems for which the greedy solution is optimal for certain optimization problems (e.g. linear objective functions, bottleneck functions). This book is an attempt to unify different approaches and to lead the reader from fundamental results in matroid theory to the current borderline of open research problems. The monograph begins by reviewing classical concepts from matroid theory and extending them to greedoids. It then proceeds to the discussion of subclasses like interval greedoids, antimatroids or convex geometries, greedoids on partially ordered sets and greedoid intersections. Emphasis is placed on optimization problems in greedois. An algorithmic characterization of greedoids in terms of the greedy algorithm is derived, the behaviour with respect to linear functions is investigated, the shortest path problem for graphs is extended to a class of greedoids, linear descriptions of antimatroid polyhedra and complexity results are given and the Rado-Hall theorem on transversals is generalized. The self-contained volume which assumes only a basic familarity with combinatorial optimization ends with a chapter on topological results in connection with greedoids.

This book is a sequel to my Beginner's Guide to Mathematical Logic.The previous volume deals with elements of propositional and first-order logic, contains a bit on formal systems and recursion, and concludes with chapters on Gödel's famous incompleteness theorem, along with related results.The present volume begins with a bit more on propositional and first-order logic, followed by what I would call a "fein" chapter, which simultaneously generalizes some results from recursion theory, first-order arithmetic systems, and what I dub a "decision machine." Then come five chapters on formal systems, recursion theory and metamathematical applications in a general setting. The concluding five chapters are on the beautiful subject of combinatory logic, which is not only intriguing in its own right, but has important applications to computer science. Argonne National Laboratory is especially involved in these applications, and I am proud to say that its members have found use for some of my results in combinatory logic.This book does not cover such important subjects as set theory, model theory, proof theory, and modern developments in recursion theory, but the reader, after studying this volume, will be amply prepared for the study of these more advanced topics.

This 2nd edition, revised and extended compendium contains and explains essential mathematical formulas within an economic context. Newly added content focuses on financial mathematics, now including an international comparison between different national methods used in the calculation of interest. Further, the annuity calculation now contains unique content. A broad range of aids and supportive examples will help readers to understand the formulas and their practical applications. This mathematical formulary is presented in a practice-oriented, clear, and understandable manner, as it is needed for meaningful and relevant application in global business, as well as in the academic setting and economic practice. The topics presented include, but are not limited to: mathematical signs and symbols, logic, arithmetic, algebra, linear algebra, combinatorics, financial mathematics, optimisation of linear models, functions, differential calculus, integral calculus, elasticities, economic functions, and the Peren Theorem. Given its scope, the book offers an indispensable reference guide and is a must-read for undergraduate and graduate students, as well as managers, scholars, and lecturers in business, politics, and economics.

From the reviews of the first edition: ".... The book is a first class textbook and seems to be indispensable for everybody who has to teach combinatorial optimization. It is very helpful for students, teachers, and researchers in this area. The author finds a striking synthesis of nice and interesting mathematical results and practical applications. ... the author pays much attention to the inclusion of well-chosen exercises. The reader does not remain helpless; solutions or at least hints are given in the appendix. Except for some small basic mathematical and algorithmic knowledge the book is self-contained. ..." Mathematical Reviews 2002 This thoroughly revised new edition offers a new chapter on the network simplex algorithm and a section on the five color theorem. Moreover, numerous smaller changes and corrections have been made and several recent developments have been discussed and referenced.

Spherical buildings are certain combinatorial simplicial complexes intro duced, at first in the language of "incidence geometries," to provide a sys tematic geometric interpretation of the exceptional complex Lie groups. (The definition of a building in terms of chamber systems and definitions of the various related notions used in this introduction such as "thick," "residue," "rank," "spherical," etc. are given in Chapter 39. ) Via the notion of a BN-pair, the theory turned out to apply to simple algebraic groups over an arbitrary field. More precisely, to any absolutely simple algebraic group of positive rela tive rank £ is associated a thick irreducible spherical building of the same rank (these are the algebraic spherical buildings) and the main result of Buildings of Spherical Type and Finite BN-Pairs [101] is that the converse, for £ ::::: 3, is almost true: (1. 1) Theorem. Every thick irreducible spherical building of rank at least three is classical, algebraic' or mixed. Classical buildings are those defined in terms of the geometry of a classical group (e. g. unitary, orthogonal, etc. of finite Witt index or linear of finite dimension) over an arbitrary field or skew-field. (These are not algebraic if, for instance, the skew-field is of infinite dimension over its center. ) Mixed buildings are more exotic; they are related to groups which are in some sense algebraic groups defined over a pair of fields k and K of characteristic p, where KP eke K and p is two or (in one case) three. Erscheinungsdatum: 05.12.2010

p000-Frontmatter......Page 1 Foreword......Page 11 Conference Organization......Page 13 Additional Reviewers......Page 15 p001-Dasgupta......Page 17 p002-Donald......Page 19 Introduction......Page 21 The Previous Results......Page 22 Preliminaries......Page 23 Definitions and Set up......Page 24 The Main Lemma......Page 26 Notation and Setup......Page 28 The Lower Bound Proof......Page 32 The Upper Bounds......Page 33 Conclusions......Page 34 Introduction......Page 35 A Randomized Data Structure......Page 37 Preliminaries......Page 39 A Solution with Expected Query Time......Page 40 Generating Candidate Points......Page 41 Constructing the k Samples......Page 42 Lower Bound for Worst-Case Time with Exact Weights......Page 44 p005-Agarwal......Page 49 Introduction......Page 50 Polynomials, partitioning, and quantifier elimination......Page 51 Range spaces, VC dimension, and epsilon-samples......Page 53 The parameter space of semi-algebraic sets......Page 54 A singly-exponential algorithm......Page 55 Speeding up the algorithm using epsilon-sampling......Page 56 Point-enclosure queries......Page 57 Range searching......Page 58 Vertical ray shooting......Page 59 Cutting algebraic curves into pseudo-segments......Page 61 Introduction......Page 63 Minimum-cost partial matchings using Hungarian algorithm......Page 66 From matching to circulation......Page 67 A cost-scaling algorithm......Page 68 Fast implementation of refinement stage......Page 71 An efficient implementation......Page 73 p007-Agrawal......Page 77 Introduction......Page 78 Our Contribution......Page 79 Related Works......Page 82 NP-hardness for CM-PM......Page 84 FPT Algorithm for CM-PM......Page 85 p008-Aiger......Page 95 Introduction......Page 96 From camera positioning to approximate incidences......Page 98 The surfaces sigma_w......Page 99 Measuring proximity......Page 100 Primal-dual algorithm for geometric proximity......Page 101 Geometric proximity via canonical surfaces......Page 103 Canonizing the input surfaces......Page 104 Approximately counting {epsilon}-incidences......Page 105 Error analysis......Page 106 Experimental Results: Summary......Page 107 p009-Akitaya......Page 109 Introduction......Page 110 Large Subsets with Circumscribing Polygons......Page 111 Disjoint Segments versus Disjoint Rays......Page 118 Hardness for Circumscribing Polygons......Page 120 Simple Polygonizations of Disjoint Segments......Page 121 Higher Dimensions......Page 122 Open Problems......Page 123 p010-Angelini......Page 127 Introduction......Page 128 Definitions and Preliminaries......Page 130 From RT-Representations to Schnyder Woods......Page 131 From Schnyder Woods to RT-Representations......Page 132 Geometric Tools......Page 133 Moving a Triangle Along a Diagonal......Page 135 A Flipping Algorithm......Page 136 Morphing Representations with the same Schnyder Wood......Page 137 A Decision Algorithm......Page 139 Conclusions and Open Problems......Page 140 Introduction......Page 143 Euclidean space, distances and convex sets......Page 144 Filtrations......Page 145 Statement of results......Page 146 Building a filtration......Page 147 Perturbing filtrations......Page 148 Studying events in the filtration......Page 150 Collapsing Rips complexes......Page 153 Two geometric lemmas......Page 155 Future work......Page 157 Introduction......Page 159 Preliminaries......Page 162 The simplification transformation......Page 163 A smaller polygon......Page 165 Preprocessing the red sites......Page 167 Inserting back the red sites......Page 168 The insertion process......Page 169 Algorithmic description......Page 170 p013-Binucci......Page 173 Introduction......Page 174 Preliminaries......Page 175 NP-Completeness for kUBE (k >= 3)......Page 176 Existential Results for 2UBE......Page 181 Testing 2UBE for Plane Graphs with Special Faces......Page 182 Testing Algorithms for 2UBE Parameterized by the Branchwidth......Page 183 Conclusion and Open Problems......Page 190 p014-Bokal......Page 195 Introduction......Page 196 Graphs and the crossing number......Page 198 Crossing-critical graphs with at most 12 crossings......Page 199 Explicit 13-crossing-critical graphs with large degree......Page 202 Extended crossing-critical constructions......Page 206 Concluding remarks and open problems......Page 207 Introduction......Page 211 Our results and methods......Page 213 Graph construction......Page 214 Preserving Euclidean distance......Page 216 Generalized preconditioning framework......Page 217 Preconditioner construction......Page 219 Generating a transportation map......Page 222 p016-Braverman......Page 225 Introduction......Page 226 Preliminaries......Page 227 Technical overview......Page 229 The bounded -DTW problem......Page 235 Introduction......Page 241 Preprocessing......Page 244 Free-space diagram......Page 245 Pruning rules......Page 246 Implementation details of simple boundaries......Page 249 Effects of combined pruning rules......Page 250 Decider with filters......Page 251 Decider setting......Page 252 Query setting......Page 254 Introduction......Page 263 Contribution 2: Conditional lower bound......Page 264 Preliminaries......Page 266 Algorithms for Global-Fréchet simplification......Page 267 An O(kn^5) algorithm for Global Fréchet simplification......Page 268 An O(n^3) algorithm for Global-Fréchet simplification......Page 269 Cell Reachability......Page 271 Conditional lower bound for curve simplification......Page 272 Overview of the reduction......Page 273 Cordinate gadgets......Page 274 The vectors a', b', c', and s'......Page 276 The vectors a'',b'', c'', and s''......Page 277 Introduction......Page 279 Our results......Page 281 Expanders are reliable......Page 282 Bounding the size of the shadow......Page 283 Analysis......Page 284 Construction of theta-reliable exact spanners in one dimension......Page 287 Analysis......Page 288 Analysis......Page 290 Conclusions......Page 292 Introduction......Page 295 Preliminaries......Page 299 Convex hull......Page 300 Notation for axis-parallel problems......Page 302 Interpreting Shapley values geometrically......Page 304 Handling empty blocks......Page 305 Chains......Page 306 General point sets......Page 308 Area of the bounding box......Page 309 Conclusions......Page 310 Introduction......Page 315 Persistence Diagrams......Page 317 Bi-Lipschitz embeddings.......Page 319 Mapping into separable Hilbert spaces......Page 320 Mapping into finite-dimensional Hilbert spaces......Page 324 Experiments......Page 325 Conclusion......Page 327 p022-DeCarufel......Page 331 Introduction......Page 332 Upper bounds in the plane......Page 334 Upper bounds in higher dimensions......Page 336 Lower bounds in higher dimensions......Page 338 Convex caps......Page 341 Upper bounds......Page 343 Lower bounds......Page 344 The maximum number of weakly convex polygons......Page 345 Introduction......Page 349 An exact near-quadratic algorithm......Page 351 k-sensitive running time for smallest area......Page 353 An approximation algorithm for smallest area......Page 355 3-sided smallest k-enclosing rectangle......Page 358 Arbitrarily oriented smallest k-enclosing rectangle......Page 359 Subset sum for k-enclosing rectangle......Page 360 Conditional lower bounds......Page 361 Introduction......Page 365 Dynamic 3D Convex Hull Size......Page 368 Dynamic 2D Hausdorff Distance......Page 371 Dynamic 3D Convex Hull Queries......Page 373 Dynamic 2D Bichromatic Closest Pair......Page 375 Introduction......Page 379 Multicurves and medial graphs......Page 381 Local moves......Page 382 Equivalence of tightness......Page 383 Two-terminal plane graphs......Page 385 Smoothing lemma in the annulus......Page 386 Quadratic lower bound......Page 387 Defect......Page 388 Tangle flips......Page 389 Back to planar electrical moves......Page 391 Open problems......Page 392 p026-Agarwal......Page 395 Introduction......Page 396 Data structure and operations......Page 397 Finding the intersection point of two lower envelopes......Page 398 Maintaining the union of unit discs under insertions......Page 402 Preliminaries......Page 403 Dynamically maintaining the upper curves......Page 405 Intersection-searching of unit arcs with unit disc......Page 407 p027-Cohen-Addad......Page 411 Introduction......Page 412 Preliminaries......Page 415 The 4-regular graph tiling problem......Page 417 Multiway cut with four terminals......Page 420 Shortest cut graph......Page 422 Multiway cut with a large number of terminals......Page 423 p028-Driemel......Page 427 Distance measures......Page 428 Range spaces......Page 429 Range spaces induced by distance measures......Page 430 Our Approach......Page 431 Basic Idea: Discrete Fréchet and Hausdorff......Page 432 Geometric primitives......Page 433 Bounding the VC-Dimension......Page 434 Fréchet distance predicates......Page 435 Fréchet distance VC dimension bounds......Page 436 Lower bounds......Page 438 Implications......Page 439 Introduction......Page 443 Proof of Theorem 1......Page 445 Faces that are Touched, Pinched, and Caressed......Page 446 Auxilliary Graphs and Trees: H, H, T_0, and T_1......Page 448 Interactions Between Bad Nodes......Page 452 Really Bad Nodes......Page 453 Tree/Cycle Surgery......Page 454 Discussion......Page 457 Introduction......Page 461 Setup of the proof......Page 462 Charging scheme......Page 463 Charging scheme analysis......Page 467 Conclusion of the proof......Page 472 Introduction......Page 473 Entropy and Relative Entropy......Page 475 Fisher Information Metric......Page 477 Information and Loss......Page 479 TDA in Information Space......Page 481 Discussion......Page 485 Introduction......Page 487 Overview......Page 490 Nested polygons with no surrounded crossings......Page 491 The main result......Page 493 Series-parallel graphs......Page 494 Apex-tree graphs requiring many lines......Page 496 Drawing apex-tree graphs on few lines......Page 498 Conclusions and open problems......Page 499 Introduction......Page 503 Counting complexity......Page 504 Red–blue arrangements......Page 505 Reduction to triangulation......Page 507 From segments to polygons......Page 508 Triangulation within a gadget......Page 509 Counting within a gadget......Page 510 Global counting......Page 513 Completing the reduction......Page 515 Conclusions and open problems......Page 516 Introduction......Page 521 Background......Page 523 Simple Cycles......Page 525 Non-simple Closed Walks......Page 526 Bounding Closed Walks......Page 528 Edge Labeling......Page 530 Hyperbolic Geometry......Page 531 Triangulation......Page 532 Context-Free Grammar......Page 534 Introduction......Page 539 Preliminaries......Page 541 Boundary Packing: A Subroutine......Page 542 Ring Packing: A Subroutine......Page 543 Analysis of Phase 1 - The Recursion......Page 544 Outline of the Remaining Analysis......Page 545 Analysis of Boundary Packing......Page 546 Analysis of Ring Packing......Page 547 Analysis of the Algorithm for the Case r_1 = log^{8}n: Main Idea......Page 810 Ray Shooting for B >= log^{8}n: Fully-Dynamic Structure......Page 812 Missing Details......Page 815 Introduction......Page 819 Ortho-Radial Drawings and Representations......Page 822 Finding Monotone Cycles......Page 824 Rectangulation......Page 827 1st Improvement – Faster Validity Test......Page 828 Conclusion......Page 830 Introduction......Page 833 Families of curves......Page 834 Our results......Page 835 A bounding function for grounded curves – Proofs of Theorems 1 and 5......Page 836 Magical graphs – Proof of Theorem 2......Page 838 Bounding function for curves that intersect a vertical line–Proof of Theorem 3......Page 843 Construction of double-magical graphs–Proof of Theorem 4......Page 845 Concluding remarks......Page 847 Introduction......Page 851 Preliminaries......Page 853 Strong Collapse of a Flag complex......Page 855 A new construction......Page 858 Computational experiments......Page 861 Introduction......Page 867 Ham Sandwich Cuts in horizontal subspaces......Page 871 Application: bisecting lines in space......Page 872 Center Transversals in general subspaces......Page 873 Sections in product bundles......Page 875 Application: bisections with several cuts......Page 876 Introduction......Page 883 A Lower Bound of Omega(1/rho)......Page 884 On a Distribution of Inputs: Small Summaries for Halfspaces......Page 885 New Definitions on Range Spaces......Page 887 Analysis......Page 888 Bounding the Size......Page 889 A Remark......Page 890 Bridging distribution and finite-set disagreement coefficients......Page 891 o(1/rho)-size summaries for halfspace ranges......Page 892 A lower bound with disagreement coefficients......Page 894 Introduction......Page 897 Filtrations and interleaving distance......Page 899 Definition......Page 901 Weighted Vietoris-Rips filtrations......Page 903 The distance to measure (DTM)......Page 905 DTM-filtrations......Page 906 Stability when p=1......Page 907 Stability when p>1......Page 909 Conclusion......Page 910 Introduction......Page 913 Preliminaries......Page 916 Solving the Main Sub-Problem......Page 918 The Query Algorithm......Page 920 Reducing the Query Time to O(log n)......Page 924 Introduction......Page 927 Related work and our contributions......Page 928 Notations......Page 929 The exact algorithm......Page 930 First update......Page 933 Second update......Page 934 Putting everything together......Page 935 The approximation algorithm......Page 936 Approximate update......Page 937 Putting everything together......Page 938 Introduction......Page 941 Related work and our contributions......Page 942 Preliminaries......Page 943 Translation RCP queries for polygons......Page 944 Reduction to (co-)wedge translation queries......Page 945 Handling wedge translation queries......Page 946 Handling co-wedge translation queries......Page 949 Translation RCP queries for smooth convex bodies......Page 950 Handling short-answer queries......Page 951 Handling long-answer queries......Page 952 Proof of Theorem 20......Page 953 Introduction......Page 957 The case of pseudoplanes......Page 960 An extension of the dual version of the Lovász Lemma......Page 961 The dual version of the Crossing Lemma......Page 966 Discussion......Page 969 p063-Becker......Page 973 Introduction......Page 974 Circles in a Square: Split Packing......Page 975 Circles in a Circle: Boundary/Ring Packing......Page 976 The Video......Page 977 Introduction......Page 979 Counting Minimal-Perimeter Polyominoes......Page 980 The Video......Page 981 Introduction......Page 983 Results......Page 984 The Fréchet-distance......Page 985 The Fréchet-distance for Simple Polygons......Page 986 The k-Fréchet-distance......Page 987 Fréchet View – the Implementation......Page 988

This book deals with the theory and applications of the Reformulation- Linearization/Convexification Technique (RL T) for solving nonconvex optimization problems. A unified treatment of discrete and continuous nonconvex programming problems is presented using this approach. In essence, the bridge between these two types of nonconvexities is made via a polynomial representation of discrete constraints. For example, the binariness on a 0-1 variable x . can be equivalently J expressed as the polynomial constraint x . (1-x . ) = 0. The motivation for this book is J J the role of tight linear/convex programming representations or relaxations in solving such discrete and continuous nonconvex programming problems. The principal thrust is to commence with a model that affords a useful representation and structure, and then to further strengthen this representation through automatic reformulation and constraint generation techniques. As mentioned above, the focal point of this book is the development and application of RL T for use as an automatic reformulation procedure, and also, to generate strong valid inequalities. The RLT operates in two phases. In the Reformulation Phase, certain types of additional implied polynomial constraints, that include the aforementioned constraints in the case of binary variables, are appended to the problem. The resulting problem is subsequently linearized, except that certain convex constraints are sometimes retained in XV particular special cases, in the Linearization/Convexijication Phase. This is done via the definition of suitable new variables to replace each distinct variable-product term. The higher dimensional representation yields a linear (or convex) programming relaxation.

The advent of very large scale integrated circuit technology has enabled the construction of very complex and large interconnection networks. By most accounts, the next generation of supercomputers will achieve its gains by increasing the number of processing elements, rather than by using faster processors. The most difficult technical problem in constructing a supercom puter will be the design of the interconnection network through which the processors communicate. Selecting an appropriate and adequate topological structure of interconnection networks will become a critical issue, on which many research efforts have been made over the past decade. The book is aimed to attract the readers' attention to such an important research area. Graph theory is a fundamental and powerful mathematical tool for de signing and analyzing interconnection networks, since the topological struc ture of an interconnection network is a graph. This fact has been univer sally accepted by computer scientists and engineers. This book provides the most basic problems, concepts and well-established results on the topological structure and analysis of interconnection networks in the language of graph theory. The material originates from a vast amount of literature, but the theory presented is developed carefully and skillfully. The treatment is gen erally self-contained, and most stated results are proved. No exercises are explicitly exhibited, but there are some stated results whose proofs are left to the reader to consolidate his understanding of the material. Erscheinungsdatum: 10.12.2010

Combinatorial Engineering of Decomposable Systems presents a morphological approach to the combinatorial design/synthesis of decomposable systems. Applications involve the following: design (e.g., information systems; user's interfaces; educational courses); planning (e.g., problem-solving strategies; product life cycles; investment); metaheuristics for combinatorial optimization; information retrieval; etc.

Das Werk bietet eine Einf?hrung in die wichtigsten mathematischen Grundlagen aus den Gebieten der Linearen und Nichtlinearen Algebra, der Analysis und der Diskreten Mathematik f?r Informatiker. Besondere Schwerpunkte bilden die in den Computerwissenschaften wichtigen Methoden aus Kombinatorik, Graphentheorie und der Theorie endlicher K?rper. Damit zeichnet sich das Werk gegen?ber den klassischen Grundlagenwerken der Ingenieurmathematik durch informatikspezifische Inhalte aus. Zahlreiche durchgerechnete Beispiele und Erkl?rungen sollen die M?glichkeiten des Selbststudiuns f?rdern. Die Darstellung ist mathematisch exakt und ohne ?bertriebenen logischen Formalismus. Ziel des Werkes ist es, Studierende der Informatik in mathematischen Einf?hrungsvorlesungen zu unterst?tzen und Studenten anderer F?cher, die sich in Computermethoden spezialisieren, die spezifischen mathematischen Grundlagen zu vermitteln. Mit Band 3 ist das Werk abgeschlossen.

Computer science majors taking a non-programming-based course like discrete mathematics might ask 'Why do I need to learn this?' Written with these students in mind, this text introduces the mathematical foundations of computer science by providing a comprehensive treatment of standard technical topics while simultaneously illustrating some of the broad-ranging applications of that material throughout the field. Chapters on core topics from discrete structures – like logic, proofs, number theory, counting, probability, graphs – are augmented with around 60 'computer science connections' pages introducing their applications: for example, game trees (logic), triangulation of scenes in computer graphics (induction), the Enigma machine (counting), algorithmic bias (relations), differential privacy (probability), and paired kidney transplants (graphs). Pedagogical features include 'Why You Might Care' sections, quick-reference chapter guides and key terms and results summaries, problem-solving and writing tips, 'Taking it Further' asides with more technical details, and around 1700 exercises, 435 worked examples, and 480 figures.

"This book constitutes the refereed proceedings of the Fifth Scandinavian Workshop on Algorithm Theory, SWAT '96, held in Reykjavik, Iceland, in July 1996. The 35 full papers included in the book in revised version were carefully selected from a total of 95 submissions; also included are abstracts or full versions of three invited talks by prominent researchers. All in all, the collection of articles reports state-of-the-art results on various topics of current design and analysis of algorithms."--PUBLISHER'S WEBSITE

Dieses Buch stellt eine umfassende und leicht lesbare Einführung in die Graphentheorie dar. Das Hauptziel ist es, dem Leser, insbesondere dem Studierenden, Methoden zu übermitteln und ihn für graphentheoretisches Denken zu interessieren. Der Text enthält neben dem gesamten klassischen Bestand der Graphentheorie eine Fülle neuer und moderner Aspekte, die zum großen Teil erstmalig in dieser Form zusammengefaßt worden sind. Besonders hervorzuheben sind die Kapitel über Hamiltonsche Graphen, Turniertheorie, Faktortheorie, Dominanz und Irredundanz, Kanten- und Totalfärbung, Ramsey-Theorie und lokal-semi-vollständige Digraphen. Ausführliche Beweise, zahlreiche Beispiele und eine gelungene didaktische Aufbereitung machen das Werk durchsichtig und verständlich. Erscheinungsdatum: 22.01.1996

The main topics of this volume, dedicated to Lance Littlejohn, are operator and spectral theory, orthogonal polynomials, combinatorics, number theory, and the various interplays of these subjects. Although the event, originally scheduled as the Baylor Analysis Fest, had to be postponed due to the pandemic, scholars from around the globe have contributed research in a broad range of mathematical fields. The collection will be of interest to both graduate students and professional mathematicians. Contributors are: G.E. Andrews, B.M. Brown, D. Damanik, M.L. Dawsey, W.D. Evans, J. Fillman, D. Frymark, A.G. García, L.G. Garza, F. Gesztesy, D. Gómez-Ullate, Y. Grandati, F.A. Grünbaum, S. Guo, M. Hunziker, A. Iserles, T.F. Jones, K. Kirsten, Y. Lee, C. Liaw, F. Marcellán, C. Markett, A. Martinez-Finkelshtein, D. McCarthy, R. Milson, D. Mitrea, I. Mitrea, M. Mitrea, G. Novello, D. Ong, K. Ono, J.L. Padgett, M.M.M. Pang, T. Poe, A. Sri Ranga, K. Schiefermayr, Q. Sheng, B. Simanek, J. Stanfill, L. Velázquez, M. Webb, J. Wilkening, I.G. Wood, M. Zinchenko.

Computer science majors taking a non-programming-based course like discrete mathematics might ask 'Why do I need to learn this?' Written with these students in mind, this text introduces the mathematical foundations of computer science by providing a comprehensive treatment of standard technical topics while simultaneously illustrating some of the broad-ranging applications of that material throughout the field. Chapters on core topics from discrete structures – like logic, proofs, number theory, counting, probability, graphs – are augmented with around 60 'computer science connections' pages introducing their applications: for example, game trees (logic), triangulation of scenes in computer graphics (induction), the Enigma machine (counting), algorithmic bias (relations), differential privacy (probability), and paired kidney transplants (graphs). Pedagogical features include 'Why You Might Care' sections, quick-reference chapter guides and key terms and results summaries, problem-solving and writing tips, 'Taking it Further' asides with more technical details, and around 1700 exercises, 435 worked examples, and 480 figures.

Die vorliegenden Bände sind aus einer dreisemestrigen Einführungsvorlesung für Informatiker an der TU Wien entstanden, in der die wichtigsten Grund­ lagen aus den Gebieten Lineare und Nichtlineare Algebra, Analysis und Diskrete Mathematik behandelt werden. Zusätzlich zu den Inhalten, die in den Mathematikgrundvorlesungen der klassischen Ingenieurfächer auftreten, bilden dabei die in den Computerwissenschaften besonders wichtigen Metho­ den aus Kombinatorik, Graphentheorie und der Algebra endlicher Körper Schwerpunkte. Bei der Ausarbeitung wurde der Stoff einerseits durch Fakten und Beweise ergänzt, die auf grund ihres Umfanges in der Vorlesung nicht gebracht werden können; andererseits wurde auch eine Vielzahl von durchge­ rechneten Beispielen in den Text aufgenommen, um das Verständnis und die Möglichkeit des Selbststudiums zu fördern. Neben Beispielen, in denen es um das direkte Anwenden mathematischer'Rezepte'geht, finden sich auch zahl­ reiche solche, in denen inhaltliche Beobachtungen wichtiger Art gemacht werden. Der Stil der Darstellung wurde nach Möglichkeit mathematisch exakt gehalten, ohne einen allzu abstrakten logischen Formalismus zu verwenden. Tiefgehende Fakten, deren Beweise über den Rahmen einer solchen einfüh­ renden Darstellung für Informatiker hinausgehen, werden ohne Beweis ange­ geben, die einfacher zu führenden Beweise jedoch vorgeführt, da auch der Ingenieurstudent aus dem Verstehen von Beweisideen viel Verständnis für die von ihm verwendeten mathematischen Methoden und deren Grenzen gewin­ nen kann. Aus dem Inhalt der 3 Bände großteils ausgespart blieben Methoden, denen üblicherweise eigene Vorlesungen gewidmet sind, wie Wahrscheinlich­ keitsrechnung und Statistik, Logik und Numerische Mathematik, da ihre Aufnahmeden Gesamtumfang bei weitem gesprengt hätte.

This textbook provides a "cheap ticket" to the design of algorithms for hard computing problems, Le., for problems for which no low-degree polynomial-time algorithms1 are known. It focuses on a systematic presentation of the fundamental concepts and algorithm design techniques. The presentation of these concepts and techniques starts with some fundamental informal ideas that are later consecutively specified in detail. The algorithms used to illustrate the application of these methods are chosen with respect to their simplicity and transparency rather than with respect to their quality (complexity and reliability).

Dieses Buch ist aus Vorlesungen hervorgegangen, die der Autor regelmäßig an der RWTH Aachen für Studenten der Mathematik und Informatik gehalten hat. Folgende Themen werden ausführlich behandelt: Bäume, Euler- und Hamiltonsche Graphen, Matching- und Faktortheorie, Überdeckungen, AbsorptionsmengeÄn, planare Graphen, Kanten- und Eckenfärbungen, mehrfacher Zusammenhang und Netzwerktheorie. Das Werk bietet eine moderne und exakte Einführung in die Theorie der endlichen Graphen und Digraphen, welche nahezu alle fundamentalen Begriffsbildungen und die wichtigsten klassischen Ergebnisse enthält. Neben neuen und kurzen Beweisen bekannter Resultate findet der Leser einige aktuelle Forschungsergebnisse, die in keinem anderen Lehrbuch zu finden sind. Darüber hinaus werden eine Vielzahl von graphentheoretischen Algorithmen vorgestellt, die hochinteressante Anwendungen in Wirtschaft, Technik und Naturwissenschaften haben. Das Buch setzt außer Vertrautheit mit Elementarmathematik (vollständige Induktion, elementare Kombinatorik, Matrizen und Determinanten) keine besonderen Kenntnisse voraus. Erscheinungsdatum: 10.04.1991

Problemlösen gilt als eine der Schlüsselqualifikationen des 21. Jahrhunderts. Es geht beim Problemlösen nicht nur darum, Informationen sinnvoll zu vernetzen, dynamisch in Beziehung zu setzen, Wahrscheinlichkeiten zu berechnen und eine Kette richtiger Entscheidungen zu treffen, sondern auch vielfach darum, eine Vielzahl an Außenkriterien zu berücksichtigen und ein entsprechendes „Weltwissen“ an den Tag zu legen. Dieses Buch soll die Frage beantworten, wie man zu einem guten Problemlöser / einer guten Problemlöserin werden kann bzw. warum bestimmte Personen bei der Lösung von Problemen erfolgreicher sind als andere. Nach einer umfassenden Zusammenstellung der Literatur zu diesem Thema werden Einflussgrößen auf das Problemlösen analysiert und miteinander abgeglichen und Ideen für den Unterricht in der Primarstufe und Sekundarstufe I präsentiert. Zudem werden verschiedene Arten des Unterrichts im Hinblick auf die Wirksamkeit für unterschiedliche Personengruppen diskutiert, dies vor dem Hintergrund, dass nicht nur Problemstellungen stark variieren sondern auch die Problemlöser/innen. Eine Handreichung mit Brettspielen, die die Kriterien des Problemlösens erfüllen, ist ebenfalls Teil dieses Buches. Letztendlich wird ein Modell vorgeschlagen, welches erfolgreiches Problemlösen vielschichtig zu erklären versucht.  In der 2. Auflage wurden alle Kapitel überarbeitet und an den neuesten Kenntnisstand zum Thema Problemlösen angepasst. Die Kapitel zur schulischen und häuslichen Förderung der Problemlösekompetenz wurden stark erweitert und es werden nun eine Vielzahl an neuen Aufgaben und Spielen präsentiert, die beim Fördern von Problemlösen eine wertvolle Unterstützung leisten können. Das Kapitel zur Kombinatorik wurde nach der Rückmeldung von Studierenden und Kollegen komplett umstrukturiert und ist nun verständlicher und übersichtlicher aufgebaut.

Aus Den Rezensionen Zur 2. Auflage: ?diese Von Einem Mathematiker Geschriebene ?einf?hrung In Die Kryptographie F?llt Sofort Durch Ihren Pr?zisen Und Logisch Einwandfreien Stil Auf. Die Verwendeten Begriffe (verschl?sselungsverfahren, Perfekte Sicherheit, Hashfunktion, ...) Werden Mathematisch Sauber Und Kurz Definiert, Und Es Wird Sehr Wohl Unterschieden, Was ?bewiesen Ist [...] Und Was ?allgemeiner Glaube Der Experten Ist. Trotz Dieser Mathematischen Genauigkeit Ist Dieses Buch Insbesondere F?r Anf?nger Und Speziell Auch F?r Nicht-mathematiker Sehr Verst?ndlich Gehalten. Das N?tige Mathematische Werkzeug (restklassenringe, Matrizen Und Lineare Abbildungen, Wahrscheinlichkeit, Erzeugung Von Primzahlen, Faktorisierungsverfahren, Ausblick Auf Elliptische Kurven) Wird Vor Der Jeweiligen Anwendung Vorgestellt. Dann Wird Der Themenkreis Pr?gnant, Sehr Verst?ndlich Und Mit Beispielen Illustriert, Besprochen, Wobei Der Ge?bte Die Dahinterliegende Mathematische Idee Schnell Erkennen Kann. Dabei Werden Im Wesentlichen Alle Basistechniken Der Modernen Kryptographie Erfasst. Soweit M?glich Bzw. ?blich, Wird Das Englische Fachvokabular Der Kryptographie In Die Deutsche Sprache ?bersetzt. Ich Hoffe, Dass Die Freude Des Rezensenten Beim Lesen Dieses Buches Auch Von Vielen Anderen Lesern Ebenfalls Empfunden Werden Kann. (int. Math. Nachrichten 2002, Vol 56, Issue 189). ?[...] ?klein, Aber Fein Ist Das Charakteristikum Des Buches: Jeder Abschnitt Ist Mit (f?r Mathematiker) Sehr Anschaulichen Beispielen Und Kleinen ?bungsaufgaben Versehen. Entwickler Finden Hier Eine Reiche Quelle Der Hintergr?nde, Die Sie F?r Eine Implementation Wissen M?ssen. (amazon.de-redaktion)

This is the most comprehensive survey of the mathematical life of the legendary Paul Erdös, one of the most versatile and prolific mathematicians of our time. For the first time, all the main areas of Erdös' research are covered in a single project. Because of overwhelming response from the mathematical community, the project now occupies over 900 pages, arranged into two volumes. These volumes contain both high level research articles as well as "key" articles which survey some of the cornerstones of Erdös' work, each written by a leading world specialist in the field. A special chapter "Early Days", rare photographs, and art related to Erdös complement this striking collection. A unique contribution is the bibliography on Erdös' publications: the most comprehensive ever published.;Vol. I: Illustrations I -- Early days, II -- Number Theory. III. Randomness and Applications. Vol. II: IV. Combinatorics and Graph Theory -- V. Ramsey and Extremal Theory -- VI. Geometry -- VII. Infinity, Postscript.