This book presents methods of solving problems in three areas of elementary combinatorial mathematics: classical combinatorics, combinatorial arithmetic, and combinatorial geometry. In each topic, brief theoretical discussions are immediately followed by carefully worked-out examples of increasing degrees of difficulty, and by exercises that range from routine to rather challenging. While this book emphasizes some methods that are not usually covered in beginning university courses, it nevertheless teaches techniques and skills that are useful not only in the specific topics covered here. There are approximately 310 examples and 650 exercises. Jiri Herman is the headmaster of a prestigious secondary school (Gymnazium) in Brno, Radan Kucera is Associate Professor of Mathematics at Masaryk University in Brno, and Jaromir Simsa is a researcher at the Mathematical Institute of the Academy of Sciences of the Czech Republic. The translator, Karl Dilcher, is Professor of Mathematics at Dalhousie University in Canada. This book can be seen as a continuation of the previous book by the same authors and also translated by Karl Dilcher, Equations and Inequalities: Elementary Problems and Theorems in Algebra and Number Theory (Springer-Verlag 2000).

Lattices are discrete subgroups of maximal rank in a Euclidean space. To each such geometrical object, we can attach a canonical sphere packing which, assuming some regularity, has a density. The question of estimating the highest possible density of a sphere packing in a given dimension is a fascinating and difficult problem: the answer is known only up to dimension 3. This book thus discusses a beautiful and central problem in mathematics, which involves geometry, number theory, coding theory and group theory, centering on the study of extreme lattices, i.e. those on which the density attains a local maximum, and on the so-called perfection property. Written by a leader in the field, it is closely related to, though disjoint in content from, the classic book by J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, published in the same series as vol. 290. Every chapter except the first and the last contains numerous exercises. For simplicity those chapters involving heavy computational methods contain only few exercises. It includes appendices on Semi-Simple Algebras and Quaternions and Strongly Perfect Lattices.

Invariant, or coordinate-free methods provide a natural framework for many geometric questions. Invariant Methods in Discrete and Computational Geometry provides a basic introduction to several aspects of invariant theory, including the supersymmetric algebra, the Grassmann-Cayler algebra, and Chow forms. It also presents a number of current research papers on invariant theory and its applications to problems in geometry, such as automated theorem proving and computer vision. Audience: Researchers studying mathematics, computers and robotics.

This book is a monograph on unitals embedded in ?nite projective planes. Unitals are an interesting structure found in square order projective planes, and numerous research articles constructing and discussing these structures have appeared in print. More importantly, there still are many open pr- lems, and this remains a fruitful area for Ph.D. dissertations. Unitals play an important role in ?nite geometry as well as in related areas of mathematics. For example, unitals play a parallel role to Baer s- planes when considering extreme values for the size of a blocking set in a square order projective plane (see Section 2.3). Moreover, unitals meet the upper bound for the number of absolute points of any polarity in a square order projective plane (see Section 1.5). From an applications point of view, the linear codes arising from unitals have excellent technical properties (see 2 Section 6.4). The automorphism group of the classical unitalH =H(2, q ) is 2-transitive on the points ofH, and so unitals are of interest in group theory. In the ?eld of algebraic geometry over ?nite ?elds, H is a maximal curve that contains the largest number of F -rational points with respect to its genus, 2 q as established by the Hasse-Weil boun

This book gives the complete classification of Moufang polygons, starting from first principles. In particular, it may serve as an introduction to the various important algebraic concepts which arise in this classification including alternative division rings, quadratic Jordan division algebras of degree three, pseudo-quadratic forms, BN-pairs and norm splittings of quadratic forms. This book also contains a new proof of the classification of irreducible spherical buildings of rank at least three based on the observation that all the irreducible rank two residues of such a building are Moufang polygons. In an appendix, the connection between spherical buildings and algebraic groups is recalled.

This ambitious exposition by Malik and Mordeson on the fuzzification of discrete structures not only supplies a solid basic text on this key topic, but also serves as a viable tool for learning basic fuzzy set concepts "from the ground up" due to its unusual lucidity of exposition. While the entire presentation of this book is in a completely traditional setting, with all propositions and theorems provided totally rigorous proofs, the readability of the presentation is not compromised in any way; in fact, the many ex cellently chosen examples illustrate the often tricky concepts the authors address. The book's specific topics - including fuzzy versions of decision trees, networks, graphs, automata, etc. - are so well presented, that it is clear that even those researchers not primarily interested in these topics will, after a cursory reading, choose to return to a more in-depth viewing of its pages. Naturally, when I come across such a well-written book, I not only think of how much better I could have written my co-authored monographs, but naturally, how this work, as distant as it seems to be from my own area of interest, could nevertheless connect with such. Before presenting the briefest of some ideas in this direction, let me state that my interest in fuzzy set theory (FST) has been, since about 1975, in connecting aspects of FST directly with corresponding probability concepts. One chief vehicle in carrying this out involves the concept of random sets."

This book presents a mathematical structure modeling a physical or biological system that can be in any of a number of states. Each state is characterized by a set of binary features, and differs from some other neighbor state or states by just one of those features. The book considers the evolution of such a system over time and analyzes such a structure from algebraic and probabilistic (stochastic) standpoints.

In 1993, the Federal Ministry of Research and Education of the Federal - public of Germany (BMBF) started the ?rst of now ?ve periods of funding mathematics for industry and services. To date, its e?orts have supported approximately 280 projects investigating complex problems in industry and services.Whereasstandardproblemscanbe solvedusing standardmathem- ical methods and software o? the shelf, complex problems arising from e.g. industrial research and developments require advanced and innovative ma- ematical approaches and methods. Therefore, the BMBF funding programme focuses on the transfer of the latest developments in mathematical research to industrial applications. This initiative has proved to be highly successful in promoting mathematical modelling, simulation and optimization in science and technology. Substantial contributions to the solution of complex problems have been made in several areas of industry and services. Results from the ?rst funding period were published in "Mathematik - Schlusseltechnologie fur die Zukunft, Verbundprojekte zwischen Universitat und Industrie" (K.-H. Ho?mann, W. Jager, T. Lohmann, H. Schunck (E- tors), Springer1996).Thesecondpublication"Mathematics-KeyTechnology for the Future, Joint Projects between Universities and Industry" (W. Jager, H.-J. Krebs (Editors), Springer 2003) covered the period 1997 to 2000. Both books were out of print shortly after publication. This volume presents the results from the BMBF's fourth funding period (2004to2007)andcontainsasimilarspectrumofindustrialandmathematical problems as described in the previous publications, but with one additional new topic in the funding programme: risk management in ?nance and ins- ance. Othertopicscoveredaremathematicalmodellingandnumericalsimulation inmicroelectronics, thin?lms, biochemicalreactionsandtransport, comput- aided medicine, transport, tra?c and energy."

This book provides a broad overview of the entire field of DNA computation, tracing its history and development. It contains detailed descriptions of all major theoretical models and experimental results to date and discusses potential future developments. It concludes by outlining the challenges currently faced by researchers in the field. This book will be a useful reference for researchers and students, as well as an accessible introduction for those new to the field.

This book is a concept-oriented treatment of the structure theory of association schemes. The generalization of Sylow 's group theoretic theorems to scheme theory arises as a consequence of arithmetical considerations about quotient schemes. The theory of Coxeter schemes (equivalent to the theory of buildings) emerges naturally and yields a purely algebraic proof of Tits main theorem on buildings of spherical type.

This book contains both a synthesis and mathematical analysis of a wide set of algorithms and theories whose aim is the automatic segmen tation of digital images as well as the understanding of visual perception. A common formalism for these theories and algorithms is obtained in a variational form. Thank to this formalization, mathematical questions about the soundness of algorithms can be raised and answered. Perception theory has to deal with the complex interaction between regions and "edges" (or boundaries) in an image: in the variational seg mentation energies, "edge" terms compete with "region" terms in a way which is supposed to impose regularity on both regions and boundaries. This fact was an experimental guess in perception phenomenology and computer vision until it was proposed as a mathematical conjecture by Mumford and Shah. The third part of the book presents a unified presentation of the evi dences in favour of the conjecture. It is proved that the competition of one-dimensional and two-dimensional energy terms in a variational for mulation cannot create fractal-like behaviour for the edges. The proof of regularity for the edges of a segmentation constantly involves con cepts from geometric measure theory, which proves to be central in im age processing theory. The second part of the book provides a fast and self-contained presentation of the classical theory of rectifiable sets (the "edges") and unrectifiable sets ("fractals")."

This book is an introduction to the two closely related subjects of quantum optics and quantum information. The book gives a simple, self-contained introduction to both subjects, while illustrating the physical principles of quantum information processing using quantum optical systems. To make the book accessible to those with backgrounds other than physics, the authors also include a brief review of quantum mechanics. Furthermore, some aspects of quantum information, for example those pertaining to recent experiments on cavity QED and quantum dots, are described here for the first time in book form.

This volume contains revised papers that were presented at the international workshop entitled Computational Methods for Algebraic Spline Surfaces ("COMPASS"), which was held from September 29 to October 3, 2003, at Schloss Weinberg, Kefermarkt (A- tria). The workshop was mainly devoted to approximate algebraic geometry and its - plications. The organizers wanted to emphasize the novel idea of approximate implici- zation, that has strengthened the existing link between CAD / CAGD (Computer Aided Geometric Design) and classical algebraic geometry. The existing methods for exact implicitization (i. e., for conversion from the parametric to an implicit representation of a curve or surface) require exact arithmetic and are too slow and too expensive for industrial use. Thus the duality of an implicit representation and a parametric repres- tation is only used for low degree algebraic surfaces such as planes, spheres, cylinders, cones and toroidal surfaces. On the other hand, this duality is a very useful tool for - veloping ef?cient algorithms. Approximate implicitization makes this duality available for general curves and surfaces. The traditional exact implicitization of parametric surfaces produce global rep- sentations, which are exact everywhere. The surface patches used in CAD, however, are always de?ned within a small box only; they are obtained for a bounded parameter domain (typically a rectangle, or - in the case of "trimmed" surface patches - a subset of a rectangle). Consequently, a globally exact representation is not really needed in practice."

Hungarian mathematics has always been known for discrete mathematics, including combinatorial number theory, set theory and recently random structures, and combinatorial geometry. The recent volume contains high level surveys on these topics with authors mostly being invited speakers for the conference "Horizons of Combinatorics" held in Balatonalmadi, Hungary in 2006. The collection gives an overview of recent trends and results in a large part of combinatorics and related topics.

Substantially revised, reorganised and updated, the second edition now comprises eighteen chapters, carefully arranged in a straightforward and logical manner, with many new results and open problems.

As well as covering the theoretical aspects of the subject, with detailed proofs of many important results, the authors present a number of algorithms, and whole chapters are devoted to topics such as branchings, feedback arc and vertex sets, connectivity augmentations, sparse subdigraphs with prescribed connectivity, and also packing, covering and decompositions of digraphs. Throughout the book, there is a strong focus on applications which include quantum mechanics, bioinformatics, embedded computing, and the travelling salesman problem.

Detailed indices and topic-oriented chapters ease navigation, and more than 650 exercises, 170 figures and 150 open problems are included to help immerse the reader in all aspects of the subject.

In the 20th century philosophy of mathematics has to a great extent been dominated by views developed during the so-called foundational crisis in the beginning of that century. These views have primarily focused on questions pertaining to the logical structure of mathematics and questions regarding the justi?cation and consistency of mathematics. Paradigmatic in this - spect is Hilbert's program which inherits from Frege and Russell the project to formalize all areas of ordinary mathematics and then adds the requi- ment of a proof, by epistemically privileged means (?nitistic reasoning), of the consistency of such formalized theories. While interest in modi?ed v- sions of the original foundational programs is still thriving, in the second part of the twentieth century several philosophers and historians of mat- matics have questioned whether such foundational programs could exhaust the realm of important philosophical problems to be raised about the nature of mathematics. Some have done so in open confrontation (and hostility) to the logically based analysis of mathematics which characterized the cl- sical foundational programs, while others (and many of the contributors to this book belong to this tradition) have only called for an extension of the range of questions and problems that should be raised in connection with an understanding of mathematics. The focus has turned thus to a consideration of what mathematicians are actually doing when they produce mathematics. Questions concerning concept-formation, understanding, heuristics, changes instyle of reasoning, the role of analogies and diagrams etc.

Since the publication of the first edition of our book, geometric algorithms and combinatorial optimization have kept growing at the same fast pace as before. Nevertheless, we do not feel that the ongoing research has made this book outdated. Rather, it seems that many of the new results build on the models, algorithms, and theorems presented here. For instance, the celebrated Dyer-Frieze-Kannan algorithm for approximating the volume of a convex body is based on the oracle model of convex bodies and uses the ellipsoid method as a preprocessing technique. The polynomial time equivalence of optimization, separation, and membership has become a commonly employed tool in the study of the complexity of combinatorial optimization problems and in the newly developing field of computational convexity. Implementations of the basis reduction algorithm can be found in various computer algebra software systems. On the other hand, several of the open problems discussed in the first edition are still unsolved. For example, there are still no combinatorial polynomial time algorithms known for minimizing a submodular function or finding a maximum clique in a perfect graph. Moreover, despite the success of the interior point methods for the solution of explicitly given linear programs there is still no method known that solves implicitly given linear programs, such as those described in this book, and that is both practically and theoretically efficient. In particular, it is not known how to adapt interior point methods to such linear programs.

Lnear prediction theory and the related algorithms have matured to the point where they now form an integral part of many real-world adaptive systems. When it is necessary to extract information from a random process, we are frequently faced with the problem of analyzing and solving special systems of linear equations. In the general case these systems are overdetermined and may be characterized by additional properties, such as update and shift-invariance properties. Usually, one employs exact or approximate least-squares methods to solve the resulting class of linear equations. Mainly during the last decade, researchers in various fields have contributed techniques and nomenclature for this type of least-squares problem. This body of methods now constitutes what we call the theory of linear prediction. The immense interest that it has aroused clearly emerges from recent advances in processor technology, which provide the means to implement linear prediction algorithms, and to operate them in real time. The practical effect is the occurrence of a new class of high-performance adaptive systems for control, communications and system identification applications. This monograph presumes a background in discrete-time digital signal processing, including Z-transforms, and a basic knowledge of discrete-time random processes. One of the difficulties I have en countered while writing this book is that many engineers and computer scientists lack knowledge of fundamental mathematics and geometry."

It is general consensus that Combinatorics has developed into a full-fledged mathematical discipline whose beginnings as a charming pastime have long since been left behind and whose great signifi cance for other branches of both pure and applied mathematics is only beginning to be realized. The last ten years have witnessed a tremendous outburst of activity both in relatively new fields such as Coding Theory and the Theory of Matroids as well as in' more time honored endeavors such as Generating Functions and the Inver sion Calculus. Although the number of text books on these subjects is slowly increasing, there is also a great need for up-to-date surveys of the main lines of research designed to aid the beginner and serve as a reference for the expert. It was the aim of the Advanced Study Institute "Higher Combinatorics" in Berlin, 1976, to help fulfill this need. There were five sections: I. Counting Theory, II. Combinatorial Set Theory and Order Theory, III. Matroids, IV. Designs and V. Groups and Coding Theory, with three principal lecturers in each section. Expanded versions of most lectures form the contents of this book. The Institute was designed to offer, especially to young researchers, a comprehen sive picture of the most interesting developments currently under way. It is hoped that these proceedings will serve the same purpose for a wider audience."

Graph Theory, Combinatorics and Algorithms: Interdisciplinary Applications focuses on discrete mathematics and combinatorial algorithms interacting with real world problems in computer science, operations research, applied mathematics and engineering. The book contains eleven chapters written by experts in their respective fields, and covers a wide spectrum of high-interest problems across these discipline domains. Among the contributing authors are Richard Karp of UC Berkeley and Robert Tarjan of Princeton; both are at the pinnacle of research scholarship in Graph Theory and Combinatorics. The chapters from the contributing authors focus on "real world" applications, all of which will be of considerable interest across the areas of Operations Research, Computer Science, Applied Mathematics, and Engineering. These problems include Internet congestion control, high-speed communication networks, multi-object auctions, resource allocation, software testing, data structures, etc. In sum, this is a book focused on major, contemporary problems, written by the top research scholars in the field, using cutting-edge mathematical and computational techniques.

This volume contains survey articles based on the invited lectures given at the Twenty-first British Combinatorial Conference, held in July 2007 at the University of Reading. This biennial conference is a well-established international event and the articles are of the high quality that befits the event. By its nature this volume provides an up-to-date overview of current research activity in several areas of combinatorics, ranging from graph theory to current applications of combinatorial mathematics, including efficient approximability of NP-hard optimization problems and cryptographic key management. The authors are some of the world's foremost researchers in their fields, and here they summarize existing results, and give a unique preview of work currently being written up. The book provides a valuable survey of the present state of knowledge in combinatorics. It will be useful to research workers and advanced graduate students, primarily in mathematics but also in computer science, statistics and engineering.

""Algorithms for Games"" aims to provide a concrete example of the programming of a two-person game with complete information, and to demonstrate some of the methods of solutions; to show the reader that it is profitable not to fear a search, but rather to undertake it in a rational fashion, make a proper estimate of the dimensions of the "catastrophe," and use all suitable means to keep it down to a reasonable size. The book is dedicated to the study of methods for limiting the extent of a search. The game programming problem is very well suited to the study of the search problem, and in general for multi-step solution processes. With this in mind, the book focuses on the programming of games as the best means of developing the ideas and methods presented. While many of the examples are related to chess, only an elementary knowledge of the game is needed.

This volume contains the texts of the principal survey papers presented at ALGORITHMS -and ORDER, held. at Ottawa, Canada from June 1 to June 12, 1987. The conference was supported by grants from the N.A.T.O. Advanced Study Institute programme, the University of Ottawa, and the Natural Sciences and Engineering Research Council of Canada. We are grateful for this considerable support. Over fifty years ago, the Symposium on Lattice Theory, in Charlottesville, U.S.A., proclaimed the vitality of ordered sets. Only twenty years later the Symposium on Partially Ordered Sets and Lattice Theory, held at Monterey, U.S.A., had solved many of the problems that had been originally posed. In 1981, the Symposium on Ordered Sets held at Banff, Canada, continued this tradition. It was marked by a landmark volume containing twenty-three articles on almost all current topics in the theory of ordered sets and its applications. Three years after, Graphs and Orders, also held at Banff, Canada, aimed to document the role of graphs in the theory of ordered sets and its applications. Because of its special place in the landscape of the mathematical sciences order is especially sensitive to new trends and developments. Today, the most important current in the theory and application of order springs from theoretical computer seience. Two themes of computer science lead the way. The first is data structure. Order is common to data structures."

Polynomial extremal problems (PEP) constitute one of the most important subclasses of nonlinear programming models. Their distinctive feature is that an objective function and constraints can be expressed by polynomial functions in one or several variables. Let: e = {: e 1, ...: en} be the vector in n-dimensional real linear space Rn; n PO(: e), PI (: e), ..., Pm (: e) are polynomial functions in R with real coefficients. In general, a PEP can be formulated in the following form: (0.1) find r = inf Po(: e) subject to constraints (0.2) Pi (: e) =0, i=l, ..., m (a constraint in the form of inequality can be written in the form of equality by introducing a new variable: for example, P( x) 0 is equivalent to P(: e) + y2 = 0). Boolean and mixed polynomial problems can be written in usual form by adding for each boolean variable z the equality: Z2 - Z = O. Let a = {al, ..., a } be integer vector with nonnegative entries {a;}f=l. n Denote by R a](: e) monomial in n variables of the form: n R a](: e) = IT: ef';;=1 d(a) = 2:7=1 ai is the total degree of monomial R a]. Each polynomial in n variables can be written as sum of monomials with nonzero coefficients: P(: e) = L caR a](: e), aEA{P) IX x Nondifferentiable optimization and polynomial problems where A(P) is the set of monomials contained in polynomial P

Intended for first- or second-year undergraduates, this introduction to discrete mathematics covers the usual topics of such a course, but applies constructivist principles that promote - indeed, require - active participation by the student. Working with the programming language ISETL, whose syntax is close to that of standard mathematical language, the student constructs the concepts in her or his mind as a result of constructing them on the computer in the syntax of ISETL. This dramatically different approach allows students to attempt to discover concepts in a "Socratic" dialog with the computer. The discussion avoids the formal "definition-theorem" approach and promotes active involvement by the reader by its questioning style. An instructor using this text can expect a lively class whose students develop a deep conceptual understanding rather than simply manipulative skills. Topics covered in this book include: the propositional calculus, operations on sets, basic counting methods, predicate calculus, relations, graphs, functions, and mathematical induction.