The book is an introduction to the theory of convex polytopes and polyhedral sets, to algebraic geometry, and to the connections between these fields, known as the theory of toric varieties. The first part of the book covers the theory of polytopes and provides large parts of the mathematical background of linear optimization and of the geometrical aspects in computer science. The second part introduces toric varieties in an elementary way.

This work presents new and old constructions of nearrings. Links between properties of the multiplicative of nearrings (as regularity conditions and identities) and the structure of nearrings are studied. Primality and minimality properties of ideals are collected. Some types of simpler' nearrings are examined. Some nearrings of maps on a group are reviewed and linked with group-theoretical and geometrical questions.
Audience: Researchers working in nearring theory, group theory, semigroup theory, designs, and translation planes. Some of the material will be accessible to graduate students.

What is the "archetypal" image that comes to mind when one thinks of an infinite graph? What with a finite graph - when it is thought of as opposed to an infinite one? What structural elements are typical for either - by their presence or absence - yet provide a common ground for both? In planning the workshop on "Cycles and Rays" it had been intended from the outset to bring infinite graphs to the fore as much as possible. There never had been a graph theoretical meeting in which infinite graphs were more than "also rans", let alone one in which they were a central theme. In part, this is a matter of fashion, inasmuch as they are perceived as not readily lending themselves to applications, in part it is a matter of psychology stemming from the insecurity that many graph theorists feel in the face of set theory - on which infinite graph theory relies to a considerable extent. The result is that by and large, infinite graph theorists know what is happening in finite graphs but not conversely. Lack of knowledge about infinite graph theory can also be found in authoritative l sources. For example, a recent edition (1987) of a major mathematical encyclopaedia proposes to ". . . restrict [itself] to finite graphs, since only they give a typical theory". If anything, the reverse is true, and needless to say, the graph theoretical world knows better. One may wonder, however, by how much.

When? These are the proceedings of Finite Geometries, the Fourth Isle of Thorns Conference, which took place from Sunday 16 to Friday 21 July, 2000. It was organised by the editors of this volume. The Third Conference in 1990 was published as Advances in Finite Geometries and Designs by Oxford University Press and the Second Conference in 1980 was published as Finite Geometries and Designs by Cambridge University Press. The main speakers were A. R. Calderbank, P. J. Cameron, C. E. Praeger, B. Schmidt, H. Van Maldeghem. There were 64 participants and 42 contributions, all listed at the end of the volume. Conference web site http://www. maths. susx. ac. uk/Staff/JWPH/ Why? This collection of 21 articles describes the latest research and current state of the art in the following inter-linked areas: * combinatorial structures in finite projective and affine spaces, also known as Galois geometries, in which combinatorial objects such as blocking sets, spreads and partial spreads, ovoids, arcs and caps, as well as curves and hypersurfaces, are all of interest; * geometric and algebraic coding theory; * finite groups and incidence geometries, as in polar spaces, gener alized polygons and diagram geometries; * algebraic and geometric design theory, in particular designs which have interesting symmetric properties and difference sets, which play an important role, because of their close connections to both Galois geometry and coding theory.

This monograph extends this approach to the more general investigation of X-lattices, and these "tree lattices" are the main object of study. The authors present a coherent survey of the results on uniform tree lattices, and a (previously unpublished) development of the theory of non-uniform tree lattices, including some fundamental and recently proved existence theorems. Tree Lattices should be a helpful resource to researchers in the field, and may also be used for a graduate course on geometric methods in group theory.

This introductory text in graph theory focuses on partial cubes, which are graphs that are isometrically embeddable into hypercubes of an arbitrary dimension, as well as bipartite graphs, and cubical graphs. Currently, "Graphs and Cubes" is the only book available on the market that presents a comprehensive coverage of cubical graph and partial cube theories. Many exercises, along with historical notes, are included at the end of every chapter, and readers are encouraged to explore the exercises fully, and use them as a basis for research projects.

The prerequisites for this text include familiarity with basic mathematical concepts and methods on the level of undergraduate courses in discrete mathematics, linear algebra, group theory, and topology of Euclidean spaces. While the book is intended for lower-division graduate students in mathematics, it will be of interest to a much wider audience; because of their rich structural properties, partial cubes appear in theoretical computer science, coding theory, genetics, and even the political and social sciences."

Combinatorics is an area of mathematics involving an impressive breadth of ideas, and it encompasses topics ranging from codes and circuit design to algorithmic complexity and algebraic graph theory. In a highly distinguished career Bela Bollobas has made, and continues to make, many significant contributions to combinatorics, and this volume reflects the wide range of topics on which his work has had a major influence. It arises from a conference organized to mark his 60th birthday and the thirty-one articles contained here are of the highest calibre. That so many excellent mathematicians have contributed is testament to the very high regard in which Bela Bollobas is held. Students and researchers across combinatorics and related fields will find that this volume provides a wealth of insight to the state of the art.

From the reviews: "This is a textbook in cryptography with emphasis on algebraic methods. It is supported by many exercises (with answers) making it appropriate for a course in mathematics or computer science. ...] Overall, this is an excellent expository text, and will be very useful to both the student and researcher." Mathematical Reviews

Recent progress in research, teaching and communication has arisen from the use of new tools in visualization. To be fruitful, visualization needs precision and beauty. This book is a source of mathematical illustrations by mathematicians as well as artists. It offers examples in many basic mathematical fields including polyhedra theory, group theory, solving polynomial equations, dynamical systems and differential topology.
For a long time, arts, architecture, music and painting have been the source of new developments in mathematics. And vice versa, artists have often found new techniques, themes and inspiration within mathematics. Here, while mathematicians provide mathematical tools for the analysis of musical creations, the contributions from sculptors emphasize the role of mathematics in their work.

This classic geometry text explores the theory of 3-dimensional convex polyhedra in a unique fashion, with exceptional detail. Vital and clearly written, the book includes the basics of convex polyhedra and collects the most general existence theorems for convex polyhedra that are proved by a new and unified method. This edition includes a comprehensive bibliography by V.A. Zalgaller, and related papers as supplements to the original text.

Graph algorithms are easy to visualize and indeed there already exists a variety of packages to animate the dynamics when solving problems from graph theory. Still it can be difficult to understand the ideas behind the algorithm from the dynamic display alone.

CATBox consists of a software system for animating graph algorithms and a course book which we developed simultaneously. The software system presents both the algorithm and the graph and puts the user always in control of the actual code that is executed. In the course book, intended for readers at advanced undergraduate or graduate level, computer exercises and examples replace the usual static pictures of algorithm dynamics.

For this volume we have chosen solely algorithms for classical problems from combinatorial optimization, such as minimum spanning trees, shortest paths, maximum flows, minimum cost flows, weighted and unweighted matchings both for bipartite and non-bipartite graphs.

Find more information at http: //schliep.org/CATBox/.

Parallel and distributed computation has been gaining a great lot of attention in the last decades. During this period, the advances attained in computing and communication technologies, and the reduction in the costs of those technolo gies, played a central role in the rapid growth of the interest in the use of parallel and distributed computation in a number of areas of engineering and sciences. Many actual applications have been successfully implemented in various plat forms varying from pure shared-memory to totally distributed models, passing through hybrid approaches such as distributed-shared memory architectures. Parallel and distributed computation differs from dassical sequential compu tation in some of the following major aspects: the number of processing units, independent local dock for each unit, the number of memory units, and the programming model. For representing this diversity, and depending on what level we are looking at the problem, researchers have proposed some models to abstract the main characteristics or parameters (physical components or logical mechanisms) of parallel computers. The problem of establishing a suitable model is to find a reasonable trade-off among simplicity, power of expression and universality. Then, be able to study and analyze more precisely the behavior of parallel applications."

This solid volume discusses all the key topics in detail, including classification, orbit structure, representations, universal constructions, and abstract analogues. Open problems are discussed as they arise and many useful exercises are included.

One of the greatest scientific challenges of the 21st century is how to master, organize and extract useful knowledge from the overwhelming flow of information made available by today 's data acquisition systems and computing resources. Visualization is the premium means of taking up this challenge. This book is based on selected lectures given by leading experts in scientific visualization during a workshop held at Schloss Dagstuhl, Germany. Topics include user issues in visualization, large data visualization, unstructured mesh processing for visualization, volumetric visualization, flow visualization, medical visualization and visualization systems. The book contains more than 350 color illustrations.

Combinatorics and Matrix Theory have a symbiotic, or mutually beneficial, relationship. This relationship is discussed in my paper The symbiotic relationship of combinatorics and matrix theoryl where I attempted to justify this description. One could say that a more detailed justification was given in my book with H. J. Ryser entitled Combinatorial Matrix Theon? where an attempt was made to give a broad picture of the use of combinatorial ideas in matrix theory and the use of matrix theory in proving theorems which, at least on the surface, are combinatorial in nature. In the book by Liu and Lai, this picture is enlarged and expanded to include recent developments and contributions of Chinese mathematicians, many of which have not been readily available to those of us who are unfamiliar with Chinese journals. Necessarily, there is some overlap with the book Combinatorial Matrix Theory. Some of the additional topics include: spectra of graphs, eulerian graph problems, Shannon capacity, generalized inverses of Boolean matrices, matrix rearrangements, and matrix completions. A topic to which many Chinese mathematicians have made substantial contributions is the combinatorial analysis of powers of nonnegative matrices, and a large chapter is devoted to this topic. This book should be a valuable resource for mathematicians working in the area of combinatorial matrix theory. Richard A. Brualdi University of Wisconsin - Madison 1 Linear Alg. Applies., vols. 162-4, 1992, 65-105 2Camhridge University Press, 1991.

Computers have stretched the limits of what is possible in mathematics. More: they have given rise to new fields of mathematical study; the analysis of new and traditional algorithms, the creation of new paradigms for implementing computational methods, the viewing of old techniques from a concrete algorithmic vantage point, to name but a few. Computational Algebra and Number Theory lies at the lively intersection of computer science and mathematics. It highlights the surprising width and depth of the field through examples drawn from current activity, ranging from category theory, graph theory and combinatorics, to more classical computational areas, such as group theory and number theory. Many of the papers in the book provide a survey of their topic, as well as a description of present research. Throughout the variety of mathematical and computational fields represented, the emphasis is placed on the common principles and the methods employed. Audience: Students, experts, and those performing current research in any of the topics mentioned above.

This book introduces a new theory in Computer Vision yielding elementary techniques to analyze digital images. These techniques are a mathematical formalization of the Gestalt theory. From the mathematical viewpoint the closest field to it is stochastic geometry, involving basic probability and statistics, in the context of image analysis. The book is mathematically self-contained, needing only basic understanding of probability and calculus. The text includes more than 130 illustrations, and numerous examples based on specific images on which the theory is tested. Detailed exercises at the end of each chapter help the reader develop a firm understanding of the concepts imparted.

An in-depth look at soft computing methods and their applications in the human sciences, such as the social and the behavioral sciences. Soft computing methods - including fuzzy systems, neural networks, evolutionary computing and probabilistic reasoning - are state-of-the-art methods in theory formation and model construction. The powerful application areas of these methods in the human sciences are demonstrated, including the replacement of statistical models by simpler numerical or linguistic soft computing models and the use of computer simulations with approximate and linguistic constituents.

"Dr. Niskanen's work opens new vistas in application of soft computing, fuzzy logic and fuzzy set theory to the human sciences. This book is likely to be viewed in retrospect as a landmark in its field"
(Lotfi A. Zadeh, Berkeley)

Here is an accessible, algorithmically oriented guide to some of the most interesting techniques of complexity theory. The book shows that simple algorithms are at the heart of complexity theory. The book is organized by technique rather than by topic. Each chapter focuses on one technique: what it is, and what results and applications it yields.

This informative and exhaustive study gives a problem-solving approach to the difficult subject of analytic number theory. It is primarily aimed at graduate students and senior undergraduates. The goal is to provide a rapid introduction to analytic methods and the ways in which they are used to study the distribution of prime numbers. The book also includes an introduction to p-adic analytic methods. It is ideal for a first course in analytic number theory. The new edition has been completely rewritten, errors have been corrected, and there is a new chapter on the arithmetic progression of primes.

The need for a comprehensive survey-type exposition on formal languages and related mainstream areas of computer science has been evident for some years. In the early 1970s, when the book Formal Languages by the second mentioned editor appeared, it was still quite feasible to write a comprehensive book with that title and include also topics of current research interest. This would not be possible anymore. A standard-sized book on formal languages would either have to stay on a fairly low level or else be specialized and restricted to some narrow sector of the field. The setup becomes drastically different in a collection of contributions, where the best authorities in the world join forces, each of them concentrat ing on their own areas of specialization. The present three-volume Handbook constitutes such a unique collection. In these three volumes we present the current state of the art in formallanguage theory. We were most satisfied with the enthusiastic response given to our request for contributions by specialists representing various subfields. The need for a Handbook of Formal Languages was in many answers expressed in different ways: as an easily accessible his torical reference, a general source of information, an overall course-aid, and a compact collection of material for self-study. We are convinced that the final result will satisfy such various needs."

The primary goal of this book is unifying and making more widely accessible the vibrant stream of research - spanning more than two decades - on the theory of semi-feasible algorithms. In doing so it demonstrates the richness inherent in central notions of complexity: running time, nonuniform complexity, lowness, and NP-hardness. The book requires neither great mathematical maturity nor an extensive background in computational complexity theory or in computer science. Another aim of this book is to lay out a path along which the reader can quickly reach the frontiers of current research, and meet and engage the many exciting open problems in this area.

It is not an exaggeration to view Professor Lee's book," Software Engineer ing with Computational Intelligence," or SECI for short, as a pioneering contribution to software engineering. Breaking with the tradition of treat ing uncertainty, imprecision, fuzziness and vagueness as issues of peripheral importance, SECI moves them much closer to the center of the stage. It is ob vious, though still not widely accepted, that this is where these issues should be, since the real world is much too complex and much too ill-defined to lend itself to categorical analysis in the Cartesian spirit. As its title suggests, SECI employs the machineries of computational intel ligence (CI) and, more or less equivalently, soft computing (SC), to deal with the foundations and principal issues in software engineering. Basically, CI and SC are consortia of methodologies which collectively provide a body of con cepts and techniques for conception, design, construction and utilization of intelligent systems. The principal constituents of CI and SC are fuzzy logic, neurocomputing, evolutionary computing, probabilistic computing, chaotic computing and machine learning. The leitmotif of CI and SC is that, in general, better performance can be achieved by employing the constituent methodologies of CI and SC in combination rat her than in a stand-alone mode. In what follows, I will take the liberty of focusing my attention on fuzzy logic and fuzzy set theory, and on their roles in software engineering. But first, a couple of points of semantics which are in need of clarification."

These notes were first used in an introductory course team taught by the authors at Appalachian State University to advanced undergraduates and beginning graduates. The text was written with four pedagogical goals in mind: offer a variety of topics in one course, get to the main themes and tools as efficiently as possible, show the relationships between the different topics, and include recent results to convince students that mathematics is a living discipline.

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.