The mathematical theory of computation has given rise to two important ap proaches to the informal notion of "complexity" Kolmogorov complexity, usu ally a complexity measure for a single object such as a string, a sequence etc., measures the amount of information necessary to describe the object. Compu tational complexity, usually a complexity measure for a set of objects, measures the compuational resources necessary to recognize or produce elements of the set. The relation between these two complexity measures has been considered for more than two decades, and may interesting and deep observations have been obtained. In March 1990, the Symposium on Theory and Application of Minimal Length Encoding was held at Stanford University as a part of the AAAI 1990 Spring Symposium Series. Some sessions of the symposium were dedicated to Kolmogorov complexity and its relations to the computational complexity the ory, and excellent expository talks were given there. Feeling that, due to the importance of the material, some way should be found to share these talks with researchers in the computer science community, I asked the speakers of those sessions to write survey papers based on their talks in the symposium. In response, five speakers from the sessions contributed the papers which appear in this book."

A gentle introduction to the highly sophisticated world of discrete mathematics, Mathematical Problems and Proofs presents topics ranging from elementary definitions and theorems to advanced topics -- such as cardinal numbers, generating functions, properties of Fibonacci numbers, and Euclidean algorithm. This excellent primer illustrates more than 150 solutions and proofs, thoroughly explained in clear language. The generous historical references and anecdotes interspersed throughout the text create interesting intermissions that will fuel readers' eagerness to inquire further about the topics and some of our greatest mathematicians. The author guides readers through the process of solving enigmatic proofs and problems, and assists them in making the transition from problem solving to theorem proving. At once a requisite text and an enjoyable read, Mathematical Problems and Proofs is an excellent entree to discrete mathematics for advanced students interested in mathematics, engineering, and science.

This volume contains survey articles based on the invited lectures given at the Twenty-second British Combinatorial Conference, held in July 2009 at the University of St Andrews. This biennial conference is a well-established international event, with speakers from all over the world. By its nature this volume provides an up-to-date overview of current research activity in several areas of combinatorics, including graph theory, design theory and packing problems. Each article is clearly written and assumes little prior knowledge on the part of the reader. 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 the most recent developments. 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.

The use of topological ideas to explore various aspects of graph theory, and vice versa, is a fruitful area of research. There are links with other areas of mathematics, such as design theory and geometry, and increasingly with such areas as computer networks where symmetry is an important feature. Other books cover portions of the material here, but there are no other books with such a wide scope. This book contains fifteen expository chapters written by acknowledged international experts in the field. Their well-written contributions have been carefully edited to enhance readability and to standardize the chapter structure, terminology and notation throughout the book. To help the reader, there is an extensive introductory chapter that covers the basic background material in graph theory and the topology of surfaces. Each chapter concludes with an extensive list of references.

The theory of Petri nets is a part of computer science whose importance is increasingly acknowledged. Many papers and anthologies, whose subject matter is net theory and its applications, have appeared to date. There exist at least seven introductory textbooks on the theory. The present monograph augments this literature by offering a mathematical treatment of one of the central aspects of net theory: the modelling of concur rency by partially ordered sets. Occurrence nets - which are special nets as well as special partial orders - are proposed by net theory for this purpose. We study both the general properties of occurrence nets and their use in describing the concurrent behaviour of systems. Occurrence nets may be contrasted with a more language-oriented approach to the modelling of concurrency known as arbitrary interleaving. We will dis cuss some connections between these' two approaches. Other approaches based on partially ordered sets - such as the theory of traces, the theory of event structures and the theory of semi words - are not considered in this book, in spite of the strong links between them and net theory."

Food webs hold a central place in ecology. They describe which organisms feed on which others in natural habitats. This book describes recently discovered empirical regularities in real food webs: it proposes a novel theory unifying many of these regularities, as well as extensive empirical data. After a general introduction, reviewing the empirical and theoretical discoveries about food webs, the second portion of the book shows that community food webs obey several striking phenomenological regularities. Some of these unify, regardless of habitat. Others differentiate, showing that habitat significantly influences structure. The third portion of the book presents a theoretical analysis of some of the unifying empirical regularities. The fourth portion of the book presents 13 community food webs. Collected from scattered sources and carefully edited, they are the empirical basis for the results in the volume. The largest available set of data on community food webs provides a valuable foundation for future studies of community food webs. The book is intended for graduate students, teachers and researchers primarily in ecology. The theoretical portions of the book provide materials useful to teachers of applied combinatorics, in particular, random graphs. Researchers in random graphs will find here unsolved mathematical problems.

Ever since the discovery of the five platonic solids in ancient times, the study of symmetry and regularity has been one of the most fascinating aspects of mathematics. Quite often the arithmetical regularity properties of an object imply its uniqueness and the existence of many symmetries. This interplay between regularity and symmetry properties of graphs is the theme of this book. Starting from very elementary regularity properties, the concept of a distance-regular graph arises naturally as a common setting for regular graphs which are extremal in one sense or another. Several other important regular combinatorial structures are then shown to be equivalent to special families of distance-regular graphs. Other subjects of more general interest, such as regularity and extremal properties in graphs, association schemes, representations of graphs in euclidean space, groups and geometries of Lie type, groups acting on graphs, and codes are covered independently. Many new results and proofs and more than 750 references increase the encyclopaedic value of this book.

In the late forties, Mathematical Programming became a scientific discipline in its own right. Since then it has experienced a tremendous growth. Beginning with economic and military applications, it is now among the most important fields of applied mathematics with extensive use in engineering, natural sciences, economics, and biological sciences. The lively activity in this area is demonstrated by the fact that as early as 1949 the first "Symposium on Mathe- matical Programming" took place in Chicago. Since then mathematical programmers from all over the world have gath- ered at the intfrnational symposia of the Mathematical Programming Society roughly every three years to present their recent research, to exchange ideas with their colleagues and to learn about the latest developments in their own and related fields. In 1982, the XI. International Symposium on Mathematical Programming was held at the University of Bonn, W. Germany, from August 23 to 27. It was organized by the Institut fUr Okonometrie und Operations Re- search of the University of Bonn in collaboration with the Sonderforschungs- bereich 21 of the Deutsche Forschungsgemeinschaft. This volume constitutes part of the outgrowth of this symposium and docu- ments its scientific activities. Part I of the book contains information about the symposium, welcoming addresses, lists of committees and sponsors and a brief review about the Ful- kerson Prize and the Dantzig Prize which were awarded during the opening ceremony.

This book, dedicated to the memory of Gian-Carlo Rota, is the result of a collaborative effort by his friends, students and admirers. Rota was one of the great thinkers of our times, innovator in both mathematics and phenomenology. I feel moved, yet touched by a sense of sadness, in presenting this volume of work, despite the fear that I may be unworthy of the task that befalls me. Rota, both the scientist and the man, was marked by a generosity that knew no bounds. His ideas opened wide the horizons of fields of research, permitting an astonishing number of students from all over the globe to become enthusiastically involved. The contagious energy with which he demonstrated his tremendous mental capacity always proved fresh and inspiring. Beyond his renown as gifted scientist, what was particularly striking in Gian-Carlo Rota was his ability to appreciate the diverse intellectual capacities of those before him and to adapt his communications accordingly. This human sense, complemented by his acute appreciation of the importance of the individual, acted as a catalyst in bringing forth the very best in each one of his students. Whosoever was fortunate enough to enjoy Gian-Carlo Rota's longstanding friendship was most enriched by the experience, both mathematically and philosophically, and had occasion to appreciate son cote de bon vivant. The book opens with a heartfelt piece by Henry Crapo in which he meticulously pieces together what Gian-Carlo Rota's untimely demise has bequeathed to science.

1. Introduction . 1 2. Areas and Angles . . 6 3. Tessellations and Symmetry 14 4. The Postulate of Closest Approach 28 5. The Coexistence of Rotocenters 36 6. A Diophantine Equation and its Solutions 46 7. Enantiomorphy. . . . . . . . 57 8. Symmetry Elements in the Plane 77 9. Pentagonal Tessellations . 89 10. Hexagonal Tessellations 101 11. Dirichlet Domain 106 12. Points and Regions 116 13. A Look at Infinity . 122 14. An Irrational Number 128 15. The Notation of Calculus 137 16. Integrals and Logarithms 142 17. Growth Functions . . . 149 18. Sigmoids and the Seventh-year Trifurcation, a Metaphor 159 19. Dynamic Symmetry and Fibonacci Numbers 167 20. The Golden Triangle 179 21. Quasi Symmetry 193 Appendix I: Exercise in Glide Symmetry . 205 Appendix II: Construction of Logarithmic Spiral . 207 Bibliography . 210 Index . . . . . . . . . . . . . . . . . . . . 225 Concepts and Images is the result of twenty years of teaching at Harvard's Department of Visual and Environmental Studies in the Carpenter Center for the Visual Arts, a department devoted to turning out students articulate in images much as a language department teaches reading and expressing one self in words. It is a response to our students' requests for a "handout" and to l our colleagues' inquiries about the courses: Visual and Environmental Studies 175 (Introduction to Design Science), YES 176 (Synergetics, the Structure of Ordered Space), Studio Arts 125a (Design Science Workshop, Two-Dimension al), Studio Arts 125b (Design Science Workshop, Three-Dimensional),2 as well as my freshman seminars on Structure in Science and Art."

With the advent of digital computers more than half a century ago, - searchers working in a wide range of scienti?c disciplines have obtained an extremely powerful tool to pursue deep understanding of natural processes in physical, chemical, and biological systems. Computers pose a great ch- lenge to mathematical sciences, as the range of phenomena available for rigorous mathematical analysis has been enormously expanded, demanding the development of a new generation of mathematical tools. There is an explosive growth of new mathematical disciplines to satisfy this demand, in particular related to discrete mathematics. However, it can be argued that at large mathematics is yet to provide the essential breakthrough to meet the challenge. The required paradigm shift in our view should be compa- ble to the shift in scienti?c thinking provided by the Newtonian revolution over 300 years ago. Studies of large-scale random graphs and networks are critical for the progress, using methods of discrete mathematics, probabil- tic combinatorics, graph theory, and statistical physics. Recent advances in large scale random network studies are described in this handbook, which provides a signi?cant update and extension - yond the materials presented in the "Handbook of Graphs and Networks" published in 2003 by Wiley. The present volume puts special emphasis on large-scale networks and random processes, which deemed as crucial for - tureprogressinthe?eld. Theissuesrelatedtorandomgraphsandnetworks pose very di?cult mathematical questions.

Discrete probability theory and the theory of algorithms have become close partners over the last ten years, though the roots of this partnership go back much longer. The papers in this volume address the latest developments in this active field. They are from the IMA Workshops "Probability and Algorithms" and "The Finite Markov Chain Renaissance." They represent the current thinking of many of the world's leading experts in the field. Researchers and graduate students in probability, computer science, combinatorics, and optimization theory will all be interested in this collection of articles. The techniques developed and surveyed in this volume are still undergoing rapid development, and many of the articles of the collection offer an expositionally pleasant entree into a research area of growing importance.

How do you convey to your students, colleagues and friends some of the beauty of the kind of mathematics you are obsessed with? If you are a mathematician interested in finite or topological geometry and combinatorial designs, you could start by showing them some of the (400+) pictures in the "picture book". Pictures are what this book is all about; original pictures of everybody's favorite geometries such as configurations, projective planes and spaces, circle planes, generalized polygons, mathematical biplanes and other designs which capture much of the beauty, construction principles, particularities, substructures and interconnections of these geometries. The level of the text is suitable for advanced undergraduates and graduate students. Even if you are a mathematician who just wants some interesting reading you will enjoy the author's very original and comprehensive guided tour of small finite geometries and geometries on surfaces This guided tour includes lots of sterograms of the spatial models, games and puzzles and instructions on how to construct your own pictures and build some of the spatial models yourself.

Combinatory logic started as a programme in the foundation of mathematics and in an historical context at a time when such endeavours attracted the most gifted among the mathematicians. This small volume arose under quite differ ent circumstances, namely within the context of reworking the mathematical foundations of computer science. I have been very lucky in finding gifted students who agreed to work with me and chose, for their Ph. D. theses, subjects that arose from my own attempts 1 to create a coherent mathematical view of these foundations. The result of this collaborative work is presented here in the hope that it does justice to the individual contributor and that the reader has a chance of judging the work as a whole. E. Engeler ETH Zurich, April 1994 lCollected in Chapter III, An Algebraization of Algorithmics, in Algorithmic Properties of Structures, Selected Papers of Erwin Engeler, World Scientific PubJ. Co., Singapore, 1993, pp. 183-257. I Historical and Philosophical Background Erwin Engeler In the fall of 1928 a young American turned up at the Mathematical Institute of Gottingen, a mecca of mathematicians at the time; he was a young man with a dream and his name was H. B. Curry. He felt that he had the tools in hand with which to solve the problem of foundations of mathematics mice and for all. His was an approach that came to be called "formalist" and embodied that later became known as Combinatory Logic."

This fascinating look at combinatorial games, that is, games not involving chance or hidden information, offers updates on standard games such as Go and Hex, on impartial games such as Chomp and Wythoff's Nim, and on aspects of games with infinitesimal values, plus analyses of the complexity of some games and puzzles and surveys on algorithmic game theory, on playing to lose, and on coping with cycles. The volume is rounded out with an up-to-date bibliography by Fraenkel and, for readers eager to get their hands dirty, a list of unsolved problems by Guy and Nowakowski. Highlights include some of Siegel's groundbreaking work on loopy games, the unveiling by Friedman and Landsberg of the use of renormalization to give very intriguing results about Chomp, and Nakamura's 'Counting Liberties in Capturing Races of Go'. Like its predecessors, this book should be on the shelf of all serious games enthusiasts.

The main purpose of these lectures is first to briefly survey the fundamental con nection between the representation theory of the symmetric group Sn and the theory of symmetric functions and second to show how combinatorial methods that arise naturally in the theory of symmetric functions lead to efficient algorithms to express various prod ucts of representations of Sn in terms of sums of irreducible representations. That is, there is a basic isometry which maps the center of the group algebra of Sn, Z(Sn), to the space of homogeneous symmetric functions of degree n, An. This basic isometry is known as the Frobenius map, F. The Frobenius map allows us to reduce calculations involving characters of the symmetric group to calculations involving Schur functions. Now there is a very rich and beautiful theory of the combinatorics of symmetric functions that has been developed in recent years. The combinatorics of symmetric functions, then leads to a number of very efficient algorithms for expanding various products of Schur functions into a sum of Schur functions. Such expansions of products of Schur functions correspond via the Frobenius map to decomposing various products of irreducible representations of Sn into their irreducible components. In addition, the Schur functions are also the characters of the irreducible polynomial representations of the general linear group over the complex numbers GLn(C).

This book is written by a mathematician and a theoretical biologist who have arrived at a good mutual understanding and a well worked-out common notation. The reader need hardly be convinced of the necessity of such a mutual understanding, not only for the two investigators, but also for the sciences they represent. Like Moliere's hero, geneticists are gradually beginning to understand that, unknowingly, they have been speaking in the language of cybernetics. Mathematicians are unexpec tedly discovering that many past and present problems and methods of genetics can be naturally formulated in the language of graph theory. In this way a powerful abstract mathematical theory suddenly finds a productive application. Moreover, in its turn, such an application be gins to "feed" the mathematical theory by presenting it with a number of new problems. The reader may judge for himself the fruitfulness of such mutual interaction. At the same time several important circumstances need to be men tioned. The formalization and rigorous formulation given here embraces not only the older problems, known by geneticists for many decades (the construction of genetic maps, the analysis of complementation, etc. ), but also comparatively new problems: the construction of partial com plementation maps, phylogenetic trees of proteins, etc."

This volume is dedicated to the memory of Rudolf Ahlswede, who passed away in December 2010. The Festschrift contains 36 thoroughly refereed research papers from a memorial symposium, which took place in July 2011.

Thefour macro-topics of this workshop: theory of games and strategic planning; combinatorial group testing and database mining; computational biology and string matching; information coding and spreading and patrolling on networks; provide a comprehensive picture of the vision Rudolf Ahlswede put forward of a broad and systematic theory of search.

In January 1989 a Workshop on Algorithms, Word Problems and Classi- fication in Combinatorial Group Theory was held at MSRl. This was part of a year-long program on Geometry and Combinatorial Group Theory or- ganised by Adyan, Brown, Gersten and Stallings. The organisers of the workshop were G. Baumslag, F.B. Cannonito and C.F. Miller III. The pa- pers in this volume are an outgrowth of lectures at this conference. The first three papers are concerned with decision problems and the next two with finitely presented simple groups. These are followed by two papers dealing with combinatorial geometry and homology. The remaining papers are about automatic groups and related topics. Some of these papers are, in essence, announcements of new results. The complexity of some of them are such that neither the Editors nor the Reviewers feel that they can take responsibility for vouching for the completeness of the proofs involved. We wish to thank the staff at MSRl for their help in organising the workshop and this volume.

This IMA Volume in Mathematics and its Appllcations GRAPH THEORY AND SPARSE MATRIX COMPUTATION is based on the proceedings of a workshop that was an integraI part of the 1991- 92 IMA program on "Applied Linear AIgebra." The purpose of the workshop was to bring together people who work in sparse matrix computation with those who conduct research in applied graph theory and grl: l, ph algorithms, in order to foster active cross-fertilization. We are grateful to Richard Brualdi, George Cybenko, Alan Geo ge, Gene Golub, Mitchell Luskin, and Paul Van Dooren for planning and implementing the year-Iong program. We espeeially thank Alan George, John R. Gilbert, and Joseph W.H. Liu for organizing this workshop and editing the proceedings. The finaneial support of the National Science Foundation made the workshop possible. A vner Friedman Willard Miller. Jr. PREFACE When reality is modeled by computation, linear algebra is often the con nec tiori between the continuous physical world and the finite algorithmic one. Usually, the more detailed the model, the bigger the matrix, the better the answer. Efficiency demands that every possible advantage be exploited: sparse structure, advanced com puter architectures, efficient algorithms. Therefore sparse matrix computation knits together threads from linear algebra, parallei computing, data struetures, geometry, and both numerieal and discrete algorithms."

This volume represents the refereed proceedings of the "Sixth International Conference on Finite Fields and Applications (Fq6)" held in the city of Oaxaca, Mexico, between 22-26 May 200l. The conference was hosted by the Departmento do Matermiticas of the U niversidad Aut6noma Metropolitana- Iztapalapa, Nlexico. This event continued a series of biennial international conferences on Finite Fields and Applications, following earlier meetings at the University of Nevada at Las Vegas (USA) in August 1991 and August 1993, the University of Glasgow (Scotland) in July 1995, the University of Waterloo (Canada) in August 1997, and at the University of Augsburg (Ger- many) in August 1999. The Organizing Committee of Fq6 consisted of Dieter Jungnickel (University of Augsburg, Germany), Neal Koblitz (University of Washington, USA), Alfred }. lenezes (University of Waterloo, Canada), Gary Mullen (The Pennsylvania State University, USA), Harald Niederreiter (Na- tional University of Singapore, Singapore), Vera Pless (University of Illinois, USA), Carlos Renteria (lPN, Mexico). Henning Stichtenoth (Essen Univer- sity, Germany). and Horacia Tapia-Recillas, Chair (Universidad Aut6noma l'vIetropolitan-Iztapalapa. Mexico). The program of the conference consisted of four full days and one half day of sessions, with 7 invited plenary talks, close to 60 contributed talks, basic courses in finite fields. cryptography and coding theory and a series of lectures at local educational institutions. Finite fields have an inherently fascinating structure and they are im- portant tools in discrete mathematics.

The Probability Theory of Patterns and Runs has had a long and distinguished history, starting with the work of de Moivre in the 18th century and that of von Mises in the early 1920's, and continuing with the renewal-theoretic results in Feller's classic text An Introduction to Probability Theory and its Applications, Volume 1. It is worthwhile to note, in particular, that de Moivre, in the third edition of The Doctrine of Chances (1756, reprinted by Chelsea in 1967, pp. 254-259), provides the generating function for the waiting time for the appearance of k consecutive successes. During the 1940's, statisticians such as Mood, Wolfowitz, David and Mosteller studied the distribution theory, both exact and asymptotic, of run-related statistics, thereby laying the foundation for several exact run tests. In the last two decades or so, the theory has seen an impressive re-emergence, primarily due to important developments in Molecular Biology, but also due to related research thrusts in Reliability Theory, Distribution Theory, Combinatorics, and Statistics.

One service mathematics has rendered the 'Et moi, ..., si j'avait su comment en revenir, It has put common sense back je n'y serais point al e.' human race. Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded n- sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell o. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puterscience .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series."

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."

"La narraci6n literaria es la evocaci6n de las nostalgias. " ("Literary narration is the evocation of nostalgia. ") G. G. Marquez, interview in Puerta del Sol, VII, 4, 1996. A Personal Prehistory In 1972 I started cooperating with members of the Biodynamics Research Unit at the Mayo Clinic in Rochester, Minnesota, which was under the direction of Earl H. Wood. At that time, their ambitious (and eventually realized) dream was to build the Dynamic Spatial Reconstructor (DSR), a device capable of collecting data regarding the attenuation of X-rays through the human body fast enough for stop-action imaging the full extent of the beating heart inside the thorax. Such a device can be applied to study the dynamic processes of cardiopulmonary physiology, in a manner similar to the application of an ordinary cr (computerized tomography) scanner to observing stationary anatomy. The standard method of displaying the information produced by a cr scanner consists of showing two-dimensional images, corresponding to maps of the X-ray attenuation coefficient in slices through the body. (Since different tissue types attenuate X-rays differently, such maps provide a good visualization of what is in the body in those slices; bone - which attenuates X-rays a lot - appears white, air appears black, tumors typically appear less dark than the surrounding healthy tissue, etc. ) However, it seemed to me that this display mode would not be appropriate for the DSR.