This book is concerned with cardinal number valued functions defined for any Boolean algebra. Examples of such functions are independence, which assigns to each Boolean algebra the supremum of the cardinalities of its free subalgebras, and cellularity, which gives the supremum of cardinalities of sets of pairwise disjoint elements. Twenty-one such functions are studied in detail, and many more in passing. The questions considered are the behaviour of these functions under algebraic operations such as products, free products, ultraproducts, and their relationships to one another. Assuming familiarity with only the basics of Boolean algebras and set theory, through to simple infinite combinatorics and forcing, the book reviews current knowledge about these functions, giving complete proofs for most facts. A special feature of the book is the attention given to open problems, of which 97 are formulated. Based on Cardinal Functions on Boolean Algebras (1990) by the same author, the present work is nearly twice the size of the original work. It contains solutions to many of the open problems which are discussed in greater detail than before. Among the new topics considered are ultraproducts and Fedorchukís theorem, and there is a more complete treatment of the cellularity of free products. Diagrams at the end of the book summarize the relationships between the functions for many important classes of Boolean algebras, including tree algebras and superatomic algebras.

Fibonacci and Lucas sequences are “two shining stars in the vast array of integer sequences,” and because of their ubiquitousness, tendency to appear in quite unexpected and unrelated places, abundant applications, and intriguing properties, they have fascinated amateurs and mathematicians alike. However, Catalan numbers are even more fascinating. Like the North Star in the evening sky, they are a beautiful and bright light in the mathematical heavens. They continue to provide a fertile ground for number theorists, especially, Catalan enthusiasts and computer scientists. Since the publication of Euler's triangulation problem (1751) and Catalan's parenthesization problem (1838), over 400 articles and problems on Catalan numbers have appeared in various periodicals. As Martin Gardner noted, even though many amateurs and mathematicians may know the abc's of Catalan sequence, they may not be familiar with their myriad unexpected occurrences, delightful applications, properties, or the beautiful and surprising relationships among numerous examples. Like Fibonacci and Lucas numbers, Catalan numbers are also an excellent source of fun and excitement. They can be used to generate interesting dividends for students, such as intellectual curiosity, experimentation, pattern recognition, conjecturing, and problem-solving techniques. The central character in the nth Catalan number is the central binomial coefficient. So, Catalan numbers can be extracted from Pascal's triangle. In fact, there are a number of ways they can be read from Pascal's triangle; every one of them is described and exemplified. This brings Catalan numbers a step closer to number-theory enthusiasts, especially.

Srinivasa Ramanujan Aiyangar,Godfrey Harold Hardy,P. Veṅkatesvara Seshu Aiyar,Bertram Martin Wilson

American Mathematical Society
May 2000

Description

The influence of Ramanujan on number theory is without parallel in mathematics. His papers, problems and letters have spawned a remarkable number of later results by many different mathematicians. Here, his 37 published papers, most of his first two and last letters to Hardy, the famous 58 problems submitted to the Journal of the Indian Mathematical Society, and the commentary of the original editors (Hardy, Seshu Aiyar and Wilson) are reprinted again, after having been unavailable for some time. In this, the third printing of Ramanujan's collected papers, Bruce Berndt provides an annotated guide to Ramanujan's work and to the mathematics it inspired over the last three-quarters of a century. The historical development of ideas is traced in the commentary and by citations to the copious references. The editor has done the mathematical world a tremendous service that few others would be qualified to do.

This textbook thoroughly outlines combinatorial algorithms for generation, enumeration, and search. Topics include backtracking and heuristic search methods applied to various combinatorial structures, such as:·Combinations·Permutations·Graphs·Designs·Many classical areas are covered as well as new research topics not included in most existing texts, such as:·Group algorithms·Graph isomorphism·Hill-climbing·Heuristic search algorithms·This work serves as an exceptional textbook for a modern course in combinatorial algorithms, providing a unified and focused collection of recent topics of interest in the area. The authors, synthesizing material that can only be found scattered through many different sources, introduce the most important combinatorial algorithmic techniques - thus creating an accessible, comprehensive text that students of mathematics, electrical engineering, and computer science can understand without needing a prior course on combinatorics.

During the past few decades, the gradual merger of Discrete Geometry and the newer discipline of Computational Geometry has provided enormous impetus to mathematicians and computer scientists interested in geometric problems. This volume, which contains 32 papers on a broad range of topics of current interest in the field, is an outgrowth of that synergism. It includes surveys and research articles exploring geometric arrangements, polytopes, packing, covering, discrete convexity, geometric algorithms and their complexity, and the combinatorial complexity of geometric objects, particularly in low dimension. There are points of contact with many applied areas such as mathematical programming, visibility problems, kinetic data structures, and biochemistry, as well as with algebraic topology, geometric probability, real algebraic geometry, and combinatorics. ? Over 30 surveys and research papers by current leaders in the field ? It contains several articles on new research areas, such as Geometric Graph Theory and the Theory of Core Sets and their applications

During the past few decades, the gradual merger of Discrete Geometry and the newer discipline of Computational Geometry has provided enormous impetus to mathematicians and computer scientists interested in geometric problems. This 2005 volume, which contains 32 papers on a broad range of topics of current interest in the field, is an outgrowth of that synergism. It includes surveys and research articles exploring geometric arrangements, polytopes, packing, covering, discrete convexity, geometric algorithms and their complexity, and the combinatorial complexity of geometric objects, particularly in low dimension. There are points of contact with many applied areas such as mathematical programming, visibility problems, kinetic data structures, and biochemistry, as well as with algebraic topology, geometric probability, real algebraic geometry, and combinatorics.

Ergodic theory studies measure-preserving transformations of measure spaces. These objects are intrinsically infinite, and the notion of an individual point or of an orbit makes no sense. Still there are a variety of situations when a measure-preserving transformation (and its asymptotic behavior) can be well described as a limit of certain finite objects (periodic processes).

This text provides an introduction to the theory of convex polytopes and polyhedral sets, to algebraic geometry and to the fascinating connections between these fields: the theory of toric varieties (or torus embeddings). The fist part of the book contains an introduction to the theory of polytopes - one of the most important parts of classical geometry in n-dimensional Euclidean space. Since the discussion here is independent of any applications to algebraic geometry, it would also be suitable for a course in geometry. This part also 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, building on the concepts of combinatorial geometry introduced in the first part. Many of the general concepts of algebraic geometry arise in this treatment and can be dealt with concretely. This part of the book can thus serve for a one-semester introduction to algebraic geometry, with the first part serving as a reference for combinatorial geometry.