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This book treats the interaction between discrete convex geometry, commutative ring theory, algebraic K-theory, and algebraic geometry. The basic mathematical objects are lattice polytopes, rational cones, affine monoids, the algebras derived from them, and toric varieties. The book discusses several properties and invariants of these objects, such as efficient generation, unimodular triangulations and covers, basic theory of monoid rings, isomorphism problems and automorphism groups, homological properties and enumerative combinatorics. The last part is an extensive treatment of the K-theory of monoid rings, with extensions to toric varieties and their intersection theory. This monograph has been written with a view towards graduate students and researchers who want to study the cross-connections of algebra and discrete convex geometry. While the text has been written from an algebraist's view point, also specialists in lattice polytopes and related objects will find an up-to-date discussion of affine monoids and their combinatorial structure. Though the authors do not explicitly formulate algorithms, the book takes a constructive approach wherever possible. Winfried Bruns is Professor of Mathematics at Universität Osnabrück. Joseph Gubeladze is Professor of Mathematics at San Francisco State University.

K-theory. --- Polytopes. --- Rings (Algebra). --- Polytopes --- K-theory --- Rings (Algebra) --- Geometry --- Mathematics --- Physical Sciences & Mathematics --- Algebraic rings --- Ring theory --- Mathematics. --- Algebra. --- Commutative algebra. --- Commutative rings. --- Convex geometry. --- Discrete geometry. --- Commutative Rings and Algebras. --- K-Theory. --- Convex and Discrete Geometry. --- Algebraic topology --- Homology theory --- Combinatorial geometry --- Algebra --- Mathematical analysis --- Math --- Science --- Algebraic fields --- Hyperspace --- Topology --- Discrete groups. --- Groups, Discrete --- Infinite groups --- Discrete mathematics --- Convex geometry .

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Polycyclic groups are built from cyclic groups in a specific way. They arise in many contexts within group theory itself but also more generally in algebra, for example in the theory of Noetherian rings. They also touch on some aspects of topology, geometry and number theory. The first half of this book develops the standard group theoretic techniques for studying polycyclic groups and the basic properties of these groups. The second half then focuses specifically on the ring theoretic properties of polycyclic groups and their applications, often to purely group theoretic situations. The book is not intended to be encyclopedic. Instead, it is a study manual for graduate students and researchers coming into contact with polycyclic groups, where the main lines of the subject can be learned from scratch by any reader who has been exposed to some undergraduate algebra, especially groups, rings and vector spaces. Thus the book has been kept short and readable with a view that it can be read and worked through from cover to cover. At the end of each topic covered there is a description without proofs, but with full references, of further developments in the area. The book then concludes with an extensive bibliography of items relating to polycyclic groups.

Graph theory. --- Polycyclic groups. --- Rings (Algebra). --- Solvable groups. --- Polycyclic groups --- Solvable groups --- Graph theory --- Rings (Algebra) --- Mathematics --- Algebra --- Physical Sciences & Mathematics --- Algebraic rings --- Ring theory --- Graphs, Theory of --- Theory of graphs --- Extremal problems --- Mathematics. --- Associative rings. --- Commutative algebra. --- Commutative rings. --- Group theory. --- Group Theory and Generalizations. --- Associative Rings and Algebras. --- Commutative Rings and Algebras. --- Algebraic fields --- Combinatorial analysis --- Topology --- Infinite groups --- Nilpotent groups --- Algebra. --- Mathematical analysis --- Groups, Theory of --- Substitutions (Mathematics)

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This commemorative book contains the 28 major articles that appeared in the 2008 Twentieth Anniversary Issue of the journal Discrete & Computational Geometry, and presents a comprehensive picture of the current state of the field. Formed during the past few decades by the merger of the classical discipline of combinatorial and discrete geometry with the new field of computational geometry that sprang up in the 1970s, discrete and computational geometry now claims the allegiance of a sizeable number of mathematicians and computer scientists all over the world, whose most important work has been appearing since 1986 in the pages of the journal. The articles in this volume, a number of which solve long-outstanding problems in the field, were chosen by the editors of DCG for the importance of their results, for the breadth of their scope, and to show the intimate connections that have arisen between discrete and computational geometry and other areas of both computer science and mathematics. Apart from the articles, the editors present an expanded preface, along with a set of photographs of groups and individuals who have played a major role in the history of the field during the past twenty years. Contributors include: E. Ackerman P.K. Agarwal I. Aliev I. Bárány A. Barvinok S. Basu L.J. Billera J.-D. Boissonnat C. Borcea E. Boros K. Borys B. Braun K. Buchin O. Cheong D. Cohen-Steiner M. Damian K. Elbassioni R. Flatland T. Gerken J.E. Goodman X. Goaoc P. Gronchi V. Gurvich S. Har-Peled J. Hershberger A. Holmsen S.K. Hsiao A. Hubard J. Jerónimo L. Khachiyan R. Klein C. Knauer S. Langerman J.-Y. Lee M. Longinetti E. Miller P. Morin U. Nagel E. Nevo P. Niyogi I. Novik J. O’Rourke J. Pach I. Pak M.J. Pelsmajer S. Petitjean F. Pfender R. Pinchasi R. Pollack J.S. Provan K. Przeslawski R.M. Richardson G. Rote M. Schaefer Y. Schreiber M. Sharir J.R. Shewchuk S. Smale B. Solomyak M. Soss D. Štefankovic G. Vegter V.H. Vu S. Weinberger L. Wu D. Yost H. Yu T. Zell.

Discrete geometry. --- Geometry --Data processing. --- Geometry --- Discrete geometry --- Mathematics --- Physical Sciences & Mathematics --- Data processing --- Data processing. --- Mathematics. --- Computer science --- Computer graphics. --- Algebraic geometry. --- Computer mathematics. --- Convex geometry. --- Discrete mathematics. --- Convex and Discrete Geometry. --- Discrete Mathematics. --- Algebraic Geometry. --- Discrete Mathematics in Computer Science. --- Computer Graphics. --- Computational Mathematics and Numerical Analysis. --- Combinatorial geometry --- Discrete groups. --- Geometry, algebraic. --- Computational complexity. --- Groups, Discrete --- Infinite groups --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Automatic drafting --- Graphic data processing --- Graphics, Computer --- Computer art --- Graphic arts --- Engineering graphics --- Image processing --- Complexity, Computational --- Machine theory --- Algebraic geometry --- Digital techniques --- Convex geometry . --- Computer science—Mathematics. --- Discrete mathematical structures --- Mathematical structures, Discrete --- Structures, Discrete mathematical --- Numerical analysis