Found inside – Page iThis book will provide a unique and invaluable source for researchers and graduate students. From the Reviews: "This book is definitely a milestone in the literature of integer programming and combinatorial optimization. Found inside – Page 24[2] J. A. De Loera, The many aspects of counting lattice points in polytopes, Mathematische Semesterberichte, 52(2005), No. 2, 175-195. [3] J. De Loeram, ... A self-contained introduction to the geometry of numbers. 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. "This book revives and vastly expands the classical theory of resultants and discriminants. Most of the main new results of the book have been published earlier in more than a dozen joint papers of the authors. For scientists, this text can be utilized as a self-contained tooling device. Found inside – Page 62[5] M. Brion and M. Vergne, Lattice points in simple polytopes, J. Amer. Math. ... Genera of algebraic varieties and counting lattice points, Bull. Amer. Found inside – Page 160Brion, M., Vergne, M. Lattice points in simple polytopes. ... R., Tauzer, J., Yoshida, R. Effective lattice point counting in rational convex polytopes. Found inside – Page 229This chapter is concerned with counting the lattice points in a convex polytope 'P. If the vertices of the polytope are lattice points themselves, ... Based on a graduate course at the Technische Universität, Berlin, these lectures present a wealth of material on the modern theory of convex polytopes. Found inside – Page 39417.4.3 RELATIONSHIPS TO COUNTING LATTICE POINTS Triangulations of polytopes can be used to enumerate lattice points in polytopes |Stagd]. This book constitutes the refereed proceedings of the 7th International Conference on Integer Programming and Combinatorial Optimization, IPCO'99, held in Graz, Austria, in June 1999. Found inside – Page 64Here is the fundamental theorem concerning the lattice-point count in an integral convex polytope. Theorem 3.8 (Ehrhart's theorem). Found insideUsing the combinatorial reciprocity theorems as a leitmotif, this book unfolds central ideas and techniques in enumerative and geometric combinatorics. Found insideThis volume provides accessible and self-contained research problems designed for undergraduate student projects, and simultaneously promotes the development of sustainable undergraduate research programs. Found insideThis book offers a self-contained introduction to tropical geometry, suitable as a course text for beginning graduate students. Proofs are provided for the main results, such as the Fundamental Theorem and the Structure Theorem. Found inside – Page 143... this useful parametrization makes it possible to interpret the Littlewood–Richardson coefficients as counting lattice points in polytopes, so that ideas ... The book will also be of interest to research mathematicians, who will find some results that are recent, some that are new, and many known results that are discussed from a new perspective. Found inside – Page 947A first set of results relates lattice point counting functions to the volume of an integer polytope. In 1899, G. Pick (according to Coxeter 1969) proved ... Found inside – Page 947A first set of results relates lattice point counting functions to the volume of an integer polytope. In 1899, G. Pick (according to Coxeter 1969) proved ... Found inside – Page 91Efficient counting of lattice points . A polynomial time algorithms for counting integer points in rational polytopes in fixed dimension is presented . Found inside – Page 788Given an 36-polytope P and a vectory e Q", is y e Iza(P)? The following ... The problem of counting lattice points of polytopes in fixed dimensions was ... Found inside – Page 375The many aspects of counting lattice points in polytopes. Math. Semesterber., 52(2):175–195, 2005. Jesús A. De Loera, Raymond Hemmecke, Jeremiah Tauzer, ... The Handbook of Discrete and Computational Geometry is intended as a reference book fully accessible to nonspecialists as well as specialists, covering all major aspects of both fields. Found inside – Page 611Barvinok, A.: Integer Points in Polyhedra. ... Los Alamitos, CA, (2012) De Loera, J. A.: The many aspects of counting lattice points in polytopes. Found inside – Page 190A lattice point or an integral point is a point in ZD. Counting lattice points inside polytopes is a fundamental and useful step in many mathematical ... Found inside – Page 27They use this to extend previous results on counting lattice points in polytopes , including a version of the Euler - Maclaurin formula for lattice polytopes . In ( 60 ) , Brion discusses the algebraic equivariant Chow groups of Edidin and Graham ... The text concludes with Stanley's theorem characterizing the numbers of simplicies in each dimension in a convex simplicial polytope. Found insideThese notes from Hans Rademacher's 1963 Hedrick Lectures have been gently polished and augmented by Emil Grosswald. Found insideThis volume consists of research papers and expository survey articles presented by the invited speakers of the Summer Workshop on Lattice Polytopes. Counting lattice points in a polytope is a classical problem in computer science and combinatorics. Richard Stanley's two-volume basic introduction to enumerative combinatorics has become the standard guide to the topic for students and experts alike. This book provides a comprehensive overview of major results, methods and ideas of convex and discrete geometry and its applications. Found inside – Page 381Beck, M.: Counting Lattice Points by means of the Residue Theorem. ... Beck, M., Diaz, R., Robins, S.: The Frobenius problem, rational polytopes, ... Found insideThis book constitutes the refereed proceedings of the 7th International Conference on Mathematical Aspects of Computer and Information Sciences, MACIS 2017, held in Vienna, Austria, in November 2017. This book presents these recent advances, particularly those that arise from algebraic geometry, commutative algebra, convex and discrete geometry, generating functions, and other tools normally considered outside of the standard curriculum ... This book studies the subdivisions and triangulations of polyhedral regions and point sets and presents the first comprehensive treatment of the theory of secondary polytopes and related topics. Found insideLattices are geometric objects that can be pictorially described as the set of intersection points of an infinite, regular n-dimensional grid. Found insideJ. Billera and C. W. Lee, 1980) and necessary (R. P. Stanley, 1980) go beyond the scope of the book. Prerequisites for reading the book are modest: standard linear algebra and elementary point set topology in [R1d will suffice. Found inside – Page 102In a different direction, for “cubical” polytopes P whose proper faces are ... we are interested in “measuring” a polytope P by counting the lattice points ... Found inside – Page 36(2008) have independently pointed out that there exists an established mathematical theory of counting lattice points in polytopes. This book examines interactions of polyhedral discrete geometry and algebra. Found inside – Page 23Beck, M.: Counting lattice points by means of the residue theorem. ... Beck, M., Pixton, D.: The ehrhart polynomial of the birkhoff polytope. Found inside – Page 283.1 Counting lattice points in polytopes A polytope P in Rk is a convex hull of a finite set of points. If these points have rational, respectively integral ... This is a self-contained exposition of several core aspects of the theory of rational polyhedra with a view towards algorithmic applications to efficient counting of integer points, a problem arising in many areas of pure and applied ... For scientists, this text can be utilized as a self-contained tooling device. The papers in this volume give a good snapshot of the ideas discussed at a Workshop on Polyhedral Computation held at the CRM in Montreal in October 2006 and, with one exception, the current state of affairs in this area. Found inside – Page 947A first set of results relates lattice point counting functions to the volume of an integer polytope. In 1899, G. Pick (according to Coxeter 1969) proved ... The Geometry of Geodesics It is for such people that this text is designed. To put it another way, we intend this as a book for beginners to learn from and not as a reference. This idea essentially determines the choice of material covered here. 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