Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra

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Springer Science & Business Media, Nov 19, 2007 - Mathematics - 227 pages
The world is continuous, but the mind is discrete. David Mumford We seek to bridge some critical gaps between various ?elds of mathematics by studying the interplay between the continuous volume and the discrete v- ume of polytopes. Examples of polytopes in three dimensions include crystals, boxes, tetrahedra, and any convex object whose faces are all ?at. It is amusing to see how many problems in combinatorics, number theory, and many other mathematical areas can be recast in the language of polytopes that exist in some Euclidean space. Conversely, the versatile structure of polytopes gives us number-theoretic and combinatorial information that ?ows naturally from their geometry. Fig. 0. 1. Continuous and discrete volume. The discrete volume of a body P can be described intuitively as the number of grid points that lie inside P, given a ?xed grid in Euclidean space. The continuous volume of P has the usual intuitive meaning of volume that we attach to everyday objects we see in the real world. VIII Preface Indeed, the di?erence between the two realizations of volume can be thought of in physical terms as follows. On the one hand, the quant- level grid imposed by the molecular structure of reality gives us a discrete notion of space and hence discrete volume. On the other hand, the N- tonian notion of continuous space gives us the continuous volume.
 

Contents

Notes
48
Open Problems
54
Notes
83
Exercises
90
Exercises
103
Finite Fourier Analysis
123
Dedekind Sums the Building Blocks of Latticepoint
138
The Decomposition of a Polytope into Its Cones
155
EulerMaclaurin Summation in Rd 167
166
Solid Angles
179
A Discrete Version of Greens Theorem Using Elliptic
191
A Vertex and Hyperplane Descriptions of Polytopes
199
Hints for Exercises
209
References
217
List of Symbols
227
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