Computing the Continuous Discretely

The Essentials of Discrete Volume Computations.- The Coin-Exchange Problem of Frobenius.- A Gallery of Discrete Volumes.- Counting Lattice Points in Polytopes:The Ehrhart Theory.- Reciprocity.- Face Numbers and the Dehn—Sommerville Relations in Ehrhartian Terms.- Magic Squares.- Beyond the Basics.- Finite Fourier Analysis.- Dedekind Sums, the Building Blocks of Lattice-point Enumeration.- The Decomposition of a Polytope into Its Cones.- Euler—Maclaurin Summation in ?d.- Solid Angles.- A Discrete Version of Green’s Theorem Using Elliptic Functions.

From the reviews:"You owe it to yourself to pick up a copy … to read about a number of interesting problems in geometry, number theory, and combinatorics … . Even people who are familiar with the material would almost certainly learn something from the clear and engaging exposition … . It contains a large number of exercises … . Each chapter also ends with a series of relevant open problems … . it is also full of mathematics that is self-contained and worth reading on its own." (Darren Glass, MathDL, February, 2007)"This beautiful book presents, at a level suitable for advanced undergraduates, a fairly complete introduction to the problem of counting lattice points inside a convex polyhedron. … Most importantly the book gives a complete presentation of the use of generating functions of various kinds to enumerate lattice points, as well as an introduction to the theory of Erhart quasipolynomials. … This book provides many well-crafted exercises, and even a list of open problems in each chapter." (Jesús A. De Loera, Mathematical Reviews, Issue 2007 h)"All mathematics majors study the topics they will need to know, should they want to go to graduate school. But most will not, and many departments recognize the need for capstone courses in which graduating students can see the tools they have acquired come together in some satisfying way. Beck (San Francisco State Univ.) and Robins (Temple Univ.) have written the perfect text for such a course. … Summing Up: Highly recommended. General readers; lower-division undergraduates through faculty." (D. V. Feldman, CHOICE, Vol. 45 (2), 2007)"This book is concerned with the mathematics of that connection between the discrete and the continuous, with significance for geometry, number theory and combinatorics. … The book is written as an accessible and engaging textbook, with many examples, historical notes, pithy quotes, commentary integrating the material, exercises, open problems and an extensive bibliography." (Margaret M. Bayer, Zentralblatt MATH, Vol. 1114 (16), 2007)"The main topic of the book is initiated by a theorem of Ehrhart … . This is a wonderful book for various readerships. Students, researchers, lecturers in enumeration, geometry and number theory all find it very pleasing and useful. The presentation is accessible for mature undergraduates. … it is a clear introduction to graduate students and researchers with many exercises and with a list of open problems at the end of each chapter." (Péter Hajnal, Acta Scientiarum Mathematicarum, Vol. 75, 2009)“The theme of this textbook … is the relation between the continuous and the discrete, namely, the interplay between the volume of a polytope and the number of grid points contained in it. … The text contains many exercises, some of which present material needed later (for these exercises hints are provided), and lists also many open research problems. – The book can be recommended both for its style and for its interesting … content.” (P. Schmitt, Monatshefte für Mathematik, Vol. 155 (2), October, 2008)