Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra

ISBN-13: 9781441921192 / Angielski / Miękka / 2010 / 227 str.

Matthias Beck; Sinai Robins

Opis

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.

Recenzje

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)

Spis treści

Preface.- 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.- Finite Fourier Analysis.- Dedekind Sums.- The Decomposition of a Polytope into Its Cones.- Euler-MacLaurin Summation in Rd.- Solid Angles.- A Discrete Version of Green's Theorem Using Elliptic Functions.- Appendix A: Triangulations of Polytopes.- Appendix B: Hints for Selected Exercises.- References.- Index.- List of Symbols.-

Blurb

This much-anticipated textbook illuminates the field of discrete mathematics with examples, theory, and applications of the discrete volume of a polytope. The authors have weaved a unifying thread through basic yet deep ideas in discrete geometry, combinatorics, and number theory. Because there is no other book that puts together all of these ideas in one place, this text is truly a service to the mathematical community.

We encounter here a friendly invitation to the field of "counting integer points in polytopes," also known as Ehrhart theory, and its various connections to elementary finite Fourier analysis, generating functions, the Frobenius coin-exchange problem, solid angles, magic squares, Dedekind sums, computational geometry, and more. With 250 exercises and open problems, the reader feels like an active participant, and the authors' engaging style encourages such participation. The many compelling pictures that accompany the proofs and examples add to the inviting style.

For teachers, this text is ideally suited as a capstone course for undergraduate students or as a compelling text in discrete mathematical topics for beginning graduate students. For scientists, this text can be utilized as a quick tooling device, especially for those who want a self-contained, easy-to-read introduction to these topics.