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Undergraduate Texts in Mathematics

Computing the Continuous Discretely

Integer-point Enumeration in Polyhedra

Authors: Beck, Matthias, Robins, Sinai

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  • The authors write with flair and have chosen a unique set of topics
  • Places a strong emphasis on computational techniques
  • Contains more than 200 exercises and has been heavily class-tested
  • Includes hints to selected exercises
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eBook $39.99
price for USA in USD (gross)
  • ISBN 978-0-387-46112-0
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $54.99
price for USA in USD
  • ISBN 978-1-4419-2119-2
  • Free shipping for individuals worldwide
  • Immediate ebook access, if available*, with your print order
  • Usually dispatched within 3 to 5 business days.
About this Textbook

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.

 

Reviews

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)


Table of contents (12 chapters)

Table of contents (12 chapters)
  • The Coin-Exchange Problem of Frobenius

    Pages 3-24

  • A Gallery of Discrete Volumes

    Pages 25-55

  • Counting Lattice Points in Polytopes:The Ehrhart Theory

    Pages 57-82

  • Reciprocity

    Pages 83-93

  • Face Numbers and the Dehn—Sommerville Relations in Ehrhartian Terms

    Pages 95-104

Buy this book

eBook $39.99
price for USA in USD (gross)
  • ISBN 978-0-387-46112-0
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $54.99
price for USA in USD
  • ISBN 978-1-4419-2119-2
  • Free shipping for individuals worldwide
  • Immediate ebook access, if available*, with your print order
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Computing the Continuous Discretely
Book Subtitle
Integer-point Enumeration in Polyhedra
Authors
Series Title
Undergraduate Texts in Mathematics
Copyright
2007
Publisher
Springer-Verlag New York
Copyright Holder
Springer-Verlag New York
eBook ISBN
978-0-387-46112-0
DOI
10.1007/978-0-387-46112-0
Softcover ISBN
978-1-4419-2119-2
Series ISSN
0172-6056
Edition Number
1
Number of Pages
XVIII, 227
Topics

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