Matthias Beck, Sinai Robins

Computing the Continuous Discretely (eBook, PDF)

Integer-Point Enumeration in Polyhedra

• Format: PDF

Produktbeschreibung

This richly illustrated textbook explores the amazing interaction between combinatorics, geometry, number theory, and analysis which arises in the interplay between polyhedra and lattices. Highly accessible to advanced undergraduates, as well as beginning graduate students, this second edition is perfect for a capstone course, and adds two new chapters, many new exercises, and updated open problems. For scientists, this text can be utilized as a self-contained tooling device.

The topics include a friendly invitation to Ehrhart's theory of counting lattice points in polytopes, finite Fourier analysis, the Frobenius coin-exchange problem, Dedekind sums, solid angles, Euler-Maclaurin summation for polytopes, computational geometry, magic squares, zonotopes, and more.

With more than 300 exercises and open research problems, the reader is an active participant, carried through diverse but tightly woven mathematical fields that are inspired by an innocently elementary question: What are the relationships between the continuous volume of a polytope and its discrete volume?

Reviews of the first edition:

"You owe it to yourself to pick up a copy of Computing the Continuous Discretely to read about a number of interesting problems in geometry, number theory, and combinatorics."

- MAA Reviews

"The book is written as an accessible and engaging textbook, with many examples, historical notes, pithy quotes, commentary integrating the mate

rial, exercises, open problems and an extensive bibliography."

- Zentralblatt MATH

"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."

- Mathematical Reviews

"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 and Robinshave written the perfect text for such a course."

- CHOICE

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Produktdetails

• Verlag: Springer New York
• Seitenzahl: 285
• Erscheinungstermin: 14. November 2015
• Englisch
• ISBN-13: 9781493929696
• Artikelnr.: 44616058

Autorenporträt

Matthias Beck is Professor of Mathematics at San Francisco State University. Sinai Robins is Associate Professor of Mathematics at Nanyang Technological University, Singapore.

Inhaltsangabe

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.- Zonotopes.- h-Polynomials and h*-Polynomials.- 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.

Rezensionen

"This book is an outstanding book on counting integer points of polytopes ... . The book contains lots of exercises with very helpful hints. Another essential feature of the book is a vast collection of open problems on different aspects of integer point counting and related areas. ... The book is reader-friendly written, self-contained and contains numerous beautiful illustrations. The reader is always accompanied with deep research jokes by famous researchers and valuable historical notes." (Oleg Karpenkov, zbMATH 1339.52002, 2016)

Reviews of the first edition:

"You owe it to yourself to pick up a copy of Computing the Continuous Discretely to read about a number of interesting problems in geometry, number theory, and combinatorics."

- MAA Reviews

"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."

- Zentralblatt MATH

"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."

- Mathematical Reviews

"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 and Robins have written the perfect text for such a course."

- CHOICE

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)