Computing the Continuous Discretely Integer-Point Enumeration in Polyhedra için kapak resmi
Başlık:
Computing the Continuous Discretely Integer-Point Enumeration in Polyhedra
Dil:
English
ISBN:
9780387461120
Yayın Bilgileri:
New York, NY : Springer New York, 2007.
Fiziksel Tanımlama:
XVIII, 227 p. online resource.
İçerik:
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.
Özet:
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. .
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Özet

Özet

This 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. We encounter here a friendly invitation to the field of "counting integer points in polytopes", 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.


İncelemeler 1

İnceleme Seç

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. This material reinforces and extends ideas from calculus (partial fractions), advanced calculus (Green's theorem), numerical analysis (Euler-Maclauren summation), analysis (Fourier transform, but in the finite setting), complex analysis (elliptic functions), combinatorics (generating functions), and even recreational mathematics (magic squares). Students will also likely meet for the first time some basic topics (Dedekind sums). But no hodgepodge; the book develops a consistent theme--the relation between the (continuous) volume of a polytope (a notion that generalizes polygon and polyhedron) and its discrete volume, namely, the number of integer points that lie inside it. With these familiar tools, the authors lead students to a host of results that will seem simultaneously concrete and magical and almost certainly unfamiliar. Summing Up: Highly recommended. General readers; lower-division undergraduates through faculty. D. V. Feldman University of New Hampshire