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E-raamat: Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra

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Computing the Continuous Discretely: Integer-Point Enumeration in PolyhedraSuurem pilt
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IntegerPoint Enumeration in Polyhedra.

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 Robins have written the perfect text for such a course.”

— CHOICE

Arvustused

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

Part I The Essentials of Discrete Volume Computations
1 The Coin-Exchange Problem of Frobenius
3(24)
1.1 Why Use Generating Functions?
3(2)
1.2 Two Coins
5(3)
1.3 Partial Fractions and a Surprising Formula
8(4)
1.4 Sylvester's Result
12(2)
1.5 Three and More Coins
14(13)
Notes
16(3)
Exercises
19(7)
Open Problems
26(1)
2 A Gallery of Discrete Volumes
27(32)
2.1 The Language of Polytopes
27(2)
2.2 The Unit Cube
29(2)
2.3 The Standard Simplex
31(3)
2.4 The Bernoulli Polynomials as Lattice-Point Enumerators of Pyramids
34(4)
2.5 The Lattice-Point Enumerators of the Cross-Polytopes
38(2)
2.6 Pick's Theorem
40(3)
2.7 Polygons with Rational Vertices
43(5)
2.8 Euler's Generating Function for General Rational Polytopes
48(11)
Notes
51(2)
Exercises
53(5)
Open Problems
58(1)
3 Counting Lattice Points in Polytopes: The Ehrhart Theory
59(30)
3.1 Triangulations
59(3)
3.2 Cones
62(2)
3.3 Integer-Point Transforms for Rational Cones
64(4)
3.4 Expanding and Counting Using Ehrhart's Original Approach
68(3)
3.5 The Ehrhart Series of an Integral Polytope
71(5)
3.6 From the Discrete to the Continuous Volume of a Polytope
76(2)
3.7 Interpolation
78(2)
3.8 Rational Polytopes and Ehrhart Quasipolynomials
80(1)
3.9 Reflections on the Coin-Exchange Problem and the Gallery of
Chapter 2
81(8)
Notes
81(1)
Exercises
82(6)
Open Problems
88(1)
4 Reciprocity
89(12)
4.1 Generating Functions for Somewhat Irrational Cones
90(2)
4.2 Stanley's Reciprocity Theorem for Rational Cones
92(1)
4.3 Ehrhart-Macdonald Reciprocity for Rational Polytopes
93(1)
4.4 The Ehrhart Series of Reflexive Polytopes
94(2)
4.5 More "Reflections" on the Coin-Exchange Problem and the Gallery of
Chapter 2
96(5)
Notes
96(2)
Exercises
98(2)
Open Problems
100(1)
5 Face Numbers and the Dehn-Sommerville Relations in Ehrhartian Terms
101(12)
5.1 Face It!
101(2)
5.2 Dehn-Sommerville Extended
103(2)
5.3 Applications to the Coefficients of an Ehrhart Polynomial
105(1)
5.4 Relative Volume
106(7)
Notes
108(1)
Exercises
109(4)
6 Magic Squares
113(20)
6.1 It's a Kind of Magic
114(1)
6.2 Semimagic Squares: Points in the Birkhoff-von Neumann Polytope
115(4)
6.3 Magic Generating Functions and Constant-Term Identities
119(5)
6.4 The Enumeration of Magic Squares
124(9)
Notes
125(2)
Exercises
127(1)
Open Problems
128(5)
Part II Beyond the Basics
7 Finite Fourier Analysis
133(16)
7.1 A Motivating Example
133(2)
7.2 Finite Fourier Series for Periodic Functions on Z
135(4)
7.3 The Finite Fourier Transform and Its Properties
139(2)
7.4 The Parseval Identity
141(2)
7.5 The Convolution of Finite Fourier Series
143(6)
Notes
145(1)
Exercises
146(3)
8 Dedekind Sums, the Building Blocks of Lattice-Point Enumeration
149(18)
8.1 Fourier-Dedekind Sums and the Coin-Exchange Problem Revisited
149(4)
8.2 The Dedekind Sum and Its Reciprocity and Computational Complexity
153(1)
8.3 Rademacher Reciprocity for the Fourier-Dedekind Sum
154(4)
8.4 The Mordell-Pommersheim Tetrahedron
158(9)
Notes
160(1)
Exercises
161(3)
Open Problems
164(3)
9 Zonotopes
167(16)
9.1 Definitions and Examples
167(3)
9.2 Paving a Zonotope
170(2)
9.3 The Permutahedron
172(4)
9.4 The Ehrhart Polynomial of a Zonotope
176(7)
Notes
178(1)
Exercises
179(2)
Open Problems
181(2)
10 h-Polynomials and h*-Polynomials
183(16)
10.1 Simplicial Polytopes and (Unimodular) Triangulations
183(4)
10.2 Fundamental Parallelepipeds Open Up, with an h-Twist
187(2)
10.3 Palindromic Decompositions of h*-Polynomials
189(2)
10.4 Inequalities for h*-Polynomials
191(8)
Notes
192(1)
Exercises
193(3)
Open Problems
196(3)
11 The Decomposition of a Polytope into Its Cones
199(14)
11.1 The Identity "σmz zm = 0" ... Or "Much Ado About Nothing"
199(2)
11.2 Technically Speaking
201(3)
11.3 Tangent Cones and Their Rational Generating Functions
204(1)
11.4 Brion's Theorem
205(2)
11.5 Brion Implies Ehrhart
207(6)
Notes
209(1)
Exercises
210(3)
12 Euler-Maclaurin Summation in Rd
213(14)
12.1 Todd Operators and Bernoulli Numbers
214(2)
12.2 A Continuous Version of Brion's Theorem
216(2)
12.3 Polytopes Have Their Moments
218(3)
12.4 Computing the Discrete Continuously
221(6)
Notes
223(1)
Exercises
224(1)
Open Problems
225(2)
13 Solid Angles
227(14)
13.1 A New Discrete Volume Using Solid Angles
227(4)
13.2 Solid-Angle Generating Functions and a Brion-Type Theorem
231(1)
13.3 Solid-Angle Reciprocity and the Brianchon-Gram Relations
232(4)
13.4 The Generating Function of Macdonald's Solid-Angle Polynomials
236(5)
Notes
237(1)
Exercises
238(1)
Open Problems
239(2)
14 A Discrete Version of Green's Theorem Using Elliptic Functions
241(8)
14.1 The Residue Theorem
241(2)
14.2 The Weierstraß ℘- and ζ-Functions
243(2)
14.3 A Contour-Integral Extension of Pick's Theorem
245(4)
Notes
246(1)
Exercises
247(1)
Open Problems
248(1)
Appendix: Vertex and Hyperplane Descriptions of Polytopes
249(6)
A.1 Every h-Cone Is a v-Cone
250(2)
A.2 Every v-Cone Is an h-Cone
252(3)
Hints for ♣ Exercises 255(12)
References 267(12)
List of Symbols 279(2)
Index 281
Matthias Beck is Professor of Mathematics at San Francisco State University. Sinai Robins is Associate Professor of Mathematics at Nanyang Technological University, Singapore.
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