''Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra'' is an undergraduate-level textbook in

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, on the interplay between the

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...

convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

s and the number of

lattice point In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate wise addition or subtraction of two points in the lattice produces another lattice poi ...

s they contain. It was written by Matthias Beck and Sinai Robins, and published in 2007 by

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

in their

Undergraduate Texts in Mathematics Undergraduate Texts in Mathematics (UTM) (ISSN 0172-6056) is a series of undergraduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are small yellow boo ...

series (Vol. 154). A second edition was published in 2015, and a German translation of the first edition by Kord Eickmeyer, ''Das Kontinuum diskret berechnen'', was published by Springer in 2008.

Topics

The book begins with a motivating problem, the coin problem of determining which amounts of money can be represented (and what is the largest non-representable amount of money) for a given system of coin values. Other topics touched on include face lattices of polytopes and the

Dehn–Sommerville equations In mathematics, the Dehn–Sommerville equations are a complete set of linear relations between the numbers of faces of different dimension of a simplicial polytope. For polytopes of dimension 4 and 5, they were found by Max Dehn in 1905. Their gen ...

relating numbers of faces;

Pick's theorem In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 18 ...

and the Ehrhart polynomials, both of which relate lattice counting to volume;

generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

s, and

Dedekind sum In mathematics, Dedekind sums are certain sums of products of a sawtooth function, and are given by a function ''D'' of three integer variables. Dedekind introduced them to express the functional equation of the Dedekind eta function. They have sub ...

s, different ways of encoding sequences of numbers into mathematical objects;

Green's theorem In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem. Theorem Let be a positively orient ...

and its discretization;

Bernoulli polynomials In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula. These polynomials occur in ...

; the

Euler–Maclaurin formula In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using ...

for the difference between a sum and the corresponding integral; special polytopes including zonotopes, the

Birkhoff polytope The Birkhoff polytope ''B'n'' (also called the assignment polytope, the polytope of doubly stochastic matrices, or the perfect matching polytope of the complete bipartite graph K_) is the convex polytope in R''N'' (where ''N'' = ''n''2) who ...

, and permutohedra; and the enumeration of

magic square In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number ...

s. In this way, the topics of the book connect together geometry,

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

, and

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...

Audience and reception

This book is written at an undergraduate level, and provides many exercises, making it suitable as an undergraduate textbook. Little mathematical background is assumed, except for some

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

towards the end of the book. The book also includes open problems, of more interest to researchers in these topics. As reviewer Darren Glass writes, "Even people who are familiar with the material would almost certainly learn something from the clear and engaging exposition that these two authors use." Reviewer

Margaret Bayer Margaret M. Bayer is an American mathematician working in polyhedral combinatorics. She is a professor of mathematics at the University of Kansas. Education Bayer earned her Ph.D. in 1983 from Cornell University. Her dissertation, ''Facial Enumer ...

calls the book "coherent and tightly developed ... accessible and engaging", and reviewer Oleg Karpenkov calls it "outstanding".

References

{{reflist, refs= {{citation, title=Review of ''Computing the Continuous Discretely'', last=Bayer, first=Margaret M., authorlink= Margaret Bayer , journal= zbMATH, zbl=1114.52013 {{citation, title=Review of ''Computing the Continuous Discretely'', last=De Loera, first=Jesús A., authorlink=Jesús A. De Loera, journal=

Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also pu ...

, year=2007, mr=2271992 {{citation, title=Review of ''Computing the Continuous Discretely'', last=Glass, first=Darren, journal=MAA Reviews, publisher=

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...

, date=February 2007, url=https://www.maa.org/press/maa-reviews/computing-the-continuous-discretely-integer-point-enumeration-in-polyhedra {{citation, title=Review of ''Computing the Continuous Discretely'', last=Karpenkov, first=Oleg, journal= zbMATH, zbl=1339.52002 Polytopes Lattice points Volume Mathematics textbooks 2007 non-fiction books 2015 non-fiction books Springer Science+Business Media books

Topics

Audience and reception

See also

References