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Problems from the Discrete to the Continuous

Ross G. Pinsky
Publication Date: 
Number of Pages: 
[Reviewed by
Miklós Bóna
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This is a collection of interesting problems that can be solved in several ways, and at least one of those ways belongs to discrete mathematics, and at least one of those ways belongs to a field that uses continuous tools, like analysis or probability.

A good example is a pair of facts from number theory. Select two integers from the set \(\{1,2,...n\}\) uniformly at random. What is the probability that they are relatively prime? Now select just one integer from the same set uniformly at random. What is the probability that it does not have a divisor that is larger than one and is a perfect square? It turns out that the answer to both questions is \(6/\pi^2\), and the author proves this fact in numerous ways, some of which are his original ideas.

Other chapters in the book discuss questions from graph theory, Ramsey theory, the enumerative combinatorics of permutations, and more questions from number theory. The problems are well chosen and interesting.

My only critical remark is that the book would have deserved better editing. Many sentences in the book are simply too long. Somebody could have told the author that it is bad style to start a theorem with a formula. And there are some weird statements, like when on page 6, the author talks about Herb Wilf's “little book.” That book, generatingfunctionology, is more than 50 percent longer than the book under review! It also deserved a proper citation of its title, not a misspelled one. These very avoidable flaws are an unnecessary distraction.

Miklós Bóna is Professor of Mathematics at the University of Florida.

Partitions With Restricted Summands or "The Money Changing Problem"

The Asymptotic Density of Relatively Prime Pairs and of Square-Free Numbers

A One-Dimensional Probabilistic Packing Problem

The Arcsine Laws for the One-Dimensional Simple Symmetric Random Walk

The Distribution of Cycles in Random Permutations

Chebyshev's Theorem on the Asymptotic Density of the Primes

Mertens' Theorems on the Asymptotic Behavior of the Primes

The Hardy-Ramanujan Theorem on the Number of Distinct Prime Divisors

The Largest Clique in a Random Graph and Applications to Tampering Detection and Ramsey Theory

The Phase Transition Concerning the Giant Component in a Sparse Random Graph–a Theorem of Erdős and Rényi