Problems from the discrete to the continuous : probability, number theory, graph theory, and combinatorics

Includes index

Includes bibliography (p. 151-152) and references

1 Partitions With Restricted Summands or "The Money Changing Problem"
2 The Asymptotic Density of Relatively Prime Pairs and of Square-Free Numbers
3 A One-Dimensional Probabilistic Packing Problem The
4 Arcsine Laws for the One-Dimensional Simple Symmetric Random Walk
5 The Distribution of Cycles in Random Permutations
6 Chebyshev's Theorem on the Asymptotic Density of the Primes
7 Mertens' Theorems on the Asymptotic Behavior of the Primes
8 The Hardy-Ramanujan Theorem on the Number of Distinct Prime Divisors
9 The Largest Clique in a Random Graph and Applications to Tampering Detection and Ramsey Theory
10 The Phase Transition Concerning the Giant Component in a Sparse Random Graph-a Theorem of Erdős and Rényi.

The primary intent of the book is to introduce an array of beautiful problems in a variety of subjects quickly, pithily and completely rigorously to graduate students and advanced undergraduates. The book takes a number of specific problems and solves them, the needed tools developed along the way in the context of the particular problems. It treats a mélange of topics from combinatorial probability theory, number theory, random graph theory and combinatorics. The problems in this book involve the asymptotic analysis of a discrete construct as some natural parameter of the system tends to infinity. Besides bridging discrete mathematics and mathematical analysis, the book makes a modest attempt at bridging disciplines. The problems were selected with an eye toward accessibility to a wide audience, including advanced undergraduate students. The book could be used for a seminar course in which students present the lectures.