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Full Description
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 melange 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.
Contents
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.
