Pinsky R.G. Problems from the discrete to the continuous: probability, number theory, graph theory, and combinatorics

Springer, 2014. — xiv, 154 p. — (Universitext). — ISBN: 3-319-07964-6, 3-319-07965-4, 978-3-319-07964-6, 978-3-319-07965-3.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 several 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.Partitions:
With Restricted Summands or "The Money Changing Problem"
The Asymptotic Density of Relatively Prime Pairs and 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 RényiTrue PDF