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Asymptopia
Joel Spencer, New York University, NY
with Laura Florescu, New York University, NY
 Student Mathematical Library 2014; approx. 195 pp; softcover Volume: 71 ISBN-10: 1-4704-0904-6 ISBN-13: 978-1-4704-0904-3 List Price: US$39 Member Price: US$31.20 Order Code: STML/71 Not yet published.Expected publication date is June 12, 2014. See also: Complex Graphs and Networks - Fan Chung and Linyuan Lu Combinatorial Problems and Exercises: Second Edition - Laszlo Lovasz Geometric Approximation Algorithms - Sariel Har-Peled Asymptotics in one form or another are part of the landscape for every mathematician. The objective of this book is to present the ideas of how to approach asymptotic problems that arise in discrete mathematics, analysis of algorithms, and number theory. A broad range of topics is covered, including distribution of prime integers, Erdős Magic, random graphs, Ramsey numbers, and asymptotic geometry. The author is a disciple of Paul Erdős, who taught him about Asymptopia. Primes less than $$n$$, graphs with $$v$$ vertices, random walks of $$t$$ steps--Erdős was fascinated by the limiting behavior as the variables approached, but never reached, infinity. Asymptotics is very much an art. The various functions $$n\ln n$$, $$n^2$$, $$\frac{\ln n}{n}$$, $$\sqrt{\ln n}$$, $$\frac{1}{n\ln n}$$ all have distinct personalities. Erdős knew these functions as personal friends. It is the author's hope that these insights may be passed on, that the reader may similarly feel which function has the right temperament for a given task. This book is aimed at strong undergraduates, though it is also suitable for particularly good high school students or for graduates wanting to learn some basic techniques. Asymptopia is a beautiful world. Enjoy! Readership Undergraduate and graduate students interested in asymptotic techniques. Table of Contents An infinity of primes Stirling's formula Big Oh, little Oh and all that Integration in Asymptopia From integrals to sums Asymptotics of binomial coefficients $$\binom{n}{k}$$ Unicyclic graphs Ramsey numbers Large deviations Primes Asymptotic geometry Algorithms Potpourri Really Big Numbers! Bibliography Index