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Asymptopia

By (author) Joel Spencer

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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, ErdÅ‘s Magic, random graphs, Ramsey numbers, and asymptotic geometry. The author is a disciple of Paul ErdÅ‘s, who taught him about Asymptopia. Primes less than n , graphs with v vertices, random walks of t steps - ErdÅ‘s was fascinated by the limiting behavior as the variables approached, but never reached, infinity. Asymptotics is very much an art. The various functions nlnn , n 2 , lnn n , lnn ? ? ? ? , 1 nlnn all have distinct personalities. ErdÅ‘s 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!

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