Problems from the Discrete to the Continuous: Probability, Number Theory, Graph Theory, and Combinatorics

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Springer, Aug 9, 2014 - Mathematics - 154 pages

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.

 

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Contents

1 Partitions with Restricted Summands or the Money Changing Problem
1
2 The Asymptotic Density of Relatively Prime Pairs and of SquareFree Numbers
7
3 A OneDimensional Probabilistic Packing Problem
21
4 The Arcsine Laws for the OneDimensional Simple Symmetric Random Walk
35
5 The Distribution of Cycles in Random Permutations
49
6 Chebyshevs Theorem on the Asymptotic Density of the Primes
67
7 Mertens Theorems on the Asymptotic Behavior of the Primes
75
8 The HardyRamanujan Theorem on the Number of Distinct Prime Divisors
81
A Theorem of Erdős and Rényi
109
Appendix A A Quick Primer on Discrete Probability
133
Appendix B Power Series and Generating Functions
141
Appendix C A Proof of Stirlings Formula
145
Appendix D An Elementary Proof of n11n2π26
149
References
151
Index
153
Copyright

9 The Largest Clique in a Random Graph and Applications to Tampering Detection and Ramsey Theory
89

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About the author (2014)

Ross Pinsky is a Professor in the Department of Mathematics at Technion-Israel Institute of Technology.

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