## Problems from the Discrete to the Continuous: Probability, Number Theory, Graph Theory, and CombinatoricsThe 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 | |

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 |

151 | |

153 | |

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

### Other editions - View all

Problems from the Discrete to the Continuous: Probability, Number Theory ... Ross Pinsky No preview available - 2014 |

Problems from the Discrete to the Continuous: Probability, Number Theory ... Ross Pinsky No preview available - 2014 |