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

4.50/5 (4 ratings by Goodreads)
  • Formatas: PDF+DRM
  • Serija: Universitext
  • Išleidimo metai: 09-Aug-2014
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319079653
Kitos knygos pagal šią temą:
Problems from the Discrete to the Continuous: Probability, Number Theory, Graph Theory, and CombinatoricsDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Universitext
  • Išleidimo metai: 09-Aug-2014
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319079653
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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 melange 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.

Recenzijos

"The book under review consists of ten well-written chapters each devoted to a single topic, or rather a single problem. ... The book is expected to appeal to a wide audience, including graduate and advanced undergraduate students, and can indeed be used for a seminar course in which students may present the lectures. ... Each chapter contains a set of exercises and ends with illuminating historical notes." (M. Hajja, Mathematical Reviews, November, 2015) "The book under review collects a number of problems of discrete nature and with solutions utilizing continuous and analytic tools. ... the book is suitable for undergraduate students to have an excursion on some selected problems and interesting theorems, and also it is suitable for instructors to use it to introduce some good and meaningful examples." (Mehdi Hassani, zbMATH 1311.11002, 2015)

1 Partitions with Restricted Summands or "the Money Changing Problem"
1(6)
2 The Asymptotic Density of Relatively Prime Pairs and of Square-Free Numbers
7(14)
3 A One-Dimensional Probabilistic Packing Problem
21(14)
4 The Arcsine Laws for the One-Dimensional Simple Symmetric Random Walk
35(14)
5 The Distribution of Cycles in Random Permutations
49(18)
6 Chebyshev's Theorem on the Asymptotic Density of the Primes
67(8)
7 Mertens' Theorems on the Asymptotic Behavior of the Primes
75(6)
8 The Hardy--Ramanujan Theorem on the Number of Distinct Prime Divisors
81(8)
9 The Largest Clique in a Random Graph and Applications to Tampering Detection and Ramsey Theory
89(20)
9.1 Graphs and Random Graphs: Basic Definitions
89(2)
9.2 The Size of the Largest Clique in a Random Graph
91(8)
9.3 Detecting Tampering in a Random Graph
99(5)
9.4 Ramsey Theory
104(5)
10 The Phase Transition Concerning the Giant Component in a Sparse Random Graph: A Theorem of Erdos and Renyi
109(24)
10.1 Introduction and Statement of Results
109(2)
10.2 Construction of the Setup for the Proofs of Theorems 10.1 and 10.2
111(2)
10.3 Some Basic Large Deviations Estimates
113(2)
10.4 Proof of Theorem 10.1
115(1)
10.5 The Galton--Watson Branching Process
116(4)
10.6 Proof of Theorem 10.2
120(13)
Appendix A A Quick Primer on Discrete Probability 133(8)
Appendix B Power Series and Generating Functions 141(4)
Appendix C A Proof of Stirling's Formula 145(4)
Appendix D An Elementary Proof of Σ∞n=1 1/n2 = π2/6 149(2)
References 151(2)
Index 153
Ross Pinsky is a Professor in the Department of Mathematics at Technion-Israel Institute of Technology.
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