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

ISBN-13: 9783319079646 / Angielski / Miękka / 2014 / 154 str.

Ross Pinsky

Opis

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.

Recenzje

"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)

Spis treści

Partitions With Restricted Summands or "The Money Changing Problem".- The Asymptotic Density of Relatively Prime Pairs and of Square-Free Numbers.- A One-Dimensional Probabilistic Packing Problem.- The Arcsine Laws for the One-Dimensional Simple Symmetric Random Walk.- The Distribution of Cycles in Random Permutations.- Chebyshev's Theorem on the Asymptotic Density of the Primes.- Mertens' Theorems on the Asymptotic Behavior of the Primes.- The Hardy-Ramanujan Theorem on the Number of Distinct Prime Divisors.- The Largest Clique in a Random Graph and Applications to Tampering Detection and Ramsey Theory.- The Phase Transition Concerning the Giant Component in a Sparse Random Graph–a Theorem of Erdős and Rényi.

O autorze

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

Blurb

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.