This book is mostly based on the author's 25 years of teaching combinatorics to two distinct sets of students: first-year students and seniors from all backgrounds. The prerequisites are kept to a minimum; essentially, only high school algebra is required. The design is to go quickly from zero knowledge to advanced themes and various applications with a lot of topics intended for additional reading and research projects. It contains an all-inclusive collection of 135 problems and 275 exercises with four difficulty levels: solutions, hints and answers are provided.Some themes of the book:Enumerative combinatorics and basic graph theory: Introduction to dimers, tilings, magic and Latin squares, permutations, combinations, generating functions, games of chance, random walks, binomial and Poisson distributions. Catalan numbers, their generalizations and applications, including roulette and pricing derivatives. Euler and Hamiltonian paths, linear and planar graphs, labeled trees and other topics on graphs; many of them are presented as exercises.Modeling: Linear recurrence relations, Fibonacci rabbits, population growth, tree growth, epidemic spread and reinfections, resonances and nuclear reactors, predator-prey relationships and stopping times.Elementary number theory: Residues, finite fields, Pisano periods, quadratic reciprocity, Pell's equation, continued fractions, and Frobenius coin problem. Applications to cryptography, designs and magic squares, error-correcting codes and nonattacking queens.

N� de ref. del art�culo LU-9789811265396

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Detalles bibliogr�ficos

T�tulo

Combinatorics, Modeling, Elementary Number Theory: From Basic To Advanced

Editor

World Scientific Publishing Co Pte Ltd, SG

Fecha de publicaci�n

2023

Idioma

Ingl�s

ISBN 10

9811265399

ISBN 13

9789811265396

Encuadernaci�n

Hardback

Estado

New

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Sinopsis

This book is mostly based on the author's 25 years of teaching combinatorics to two distinct sets of students: first-year students and seniors from all backgrounds, not just limited to only those majoring in mathematics and physics. The prerequisites are kept to a minimum; essentially, only high school algebra is required. The design is to go from zero knowledge to advanced themes and various applications during a semester of three or three and a half months with quite a few topics intended for research projects and additional reading. This unique book features the key themes of classical introductory combinatorics, modeling (mainly linear), and elementary number theory with a constant focus on applications in statistics, physics, biology, economics, and computer science. These applications include dimers, random walks, binomial and Poisson distributions, games of chance (lottery, dice, poker, roulette), pricing options, population growth, tree growth, modeling epidemic spread, invasion ecology, fission reactors, and networks. A lot of material is provided in the form of relatively self-contained problems, about 135, and exercises, about 270, which are almost always with hints and answers. A systematic introduction to number theory (with complete justifications) is a significant part of the book, including finite fields, Pell's equations, continued fractions, quadratic reciprocity, the Frobenius coin problem, Pisano periods, applications to magic and Latin squares and elements of cryptography. The recurrence relations and modeling play a very significant role, including the usage of Bessel functions for motivated readers. The book contains a lot of history of mathematics and recreational mathematics.

Acerca del autor

Ivan Cherednik is a distinguished professor of mathematics of the University of North Carolina at Chapel Hill. His scientific interests are extremely broad: in the range from geometry, combinatorics, representation theory, number theory, topology, harmonic analysis and special functions to mathematical and theoretical physics and applied mathematics. He proved the celebrated Macdonald conjectures and introduced Cherednik's algebras (aka DAHA), a powerful tool of modern mathematics and physics. His earlier results in number theory were used in the proof of the Last Fermat Theorem. His recent papers are on modeling epidemics, econophysics, and some aspects of artificial intelligence. His two books are in soliton theory and on Cherednik's algebras. Some of his accomplishments are: an ICM-lecture, Guggenheim fellowship, Distinguished Fulbright Chair in Natural Sciences and Engineering, two Simons fellowships, two CMI fellowships, many other fellowships and grants, including 10 individual NSF grants, 4 CNRS-France grants, 3 COE-Japan grants, 2 NWO-Netherlands grants, and more than 60 visiting positions, including a recent one-year position at ETH (Zurich).

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