Combinatorial Number Theory and Additive Group Theory

Combinatorial Number Theory and Additive Group Theory
Author: Alfred Geroldinger
Publisher: Springer Science & Business Media
Total Pages: 324
Release: 2009-04-15
Genre: Mathematics
ISBN: 3764389613

Additive combinatorics is a relatively recent term coined to comprehend the developments of the more classical additive number theory, mainly focussed on problems related to the addition of integers. Some classical problems like the Waring problem on the sum of k-th powers or the Goldbach conjecture are genuine examples of the original questions addressed in the area. One of the features of contemporary additive combinatorics is the interplay of a great variety of mathematical techniques, including combinatorics, harmonic analysis, convex geometry, graph theory, probability theory, algebraic geometry or ergodic theory. This book gathers the contributions of many of the leading researchers in the area and is divided into three parts. The two first parts correspond to the material of the main courses delivered, Additive combinatorics and non-unique factorizations, by Alfred Geroldinger, and Sumsets and structure, by Imre Z. Ruzsa. The third part collects the notes of most of the seminars which accompanied the main courses, and which cover a reasonably large part of the methods, techniques and problems of contemporary additive combinatorics.

An Introduction to Commutative Algebra and Number Theory

An Introduction to Commutative Algebra and Number Theory
Author: Sukumar Das Adhikari
Publisher: CRC Press
Total Pages: 176
Release: 2001-11
Genre: Mathematics
ISBN: 9780849309908

This is an elementary introduction to algebra and number theory. The text begins by a review of groups, rings, and fields. The algebra portion addresses polynomial rings, UFD, PID, and Euclidean domains, field extensions, modules, and Dedckind domains. The number theory portion reviews elementary congruence, quadratic reciprocity, algebraic number fields, and a glimpse into the various aspects of that subject. This book could be used as a one semester course in graduate mathematics.

Combinatorial and Additive Number Theory III

Combinatorial and Additive Number Theory III
Author: Melvyn B. Nathanson
Publisher: Springer Nature
Total Pages: 237
Release: 2019-12-10
Genre: Mathematics
ISBN: 3030311066

Based on talks from the 2017 and 2018 Combinatorial and Additive Number Theory (CANT) workshops at the City University of New York, these proceedings offer 17 peer-reviewed and edited papers on current topics in number theory. Held every year since 2003, the workshop series surveys state-of-the-art open problems in combinatorial and additive number theory and related parts of mathematics. Topics featured in this volume include sumsets, partitions, convex polytopes and discrete geometry, Ramsey theory, commutative algebra and discrete geometry, and applications of logic and nonstandard analysis to number theory. Each contribution is dedicated to a specific topic that reflects the latest results by experts in the field. This selection of articles will be of relevance to both researchers and graduate students interested in current progress in number theory.

Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory

Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory
Author: Mauro Di Nasso
Publisher: Springer
Total Pages: 211
Release: 2019-05-23
Genre: Mathematics
ISBN: 3030179567

The goal of this monograph is to give an accessible introduction to nonstandard methods and their applications, with an emphasis on combinatorics and Ramsey theory. It includes both new nonstandard proofs of classical results and recent developments initially obtained in the nonstandard setting. This makes it the first combinatorics-focused account of nonstandard methods to be aimed at a general (graduate-level) mathematical audience. This book will provide a natural starting point for researchers interested in approaching the rapidly growing literature on combinatorial results obtained via nonstandard methods. The primary audience consists of graduate students and specialists in logic and combinatorics who wish to pursue research at the interface between these areas.

Elementary Number Theory in Nine Chapters

Elementary Number Theory in Nine Chapters
Author: James J. Tattersall
Publisher: Cambridge University Press
Total Pages: 420
Release: 1999-10-14
Genre: Mathematics
ISBN: 9780521585316

This book is intended to serve as a one-semester introductory course in number theory. Throughout the book a historical perspective has been adopted and emphasis is given to some of the subject's applied aspects; in particular the field of cryptography is highlighted. At the heart of the book are the major number theoretic accomplishments of Euclid, Fermat, Gauss, Legendre, and Euler, and to fully illustrate the properties of numbers and concepts developed in the text, a wealth of exercises have been included. It is assumed that the reader will have 'pencil in hand' and ready access to a calculator or computer. For students new to number theory, whatever their background, this is a stimulating and entertaining introduction to the subject.

Principles of Combinatorics

Principles of Combinatorics
Author: Berge
Publisher: Academic Press
Total Pages: 189
Release: 1971-04-20
Genre: Computers
ISBN: 0080955819

Berge's Principles of Combinatorics is now an acknowledged classic work of the field. Complementary to his previous books, Berge's introduction deals largely with enumeration. The choice of topics is balanced, the presentation elegant, and the text can be followed by anyone with an interest in the subject with only a little algebra required as a background. Some topics were here described for the first time, including Robinston-Shensted theorum, the Eden-Schutzenberger theorum, and facts connecting Young diagrams, trees, and the symmetric group.

Combinatorics

Combinatorics
Author: Pavle Mladenović
Publisher: Springer
Total Pages: 372
Release: 2019-03-13
Genre: Mathematics
ISBN: 3030008312

This text provides a theoretical background for several topics in combinatorial mathematics, such as enumerative combinatorics (including partitions and Burnside's lemma), magic and Latin squares, graph theory, extremal combinatorics, mathematical games and elementary probability. A number of examples are given with explanations while the book also provides more than 300 exercises of different levels of difficulty that are arranged at the end of each chapter, and more than 130 additional challenging problems, including problems from mathematical olympiads. Solutions or hints to all exercises and problems are included. The book can be used by secondary school students preparing for mathematical competitions, by their instructors, and by undergraduate students. The book may also be useful for graduate students and for researchers that apply combinatorial methods in different areas.

Analytic Combinatorics

Analytic Combinatorics
Author: Philippe Flajolet
Publisher: Cambridge University Press
Total Pages: 825
Release: 2009-01-15
Genre: Mathematics
ISBN: 1139477161

Analytic combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures. The theory has emerged over recent decades as essential both for the analysis of algorithms and for the study of scientific models in many disciplines, including probability theory, statistical physics, computational biology, and information theory. With a careful combination of symbolic enumeration methods and complex analysis, drawing heavily on generating functions, results of sweeping generality emerge that can be applied in particular to fundamental structures such as permutations, sequences, strings, walks, paths, trees, graphs and maps. This account is the definitive treatment of the topic. The authors give full coverage of the underlying mathematics and a thorough treatment of both classical and modern applications of the theory. The text is complemented with exercises, examples, appendices and notes to aid understanding. The book can be used for an advanced undergraduate or a graduate course, or for self-study.