Additive Number Theory The Classical Bases

Author : Melvyn B. Nathanson
Publisher : Springer Science & Business Media
Page : 350 pages
File Size : 54,5 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9781475738452

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Additive Number Theory The Classical Bases by Melvyn B. Nathanson Pdf

[Hilbert's] style has not the terseness of many of our modem authors in mathematics, which is based on the assumption that printer's labor and paper are costly but the reader's effort and time are not. H. Weyl [143] The purpose of this book is to describe the classical problems in additive number theory and to introduce the circle method and the sieve method, which are the basic analytical and combinatorial tools used to attack these problems. This book is intended for students who want to lel?Ill additive number theory, not for experts who already know it. For this reason, proofs include many "unnecessary" and "obvious" steps; this is by design. The archetypical theorem in additive number theory is due to Lagrange: Every nonnegative integer is the sum of four squares. In general, the set A of nonnegative integers is called an additive basis of order h if every nonnegative integer can be written as the sum of h not necessarily distinct elements of A. Lagrange 's theorem is the statement that the squares are a basis of order four. The set A is called a basis offinite order if A is a basis of order h for some positive integer h. Additive number theory is in large part the study of bases of finite order. The classical bases are the squares, cubes, and higher powers; the polygonal numbers; and the prime numbers. The classical questions associated with these bases are Waring's problem and the Goldbach conjecture.

Author : Anonim
Publisher : Unknown
Page : 342 pages
File Size : 44,5 Mb
Release : 2012
Category : Number theory
ISBN : 751004409X

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加{OCLCbr#E6}?{OCLCbr#A7}数{OCLCbr#E8}{OCLCbr#AE}? by Anonim Pdf

Author : Alfred Geroldinger,Imre Ruzsa
Publisher : Springer Science & Business Media
Page : 324 pages
File Size : 49,6 Mb
Release : 2009-04-15
Category : Mathematics
ISBN : 9783764389611

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Combinatorial Number Theory and Additive Group Theory by Alfred Geroldinger,Imre Ruzsa Pdf

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.

Author : David Chudnovsky,Gregory Chudnovsky
Publisher : Springer Science & Business Media
Page : 361 pages
File Size : 50,5 Mb
Release : 2010-08-26
Category : Mathematics
ISBN : 9780387683614

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Additive Number Theory by David Chudnovsky,Gregory Chudnovsky Pdf

This impressive volume is dedicated to Mel Nathanson, a leading authoritative expert for several decades in the area of combinatorial and additive number theory. For several decades, Mel Nathanson's seminal ideas and results in combinatorial and additive number theory have influenced graduate students and researchers alike. The invited survey articles in this volume reflect the work of distinguished mathematicians in number theory, and represent a wide range of important topics in current research.

Author : Anonim
Publisher : Unknown
Page : 128 pages
File Size : 43,7 Mb
Release : 1996
Category : Electronic
ISBN : OCLC:923952419

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Additive Number Theory by Anonim Pdf

Author : Melvyn B. Nathanson
Publisher : Springer Nature
Page : 445 pages
File Size : 52,7 Mb
Release : 2021-08-12
Category : Mathematics
ISBN : 9783030679965

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Combinatorial and Additive Number Theory IV by Melvyn B. Nathanson Pdf

This is the fourth in a series of proceedings of the Combinatorial and Additive Number Theory (CANT) conferences, based on talks from the 2019 and 2020 workshops at the City University of New York. The latter was held online due to the COVID-19 pandemic, and featured speakers from North and South America, Europe, and Asia. The 2020 Zoom conference was the largest CANT conference in terms of the number of both lectures and participants. These proceedings contain 25 peer-reviewed and edited papers on current topics in number theory. Held every year since 2003 at the CUNY Graduate Center, the workshop 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, zero-sum sequences, minimal complements, analytic and prime number theory, Hausdorff dimension, combinatorial and discrete geometry, and Ramsey theory. This selection of articles will be of relevance to both researchers and graduate students interested in current progress in number theory.

Author : Gove W. Effinger,David R. Hayes
Publisher : Unknown
Page : 184 pages
File Size : 42,7 Mb
Release : 1991
Category : Mathematics
ISBN : UOM:39015022029501

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Additive Number Theory of Polynomials Over a Finite Field by Gove W. Effinger,David R. Hayes Pdf

This book helps gather the sum of additive number theory.

Author : Melvyn B. Nathanson
Publisher : Springer Science & Business Media
Page : 320 pages
File Size : 51,9 Mb
Release : 1996-08-22
Category : Mathematics
ISBN : 0387946551

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Additive Number Theory: Inverse Problems and the Geometry of Sumsets by Melvyn B. Nathanson Pdf

Many classical problems in additive number theory are direct problems, in which one starts with a set A of natural numbers and an integer H -> 2, and tries to describe the structure of the sumset hA consisting of all sums of h elements of A. By contrast, in an inverse problem, one starts with a sumset hA, and attempts to describe the structure of the underlying set A. In recent years there has been ramrkable progress in the study of inverse problems for finite sets of integers. In particular, there are important and beautiful inverse theorems due to Freiman, Kneser, Plünnecke, Vosper, and others. This volume includes their results, and culminates with an elegant proof by Ruzsa of the deep theorem of Freiman that a finite set of integers with a small sumset must be a large subset of an n-dimensional arithmetic progression.

Author : Luogeng Hua
Publisher : American Mathematical Soc.
Page : 206 pages
File Size : 51,6 Mb
Release : 2009-12-04
Category : Mathematics
ISBN : 9780821849422

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Additive Theory of Prime Numbers by Luogeng Hua Pdf

Loo-Keng Hua was a master mathematician, best known for his work using analytic methods in number theory. In particular, Hua is remembered for his contributions to Waring's Problem and his estimates of trigonometric sums. Additive Theory of Prime Numbers is an exposition of the classic methods as well as Hua's own techniques, many of which have now also become classic. An essential starting point is Vinogradov's mean-value theorem for trigonometric sums, which Hua usefully rephrases and improves. Hua states a generalized version of the Waring-Goldbach problem and gives asymptotic formulas for the number of solutions in Waring's Problem when the monomial $x^k$ is replaced by an arbitrary polynomial of degree $k$. The book is an excellent entry point for readers interested in additive number theory. It will also be of value to those interested in the development of the now classic methods of the subject.

Author : Melvyn B. Nathanson
Publisher : Springer Science & Business Media
Page : 514 pages
File Size : 55,7 Mb
Release : 2008-01-11
Category : Mathematics
ISBN : 9780387227382

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Elementary Methods in Number Theory by Melvyn B. Nathanson Pdf

This basic introduction to number theory is ideal for those with no previous knowledge of the subject. The main topics of divisibility, congruences, and the distribution of prime numbers are covered. Of particular interest is the inclusion of a proof for one of the most famous results in mathematics, the prime number theorem. With many examples and exercises, and only requiring knowledge of a little calculus and algebra, this book will suit individuals with imagination and interest in following a mathematical argument to its conclusion.

Author : Melvyn B. Nathanson
Publisher : Springer
Page : 309 pages
File Size : 52,6 Mb
Release : 2014-10-18
Category : Mathematics
ISBN : 9781493916016

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Combinatorial and Additive Number Theory by Melvyn B. Nathanson Pdf

This proceedings volume is based on papers presented at the Workshops on Combinatorial and Additive Number Theory (CANT), which were held at the Graduate Center of the City University of New York in 2011 and 2012. The goal of the workshops is to survey recent progress in combinatorial number theory and related parts of mathematics. The workshop attracts researchers and students who discuss the state-of-the-art, open problems and future challenges in number theory.

Author : Melvyn B. Nathanson
Publisher : Springer
Page : 310 pages
File Size : 46,5 Mb
Release : 2018-01-13
Category : Mathematics
ISBN : 9783319680323

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Combinatorial and Additive Number Theory II by Melvyn B. Nathanson Pdf

Based on talks from the 2015 and 2016 Combinatorial and Additive Number Theory (CANT) workshops at the City University of New York, these proceedings offer 19 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. Sumsets, partitions, convex polytopes and discrete geometry, Ramsey theory, primality testing, and cryptography are among the topics featured in this volume. Each contribution is dedicated to a specific topic that reflects the latest results by experts in the field. Researchers and graduate students interested in the current progress in number theory will find this selection of articles relevant and compelling.

Author : Cheon Seoung Ryoo
Publisher : BoD – Books on Demand
Page : 220 pages
File Size : 50,9 Mb
Release : 2020-11-04
Category : Mathematics
ISBN : 9781839680502

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Number Theory and Its Applications by Cheon Seoung Ryoo Pdf

Number theory and its applications are well known for their proven properties and excellent applicability in interdisciplinary fields of science. Until now, research on number theory and its applications has been done in mathematics, applied mathematics, and the sciences. In particular, number theory plays a fundamental and important role in mathematics and applied mathematics. This book is based on recent results in all areas related to number theory and its applications.

Author : Melvyn B. Nathanson
Publisher : Springer
Page : 400 pages
File Size : 51,5 Mb
Release : 2008-03-01
Category : Mathematics
ISBN : 0387709983

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Additive Number Theory by Melvyn B. Nathanson Pdf

A central problem in additive number theory is the growth of sumsets. If A is a finite or infinite subset of the integers and the lattice points, or more generally, of any abelian group or semigroup G, then the h-fold sumset of A is the set LA consisting of all elements of G that can be represented as the sum of L not necessarily distinct elements of A. The goal is to understand the asymptotics of the sumsets LA, that is, the size and structure of LA, as L tends to infinity. If A is finite, then the size of LA is its cardinality. If A is infinite, then the size of LA is measured by various duality functions. These problems have natural analogues when A is a subset of a nonabelian group. Additive Number Theory: Density Theorems and the Growth of Sumsets presents material that deals with the above problem. Ideas and techniques from many parts of mathematics are used to prove theorems in this subject. For example, the authors use number theory, combinatorics, commutative algebra, ultrafilters and logic, and nonstandard analysis. The book is self-contained, and includes short introductions to the various techniques that are not standard in this field. Graduate students and researchers in mathematics will find this book useful. Prerequisites include undergraduate elementary number theory and algebra.

Author : Henri Cohen
Publisher : Springer Science & Business Media
Page : 673 pages
File Size : 54,5 Mb
Release : 2007-05-23
Category : Mathematics
ISBN : 9780387499222

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Number Theory by Henri Cohen Pdf

The central theme of this book is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects. The book contains more than 350 exercises and the text is largely self-contained. Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included five appendices on these techniques.