The Experts below are selected from a list of 285 Experts worldwide ranked by ideXlab platform
Song Y. Yan - One of the best experts on this subject based on the ideXlab platform.
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Review of Applied Number Theory
ACM SIGACT News, 2020Co-Authors: Song Y. YanAbstract:Number Theory (or the Theory of Numbers [1]) starts from elementary (not necessarily simple and easy, and in fact, it is often very hard) Number Theory and grows up with analytic Number Theory (including additive and multiplicative Number Theory), algebraic Number Theory, geometric Number Theory (including arithmetic algebraic geometry), combinatorial Number Theory, computational Number Theory (including algorithmic Number Theory), to name just a few. Applied Number Theory, on the other hand, is involved in the application of various branches of Number Theory to a wide range of areas including, e.g., physics, chemistry, biology, graphics, arts, music, and particularly computing and digital communications [2]. Number Theory was once viewed as the purest of the pure mathematics, with little application to other areas. However, with the advent of modern computers and digital communications, Number Theory becomes increasingly important and applicable to many areas ranging from natural sciences, engineering to social sciences.
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Elementary Number Theory
Number Theory for Computing, 2002Co-Authors: Song Y. YanAbstract:This chapter introduces the basic concepts and results of the elementary Theory of Numbers. Its purpose is twofold: Provide a solid foundation of elementary Number Theory for Computational, Algorithmic, and Applied Number Theory of the next two chapters of the book. Provide independently a self-contained text of Elementary Number Theory for Computing, or in part a text of Mathematics for Computing.
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Computational/Algorithmic Number Theory
Number Theory for Computing, 2002Co-Authors: Song Y. YanAbstract:Computational and algorithmic Number Theory are two very closely related subjects; they are both concerned with, among many others, computer algorithms, particularly efficient algorithms (including parallel and distributed algorithms, sometimes also including computer architectures), for solving different sorts of problems in Number Theory and in other areas, including computing and cryptography. Primality testing, integer factorization and discrete logarithms are, amongst many others, the most interesting, difficult and useful problems in Number Theory, computing and cryptography. In this chapter, we shall study both computational and algorithmic aspects of Number Theory. More specifically, we shall study various algorithms for primality testing, integer factorization and discrete logarithms that are particularly applicable and useful in computing and cryptography, as well as methods for many other problems in Number Theory, such as the Goldbach conjecture and the odd perfect Number problem.
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Algorithmic Number Theory
Number Theory for Computing, 2000Co-Authors: Song Y. YanAbstract:Algorithmic (or computational) Number Theory is mainly concerned with computer algorithms (sometimes also including computer architectures), in particular efficient algorithms, for solving different sorts of problems in Number Theory. Primality testing, integer factorization and discrete logarithms are, amongst many others, the most interesting, difficult and useful problems in Number Theory. In this chapter, we shall study the algorithmic aspects of Number Theory, and more specifically, we shall introduce various algorithms for solving these three types of Number-theoretic problems.
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Applied Number Theory
Number Theory for Computing, 2000Co-Authors: Song Y. YanAbstract:The aim of this chapter is to introduce some novel applications of elementary and particularly algorithmic Number Theory to the design of computer (both hardware and software) systems, coding and cryptography, and information security, especially network/communication security.
Melvyn B. Nathanson - One of the best experts on this subject based on the ideXlab platform.
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Combinatorial and Additive Number Theory II - Combinatorial and Additive Number Theory II
Springer Proceedings in Mathematics & Statistics, 2014Co-Authors: Melvyn B. NathansonAbstract: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
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Addictive Number Theory
Additive Number Theory, 2010Co-Authors: Melvyn B. NathansonAbstract:In 1996, just after Springer-Verlag published my books Additive Number Theory: The Classical Bases [34] and Additive Number Theory: Inverse Problems and the Geometry of Sumsets [35], I went into my local Barnes and Noble superstore on Route 22 in Springfield, New Jersey, and looked for them on the shelves. Suburban bookstores do not usually stock technical mathematical books, and, of course, the books were not there. As an experiment, I asked if they could be ordered. The person at the information desk typed in the titles, and told me that his computer search reported that the books did not exist. However, when I gave him the ISBN Numbers, he did find them in the Barnes and Noble database. The problem was that the book titles had been cataloged incorrectly. The data entry person had written Addictive Number Theory.
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elementary methods in Number Theory
1999Co-Authors: Melvyn B. NathansonAbstract:Elementary Methods in Number Theory begins with "a first course in Number Theory" for students with no previous knowledge of the subject. The main topics are divisibility, prime Numbers, and congruences. There is also an introduction to Fourier analysis on finite abelian groups, and a discussion on the abc conjecture and its consequences in elementary Number Theory. In the second and third parts of the book, deep results in Number Theory are proved using only elementary methods. Part II is about multiplicative Number Theory, and includes two of the most famous results in mathematics: the Erdos-Selberg elementary proof of the prime Number theorem, and Dirichlets theorem on primes in arithmetic progressions. Part III is an introduction to three classical topics in additive Number Theory: Warings problems for polynomials, Liouvilles method to determine the Number of representations of an integer as the sum of an even Number of squares, and the asymptotics of partition functions. Melvyn B. Nathanson is Professor of Mathematics at the City University of New York (Lehman College and the Graduate Center). He is the author of the two other graduate texts: Additive Number Theory: The Classical Bases and Additive Number Theory: Inverse Problems and the Geometry of Sumsets.
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additive Number Theory the classical bases
1996Co-Authors: Melvyn B. NathansonAbstract: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 to attack these problems. This book is intended for students who want to learn additive Number Theory, not for experts who already know it. The prerequisites for this book are undergraduate courses in Number Theory and real analysis.
Arne Winterhof - One of the best experts on this subject based on the ideXlab platform.
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Applied Number Theory
2015Co-Authors: Harald Niederreiter, Arne WinterhofAbstract:This textbook effectively builds a bridge from basic Number Theory to recent advances in applied Number Theory. It presents the first unified account of the four major areas of application where Number Theory plays a fundamental role, namely cryptography, coding Theory, quasi-Monte Carlo methods, and pseudorandom Number generation, allowing the authors to delineate the manifold links and interrelations between these areas. Number Theory, which Carl-Friedrich Gauss famously dubbed the queen of mathematics, has always been considered a very beautiful field of mathematics, producing lovely results and elegant proofs. While only very few real-life applications were known in the past, today Number Theory can be found in everyday life: in supermarket bar code scanners, in our cars GPS systems, in online banking, etc. Starting with a brief introductory course on Number Theory in Chapter 1, which makes the book more accessible for undergraduates, the authors describe the four main application areas in Chapters 2-5 and offer a glimpse of advanced results that are presented without proofs and require more advanced mathematical skills. In the last chapter they review several further applications of Number Theory, ranging from check-digit systems to quantum computation and the organization of raster-graphics memory. Upper-level undergraduates, graduates and researchers in the field of Number Theory will find this book to be a valuable resource.
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A Review of Number Theory and Algebra
Applied Number Theory, 2015Co-Authors: Harald Niederreiter, Arne WinterhofAbstract:Elementary Number Theory may be regarded as a prerequisite for this book, but since we, the authors, want to be nice to you, the readers, we provide a brief review of this Theory for those who already have some background on Number Theory and a crash course on elementary Number Theory for those who have not. Apart from trying to be friendly, we also follow good practice when we prepare the ground for the coming attractions by collecting some basic notation, terminology, and facts in an introductory chapter, like a playwright who presents the main characters of the play in the first few scenes. Basically, we cover only those results from elementary Number Theory that are actually needed in this book. For more information, there is an extensive expository literature on Number Theory, and if you want to read the modern classics, then we recommend the books of Hardy and Wright [61] and of Niven, Zuckerman, and Montgomery [151].
Nicolas Bourbaki - One of the best experts on this subject based on the ideXlab platform.
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Commutative Algebra. Algebraic Number Theory
Elements of the History of Mathematics, 1994Co-Authors: Nicolas BourbakiAbstract:“Abstract” commutative algebra is a recent creation, but its development can not be understood except in relation to that of algebraic Number Theory and algebraic geometry which gave it birth.
Asghar Ali - One of the best experts on this subject based on the ideXlab platform.
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LibGuides: Algebraic Number Theory: Home
2016Co-Authors: Asghar AliAbstract:Algebraic Number Theory is a major branch of Number Theory that studies algebraic structures related to algebraic integers.
