Algebraic, Analytic, and Computational Number Theory and Its Applications PDF Download
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Author: Diana Savin
Publisher: Mdpi AG
ISBN: 9783036598598
Category : Computers
Languages : en
Pages : 0
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Book Description
Analytic number theory is a branch of number theory which uses methods from mathematical analysis in order to solve difficult problems about integers. Analytic number theory can be split into two major areas: multiplicative number theory and additive number theory. Bernhard Riemann made some very important contributions to the field of analytic number theory; among others, he investigated the Riemann zeta function, and he established its importance for understanding the distribution of prime numbers. A typical problem of analytic number theory is the enumeration of number-theoretic objects like primes, solutions of Diophantine equations, etc. Algebraic number theory on the other hand studies the arithmetic of algebraic number fields, i.e., the ring of integers of arbitrary number fields. It embraces, among others, the study of the ideals and of the group of units in the ring of integers and the extent to which unique factorization holds. The purpose and scope of this ''Special Issue" were to collect new results in algebraic number theory and analytic number theory (namely in the areas of ramification theory in algebraic number fields, class field theory, arithmetic functions, L-functions, modular forms and elliptic curves) and in some similar research areas (namely associative algebras, logical algebras, elementary number theory, combinatorics, difference equations, group rings and algebraic hyper-structures).
Author: Diana Savin
Publisher: Mdpi AG
ISBN: 9783036598598
Category : Computers
Languages : en
Pages : 0
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Book Description
Analytic number theory is a branch of number theory which uses methods from mathematical analysis in order to solve difficult problems about integers. Analytic number theory can be split into two major areas: multiplicative number theory and additive number theory. Bernhard Riemann made some very important contributions to the field of analytic number theory; among others, he investigated the Riemann zeta function, and he established its importance for understanding the distribution of prime numbers. A typical problem of analytic number theory is the enumeration of number-theoretic objects like primes, solutions of Diophantine equations, etc. Algebraic number theory on the other hand studies the arithmetic of algebraic number fields, i.e., the ring of integers of arbitrary number fields. It embraces, among others, the study of the ideals and of the group of units in the ring of integers and the extent to which unique factorization holds. The purpose and scope of this ''Special Issue" were to collect new results in algebraic number theory and analytic number theory (namely in the areas of ramification theory in algebraic number fields, class field theory, arithmetic functions, L-functions, modular forms and elliptic curves) and in some similar research areas (namely associative algebras, logical algebras, elementary number theory, combinatorics, difference equations, group rings and algebraic hyper-structures).
Author: Henri Cohen
Publisher: Springer Science & Business Media
ISBN: 3662029456
Category : Mathematics
Languages : en
Pages : 556
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Book Description
A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.
Author: Hailong Li
Publisher: World Scientific Publishing Company
ISBN: 9813231610
Category : Mathematics
Languages : en
Pages : 289
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Book Description
The aim of the book is to give a smooth analytic continuation from basic subjects including linear algebra, group theory, Hilbert space theory, etc. to number theory. With plenty of practical examples and worked-out exercises, and the scope ranging from these basic subjects made applicable to number-theoretic settings to advanced number theory, this book can then be read without tears. It will be of immense help to the reader to acquire basic sound skills in number theory and its applications.Number theory used to be described as the queen of mathematics, that is, there is no practical use. However, with the development of computers and the security of internet communications, the importance of number theory has been exponentially increasing daily. The raison d'être of the present book in this situation is that it is extremely reader-friendly while keeping the rigor of serious mathematics and in-depth analysis of practical applications to various subjects including control theory and pseudo-random number generation. The use of operators is prevailing rather abundantly in anticipation of applications to electrical engineering, allowing the reader to master these skills without much difficulty. It also delivers a very smooth bridging between elementary subjects including linear algebra and group theory (and algebraic number theory) for the reader to be well-versed in an efficient and effortless way. One of the main features of the book is that it gives several different approaches to the same topic, helping the reader to gain deeper insight and comprehension. Even just browsing through the materials would be beneficial to the reader.
Author: Richard A. Mollin
Publisher: CRC Press
ISBN: 9780849339875
Category : Mathematics
Languages : en
Pages : 472
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Book Description
Beginning with the arithmetic of the rational integers and proceeding to an introduction of algebraic number theory via quadratic orders, Fundamental Number Theory with Applications reveals intriguing new applications of number theory. This text details aspects of computer science related to cryptography factoring primality testing complexity analysis computer arithmetic computational number theory Fundamental Number Theory with Applications also covers: Carmichael numbers Dirichlet products Jacobsthal sums Mersenne primes perfect numbers powerful numbers self-contained numbers Numerous exercises are included, testing the reader's knowledge of the concepts covered, introducing new and interesting topics, and providing a venue to learn background material. Written by a professor and author who is an accomplished scholar in this field, this book provides the material essential for an introduction to the fundamentals of number theory.
Author: Jaime Gutierrez
Publisher: Springer
ISBN: 3319150812
Category : Computers
Languages : en
Pages : 222
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Book Description
Algebra and number theory have always been counted among the most beautiful mathematical areas with deep proofs and elegant results. However, for a long time they were not considered that important in view of the lack of real-life applications. This has dramatically changed: nowadays we find applications of algebra and number theory frequently in our daily life. This book focuses on the theory and algorithms for polynomials over various coefficient domains such as a finite field or ring. The operations on polynomials in the focus are factorization, composition and decomposition, basis computation for modules, etc. Algorithms for such operations on polynomials have always been a central interest in computer algebra, as it combines formal (the variables) and algebraic or numeric (the coefficients) aspects. The papers presented were selected from the Workshop on Computer Algebra and Polynomials, which was held in Linz at the Johann Radon Institute for Computational and Applied Mathematics (RICAM) during November 25-29, 2013, at the occasion of the Special Semester on Applications of Algebra and Number Theory.
Author: Henri Cohen
Publisher: Springer Science & Business Media
ISBN: 1441984895
Category : Mathematics
Languages : en
Pages : 591
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Book Description
Written by an authority with great practical and teaching experience in the field, this book addresses a number of topics in computational number theory. Chapters one through five form a homogenous subject matter suitable for a six-month or year-long course in computational number theory. The subsequent chapters deal with more miscellaneous subjects.
Author: Wieb Bosma
Publisher: Springer Science & Business Media
ISBN: 9401711089
Category : Mathematics
Languages : en
Pages : 326
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Book Description
Computers have stretched the limits of what is possible in mathematics. More: they have given rise to new fields of mathematical study; the analysis of new and traditional algorithms, the creation of new paradigms for implementing computational methods, the viewing of old techniques from a concrete algorithmic vantage point, to name but a few. Computational Algebra and Number Theory lies at the lively intersection of computer science and mathematics. It highlights the surprising width and depth of the field through examples drawn from current activity, ranging from category theory, graph theory and combinatorics, to more classical computational areas, such as group theory and number theory. Many of the papers in the book provide a survey of their topic, as well as a description of present research. Throughout the variety of mathematical and computational fields represented, the emphasis is placed on the common principles and the methods employed. Audience: Students, experts, and those performing current research in any of the topics mentioned above.
Author: Robert Rolland
Publisher: World Scientific
ISBN: 9814471666
Category : Mathematics
Languages : en
Pages : 530
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Book Description
This volume covers many topics, including number theory, Boolean functions, combinatorial geometry, and algorithms over finite fields. It contains many new, theoretical and applicable results, as well as surveys that were presented by the top specialists in these areas. New results include an answer to one of Serre's questions, posted in a letter to Top; cryptographic applications of the discrete logarithm problem related to elliptic curves and hyperelliptic curves; construction of function field towers; construction of new classes of Boolean cryptographic functions; and algorithmic applications of algebraic geometry.
Author: Marie José Bertin
Publisher: Springer
ISBN: 331917987X
Category : Mathematics
Languages : en
Pages : 215
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Book Description
Covering topics in graph theory, L-functions, p-adic geometry, Galois representations, elliptic fibrations, genus 3 curves and bad reduction, harmonic analysis, symplectic groups and mould combinatorics, this volume presents a collection of papers covering a wide swath of number theory emerging from the third iteration of the international Women in Numbers conference, “Women in Numbers - Europe” (WINE), held on October 14–18, 2013 at the CIRM-Luminy mathematical conference center in France. While containing contributions covering a wide range of cutting-edge topics in number theory, the volume emphasizes those concrete approaches that make it possible for graduate students and postdocs to begin work immediately on research problems even in highly complex subjects.
Author: M. Pohst
Publisher: Cambridge University Press
ISBN: 9780521596695
Category : Mathematics
Languages : en
Pages : 520
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Book Description
Now in paperback, this classic book is addresssed to all lovers of number theory. On the one hand, it gives a comprehensive introduction to constructive algebraic number theory, and is therefore especially suited as a textbook for a course on that subject. On the other hand many parts go beyond an introduction an make the user familliar with recent research in the field. For experimental number theoreticians new methods are developed and new results are obtained which are of great importance for them. Both computer scientists interested in higher arithmetic and those teaching algebraic number theory will find the book of value.
