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Author: L.-K. Hua
Publisher: Springer Science & Business Media
ISBN: 3642678297
Category : Mathematics
Languages : en
Pages : 252
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Book Description
Owing to the developments and applications of computer science, ma thematicians began to take a serious interest in the applications of number theory to numerical analysis about twenty years ago. The progress achieved has been both important practically as well as satisfactory from the theoretical view point. It'or example, from the seventeenth century till now, a great deal of effort was made in developing methods for approximating single integrals and there were only a few works on multiple quadrature until the 1950's. But in the past twenty years, a number of new methods have been devised of which the number theoretic method is an effective one. The number theoretic method may be described as follows. We use num ber theory to construct a sequence of uniformly distributed sets in the s dimensional unit cube G , where s ~ 2. Then we use the sequence to s reduce a difficult analytic problem to an arithmetic problem which may be calculated by computer. For example, we may use the arithmetic mean of the values of integrand in a given uniformly distributed set of G to ap s proximate the definite integral over G such that the principal order of the s error term is shown to be of the best possible kind, if the integrand satis fies certain conditions.
Author: L.-K. Hua
Publisher: Springer Science & Business Media
ISBN: 3642678297
Category : Mathematics
Languages : en
Pages : 252
Get Book Here
Book Description
Owing to the developments and applications of computer science, ma thematicians began to take a serious interest in the applications of number theory to numerical analysis about twenty years ago. The progress achieved has been both important practically as well as satisfactory from the theoretical view point. It'or example, from the seventeenth century till now, a great deal of effort was made in developing methods for approximating single integrals and there were only a few works on multiple quadrature until the 1950's. But in the past twenty years, a number of new methods have been devised of which the number theoretic method is an effective one. The number theoretic method may be described as follows. We use num ber theory to construct a sequence of uniformly distributed sets in the s dimensional unit cube G , where s ~ 2. Then we use the sequence to s reduce a difficult analytic problem to an arithmetic problem which may be calculated by computer. For example, we may use the arithmetic mean of the values of integrand in a given uniformly distributed set of G to ap s proximate the definite integral over G such that the principal order of the s error term is shown to be of the best possible kind, if the integrand satis fies certain conditions.
Author: S. K. Zaremba
Publisher: Academic Press
ISBN: 1483265161
Category : Mathematics
Languages : en
Pages : 504
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Book Description
Applications of Number Theory to Numerical Analysis contains the proceedings of the Symposium on Applications of Number Theory to Numerical Analysis, held in Quebec, Canada, on September 9-14, 1971, under the sponsorship of the University of Montreal's Center for Research in Mathematics. The symposium provided a forum for discussing number theory and its applications to numerical analysis, tackling topics ranging from methods used in estimating discrepancy to the structure of linear congruential sequences. Comprised of 17 chapters, this book begins by considering some combinatorial problems studied experimentally on computing machines. The discussion then turns to experiments on optimal coefficients; a distribution problem in finite sets; and the statistical interdependence of pseudo-random numbers generated by the linear congruential method. Subsequent chapters deal with lattice structure and reduced bases of random vectors generated by linear recurrences; modulo optimization problems and integer linear programming; equivalent forms of zero-one programs; and number theoretic foundations of finite precision arithmetic. This monograph will be of interest to students and practitioners in the field of applied mathematics.
Author: G. M. Phillips
Publisher: Elsevier
ISBN: 0080519121
Category : Mathematics
Languages : en
Pages : 461
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Book Description
Theory and Applications of Numerical Analysis is a self-contained Second Edition, providing an introductory account of the main topics in numerical analysis. The book emphasizes both the theorems which show the underlying rigorous mathematics andthe algorithms which define precisely how to program the numerical methods. Both theoretical and practical examples are included. - a unique blend of theory and applications - two brand new chapters on eigenvalues and splines - inclusion of formal algorithms - numerous fully worked examples - a large number of problems, many with solutions
Author: Michal Křížek
Publisher: Springer Nature
ISBN: 3030838994
Category : Mathematics
Languages : en
Pages : 342
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Book Description
This book provides an overview of many interesting properties of natural numbers, demonstrating their applications in areas such as cryptography, geometry, astronomy, mechanics, computer science, and recreational mathematics. In particular, it presents the main ideas of error-detecting and error-correcting codes, digital signatures, hashing functions, generators of pseudorandom numbers, and the RSA method based on large prime numbers. A diverse array of topics is covered, from the properties and applications of prime numbers, some surprising connections between number theory and graph theory, pseudoprimes, Fibonacci and Lucas numbers, and the construction of Magic and Latin squares, to the mathematics behind Prague’s astronomical clock. Introducing a general mathematical audience to some of the basic ideas and algebraic methods connected with various types of natural numbers, the book will provide invaluable reading for amateurs and professionals alike.
Author: Harald Niederreiter
Publisher: Springer
ISBN: 3319223216
Category : Mathematics
Languages : en
Pages : 452
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Book Description
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.
Author: Kai-Tai Fang
Publisher: CRC Press
ISBN: 9780412465208
Category : Mathematics
Languages : en
Pages : 356
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Book Description
This book is a survey of recent work on the application of number theory in statistics. The essence of number-theoretic methods is to find a set of points that are universally scattered over an s-dimensional unit cube. In certain circumstances this set can be used instead of random numbers in the Monte Carlo method. The idea can also be applied to other problems such as in experimental design. This book will illustrate the idea of number-theoretic methods and their application in statistics. The emphasis is on applying the methods to practical problems so only part-proofs of theorems are given.
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: M.R. Schroeder
Publisher: Springer Science & Business Media
ISBN: 3540265988
Category : Mathematics
Languages : en
Pages : 390
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Book Description
Number Theory in Science and Communication introductes non-mathematicians to the fascinating and diverse applications of number theory. This best-selling book stresses intuitive understanding rather than abstract theory. This revised fourth edition is augmented by recent advances in primes in progressions, twin primes, prime triplets, prime quadruplets and quintruplets, factoring with elliptic curves, quantum factoring, Golomb rulers and "baroque" integers.
Author: Brian Sutton
Publisher: SIAM
ISBN: 1611975700
Category : Mathematics
Languages : en
Pages : 448
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Book Description
This textbook develops the fundamental skills of numerical analysis: designing numerical methods, implementing them in computer code, and analyzing their accuracy and efficiency. A number of mathematical problems?interpolation, integration, linear systems, zero finding, and differential equations?are considered, and some of the most important methods for their solution are demonstrated and analyzed. Notable features of this book include the development of Chebyshev methods alongside more classical ones; a dual emphasis on theory and experimentation; the use of linear algebra to solve problems from analysis, which enables students to gain a greater appreciation for both subjects; and many examples and exercises. Numerical Analysis: Theory and Experiments is designed to be the primary text for a junior- or senior-level undergraduate course in numerical analysis for mathematics majors. Scientists and engineers interested in numerical methods, particularly those seeking an accessible introduction to Chebyshev methods, will also be interested in this book.
Author: Ellina Grigorieva
Publisher: Birkhäuser
ISBN: 3319909150
Category : Mathematics
Languages : en
Pages : 405
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Book Description
Through its engaging and unusual problems, this book demonstrates methods of reasoning necessary for learning number theory. Every technique is followed by problems (as well as detailed hints and solutions) that apply theorems immediately, so readers can solve a variety of abstract problems in a systematic, creative manner. New solutions often require the ingenious use of earlier mathematical concepts - not the memorization of formulas and facts. Questions also often permit experimental numeric validation or visual interpretation to encourage the combined use of deductive and intuitive thinking. The first chapter starts with simple topics like even and odd numbers, divisibility, and prime numbers and helps the reader to solve quite complex, Olympiad-type problems right away. It also covers properties of the perfect, amicable, and figurate numbers and introduces congruence. The next chapter begins with the Euclidean algorithm, explores the representations of integer numbers in different bases, and examines continued fractions, quadratic irrationalities, and the Lagrange Theorem. The last section of Chapter Two is an exploration of different methods of proofs. The third chapter is dedicated to solving Diophantine linear and nonlinear equations and includes different methods of solving Fermat’s (Pell’s) equations. It also covers Fermat’s factorization techniques and methods of solving challenging problems involving exponent and factorials. Chapter Four reviews the Pythagorean triple and quadruple and emphasizes their connection with geometry, trigonometry, algebraic geometry, and stereographic projection. A special case of Waring’s problem as a representation of a number by the sum of the squares or cubes of other numbers is covered, as well as quadratic residuals, Legendre and Jacobi symbols, and interesting word problems related to the properties of numbers. Appendices provide a historic overview of number theory and its main developments from the ancient cultures in Greece, Babylon, and Egypt to the modern day. Drawing from cases collected by an accomplished female mathematician, Methods in Solving Number Theory Problems is designed as a self-study guide or supplementary textbook for a one-semester course in introductory number theory. It can also be used to prepare for mathematical Olympiads. Elementary algebra, arithmetic and some calculus knowledge are the only prerequisites. Number theory gives precise proofs and theorems of an irreproachable rigor and sharpens analytical thinking, which makes this book perfect for anyone looking to build their mathematical confidence.
