Download or read book Bernoulli Numbers and Zeta Functions written by Tsuneo Arakawa and published by Springer. This book was released on 2014-07-11 with total page 278 pages. Available in PDF, EPUB and Kindle. Book excerpt: Two major subjects are treated in this book. The main one is the theory of Bernoulli numbers and the other is the theory of zeta functions. Historically, Bernoulli numbers were introduced to give formulas for the sums of powers of consecutive integers. The real reason that they are indispensable for number theory, however, lies in the fact that special values of the Riemann zeta function can be written by using Bernoulli numbers. This leads to more advanced topics, a number of which are treated in this book: Historical remarks on Bernoulli numbers and the formula for the sum of powers of consecutive integers; a formula for Bernoulli numbers by Stirling numbers; the Clausen–von Staudt theorem on the denominators of Bernoulli numbers; Kummer's congruence between Bernoulli numbers and a related theory of p-adic measures; the Euler–Maclaurin summation formula; the functional equation of the Riemann zeta function and the Dirichlet L functions, and their special values at suitable integers; various formulas of exponential sums expressed by generalized Bernoulli numbers; the relation between ideal classes of orders of quadratic fields and equivalence classes of binary quadratic forms; class number formula for positive definite binary quadratic forms; congruences between some class numbers and Bernoulli numbers; simple zeta functions of prehomogeneous vector spaces; Hurwitz numbers; Barnes multiple zeta functions and their special values; the functional equation of the doub le zeta functions; and poly-Bernoulli numbers. An appendix by Don Zagier on curious and exotic identities for Bernoulli numbers is also supplied. This book will be enjoyable both for amateurs and for professional researchers. Because the logical relations between the chapters are loosely connected, readers can start with any chapter depending on their interests. The expositions of the topics are not always typical, and some parts are completely new.
Download or read book A Study of Bernoulli s Numbers written by Ethan Ogden Allen and published by . This book was released on 1940 with total page 58 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book A Study of Bernoulli Numbers written by Neal Douglas Moore and published by . This book was released on 1968 with total page 58 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book About Bernoulli s Numbers written by BENCZE MIHÁLY and published by Infinite Study. This book was released on with total page 4 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many methods to compute the sum ofthe same powers of the first n natural numbers (see ([4]) are well-known.
Download or read book A History of Parametric Statistical Inference from Bernoulli to Fisher 1713 1935 written by Anders Hald and published by Springer Science & Business Media. This book was released on 2008-08-24 with total page 221 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers a detailed history of parametric statistical inference. Covering the period between James Bernoulli and R.A. Fisher, it examines: binomial statistical inference; statistical inference by inverse probability; the central limit theorem and linear minimum variance estimation by Laplace and Gauss; error theory, skew distributions, correlation, sampling distributions; and the Fisherian Revolution. Lively biographical sketches of many of the main characters are featured throughout, including Laplace, Gauss, Edgeworth, Fisher, and Karl Pearson. Also examined are the roles played by DeMoivre, James Bernoulli, and Lagrange.
Download or read book Classical Theory of Algebraic Numbers written by Paulo Ribenboim and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 676 pages. Available in PDF, EPUB and Kindle. Book excerpt: The exposition of the classical theory of algebraic numbers is clear and thorough, and there is a large number of exercises as well as worked out numerical examples. A careful study of this book will provide a solid background to the learning of more recent topics.
Download or read book The Art of Conjecturing Together with Letter to a Friend on Sets in Court Tennis written by Jacob Bernoulli and published by JHU Press. This book was released on 2006 with total page 468 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Part I reprints and reworks Huygens's On Reckoning in Games of Chance. Part II offers a thorough treatment of the mathematics of combinations and permutations, including the numbers since known as "Bernoulli numbers." In Part III, Bernoulli solves more complicated problems of games of chance using that mathematics. In the final part, Bernoulli's crowning achievement in mathematical probability becomes manifest he applies the mathematics of games of chance to the problems of epistemic probability in civil, moral, and economic matters, proving what we now know as the weak law of large numbers."
Download or read book Handbook of Number Theory I written by József Sándor and published by Springer Science & Business Media. This book was released on 2005-11-17 with total page 638 pages. Available in PDF, EPUB and Kindle. Book excerpt: This handbook covers a wealth of topics from number theory, special attention being given to estimates and inequalities. As a rule, the most important results are presented, together with their refinements, extensions or generalisations. These may be applied to other aspects of number theory, or to a wide range of mathematical disciplines. Cross-references provide new insight into fundamental research. Audience: This is an indispensable reference work for specialists in number theory and other mathematicians who need access to some of these results in their own fields of research.
Download or read book Nonlocal Euler Bernoulli Beam Theories written by Jingkai Chen and published by Springer Nature. This book was released on 2021-02-27 with total page 59 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a comparative study on the static responses of the Euler-Bernoulli beam governed by nonlocal theories, including the Eringen’s stress-gradient beam theory, the Mindlin’s strain-gradient beam theory, the higher-order beam theory and the peridynamic beam theory. Benchmark examples are solved analytically and numerically using these nonlocal beam equations, including the simply-supported beam, the clamped-clamped beam and the cantilever beam. Results show that beam deformations governed by different nonlocal theories at different boundary conditions show complex behaviors. Specifically, the Eringen’s stress-gradient beam equation and the peridynamic beam equation yield a much softer beam deformation for simply-supported beam and clamped-clamped beam, while the beam governed by the Mindlin’s strain-gradient beam equation is much stiffer. The cantilever beam exhibits a completely different behavior. The higher-order beam equation can be stiffer or softer depending on the values of the two nonlocal parameters. Moreover, the deformation fluctuation of the truncated order peridynamic beam equation is observed and explained from the singularity aspect of the solution expression. This research casts light on the fundamental explanation of nonlocal beam theories in nano-electromechanical systems.
Download or read book A Classical Introduction to Modern Number Theory written by K. Ireland and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 355 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a revised and greatly expanded version of our book Elements of Number Theory published in 1972. As with the first book the primary audience we envisage consists of upper level undergraduate mathematics majors and graduate students. We have assumed some familiarity with the material in a standard undergraduate course in abstract algebra. A large portion of Chapters 1-11 can be read even without such background with the aid of a small amount of supplementary reading. The later chapters assume some knowledge of Galois theory, and in Chapters 16 and 18 an acquaintance with the theory of complex variables is necessary. Number theory is an ancient subject and its content is vast. Any intro ductory book must, of necessity, make a very limited selection from the fascinat ing array of possible topics. Our focus is on topics which point in the direction of algebraic number theory and arithmetic algebraic geometry. By a careful selection of subject matter we have found it possible to exposit some rather advanced material without requiring very much in the way oftechnical background. Most of this material is classical in the sense that is was dis covered during the nineteenth century and earlier, but it is also modern because it is intimately related to important research going on at the present time.
Download or read book Bernoulli Numbers and Zeta Functions written by Tsuneo Arakawa and published by Springer. This book was released on 2014-07-16 with total page 274 pages. Available in PDF, EPUB and Kindle. Book excerpt: Two major subjects are treated in this book. The main one is the theory of Bernoulli numbers and the other is the theory of zeta functions. Historically, Bernoulli numbers were introduced to give formulas for the sums of powers of consecutive integers. The real reason that they are indispensable for number theory, however, lies in the fact that special values of the Riemann zeta function can be written by using Bernoulli numbers. This leads to more advanced topics, a number of which are treated in this book: Historical remarks on Bernoulli numbers and the formula for the sum of powers of consecutive integers; a formula for Bernoulli numbers by Stirling numbers; the Clausen–von Staudt theorem on the denominators of Bernoulli numbers; Kummer's congruence between Bernoulli numbers and a related theory of p-adic measures; the Euler–Maclaurin summation formula; the functional equation of the Riemann zeta function and the Dirichlet L functions, and their special values at suitable integers; various formulas of exponential sums expressed by generalized Bernoulli numbers; the relation between ideal classes of orders of quadratic fields and equivalence classes of binary quadratic forms; class number formula for positive definite binary quadratic forms; congruences between some class numbers and Bernoulli numbers; simple zeta functions of prehomogeneous vector spaces; Hurwitz numbers; Barnes multiple zeta functions and their special values; the functional equation of the double zeta functions; and poly-Bernoulli numbers. An appendix by Don Zagier on curious and exotic identities for Bernoulli numbers is also supplied. This book will be enjoyable both for amateurs and for professional researchers. Because the logical relations between the chapters are loosely connected, readers can start with any chapter depending on their interests. The expositions of the topics are not always typical, and some parts are completely new.
Download or read book Number Theory written by Henri Cohen and published by Springer Science & Business Media. This book was released on 2008-12-17 with total page 619 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book deals with several aspects of what is now called "explicit number theory." The central theme 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 local aspect, global aspect, and the third aspect is the theory of zeta and L-functions. This last aspect can be considered as a unifying theme for the whole subject.
Download or read book How Euler Did Even More written by C. Edward Sandifer and published by The Mathematical Association of America. This book was released on 2014-11-19 with total page 254 pages. Available in PDF, EPUB and Kindle. Book excerpt: Sandifer has been studying Euler for decades and is one of the world’s leading experts on his work. This volume is the second collection of Sandifer’s “How Euler Did It” columns. Each is a jewel of historical and mathematical exposition. The sum total of years of work and study of the most prolific mathematician of history, this volume will leave you marveling at Euler’s clever inventiveness and Sandifer’s wonderful ability to explicate and put it all in context.
Download or read book Sums of Squares of Integers written by Carlos J. Moreno and published by CRC Press. This book was released on 2005-12-09 with total page 368 pages. Available in PDF, EPUB and Kindle. Book excerpt: Sums of Squares of Integers covers topics in combinatorial number theory as they relate to counting representations of integers as sums of a certain number of squares. The book introduces a stimulating area of number theory where research continues to proliferate. It is a book of "firsts" - namely it is the first book to combine Liouville's elementary methods with the analytic methods of modular functions to study the representation of integers as sums of squares. It is the first book to tell how to compute the number of representations of an integer n as the sum of s squares of integers for any s and n. It is also the first book to give a proof of Szemeredi's theorem, and is the first number theory book to discuss how the modern theory of modular forms complements and clarifies the classical fundamental results about sums of squares. The book presents several existing, yet still interesting and instructive, examples of modular forms. Two chapters develop useful properties of the Bernoulli numbers and illustrate arithmetic progressions, proving the theorems of van der Waerden, Roth, and Szemeredi. The book also explains applications of the theory to three problems that lie outside of number theory in the areas of cryptanalysis, microwave radiation, and diamond cutting. The text is complemented by the inclusion of over one hundred exercises to test the reader's understanding.
Download or read book Multiplicative Number Theory I written by Hugh L. Montgomery and published by Cambridge University Press. This book was released on 2007 with total page 574 pages. Available in PDF, EPUB and Kindle. Book excerpt: A 2006 text based on courses taught successfully over many years at Michigan, Imperial College and Pennsylvania State.
Download or read book Introduction to Cyclotomic Fields written by Lawrence C. Washington and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 504 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text on a central area of number theory covers p-adic L-functions, class numbers, cyclotomic units, Fermat’s Last Theorem, and Iwasawa’s theory of Z_p-extensions. This edition contains a new chapter on the work of Thaine, Kolyvagin, and Rubin, including a proof of the Main Conjecture, as well as a chapter on other recent developments, such as primality testing via Jacobi sums and Sinnott’s proof of the vanishing of Iwasawa’s f-invariant.
Download or read book Concrete Mathematics written by Ronald L. Graham and published by Addison-Wesley Professional. This book was released on 1994-02-28 with total page 811 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills - the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data. It is an indispensable text and reference not only for computer scientists - the authors themselves rely heavily on it! - but for serious users of mathematics in virtually every discipline. Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. "More concretely," the authors explain, "it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems." The subject matter is primarily an expansion of the Mathematical Preliminaries section in Knuth's classic Art of Computer Programming, but the style of presentation is more leisurely, and individual topics are covered more deeply. Several new topics have been added, and the most significant ideas have been traced to their historical roots. The book includes more than 500 exercises, divided into six categories. Complete answers are provided for all exercises, except research problems, making the book particularly valuable for self-study. Major topics include: Sums Recurrences Integer functions Elementary number theory Binomial coefficients Generating functions Discrete probability Asymptotic methods This second edition includes important new material about mechanical summation. In response to the widespread use of the first edition as a reference book, the bibliography and index have also been expanded, and additional nontrivial improvements can be found on almost every page. Readers will appreciate the informal style of Concrete Mathematics. Particularly enjoyable are the marginal graffiti contributed by students who have taken courses based on this material. The authors want to convey not only the importance of the techniques presented, but some of the fun in learning and using them.
