Series: Lecture Notes in Mathematics , Vol. 1933
2008, Approx. 220 p., Softcover
ISBN: 978-3-540-77850-9
Due: April 2, 2008
About this book
This book sets out to state computationally verifiable initial conditions for predicting the immediate appearance of the guaranteed and fast convergence of iterative root finding methods. Attention is paid to iterative methods for simultaneous determination of polynomial zeros in the spirit of Smale's point estimation theory, introduced in 1986. Some basic concepts and Smale's theory for Newton's method, together with its modifications and higher-order methods, are presented in the first two chapters. The remaining chapters contain the recent author's results on initial conditions guaranteing convergence of a wide class of iterative methods for solving algebraic equations.
These conditions are of practical interest since they depend only on available data, the information of a function whose zeros are sought and initial approximations. The convergence approach presented can be applied in designing a package for the simultaneous approximation of polynomial zeros.
Table of contents
Basic Concepts.- Iterative Processes and Point Estimation Theory.- Point Estimation of Simultaneous methods.- Families of Simultaneous methods of Higher Order: Part I.- Families of Simultaneous Methods of Higher Order: Part II.- References.- Glossary.- Index.
Series: Universitext
2009, Approx. 120 p., Softcover
ISBN: 978-1-84800-184-8
Due: November 2008
About this textbook
Catalan’s Conjecture asserts that 8 and 9 are the only two consecutive perfect powers of natural numbers. Eugene Charles Catalan made his famous conjecture in 1844 in a letter to the editor of Crelle’s mathematical journal. Preda Mihailescu proved it 158 years later. This book contains a complete proof of Catalan’s conjecture. It discusses Mihailescu’s proof as well as the earlier work it makes use of.
To read the first few sections requires little more than a basic mathematical background and some knowledge of elementary number theory. The other sections involve Galois theory, some more algebraic number theory and a little bit of commutative algebra. The basic facts from the arithmetic of cyclotomic fields are discussed in the text.
Our exposition is self-contained with one small exception. This regards our discussion of an argument of Mihailescu’s that is based on Francesco Thaine’s famous theorem. Our proof of Thaine’s Theorem involves an application of Chebotarev’s Density Theorem to the Hilbert class field of a cyclotomic field. While we do provide a proof of Chebotarev’s Theroem, we do not prove the existence and the basic properties of the Hilbert class field. A proof would involve a good deal of class field theory and this is not included in these notes. We hope instead that the present application motivates the interested reader to study class field theory. This theory is exosed in several excellent textbooks.
Table of contents
Introduction.- The case "q = 2".- The case "p = 2".- The non-trivial solution.- Runge’s method.- Cassel’s theorem.- An obstruction group.- Small p or q .- The Stickelberger ideal.- The double Wieferich criterion.- The minus argument.- The plus argument I.- Semi-simple group rings.- The plus argument II.- The density theorem.- Thaine’s theorem.- Appendix: Euler’s theorem.- Bibliography.- Index
ISBN: 9781420066593
ISBN-10: 1420066595
Publication Date: 19/02/2008
Pages: 384
Trim Size: 6-1/8 x 9-1/4
Binding(s): Hardback
About the Title
An update of the most accessible introductory number theory text available, Fundamental Number Theory with Applications, Second Edition presents a mathematically rigorous yet easy-to-follow treatment of the fundamentals and applications of the subject. The substantial amount of reorganizing makes this edition clearer and more elementary in its coverage.
New to the Second Edition
・ Removal of all advanced material to be even more accessible in scope
・ New fundamental material, including partition theory, generating functions, and combinatorial number theory
・ Expanded coverage of random number generation, Diophantine analysis, and additive number theory
・ More applications to cryptography, primality testing, and factoring
・ An appendix on the recently discovered unconditional deterministic polynomial-time algorithm for primality testing
Taking a truly elementary approach to number theory, this text supplies the essential material for a first course on the subject. Placed in highlighted boxes to reduce distraction from the main text, nearly 70 biographies focus on major contributors to the field. The presentation of over 1,300 entries in the index maximizes cross-referencing so students can find data with ease
Table of Contents
Preface. Arithmetic of the Integers. Modular Arithmetic. Primitive Roots. Quadratic Residues. Simple Continued Fractions and Diophantine Approximation. Additivity-Sums of Powers. Diophantine Equations. Appendices. Solutions to Odd-Numbered Exercises. Bibliography. List of Symbols. Index
Foundations and TrendsR in Stochastic Systems: Vol. 1: No 2, pp 77-162.
Jounal special issue
Abstract
This manuscript summarizes a line of research that maps certain classical problems of discrete mathematics -- such as the Hamiltonian Cycle and the Traveling Salesman Problems -- into convex domains where continuum analysis can be carried out. Arguably, the inherent difficulty of these, now classical, problems stems precisely from the discrete nature of domains in which these problems are posed. The convexification of domains underpinning the reported results is achieved by assigning probabilistic interpretation to key elements of the original deterministic problems.
In particular, approaches summarized here build on a technique that embeds Hamiltonian Cycle and Traveling Salesman Problems in a structured singularly perturbed Markov Decision Process. The unifying idea is to interpret subgraphs traced out by deterministic policies (including Hamiltonian Cycles, if any) as extreme points of a convex polyhedron in a space filled with randomized policies.
The topic has now evolved to the point where there are many, both theoretical and algorithmic, results that exploit the nexus between graph theoretic structures and both probabilistic and algebraic entities of related Markov chains. The latter include moments of first return times, limiting frequencies of visits to nodes, or the spectra of certain matrices traditionally associated with the analysis of Markov chains. Numerous open questions and problems are described in the presentation.
Book Version
ISBN: 978-1-60198-088-5
These lecture notes review recent results on the high-frequency analysis of nonlinear Schrodinger equations in the presence of an external potential. The book consists of two relatively independent parts: WKB analysis, and caustic crossing. In the first part, the basic linear WKB theory is constructed and then extended to the nonlinear framework. The most difficult supercritical case is discussed in detail, together with some of its consequences concerning instability phenomena. Applications of WKB analysis to functional analysis, in particular to the Cauchy problem for nonlinear Schrodinger equations, are also given. In the second part, caustic crossing is described, especially when the caustic is reduced to a point, and the link with nonlinear scattering operators is investigated.
These notes are self-contained and combine selected articles written by the author over the past ten years in a coherent manner, with some simplified proofs. Examples and figures are provided to support the intuition, and comparisons with other equations such as the nonlinear wave equation are provided.
Contents:
WKB Analysis:
Preliminary Analysis
Weak Nonlinearity
Modulated Energy Functionals
Point-wise Description
Some Instability Phenomena
Caustic Crossing: The Case of Focal Points:
Caustic Crossing: Formal Analysis
Focal Point without External Potential
Focal Point with a Potential
Some Ideas for Supercritical Cases
Readership: Pure and applied mathematicians; physicists.
250pp (approx.) Pub. date: Scheduled Spring 2008
ISBN 978-981-279-312-6