Sequences


Book Description

THIS volume is concerned with a substantial branch of number theory of which no connected account appears to exist; we describe the general nature of the constituent topics in the introduction. Although some excellent surveys dealing with limited aspects of the subject under con sideration have been published, the literature as a whole is far from easy to study. This is due in part to the extent of the literature; it is necessary to thread one's way through a maze of results, a complicated structure of inter-relationships, and many conflicting notations. In addition, however, not all the original papers are free from obscurities, and consequently some of these papers are difficult (a few even exceed ingly difficult) to master. We try to give a readable and coherent account of the subject, con taining a cross-section of the more interesting results. We felt that it would have been neither practicable nor desirable to attempt a compre hensive account; we treat each aspect of the subject from some special point of view, and select results accordingly. Needless to say, this approach entails the omission of many interesting and important results (quite apart from defects in the selection due to errors of judgement on our part). Those results selected for inclusion are, however, proved in complete detail and without the assumption of any prior knowledge on the part of the reader.




Automatic Sequences


Book Description

Uniting dozens of seemingly disparate results from different fields, this book combines concepts from mathematics and computer science to present the first integrated treatment of sequences generated by 'finite automata'. The authors apply the theory to the study of automatic sequences and their generalizations, such as Sturmian words and k-regular sequences. And further, they provide applications to number theory (particularly to formal power series and transcendence in finite characteristic), physics, computer graphics, and music. Starting from first principles wherever feasible, basic results from combinatorics on words, numeration systems, and models of computation are discussed. Thus this book is suitable for graduate students or advanced undergraduates, as well as for mature researchers wishing to know more about this fascinating subject. Results are presented from first principles wherever feasible, and the book is supplemented by a collection of 460 exercises, 85 open problems, and over 1600 citations to the literature.




Recurrence Sequences


Book Description

Recurrence sequences are of great intrinsic interest and have been a central part of number theory for many years. Moreover, these sequences appear almost everywhere in mathematics and computer science. This book surveys the modern theory of linear recurrence sequences and their generalizations. Particular emphasis is placed on the dramatic impact that sophisticated methods from Diophantine analysis and transcendence theory have had on the subject. Related work on bilinear recurrences and an emerging connection between recurrences and graph theory are covered. Applications and links to other areas of mathematics are described, including combinatorics, dynamical systems and cryptography, and computer science. The book is suitable for researchers interested in number theory, combinatorics, and graph theory.




Handbook of MRI Pulse Sequences


Book Description

Magnetic Resonance Imaging (MRI) is among the most important medical imaging techniques available today. There is an installed base of approximately 15,000 MRI scanners worldwide. Each of these scanners is capable of running many different "pulse sequences", which are governed by physics and engineering principles, and implemented by software programs that control the MRI hardware. To utilize an MRI scanner to the fullest extent, a conceptual understanding of its pulse sequences is crucial. Handbook of MRI Pulse Sequences offers a complete guide that can help the scientists, engineers, clinicians, and technologists in the field of MRI understand and better employ their scanner. Explains pulse sequences, their components, and the associated image reconstruction methods commonly used in MRI Provides self-contained sections for individual techniques Can be used as a quick reference guide or as a resource for deeper study Includes both non-mathematical and mathematical descriptions Contains numerous figures, tables, references, and worked example problems




Uniform Distribution of Sequences


Book Description

The theory of uniform distribution began with Hermann Weyl's celebrated paper of 1916. In later decades, the theory moved beyond its roots in diophantine approximations to provide common ground for topics as diverse as number theory, probability theory, functional analysis, and topological algebra. This book summarizes the theory's development from its beginnings to the mid-1970s, with comprehensive coverage of both methods and their underlying principles. A practical introduction for students of number theory and analysis as well as a reference for researchers in the field, this book covers uniform distribution in compact spaces and in topological groups, in addition to examinations of sequences of integers and polynomials. Notes at the end of each section contain pertinent bibliographical references and a brief survey of additional results. Exercises range from simple applications of theorems to proofs of propositions that expand upon results stated in the text.




A Handbook of Integer Sequences


Book Description

A Handbook of Integer Sequences contains a main table of 2300 sequences of integers that are collected from all branches of mathematics and science. This handbook describes how to use the main table and provides methods for analyzing and describing unknown and important sequences. This compilation also serves as an index to the literature for locating references on a particular problem and quickly finds numbers such as 712, number of partitions of 30, 18th Catalan number, or expansion of ? to 60 decimal places. Other topics include the method of differences, self-generating sequences, polyominoes, permutations, and puzzle sequences. This publication is a good source for students and researchers who are confronted with strange and important sequences.




Sequences and Series


Book Description

This book is addressed to all those who, after finishing the high school, wish a practical initiation in the domain of sequences and series. This is the first volume of the series "Mathematics for future engineers." To provide useful tools for (future) engineers and for specialists, in general, we put into evidence some practical applications of sequences and series (e.g., how to apply Lagrange's and Taylor's formulas to the calculus of approximations, the catenary expressed in terms of hyperbolic functions, etc.). We tried to make the involved mathematics as attractive as possible, by simplifying the presentation without loosing the mathematical rigor of the results. To increase accessibility and to encourage the reader to get a technical know-how about sequences and series, we provided for each newly introduced notion a series of applications and solved problems; each chapter ends by a section containing exercises and problems, each one of these being accompanied by hints and answers. The references contain, along with books, some links with sites which can be helpful for the reader.




Computer Analysis of Sequence Data


Book Description

DNA sequencing has become increasingly efficient over the years, resulting in an enormous increase in the amount of data gener ated. In recent years, the focus of sequencing has shifted, from being the endpoint of a project, to being a starting point. This is especially true for such major initiatives as the human genome project, where vast tracts of DNA of unknown function are sequenced. This sheer volume of available data makes advanced computer methods essen tial to analysis, and a familiarity with computers and sequence analy sis software a vital requirement for the researcher involved with DNA sequencing. Even for nonsequencers, a familiarity with sequence analysis software can be important. For instance, gene sequences already present in the databases can be extremely useful in the design of cloning and genetic manipulation experiments. This two-part work on Computer Analysis of Sequence Data is designed to be a practical aid to the researcher who uses computers for the acquisition, storage, or analysis of nucleic acid (and/or pro tein) sequences. Each chapter is written such that a competent scien tist with basic computer literacy can carry out the procedure successfully at the first attempt by simply following the detailed prac tical instructions that have been described by the author. A Notes section, which is included at the end of each chapter, provides advice on overcoming the common problems and pitfalls sometimes encoun tered by users of the sequence analysis software.




Difference Sets, Sequences and their Correlation Properties


Book Description

The explanation of the formal duality of Kerdock and Preparata codes is one of the outstanding results in the field of applied algebra in the last few years. This result is related to the discovery of large sets of quad riphase sequences over Z4 whose correlation properties are better than those of the best binary sequences. Moreover, the correlation properties of sequences are closely related to difference properties of certain sets in (cyclic) groups. It is the purpose of this book to illustrate the connection between these three topics. Most articles grew out of lectures given at the NATO Ad vanced Study Institute on "Difference sets, sequences and their correlation properties". This workshop took place in Bad Windsheim (Germany) in August 1998. The editors thank the NATO Scientific Affairs Division for the generous support of this workshop. Without this support, the present collection of articles would not have been realized.




Algorithms on Strings, Trees, and Sequences


Book Description

String algorithms are a traditional area of study in computer science. In recent years their importance has grown dramatically with the huge increase of electronically stored text and of molecular sequence data (DNA or protein sequences) produced by various genome projects. This book is a general text on computer algorithms for string processing. In addition to pure computer science, the book contains extensive discussions on biological problems that are cast as string problems, and on methods developed to solve them. It emphasises the fundamental ideas and techniques central to today's applications. New approaches to this complex material simplify methods that up to now have been for the specialist alone. With over 400 exercises to reinforce the material and develop additional topics, the book is suitable as a text for graduate or advanced undergraduate students in computer science, computational biology, or bio-informatics. Its discussion of current algorithms and techniques also makes it a reference for professionals.