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Traditionally a subject of number theory, continued fractions appear in dynamical systems, algebraic geometry, topology, and even celestial mechanics. The rise of computational geometry has resulted in renewed interest in multidimensional generalizations of continued fractions. Numerous classical theorems have been extended to the multidimensional case, casting light on phenomena in diverse areas of mathematics. This book introduces a new geometric vision of continued fractions. It covers several applications to questions related to such areas as Diophantine approximation, algebraic number theory, and toric geometry. The reader will find an overview of current progress in the geometric theory of multidimensional continued fractions accompanied by currently open problems. Whenever possible, we illustrate geometric constructions with figures and examples. Each chapter has exercises useful for undergraduate or graduate courses.

Author : Ana Escuder
ISBN : 1109800401
Genre : Approximation theory
File Size : 47.42 MB
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This paper is divided in three sections. The first deals with the basic theory and definitions of continued fractions. The second shows how to use technology, especially Dynamic Geometry Software to make the basic theory accessible to students in middle and high school. The third section gives some reflections of my experience working with students on continued fractions.

Author : Andrew M Rockett
ISBN : 9789813103412
Genre : Mathematics
File Size : 35.63 MB
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This book presents the arithmetic and metrical theory of regular continued fractions and is intended to be a modern version of A. Ya. Khintchine's classic of the same title. Besides new and simpler proofs for many of the standard topics, numerous numerical examples and applications are included (the continued fraction of e, Ostrowski representations and t-expansions, period lengths of quadratic surds, the general Pell's equation, homogeneous and inhomogeneous diophantine approximation, Hall's theorem, the Lagrange and Markov spectra, asymmetric approximation, etc). Suitable for upper level undergraduate and beginning graduate students, the presentation is self-contained and the metrical results are developed as strong laws of large numbers. Request Inspection Copy

Author : M. Iosifescu
ISBN : 9789401599405
Genre : Mathematics
File Size : 62.58 MB
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This monograph is intended to be a complete treatment of the metrical the ory of the (regular) continued fraction expansion and related representations of real numbers. We have attempted to give the best possible results known so far, with proofs which are the simplest and most direct. The book has had a long gestation period because we first decided to write it in March 1994. This gave us the possibility of essentially improving the initial versions of many parts of it. Even if the two authors are different in style and approach, every effort has been made to hide the differences. Let 0 denote the set of irrationals in I = [0,1]. Define the (reg ular) continued fraction transformation T by T (w) = fractional part of n 1/w, w E O. Write T for the nth iterate of T, n E N = {O, 1, ... }, n 1 with TO = identity map. The positive integers an(w) = al(T - (W)), n E N+ = {1,2··· }, where al(w) = integer part of 1/w, w E 0, are called the (regular continued fraction) digits of w. Writing . for arbitrary indeterminates Xi, 1 :::; i :::; n, we have w = lim [al(w),··· , an(w)], w E 0, n--->oo thus explaining the name of T. The above equation will be also written as w = lim [al(w), a2(w),···], w E O.

Author : Roger Fenn
ISBN : 1852330589
Genre : Mathematics
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Intended to introduce readers to the major geometrical topics taught at undergraduate level in a manner that is both accessible and rigorous, the author uses world measurement as a synonym for geometry - hence the importance of numbers, coordinates and their manipulation - and has included over 300 exercises, with answers to most of them.

Author : David Coeurjolly
ISBN : 9783540791256
Genre : Computers
File Size : 76.62 MB
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This book constitutes the refereed proceedings of the 14th IAPR TC-18 International Conference on Discrete Geometry for Computer Imagery, DGCI 2008, held in Lyon, France, in April 2008. The 23 revised full papers and 22 revised poster papers presented together with 3 invited papers were carefully reviewed and selected from 76 submissions. The papers are organized in topical sections on models for discrete geometry, discrete and combinatorial topology, geometric transforms, discrete shape representation, recognition and analysis, discrete tomography, morphological analysis, discrete modelling and visualization, as well as discrete and combinatorial tools for image segmentation and analysis.

Author : Benjamin Bold
ISBN : 9780486137636
Genre : Science
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Delve into the development of modern mathematics and match wits with Euclid, Newton, Descartes, and others. Each chapter explores an individual type of challenge, with commentary and practice problems. Solutions.

Author : Hubert Stanley Wall
ISBN : 9780486823690
Genre : Mathematics
File Size : 73.77 MB
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One of the most authoritative and comprehensive books on continued fractions, this monograph presents a unified theory correlating certain parts and applications of the subject within a larger analytic structure. 1948 edition.

Author : Alexander I. Bobenko
ISBN : 0198501609
Genre : Mathematics
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Recent interactions between the fields of geometry, classical and quantum dynamical systems, and visualization of geometric objects such as curves and surfaces have led to the observation that most concepts of surface theory and of the theory of integrable systems have natural discreteanalogues. These are characterized by the property that the corresponding difference equations are integrable, and has led in turn to some important applications in areas of condensed matter physics and quantum field theory, amongst others. The book combines the efforts of a distinguished team ofauthors from various fields in mathematics and physics in an effort to provide an overview of the subject. The mathematical concepts of discrete geometry and discrete integrable systems are firstly presented as fundamental and valuable theories in themselves. In the following part these concepts areput into the context of classical and quantum dynamics.

Author : Da Ruan
ISBN : 379081251X
Genre : Business & Economics
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This book is an organized edited collection of twenty-one contributed chapters covering nuclear engineering applications of fuzzy systems, neural networks, genetic algorithms and other soft computing techniques. All chapters are either updated review or original contributions by leading researchers written exclusively for this volume. The volume highlights the advantages of applying fuzzy systems and soft computing in nuclear engineering, which can be viewed as complementary to traditional methods. As a result, fuzzy sets and soft computing provide a powerful tool for solving intricate problems pertaining in nuclear engineering. Each chapter of the book is self-contained and also indicates the future research direction on this topic of applications of fuzzy systems and soft computing in nuclear engineering.

Author : A. Schrijver
ISBN : STANFORD:36105032983657
Genre : Mathematics
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Introduction and preliminaries; Problems, algorithms, and complexity. Linear algebra; Linear algebra and complexity; Notes on linear algebra; Lattices and linear diophantine equations; Theory of lattices and linear diophantine equations; Algorithms for linear diophantine equations; Diophantine approximations and basis reduction; Notes on lattices and linear diophantine equations; Polyhedra, linear inequalities, and linear programming; Fundamental concepts and results on polyhedra, linear inequalities, and linear programming; The structure of polyhedra; Polarity, and blocking and anti-blocking polyhedra; Sizes and the theoretical complexity of linear inequalities and linear programming; The simplex method; Primal-dual, elimination, and relaxation methods; Khachiyan's method for linear programming; The ellipsoid method for polyhedra more generally; Further polynomiality results in linear programming; Notes on polyhedra, linear inequalities, and linear programming; Integer linear programming; Introduction to integer linear programming; Estimates in integer linear programming; The complexity of integer linear programming; Totally unimodular matrices: fundamental properties and examples; Recognizing total unimodularity; Further theory related to total unimodularity; Integral polyhedra and total dual integrality; Cutting planes; Further methods in integer linear programming; Historical and further notes on integer linear programming; References.

Both classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces, convex sets, etc., crucial ideas and above all abstract concepts needed for attaining the results are elucidated. These are conceptual notions, each built "above" the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs: the 'ladder' in the Old Testament, that angels ascended and descended... In all this, the aim of the book is to demonstrate to readers the unceasingly renewed spirit of geometry and that even so-called "elementary" geometry is very much alive and at the very heart of the work of numerous contemporary mathematicians. It is also shown that there are innumerable paths yet to be explored and concepts to be created. The book is visually rich and inviting, so that readers may open it at random places and find much pleasure throughout according their own intuitions and inclinations. Marcel Berger is t he author of numerous successful books on geometry, this book once again is addressed to all students and teachers of mathematics with an affinity for geometry.

This volume contains a fairly complete picture of the geometry of numbers, including relations to other branches of mathematics such as analytic number theory, diophantine approximation, coding and numerical analysis. It deals with convex or non-convex bodies and lattices in euclidean space, etc. This second edition was prepared jointly by P.M. Gruber and the author of the first edition. The authors have retained the existing text (with minor corrections) while adding to each chapter supplementary sections on the more recent developments. While this method may have drawbacks, it has the definite advantage of showing clearly where recent progress has taken place and in what areas interesting results may be expected in the future.

Author : René Descartes
ISBN : 9780486158174
Genre : Mathematics
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The great work that founded analytical geometry. Includes the original French text, Descartes' own diagrams, and the definitive Smith-Latham translation. "The greatest single step ever made in the progress of the exact sciences." — John Stuart Mill.

Author : T. J. Willmore
ISBN : 9780486282107
Genre : Mathematics
File Size : 86.69 MB
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This text employs vector methods to explore the classical theory of curves and surfaces. Topics include basic theory of tensor algebra, tensor calculus, calculus of differential forms, and elements of Riemannian geometry. 1959 edition.