| Description |
1 online resource (xvii, 405 pages) : illustrations |
| Series |
Algorithms and computation in mathematics, 1431-1550 ; volume 26 |
|
Algorithms and computation in mathematics ; volume 26.
|
| Contents |
Introduction -- Part 1. Regular continued fractions. Classical notions and definitions -- On integer geometry -- Geometry of regular continued fractions -- Complete invariant of integer angles -- Integer trigonometry for integer angles -- Integer angles of integer triangles -- Continued fractions and SL(2; Z) conjugacy classes. Elements of Gauss Reduction Theory. Markoff spectrum -- Lagrange theorem -- Gauss-Kuzmin statistics -- Geometric approximation aspects -- Geometry of continued fractions with real elements and the second Kepler law -- Integer angles of polygons and global relations to toric singularities -- Part 2. Klein polyhedra. Basic notions and definitions of multidimensional integer geometry -- On empty simplices, pyramids, parallelepipeds -- Multidimensional continued fractions in the sense of Klein -- Dirichlet groups and lattice reduction -- Periodicity of Klein polyhedra. Generalization of Lagrange theorem -- Multidimensional Gauss-Kuzmin statistics -- On construction of multidimensional continued fractions -- Gauss Reduction in higher dimensions -- Decomposable forms. Relation to Littlewood and Oppenheim conjectures -- Approximation of maximal commutative subgroups -- Other generalizations of continued fractions |
| Summary |
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 |
| Bibliography |
Includes bibliographical references and index |
| Notes |
English |
|
Online resource; title from PDF title page (SpringerLink, viewed September 25, 2013) |
| In |
Springer eBooks |
| Subject |
Continued fractions.
|
|
Fracciones continuas
|
|
Continued fractions
|
| Form |
Electronic book
|
| ISBN |
9783642393686 |
|
3642393683 |
|
3642393675 |
|
9783642393679 |
|
9783642444241 |
|
3642444245 |
|
9783642393693 |
|
3642393691 |
|
