LEADER 05022nam 22006975 450 001 9910574044103321 005 20250610110404.0 010 $a9783662652770$b(electronic bk.) 010 $z9783662652763 024 7 $a10.1007/978-3-662-65277-0 035 $a(MiAaPQ)EBC7001426 035 $a(Au-PeEL)EBL7001426 035 $a(CKB)22924355500041 035 $a(PPN)26914837X 035 $a(DE-He213)978-3-662-65277-0 035 $a(MiAaPQ)EBC7255579 035 $a(EXLCZ)9922924355500041 100 $a20220528d2022 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aGeometry of Continued Fractions /$fby Oleg N. Karpenkov 205 $a2nd ed. 2022. 210 1$aBerlin, Heidelberg :$cSpringer Berlin Heidelberg :$cImprint: Springer,$d2022. 215 $a1 online resource (462 pages) 225 1 $aAlgorithms and Computation in Mathematics,$x2512-3254 ;$v26 311 08$aPrint version: Karpenkov, Oleg N. Geometry of Continued Fractions Berlin, Heidelberg : Springer Berlin / Heidelberg,c2022 9783662652763 320 $aIncludes bibliographical references and index. 327 $aPart 1. Regular continued fractions: Chapter 1. Classical notions and definitions -- Chapter 2. On integer geometry -- Chapter 3. Geometry of regular continued fractions -- Chapter 4. Complete invariant of integer angles -- Chapter 5. Integer trigonometry for integer angles -- Chapter 6. Integer angles of integer triangles -- Chapter 7. Quadratic forms and Makov spectrum. -- Chapter 8. Geometric continued fractions -- Chapter 9. Continuant representation of GL(2,Z) Matrices -- Chapter 10. Semigroup of Reduced Matrices -- Chapter 11. Elements of Gauss reduction theory -- Chapter 12. Lagrange?s theorem -- Gauss-Kuzmin statistics -- Chapter 14. Geometric aspects of approximation -- Chapter 15. Geometry of continued fractions with real elements and Kepler?s second law -- Chapter 16. Extended integer angles and their summation -- Chapter 17. Integer angles of polygons and global relations for toric singularities -- Part II. Multidimensional continued fractions -- Chapter 18. Basic notations and definitions of multidimensional integer geometry -- Chapter 19. On empty simplices, pyramids, parallelepipeds -- Chapter 20. Multidimensional continued fractions in the sense of Klein -- Chapter 21. Dirichlet groups and lattice reduction -- Chapter 22. Periodicity of Klein polyhedral. Generalization of Lagrange?s Theorem -- Chapter 23. Multidimensional Gauss-Kuzmin Statistics -- Chapter 24. On the construction of multidimensional continued fractions -- Chapter 25. Gauss reduction in higher dimensions. Chapter 26. Approximation of maximal commutative subgroups -- Capter 27. Other generalizations of continued fractions. References. Index. 330 $aThis book introduces a new geometric vision of continued fractions. It covers several applications to questions related to such areas as Diophantine approximation, algebraic number theory, and toric geometry. The second edition now includes a geometric approach to Gauss Reduction Theory, classification of integer regular polygons and some further new subjects. Traditionally a subject of number theory, continued fractions appear in dynamical systems, algebraic geometry, topology, and even celestial mechanics. The rise of computational geometry has resulted in renewed interest in multidimensional generalizations of continued fractions. Numerous classical theorems have been extended to the multidimensional case, casting light on phenomena in diverse areas of mathematics. The reader will find an overview of current progress in the geometric theory of multidimensional continued fractions accompanied by currently open problems. Whenever possible, we illustrate geometric constructions with figures and examples. Each chapter has exercises useful for undergraduate or graduate courses. 410 0$aAlgorithms and Computation in Mathematics,$x2512-3254 ;$v26 606 $aAlgebra 606 $aApproximation theory 606 $aConvex geometry 606 $aDiscrete geometry 606 $aNumber theory 606 $aAlgebra 606 $aOrder, Lattices, Ordered Algebraic Structures 606 $aApproximations and Expansions 606 $aConvex and Discrete Geometry 606 $aNumber Theory 615 0$aAlgebra. 615 0$aApproximation theory. 615 0$aConvex geometry. 615 0$aDiscrete geometry. 615 0$aNumber theory. 615 14$aAlgebra. 615 24$aOrder, Lattices, Ordered Algebraic Structures. 615 24$aApproximations and Expansions. 615 24$aConvex and Discrete Geometry. 615 24$aNumber Theory. 676 $a512 700 $aKarpenkov$b Oleg$01065194 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 912 $a9910574044103321 996 $aGeometry of Continued Fractions$92543714 997 $aUNINA