LEADER 05430nam 22008295 450 001 9910438036903321 005 20200706173027.0 010 $a3-642-39368-3 024 7 $a10.1007/978-3-642-39368-6 035 $a(CKB)3710000000015875 035 $a(EBL)1398307 035 $a(SSID)ssj0000988151 035 $a(PQKBManifestationID)11552679 035 $a(PQKBTitleCode)TC0000988151 035 $a(PQKBWorkID)10949870 035 $a(PQKB)10340267 035 $a(DE-He213)978-3-642-39368-6 035 $a(MiAaPQ)EBC6315339 035 $a(MiAaPQ)EBC1398307 035 $a(Au-PeEL)EBL1398307 035 $a(CaPaEBR)ebr10976143 035 $a(OCoLC)858924380 035 $a(PPN)172428173 035 $a(EXLCZ)993710000000015875 100 $a20130812d2013 u| 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aGeometry of Continued Fractions /$fby Oleg Karpenkov 205 $a1st ed. 2013. 210 1$aBerlin, Heidelberg :$cSpringer Berlin Heidelberg :$cImprint: Springer,$d2013. 215 $a1 online resource (409 p.) 225 1 $aAlgorithms and Computation in Mathematics,$x1431-1550 ;$v26 300 $aDescription based upon print version of record. 311 $a3-642-44424-5 311 $a3-642-39367-5 327 $aPreface -- Introduction -- Part 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. Continued fractions and SL(2; Z) conjugacy classes. Elements of Gauss Reduction Theory. Markoff spectrum -- Chapter 8. Lagrange theorem -- Chapter 9. Gauss-Kuzmin statistics -- Chapter 10. Geometric approximation aspects -- Chapter 11. Geometry of continued fractions with real elements and the second Kepler law -- Chapter 12. Integer angles of polygons and global relations to toric singularities -- Part 2. Klein polyhedra: Chapter 13. Basic notions and definitions of multidimensional integer geometry -- Chapter 14. On empty simplices, pyramids, parallelepipeds -- Chapter 15. Multidimensional continued fractions in the sense of Klein -- Chapter 16. Dirichlet groups and lattice reduction -- Chapter 17. Periodicity of Klein polyhedra. Generalization of Lagrange theorem -- Chapter 18. Multidimensional Gauss-Kuzmin statistics -- Chapter 19. On construction of multidimensional continued fractions -- Chapter 20. Gauss Reduction in higher dimensions -- Chapter 21. Decomposable forms. Relation to Littlewood and Oppenheim conjectures -- Chapter 22. Approximation of maximal commutative subgroups -- Chapter 23. Other generalizations of continued fractions -- Bibliography . 330 $aTraditionally a subject of number theory, continued fractions appear in dynamical systems, algebraic geometry, topology, and even celestial mechanics. The rise of computational geometry has resulted in renewed interest in multidimensional generalizations of continued fractions. Numerous classical theorems have been extended to the multidimensional case, casting light on phenomena in diverse areas of mathematics. This book introduces a new geometric vision of continued fractions. It covers several applications to questions related to such areas as Diophantine approximation, algebraic number theory, and toric geometry.   The reader will find an overview of current progress in the geometric theory of multidimensional continued fractions accompanied by currently open problems. Whenever possible, we illustrate geometric constructions with figures and examples. Each chapter has exercises useful for undergraduate or graduate courses. 410 0$aAlgorithms and Computation in Mathematics,$x1431-1550 ;$v26 606 $aAlgebra 606 $aOrdered algebraic structures 606 $aApproximation theory 606 $aConvex geometry  606 $aDiscrete geometry 606 $aNumber theory 606 $aAlgebra$3https://scigraph.springernature.com/ontologies/product-market-codes/M11000 606 $aOrder, Lattices, Ordered Algebraic Structures$3https://scigraph.springernature.com/ontologies/product-market-codes/M11124 606 $aApproximations and Expansions$3https://scigraph.springernature.com/ontologies/product-market-codes/M12023 606 $aConvex and Discrete Geometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M21014 606 $aNumber Theory$3https://scigraph.springernature.com/ontologies/product-market-codes/M25001 615 0$aAlgebra. 615 0$aOrdered algebraic structures. 615 0$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 $a515.243 700 $aKarpenkov$b Oleg$4aut$4http://id.loc.gov/vocabulary/relators/aut$01065194 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910438036903321 996 $aGeometry of Continued Fractions$92543714 997 $aUNINA