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Funktsional. Anal. i Prilozhen., 2004, Volume 38, Issue 2, Pages 28–37 (Mi faa105)  

This article is cited in 12 scientific papers (total in 12 papers)

On Tori Triangulations Associated with Two-Dimensional Continued Fractions of Cubic Irrationalities

O. N. Karpenkov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The notion of equivalence of multidimensional continued fractions is introduced. We consider some properties and state some conjectures related to the structure of the family of equivalence classes of two-dimensional periodic continued fractions. Our approach to the study of the family of equivalence classes of two-dimensional periodic continued fractions leads to revealing special subfamilies of continued fractions for which the triangulations of the torus (i.e., the combinatorics of their fundamental domains) are subjected to clear rules. Some of these subfamilies are studied in detail; the way to construct other similar subfamilies is indicated.

Keywords: multidimensional continued fractions, convex hulls, integer operators, cubic extensions of $\mathbb{Q}$

DOI: https://doi.org/10.4213/faa105

Full text: PDF file (229 kB)
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English version:
Functional Analysis and Its Applications, 2004, 38:2, 102–110

Bibliographic databases:

UDC: 511.9
Received: 03.02.2003

Citation: O. N. Karpenkov, “On Tori Triangulations Associated with Two-Dimensional Continued Fractions of Cubic Irrationalities”, Funktsional. Anal. i Prilozhen., 38:2 (2004), 28–37; Funct. Anal. Appl., 38:2 (2004), 102–110

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. O. N. Karpenkov, “Classification of three-dimensional multistorey completely empty convex marked pyramids”, Russian Math. Surveys, 60:1 (2005), 165–166  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. Karpenkov O., “Three examples of three-dimensional continued fractions in the sense of Klein”, C. R. Math. Acad. Sci. Paris, 343:1 (2006), 5–7  crossref  mathscinet  zmath  isi  scopus
    3. O. N. Karpenkov, “On an Invariant Möbius Measure and the Gauss–Kuzmin Face Distribution”, Proc. Steklov Inst. Math., 258 (2007), 74–86  mathnet  crossref  mathscinet  zmath  elib  elib
    4. Karpenkov O.N., “Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions”, Monatsh. Math., 152:3 (2007), 217–249  crossref  mathscinet  zmath  isi  elib  scopus
    5. Karpenkov O., “Elementary notions of lattice trigonometry”, Math. Scand., 102:2 (2008), 161–205  crossref  mathscinet  zmath  isi  scopus
    6. Karpenkov O.N., “Constructing multidimensional periodic continued fractions in the sense of Klein”, Math. Comp., 78:267 (2009), 1687–1711  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    7. O. N. Karpenkov, “Determination of Periods of Geometric Continued Fractions for Two-Dimensional Algebraic Hyperbolic Operators”, Math. Notes, 88:1 (2010), 28–38  mathnet  crossref  crossref  mathscinet  isi
    8. Karpenkov O.N., Vershik A.M., “Rational approximation of maximal commutative subgroups of GL(n, R)”, J Fixed Point Theory Appl, 7:1 (2010), 241–263  crossref  mathscinet  zmath  isi  elib  scopus
    9. Karpenkov O., “Continued fractions and the second Kepler law”, Manuscripta Math, 134:1–2 (2011), 157–169  crossref  mathscinet  zmath  isi  scopus
    10. Karpenkov O., “Multidimensional Gauss Reduction Theory for Conjugacy Classes of Sl(N, Z)”, J. Theor. Nr. Bordx., 25:1 (2013), 99–109  crossref  mathscinet  zmath  isi  scopus
    11. A. A. Illarionov, “Some properties of three-dimensional Klein polyhedra”, Sb. Math., 206:4 (2015), 510–539  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    12. A. A. Lodkin, “Parus Kleina i diofantovy priblizheniya vektora”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye i algoritmicheskie metody. XXX, Zap. nauchn. sem. POMI, 481, POMI, SPb., 2019, 63–73  mathnet
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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