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作者:Oleg Karpenkov , Matty van Son
来源:[J].Journal of Number Theory(IF 0.466), 2020
摘要:Abstract(#br)In this paper we introduce generalised Markov numbers and extend the classical Markov theory for the discrete Markov spectrum to the case of generalised Markov numbers. In particular we show recursive properties for these numbers and find corresponding values in the ...
作者:Oleg Karpenkov , Alexey Ustinov
来源:[J].Journal of Number Theory(IF 0.466), 2017, Vol.176, pp.375-419
摘要:Abstract(#br)In this paper we investigate the combinatorial structure of 3-dimensional Minkowski–Voronoi continued fractions. Our main goal is to prove the asymptotic stability of Minkowski–Voronoi complexes in special two-parametric families of rank-1 lattices. In addition ...
作者:Oleg Karpenkov
来源:[J].Manuscripta Mathematica(IF 0.392), 2011, Vol.134 (1-2), pp.157-169
摘要:Abstract(#br)In this paper we introduce a link between geometry of ordinary continued fractions and trajectories of points that moves according to the second Kepler law. We expand geometric interpretation of ordinary continued fractions to the case of continued fractions with arb...
作者:Oleg Karpenkov
来源:[J].Comptes rendus - Mathématique(IF 0.477), 2006, Vol.343 (1), pp.5-7
摘要:Abstract(#br)The problem of the investigation of the simplest n -dimensional continued fraction in the sense of Klein for n ⩾ 2 was posed by V. Arnold. The answer for the case n = 2 can be found in the works of E. Korkina (1995) and G. Lachaud (1995). In present Note we stud...
作者:Franck Doray , Oleg Karpenkov , Jan Schepers
来源:[J].Discrete & Computational Geometry(IF 0.649), 2010, Vol.43 (2), pp.436-466
摘要:Abstract(#br)Consider a graph G with n vertices. In this paper we study geometric conditions for an n -tuple of points in ℝ d to admit a nonzero self-stress with underlying graph G . We introduce and investigate a natural stratification, depending on G , of the configuration...
作者:Oleg Karpenkov , Matty van-Son
来源:[J].Journal de théorie des nombres de Bordeaux, 2019, Vol.31 (1), pp.131-144
摘要:In this paper, we generalize the Perron Identity for Markov minima. We express the values of binary quadratic forms with positive discriminant in terms of continued fractions associated to broken lines passing through the points where the values are computed.
作者:Oleg Karpenkov
来源:[J].Functional Analysis and Other Mathematics, 2007, Vol.1 (1), pp.17-35
摘要:Abstract(#br)We completely describe lattice convex polytopes in ℝ n (for any dimension n ) that are regular with respect to the group of affine transformations preserving the lattice.
作者:Oleg Karpenkov
来源:[J].Functional Analysis and Other Mathematics, 2007, Vol.1 (2), pp.175-180
摘要:Abstract(#br)V.I. Arnold recently formulated a geometric concept of monads and used it to study difference operators on the sets of {0,1}-valued sequences of length n . We consider particular examples of these monads and indicate one question that arises.
作者:Oleg Karpenkov
来源:[J].Functional Analysis and Other Mathematics, 2009, Vol.2 (2-4), pp.221-239
摘要:Abstract(#br)The aim of this paper is to generalize the notions of ordinary and expanded lattice angles and their sums studied in the work [7] by the author to the case of angles with lattice vertices but not necessarily with lattice rays. We find normal forms and extend the defi...
作者:Oleg Karpenkov
来源:[J].Arnold Mathematical Journal, 2018, Vol.4 (1), pp.19-25
摘要:Abstract(#br)In this small paper we bring together some open problems related to the study of the configuration spaces of tensegrities, i.e. graphs with stresses on edges. These problems were announced in Doray et al. (Discrete Comput Geom 43:436–466, 2010 ), Karpenkov et al...

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