144 research outputs found
On the number of return words in infinite words with complexity 2n+1
In this article, we count the number of return words in some infinite words
with complexity 2n+1. We also consider some infinite words given by codings of
rotation and interval exchange transformations on k intervals. We prove that
the number of return words over a given word w for these infinite words is
exactly k.Comment: see also http://liafa.jussieu.fr/~vuillon/articles.htm
Coding rotations on intervals
We show that the coding of rotation by on intervals with
rationally independent lengths can be recoded over Sturmian words of angle
More precisely, for a given an universal automaton is constructed such
that the edge indexed by the vector of values of the th letter on each
Sturmian word gives the value of the th letter of the coding of rotation.Comment: LIAFA repor
Combinatoire des motifs d'une suite sturmienne bidimensionnelle
RĂ©sumĂ©Nous Ă©tudions une gĂ©nĂ©ralisation des suites sturmiennes en construisant une âsurface plissĂ©eâ, donnĂ©e par l'approximation d'un plan par trois sortes de faces carrĂ©es orientĂ©es suivant les trois plans de coordonnĂ©es. Ă cette surface, on associe par projection un pavage du plan par trois sortes de losanges. On dĂ©finit sur ce pavage une fonction de complexitĂ© en comptant le nombre de motifs distincts d'une fenĂȘtre de taille donnĂ©e. Nous donnons, en Ă©tudiant les prolongements des motifs, la forme explicite de cette fonction dans le cas d'une fenĂȘtre triangulaire et d'une fenĂȘtre en forme de parallĂ©logramme.AbstractWe study a generalization of sturmian sequences by constructing a âstepped surfaceâ, given by a plane approximation with three kinds of square faces oriented according to the three coordinate planes. With a projection operation, we build a tiling of the plane by three kinds of diamonds. We define in this tiling a complexity function by counting the number of patterns in a given height window. We give the explicit form of this function in the case of triangular windows and parallelogram windows
Some special solutions to the Hyperbolic NLS equation
The Hyperbolic Nonlinear Schrodinger equation (HypNLS) arises as a model for
the dynamics of three-dimensional narrowband deep water gravity waves. In this
study, the Petviashvili method is exploited to numerically compute bi-periodic
time-harmonic solutions of the HypNLS equation. In physical space they
represent non-localized standing waves. Non-trivial spatial patterns are
revealed and an attempt is made to describe them using symbolic dynamics and
the language of substitutions. Finally, the dynamics of a slightly perturbed
standing wave is numerically investigated by means a highly acccurate Fourier
solver.Comment: 33 pages, 10 figures, 70 references. Other author's papers can be
found at http://www.denys-dutykh.com
Non lattice periodic tilings of R3 by single polycubes
International audienceIn this paper, we study a class of polycubes that tile the space by translation in a non lattice periodic way. More precisely, we construct a family of tiles indexed by integers with the property that Tk is a tile having k â„ 2 has anisohedral number. That is k copies of Tk are assembled by translation in order to form a metatile. We prove that this metatile is lattice periodic while Tk is not a lattice periodic tile
2L-CONVEX POLYOMINOES: GEOMETRICAL ASPECTS
International audienceA polyomino P is called 2L-convex if for every two cells there exists a monotone path included in P with at most two changes of direction. This paper studies the geometrical aspects of a sub-class of 2L-convex polyominoes called I0,0 and states a characterization of 2L it in terms of monotone paths. In a second part, four geometries are introduced and the tomographical point of view is investigated using the switching components (that is, the elements of this sub-class that have the same projections). Finally, some unicity results are given for the reconstruction of these polyominoes according to their projections
Non lattice periodic tilings of R3 by single polycubes
International audienceIn this paper, we study a class of polycubes that tile the space by translation in a non lattice periodic way. More precisely, we construct a family of tiles indexed by integers with the property that Tk is a tile having k â„ 2 has anisohedral number. That is k copies of Tk are assembled by translation in order to form a metatile. We prove that this metatile is lattice periodic while Tk is not a lattice periodic tile
Geometric Palindromic Closure
http://www.boku.ac.at/MATH/udt/vol07/no2/06DomVuillon13-12.pdfInternational audienceWe define, through a set of symmetries, an incremental construction of geometric objects in Z^d. This construction is directed by a word over the alphabet {1,...,d}. These objects are composed of d disjoint components linked by the origin and enjoy the nice property that each component has a central symmetry as well as the global object. This construction may be seen as a geometric palindromic closure. Among other objects, we get a 3 dimensional version of the Rauzy fractal. For the dimension 2, we show that our construction codes the standard discrete lines and is equivalent to the well known palindromic closure in combinatorics on words
How many faces can polycubes of lattice tilings by translation of R3 have?
International audienceWe construct a class of polycubes that tile the space by translation in a lattice- periodic way and show that for this class the number of surrounding tiles cannot be bounded. The first construction is based on polycubes with an L-shape but with many distinct tilings of the space. Nevertheless, we are able to construct a class of more complicated polycubes such that each polycube tiles the space in a unique way and such that the number of faces is 4k + 8 where 2k + 1 is the volume of the polycube. This shows that the number of tiles that surround the surface of a space-filler cannot be bounded
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