Перегляд за Автор "Claire David"

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  • Документ
    Bypassing dynamical systems: a simple way to get the box-counting dimension of the graph of the Weierstrass function
    (2018) Claire David
    In the following, bypassing dynamical systems tools, we propose a simple means of computing the box dimension of the graph of the classical Weierstrass function defined, for any real number~$x$, by[{mathcal W}(x)= sum_{n=0}^{+infty} lambda^n,cos left ( 2, pi,N_b^n,x right),]where $lambda$ and $N_b$ are two real numbers such that $0 1$, using a sequence a graphs that approximate the studied one.
  • Документ
    Laplacian, on the Arrowhead Curve
    (2020) Claire David
    In terms of analysis on fractals, the Sierpinski gasket stands out as one of the most studied example. The underlying aim of those studies is to determine a differential operator equivalent to the classic Laplacian. The classically adopted approach is a bidimensional one, through a sequence of so-called prefractals, i.e. a sequence of graphs that converges towards the considered domain. The Laplacian is obtained through a weak formulation, by means of Dirichlet forms, built by induction on the prefractals. It turns out that the gasket is also the image of a Peano curve, the so-called Arrowhead one, obtained by means of similarities from a starting point which is the unit line. This raises a question that appears of interest. Dirichlet forms solely depend on the topology of the domain, and not of its geometry. Which means that, if one aims at building a Laplacian on a fractal domain as the aforementioned curve, the topology of which is the same as, for instance, a line segment, one has to find a way of taking account its specific geometry. Another difference due to the geometry, is encountered may one want to build a specific measure. For memory, the sub-cells of the Kigami and Strichartz approach are triangular and closed: the similarities at stake in the building of the Curve called for semi-closed trapezoids. As far as we know, and until now, such an approach is not a common one, and does not appear in such a context. It intererestingly happens that the measure we choose corresponds, in a sense, to the natural counting measure on the curve. Also, it is in perfect accordance with the one used in the Kigami and Strichartz approach. In doing so, we make the comparison -- and the link -- between three different approaches, that enable one to obtain the Laplacian on the arrowhead curve: the natural method; the Kigami and Strichartz approach, using decimation; the Mosco approach.    
  • Документ
    On fractal properties of Weierstrass-type functions
    (2019) Claire David
    In the sequel, starting from the classical Weierstrass function defined, for any real number $x$, by $ {mathcal W}(x)=displaystyle sum_{n=0}^{+infty} lambda^n,cos left(2, pi,N_b^n,x right)$, where $lambda$ and $N_b$ are two real numbers such that~mbox{$0 <lambda<1$},~mbox{$ N_b,in,N$} and $ lambda,N_b > 1 $, we highlight intrinsic properties of curious maps which happen to constitute a new class of iterated function system. Those properties are all the more interesting, in so far as they can be directly linked to the computation of the box dimension of the curve, and to the proof of the non-differentiabilty of Weierstrass type functions.