15 research outputs found

    Dynamical Properties of Weierstrass Elliptic Functions on Square Lattices

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    In this dissertation we prove that the Julia set of a Weierstrass elliptic function on a square lattice is connected. We further show that the parameter space contains an infinite number of Mandelbrot sets. As a consequence, this proves the existence of Siegel disks and gives a description of the bifurcation locus about super-attracting parameters corresponding to super-attracting fixed points. We conclude with a description of a family of rational maps that approximate the Weierstrass elliptic function on a square lattice

    Dirac operators and spectral triples for some fractal sets built on curves

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    We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each summand is based on a curve in the space. Several fractals, like a finitely summable infinite tree and the Sierpinski gasket, fit naturally within our framework. In these cases, we show that our spectral triples do describe the geodesic distance and the Minkowski dimension as well as, more generally, the complex fractal dimensions of the space. Furthermore, in the case of the Sierpinski gasket, the associated Dixmier-type trace coincides with the normalized Hausdorff measure of dimension log3/log2\log 3/ \log 2.Comment: 48 pages, 4 figures. Elementary proofs omitted. To appear in Adv. Mat

    Iteration of holomorphic functions in the complex plane

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    Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2016, Director: Núria Fagella RabionetWhen a holomorphic function is iterated, it generates a dynamic system on the complex plane. In this project, we describe both the local and global behavior of the different orbits of a rational map on the complex plane (or the Riemann sphere). We mainly concentrate in the study of the dynamical plane (where initial conditions and orbits live) although we briefly discuss one parameter families of polynomials and their bifurcation loci, like the well known Mandelbrot set. Towards the end, we experiment with a singular perturbation of a family of cubic polynomials and explore the drastic changes that occur in the topology of their Julia sets

    MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

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    Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

    Generalized averaged Gaussian quadrature and applications

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    A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

    Acta Scientiarum Mathematicarum : Tomus 55. Fasc. 1-2.

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