We identify three structures that lie in the parameter plane of the rational map F(z) = zⁿ + λ / zᵈ, for which z is a complex number, λ a complex parameter, n ≥ 4 is even, and d ≥ 3 is odd. There exists a Sierpindelbrot arc, an infinite sequence of pairs of Mandelbrot sets and Sierpinski holes, that limits to the parameter at the end of the arc. There exists as well a qualitatively different Sierpindelbrot arc, an infinite sequence of pairs of Mandelbrot sets and Sierpinski holes, that limits to the parameter at the center of the arc. Furthermore, there exist infinitely many arcs of each type. A parameter can travel along a continuous path from the Cantor set locus, along infinitely many arcs of the first type in a successively smaller region of the parameter plane, while passing through an arc of the second type, to the parameter at the center of the latter arc. This infinite sequence of Sierpindelbrot arcs is a Sierpinski Mandelbrot spiral.

When many people think of fractals, they think of the beautiful images created by Mandelbrot’s set or the intricate dragons of Julia’s set. However, these are just the artistic stars of the fractal community. The theory behind the fractals is not necessarily pretty, but is very important to many areas outside the world of mathematics. This paper takes a closer look at various types of fractals, the fractal dimensionality of surfaces and chaotic dynamical systems. Some of the history and introduction of creating fractals is discussed. The tools used to prevent a modified Koch’s curve from overlapping itself, finding the limit of a curves length and solving for a surfaces dimensional measurement are explored. Lastly, an investigation of the theories of chaos and how they bring order into what initially appears to be random and unpredictable is presented. The practical purposes and uses of fractals throughout are also discussed. / text

Indiana University-Purdue University Indianapolis (IUPUI) / Given a parametrized family of Iterated Function System (IFS) we give sufficient conditions for a parameter on the boundary of the connectedness locus, M, to be accessible from the complement of M. Moreover, we provide a few examples of such parameters and describe how they are connected to Misiurewicz parameter in the Mandelbrot set, i.e. the connectedness locus of the quadratic family z^2+c.

Today most computer have a multicore processor and are depending on parallel execution to be able to keep up with the demanding tasks that exist today, that forces developers to write software that can take advantage of multicore systems. There are multiple programming languages and frameworks that makes it possible to execute the code in parallel on different threads, this study looks at the performance and effort required to work with two of the frameworks that are available to the C programming language, POSIX Threads(Pthreads) and OpenMP. The performance is measured by paralleling three algorithms, Matrix multiplication, Quick Sort and calculation of the Mandelbrot set using both Pthreads and OpenMP, and comparing first against a sequential version and then the parallel version against each other. The effort required to modify the sequential program using OpenMP and Pthreads is measured in number of lines the final source code has. The results shows that OpenMP does perform better than Pthreads in Matrix Multiplication and Mandelbrot set calculation but not on Quick Sort because OpenMP has problem with recursion and Pthreads does not. OpenMP wins the effort required on all the tests but because there is a large performance difference between OpenMP and Pthreads on Quick Sort OpenMP cannot be recommended for paralleling Quick Sort or other recursive programs.

Submitted by Marlene Santos (marlene.bc.ufg@gmail.com) on 2016-10-03T21:11:40Z No. of bitstreams: 2 Dissertação - Márcio Vaiz dos Reis - 2016.pdf: 2097960 bytes, checksum: 296b1790b8c8fe50c0e91d2d5ee204c4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2016-10-04T10:46:49Z (GMT) No. of bitstreams: 2 Dissertação - Márcio Vaiz dos Reis - 2016.pdf: 2097960 bytes, checksum: 296b1790b8c8fe50c0e91d2d5ee204c4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2016-10-04T10:46:49Z (GMT). No. of bitstreams: 2 Dissertação - Márcio Vaiz dos Reis - 2016.pdf: 2097960 bytes, checksum: 296b1790b8c8fe50c0e91d2d5ee204c4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-08-29 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The purpose of this dissertation is to present an introductory approach to the complex dynamics and fractal geometry, especially the Mandelbrot set. With the goal to simplify and stimulate the introduction of complex number in high school, the approach adopted was: the definition of the Mandelbrot set and its characteristics; the relationship between the Mandelbrot set and Julia set; how to find by using the Mandelbrot set. One of the tools used to help the teaching was Geogeobra, a dynamic software that allows the student to build the Mandelbrot set. Through this study, it is expected to motivate the learning of complex numbers by using fractal obtained by the study of function ( ) . Obtaining, as a result, a differentiated and motivating way of learning for a better understanding and intellectual development of the students. / Esse trabalho apresenta uma abordagem introdutória para a dinâmica complexa e a geometria fractal, em especial o conjunto de Mandelbrot. Com objetivo de facilitar e motivar a interação dos alunos com o ensino dos números complexos, a abordagem adotada foi: a definição do conjunto de Mandelbrot e suas características; a relação entre o conjunto de Mandelbrot e o conjunto de Julia; a relação do conjunto de Mandelbrot e o número . Uma das ferramentas utilizadas para auxiliar o professor foi o Geogeobra, um software dinâmico que permite o aluno a construção do conjunto de Mandelbrot. Por meio deste trabalho, espera-se motivar o ensino dos números complexos através do fractal obtido pelo estudo da função ( ) . Obtendo assim, como resultado, uma forma diferenciada e motivadora do aprendizado do aluno, garantindo um melhor entendimento e desenvolvimento intelectual.

<div>Given a parametrized family of Iterated Function System (IFS) we give sufficient conditions for a parameter on the boundary of the connectedness locus, M, to be accessible from the complement of M.</div><div>Moreover, we provide a few examples of such parameters and describe how they are connected to Misiurewicz parameter in the Mandelbrot set, i.e. the connectedness locus of the quadratic family z^2+c.<br></div>

No description available.

We characterize invariant measures for quadratic polynomial Julia sets with no interior. We prove that besides the harmonic measure —the only one that is even and invariant—, all others are generated by a suitable odd measure. / En este artículo caracterizamos medidas invariantes sobre conjuntos de Julia sin interior asociados con polinomios cuadráticos.  Probamos que más allá de la medida armónica —la única par e invariante—, el resto son generadas por su parte impar.

This Master's thesis deals with history of Fractal geometry and describes the fractal science development. In the begining there are essential Fractal science terms explained. Then description of fractal types and typical or most known examples of them are mentioned. Fractal knowledge application besides computer graphics area is discussed. Thesis informs about fractal geometry practical usage. Few present software packages or more programs which can be used for making fractal pictures are described in this work. Some of theirs capabilities are described. Thesis' practical part consists of slides, demonstrational program and poster. Electronical slides represents brief scheme usable for fractal geometry realm lectures. Program generates selected fractal types. Thesis results are projected on poster.