Aachen, Techn. Hochsch., Diss., 2002.

Aachen, Techn. Hochsch., Diss., 2002.

Honors thesis (B.A.)--Ithaca College Dept. of Mathematics, 2005. / "April 2006." Includes bibliographical references (leaf [22]). Also available in print form in the Ithaca College Archives.

Techn. Hochsch., Diss., 2002--Aachen.

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.

The goal of this thesis is to provide a unified framework in which to analyze the dynamics of two seemingly unrelated families of one-dimensional dynamical systems, namely the family of quadratic polynomials and continued fractions. We develop a combinatorial calculus to describe the bifurcation set of both families and prove they are isomorphic. As a corollary, we establish a series of results describing the behavior of entropy as a function of the parameter. One of the most important applications is the relation between the topological entropy of quadratic polynomials and the Hausdorff dimension of sets of external rays landing on principal veins of the Mandelbrot set. / Mathematics

The aim of this thesis is to implement a Mandelbrot Set Explorer that includes the functionality to create music visualizations. The Mandelbrot set is an important mathematical object, and the arguably most famous so called fractal. One of its outstanding attributes is its beauty, and therefore there are several implementations that visualize the set and allow it to navigate around it. In this thesis methods are discussed to visualize the set and create music visualizations consisting of zooms into the Mandelbrot set. For that purpose methods for analysing music are implemented, so user created zooms can react to the music that is played. Mainly the thesis deals with problems that occur during the process of developing this application to create music visualizations. Especially problems concerning performance and usability are focused. The thesis will reveal that it is in fact possible to create very aesthetically pleasing music visualizations by using zooms into the Mandelbrot set. The biggest drawback is the lack in performance, because of the high computation effort, and therefore the difficulties in rendering the visualization in real-time.

Today most computers have powerful multi core processors that can perform many calculations simultaneously. However writing programs that take full advan- tage of the processors in modern day computers can be a challenge. This is due to the challenge of managing shared resources between parallel processing threads. This report documents the development of the Majo language that aims to solve these problems by using abstractions to make parallel programming easier. The model for the abstractions is dividing the program in to what is called nodes. One node represents one thread of execution and nodes are connected to each other by thread safe communication channels. All communication channels are frst in frst out queues. The nodes communicate by pushing and popping values form these queues. The performance of the language was measured and compared to other languages such as Python, Ruby and JavaScript. The tests were based on timing how long it took to generate the Mandelbrot set as well as sorting a list of inte- gers. The language scalability was also tested by seeing how much the execution time decreased by adding more parallel threads. The results from these tests showed that the developed prototype of the language had some unforeseen bugs that slowed down the execution more then expected in some tests. However the scalability test gave encouraging results. For future development the language exe- cution time should be improved by fxing relevant bugs and a more generalized model for concurrency should be developed.

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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