Master Degree / Yüksek Lisans Tezleri

Permanent URI for this collectionhttps://hdl.handle.net/11147/3008

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Subject: search.filters.subject.Fractals

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  • Master Thesis
    Q-Periodicity, Self-Similarity and Weierstrass-Mandelbrot Function
    (Izmir Institute of Technology, 2012) Erkuş, Soner; Pashaev, Oktay
    In the present thesis we study self-similar objects by method's of the q-calculus. This calculus is based on q-rescaled finite differences and introduces the q-numbers, the qderivative and the q-integral. Main object of consideration is the Weierstrass-Mandelbrot functions, continuous but nowhere differentiable functions. We consider these functions in connection with the q-periodic functions. We show that any q-periodic function is connected with standard periodic functions by the logarithmic scale, so that q-periodicity becomes the standard periodicity. We introduce self-similarity in terms of homogeneous functions and study properties of these functions with some applications. Then we introduce the dimension of self-similar objects as fractals in terms of scaling transformation. We show that q-calculus is proper mathematical tools to study the self-similarity. By using asymptotic formulas and expansions we apply our method to Weierstrass-Mandelbrot function, convergency of this function and relation with chirp decomposition.
  • Master Thesis
    An Investigation With Fractial Geometry Analysis of Time Series
    (Izmir Institute of Technology, 2005) Kaya, Aysun; Özdemir, Serhan
    In this thesis, three kinds of fractal dimensions, correlation dimension, Hausdorff dimension and box-counting dimension were used to examine time series. To demonstrate the universality of the method, ECG (Electrocardiogram) time series were chosen. The ECG signals consisted of ECGs of three persons in four states for two applications. States are normal, walk, rapid walk and run. These three people are selected from the same age, and height group to minimize variations. First application was made for approximately 1000 samples of size of ECG signals and the second for the whole of the measured ECG signals. Fractal dimension measurements under different conditions were carried out to find out whether these dimensions could discriminate the states under question. A total of 24 ECG signals were measured to determine their corresponding fractal dimensions through the above-mentioned methods. It was expected that fractal dimension values would indicate the states related to the different activities of the persons. Results show that no direct link was found connecting a certain dimension to a certain activity in a consistent manner. Furthermore, no congruence was also found among the three dimensions that were employed. According to these results, it can be stated that fractal dimension values on their own may not be sufficient to identify distinct cases hidden in time series. Time series analysis may be facilitated when additional tools and methods are utilized as well as fractal dimensions at detecting telltale signs in signals of different states.
  • Master Thesis
    Fibonacci fractal tree antennas
    (Izmir Institute of Technology, 2004) Özbakış, Başak; Kuştepeli, Alp
    Fractal geometry is first defined by Benoit Mandelbrot. A fractal structure is generated with an iterative procedure of a simple initiator by replicating many times at different scales, positions and directions. Fractal structures generated with this method are generally self-similar and the dimensions of these structures cannot be defined with integers. Koch, Minkowski and Sierpinski structures are the most known fractal structures. These structures are commonly used as multiband and wideband antenna designs because of the self-similarity. Furthermore, their special geometry is useful to obtain small antennas which are resonant at lower frequencies. Lowering the resonant frequency has the same effect as miniaturizing the antenna at a fixed resonant frequency. Other important and interesting fractal structures used in antenna designs are the various types of the fractal trees. However, in recent studies the branch length ratios of the fractal tree antennas are taken constant. In this study fractal tree antennas with nonuniform branch length ratios are investigated. By changing the geometry and number of branches of the fractal tree structures the antenna characteristics are examined. The branch lengths and number of branches of the fractal tree antennas are determined by using the Fibonacci sequence. Leonardo Fibonacci (1170 - 1240), a famous Italian mathematician, dealt with geometry and developed a number sequence while observing the nature. Fractal tree antennas are designed with two different geometries in order to improve the resonance behavior of the antennas. The number of branches is decreased, so that less complex fractal tree antennas with the similar performance can be obtained.