Master Degree / Yüksek Lisans Tezleri
Permanent URI for this collectionhttps://hdl.handle.net/11147/3008
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Master Thesis Fourier Analysis on the Lorentz Group and Relativistic Quantum Mechanics(Izmir Institute of Technology, 2008) Ok, Zahide; Kasım, Rıfat Mir; Kasım, Rıfat Mir; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyThe non-relativistic Schrödinger and Lippman-Schwinger equations are described. The expressions of these equations are investigated in momentum and configuration spaces, using Fourier transformation. The plane wave, which is generating function for the matrix elements of three dimensional Euclidean group in spherical basis, expanded in terms of Legendre polynomials and spherical Bessel functions. Also explicit calculation of Green.s function is done.The matrix elements of the unitary irreducible representations of Lorentz group are used to introduce Fourier expansion of plane waves. And the kernel of Gelfand-Graev transformation, which is the relativistic plane wave, is expanded in to these matrix elements. Then relativistic differential difference equation in configuration space is constructed.Lippman-Schwinger equations are studied in Lobachevsky space (hyperbolic space). An analogous to the non-relativistic case, using the finite difference Schrödinger equation, one dimensional Green.s function is analyzed for the relativistic case . Also the finite difference analogue of the Heavyside step function is investigated.Master Thesis Q-Periodicity, Self-Similarity and Weierstrass-Mandelbrot Function(Izmir Institute of Technology, 2012) Erkuş, Soner; Pashaev, Oktay; Pashaev, Oktay; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyIn 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.