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Applications Of Boundary Value Problems, Annie Nguyen

Master's Theses

In this thesis, we solved the Saint-Venant's torsion problem for beams with different cross sections bounded by simple closed curves using various methods. In addition, we solved the flexure problem of beams with certain curvilinear cross sections. The first method was derived by Bassali and Obaid. We focused on cross sections bounded by hyperbolas, circular groves, lemniscate of Booth, and sectorial cross sections. The second method used Tchebycheff polynomials to solve the torsion problem corresponding to the circle and ellipse. The third method used conformal mapping to derive the solution of different cross sections bounded by curvilinear edges. The flexure …

Family Of Circulant Graphs And Its Expander Properties, Vinh Kha Nguyen Tran Nguyen

Master's Theses

In this thesis, we apply spectral graph theory to show the non-existence of an

expander family within the class of circulant graphs. Using the adjacency matrix and its properties, we prove Cheeger's inequalities and determine when the equalities hold. In order to apply Cheeger's inequalities, we compute the spectrum of a general circulant graph and approximate its second largest eigenvalue. Finally, we show that circulant graphs do not contain an expander family.

Elliptic Curves And Cryptography, Senorina Ramos Vazquez

Master's Theses

In this expository thesis we study elliptic curves and their role in cryptography. In doing so we examine an intersection of linear algebra, abstract algebra, number theory, and algebraic geometry, all of which combined provide the necessary background. First we present background information on rings, fields, groups, group actions, and linear algebra. Then we delve into the structure and classification of finite fields as well as construction of finite fields and computation in finite fields. We next explore logarithms in finite fields and introduce the Diffie-Hellman key exchange system. Subsequently, we take a look at the projective and affine planes …