Physical Sciences and Mathematics Commons^™

Cayley Graphs Of Semigroups And Applications To Hashing, Bianca Sosnovski

Dissertations, Theses, and Capstone Projects

In 1994, Tillich and Zemor proposed a scheme for a family of hash functions that uses products of matrices in groups of the form $SL_2(F_{2^n})$. In 2009, Grassl et al. developed an attack to obtain collisions for palindromic bit strings by exploring a connection between the Tillich-Zemor functions and maximal length chains in the Euclidean algorithm for polynomials over $F_2$.

In this work, we present a new proposal for hash functions based on Cayley graphs of semigroups. In our proposed hash function, the noncommutative semigroup of linear functions under composition is considered as platform for the scheme. We will also …

Stochastic Processes And Their Applications To Change Point Detection Problems, Heng Yang

Dissertations, Theses, and Capstone Projects

This dissertation addresses the change point detection problem when either the post-change distribution has uncertainty or the post-change distribution is time inhomogeneous. In the case of post-change distribution uncertainty, attention is drawn to the construction of a family of composite stopping times. It is shown that the proposed composite stopping time has third order optimality in the detection problem with Wiener observations and also provides information to distinguish the different values of post-change drift. In the case of post-change distribution uncertainty, a computationally efficient decision rule with low-complexity based on Cumulative Sum (CUSUM) algorithm is also introduced. In the time …

An Averaging Method For Advection-Diffusion Equations, Nicholas Spizzirri

Dissertations, Theses, and Capstone Projects

Many models for physical systems have dynamics that happen over various different time scales. For example, contrast the everyday waves in the ocean with the larger, slowly moving global currents. The method of multiple scales is an approach for approximating the solutions of differential equations by separating out the dynamics at slower and faster time scales. In this work, we apply the method of multiple scales to generic advection-diffusion equations (both linear and non-linear, and in arbitrary spatial dimensions) and develop a method for 'averaging out' the faster scale phenomena, giving us an 'effective' solution for the slower scale dynamics. …