Physical Sciences and Mathematics Commons^™

Birkhoff Summation Of Irrational Rotations: A Surprising Result For The Golden Mean, Heather Moore

University Honors Theses

This thesis presents a surprising result that the difference in a certain sums of constant rotations by the golden mean approaches exactly 1/5. Specifically, we focus on the Birkhoff sums of these rotations, with the number of terms equal to squared Fibonacci numbers. The proof relies on the properties of continued fraction approximants, Vajda's identity and the explicit formula for the Fibonacci numbers.

Policing By Proxy: Interrogating Big Tech's Role In Law Enforcement, Claire Elizabeth

University Honors Theses

Predictive policing, sometimes referred to as data-driven or actuarial policing, is a method of policing that uses a risk-based approach to law enforcement. For-profit technology companies market proprietary risk assessment algorithms to law enforcement organizations as tools meant to proactively mitigate crime. Using data collected from a vast array of sources, both personal and public, police are able to "predict" the likelihood of criminal activity in a given area using these algorithms. Proponents claim that risk assessment tools have the potential to fight crime with unbiased accuracy and speed by predicting when, where, and whom to police by relying on …

Minimality Of Integer Bar Visibility Graphs, Emily Dehoff

University Honors Theses

A visibility representation is an association between the set of vertices in a graph and a set of objects in the plane such that two objects have an unobstructed, positive-width line of sight between them if and only if their two associated vertices are adjacent. In this paper, we focus on integer bar visibility graphs (IBVGs), which use horizontal line segments with integer endpoints to represent the vertices of a given graph. We present results on the exact widths of IBVGs of paths, cycles, and stars, and lower bounds on trees and general graphs. In our main results, we find …

Functional Role Of The N-Terminal Domain In Connexin 46/50 By In Silico Mutagenesis And Molecular Dynamics Simulation, Umair Khan

University Honors Theses

Connexins form intercellular channels known as gap junctions that facilitate diverse physiological roles, from long-range electrical and chemical coupling to nutrient exchange. Recent structural studies on Cx46 and Cx50 have defined a novel and stable open state and implicated the amino-terminal (NT) domain as a major contributor to functional differences between connexin isoforms. This thesis presents two studies which use molecular dynamics simulations with these new structures to provide mechanistic insight into the function and behavior of the NTH in Cx46 and Cx50. In the first, residues in the NTH that differ between Cx46 and Cx50 are swapped between the …

Spanning Trees Of Complete Graphs And Cycles, Minjin Enkhjargal

University Honors Theses

Spanning trees are typically used to solve least path problems for finding the minimal spanning tree of a graph. Given a number t ≥ 3 what is the least number n = α(t) such that there exists a graph on n vertices having precisely t spanning trees? Specifically, how will the factoring of t with the use of cycles connected by one vertex affect α(t)? Lower and upper bounds of α(t) are graphed by using properties of cycles and complete graphs. The upper bound of α(t) is then improved by constructing a graph of connected cycles {C_p1, C­ …

Group Theory Visualized Through The Rubik's Cube, Ashlyn Okamoto

University Honors Theses

In my thesis, I describe the work done to implement several Group Theory concepts in the context of the Rubik’s cube. A simulation of the cube was constructed using Processing-Java and with help from a YouTube series done by TheCodingTrain. I reflect on the struggles and difficulties that came with creating this program along with the inspiration behind the project. The concepts that are currently implemented at this time are: Identity, Associativity, Order, and Inverses. The functionality of the cube is described as it moves like a regular cube but has extra keypresses that demonstrate the concepts listed. Each concept …

On Dc And Local Dc Functions, Liam Jemison

University Honors Theses

In this project we investigate the class of functions which can be represented by a difference of convex functions, hereafter referred to simply as 'DC' functions. DC functions are of interest in optimization because they allow the use of convex optimization techniques in certain non-convex problems. We present known results about DC and locally DC functions, including detailed proofs of important theorems by Hartman and Vesely.

We also investigate the DCA algorithm for optimizing DC functions and implement it to solve the support vector machine problem.

Laurent Series Expansion And Its Applications, Anna Sobczyk

University Honors Theses

The Laurent expansion is a well-known topic in complex analysis for its application in obtaining residues of complex functions around their singularities. Computing the Laurent series of a function around its singularities turns out to be an efficient way to determine the residue of the function as well as to compute the integral of the function along any closed curves around its singularities. Based on the theory of the Laurent series, this paper provides several working examples where the Laurent series of a function is determined and then used to calculate the integral of the function along any closed curve …

Modeling And Visualizing Power Amplification In Fiber Optic Cables, Gil Parnon

University Honors Theses

Transverse mode instability in fiber optic cables causes power amplification to exhibit chaotic behavior. Due to this, numerical modeling of fiber optic power amplification is extremely computationally expensive. In this paper I work through modeling similar behavior in a simpler system. I also visualize the three-dimensional phase portrait of the system in order to better understand the behavior and hopefully relate it to more well-understood problems.

Dictionary Learning For Image Reconstruction Via Numerical Non-Convex Optimization Methods, Lewis M. Hicks

University Honors Theses

This thesis explores image dictionary learning via non-convex (difference of convex, DC) programming and its applications to image reconstruction. First, the image reconstruction problem is detailed and solutions are presented. Each such solution requires an image dictionary to be specified directly or to be learned via non-convex programming. The solutions explored are the DCA (DC algorithm) and the boosted DCA. These various forms of dictionary learning are then compared on the basis of both image reconstruction accuracy and number of iterations required to converge.