Physical Sciences and Mathematics Commons^™

Enhancing The Quandle Coloring Invariant For Knots And Links, Karina Elle Cho

HMC Senior Theses

Quandles, which are algebraic structures related to knots, can be used to color knot diagrams, and the number of these colorings is called the quandle coloring invariant. We strengthen the quandle coloring invariant by considering a graph structure on the space of quandle colorings of a knot, and we call our graph the quandle coloring quiver. This structure is a categorification of the quandle coloring invariant. Then, we strengthen the quiver by decorating it with Boltzmann weights. Explicit examples of links that show that our enhancements are proper are provided, as well as background information in quandle theory.

Radial Solutions To Semipositone Dirichlet Problems, Ethan Sargent

HMC Senior Theses

We study a Dirichlet problem, investigating existence and uniqueness for semipositone and superlinear nonlinearities. We make use of Pohozaev identities, energy arguments, and bifurcation from a simple eigenvalue.

Mathematical Modeling Of Type 1 Diabetes, Gianna Wu

HMC Senior Theses

Type 1 Diabetes (T1D) is an autoimmune disease where the pancreas produces little to no insulin, which is a hormone that regulates blood glucose levels. This happens because the immune system attacks (and kills) the beta cells of the pancreas, which are responsible for insulin production. Higher levels of glucose in the blood could have very negative, long term effects such as organ damage and blindness.

To date, T1D does not have a defined cause nor cure, and research for this disease is slow and difficult due to the invasive nature of T1D experimentation. Mathematical modeling provides an alternative approach …

Gait And Postural Analysis In Healthy Young Adults And People With Parkinson's Disease, Aisha Joy Chen

CGU Theses & Dissertations

Postural analysis is the study of how the position of the body in any mode interacts with internal and external forces. This type of analysis is typically used to assess potential abnormalities in the balance control system and to understand how the balance control system changes with time. However, compared to other medical fields of study, postural analysis is relatively new [1]. In fact, although widely used in clinical and research studies, postural assessment methods are scientifically inaccurate, and some data collection methods are relatively expensive. A better understanding of the human balance control system could lead to more accurate …

Free Market On The Free Way, Yuan Cheng

CGU Theses & Dissertations

Self-driving cars have the potential to decrease congestion and will probably become the future of efficient transportation. This dissertation presents a unique approach to implement sharing lanes on a freeway using the idea of option pricing. A macroscopic physical model (LWR) is implemented by adding noise to the speed which accounts for unexpected events. We then proceed to provide a fair price for lanes in real time.

Procuring Pediatric Vaccines In A Two-Economy Duopoly, Seongeun Lee, Susan E. Martonosi

Scripps Senior Theses

In this work, we aim to present an optimization model for vaccine pricing in a two-economy duopoly. This model observes the price dynamics between a high income country and a low income country that procure vaccinations through PAHO. This model is formulated to provide insights on optimal pricing strategy for PAHO to ultimately increase vaccine accessibility to low income countries. The objective is to satisfy the public demand at the lowest price possible, while providing enough profit for the vaccine manufacturers to stay in business. Using non-linear integer programming, the model results show that cross-subsidization occurs in PAHO vaccine procurement.

Eigenvalues And Approximation Numbers, Ryan Chakmak

CMC Senior Theses

While the spectral theory of compact operators is known to many, knowledge regarding the relationship between eigenvalues and approximation numbers might be less known. By examining these numbers in tandem, one may develop a link between eigenvalues and l^p spaces. In this paper, we develop the background of this connection with in-depth examples.