The Cheeger’s constant, also known as the isoperimetric number, is a constant that helps describe the bottleneck present in a graph, if any. Some fields, such as computer networks, have an interest in this constant due to the application of the constant in their field. We examined randomly generated connected graphs and their isoperimetric numbers by developing algorithms to calculate it.

One of the major challenges in education is accurately quantifying a student’s knowledge and skills. Since we cannot directly measure a student’s true intelligence, we rely on test performance, which serves as an imperfect representation of their abilities. This issue arises in many statistical applications where the key problem involves a population in which each individual possesses an underlying ability or trait that cannot be directly observed but can only be inferred through proxy variables. However, these proxies are often contaminated, providing only a noisy or imperfect approximation of the true latent variable.
This project focuses on techniques for recovering latent variables from noisy data. In this context, "recovery" refers to estimating the latent variable using indirect observations. Assuming a linear relationship between the latent trait and the observed proxy variables, we can estimate model parameters and subsequently recover the values of the latent variables.
Specifically, we will examine statistical approaches to latent variable recovery when the test contains items that exhibit differential item functioning (DIF). This means that certain test items do not solely measure the intended knowledge or ability but are also biased toward specific groups. The objective is to develop methods that detect the presence of DIF and adjust for it, allowing for a more accurate estimation of the underlying abilities.
To illustrate these methods, we will use the Holzinger-Swineford dataset, a well-known dataset in psychometrics used to analyze cognitive abilities across multiple domains. This dataset includes 88 observations with scores in five areas: Mechanical Comprehension, Verbal or Visual Comprehension, Algebra Operations, Analytical Operations, and Statistical Reasoning. By applying a linear contamination model, we aim to recover each student's latent ability while accounting for DIF.

Oral Reading Accuracy (ORA) is an important metric for evaluating a student's reading proficiency, measuring how accurately a reader can read words aloud. Traditional ORA evaluations performed by human assessors often require significant time and labor. This study explores the potential of integrating a speech recognition system into ORA assessments to improve efficiency. We analyzed ORA data from 507 elementary school students across ten passages of different lengths and difficulties. Both human evaluators and AI systems recorded the number of words read correctly. The misclassification rates of these scores are divided into two components: True Positive (correct words are identified as correct), and True Negative (incorrect words are identified as correct). This second study expands upon Method of Moments method to estimate these misclassification rates. We apply Generalized Method of Moments which incorporates additional variance information. To compare the two approaches' accuracy, we apply the m-out-of-n Bootstrap method to estimate their standard errors and compare their reductions in estimator variance. Additionally, we introduce a Contaminated Data Solution to address real-world scenarios where true count data is unavailable and only contaminated observed data is observed.

Several viruses can cause cells to fuse into large multinucleated cells called syncytia. Syncytia formation allows the virus to spread without entering the extracellular space, where it might be exposed to immune responses. However, there is evidence that antibodies can also hinder the fusion process. This project uses mathematical analysis to find different possible infection outcomes. A stability analysis of the coinfection model is used to find the fixed points of the model and their stability. This gives us parameter space regions that lead to different possible infection outcomes. Simulations were made to verify the mathematical analysis and see how different syncytia formation properties affect the resulting dynamics. These findings could help develop strategies for controlling viral spread.

Graphene Quantum Dots (GQDs) are nanoscale carbon based graphene sheets that exhibit unique fluorescent properties throughout a wide range of wavelengths. Given their uniquely small size, low toxicity, biocompatibility, and fluorescent capabilities, GQDs have many unique and important roles. To name a few, GQDs are used in drug delivery, fluorescent imaging, and biosensing thanks to their unique ability to fluoresce under different wavelengths of light. Furthermore, there are different types of GQDs with their own unique properties. Knowing this, five amphipathic molecules, called surfactants, were added to two different types of GQDs to test if they would impact the resulting fluorescence. Furthermore, concentrations of these added surfactants were varied to test how different concentrations of a given surfactant might affect the fluorescence for a given GQD. We observed that some of these surfactants provided a beneficial boost to GQDs fluorescence, while others slightly inhibited the fluorescence. Moreover, we saw that the increase in fluorescence varied based on the concentration of surfactant added yielding lower fluorescence for extremely low and high concentrations, while increasing the fluorescence at a more moderate concentration.