We propose two neural network based and data-driven supply and demand models to analyze the efficiency, identify service gaps, and determine the significant predictors of demand, in the bus system for the Department of Public Transportation (HDPT) in Harrisonburg City, Virginia, which is the home to James Madison University (JMU). The supply and demand models, one temporal and one spatial, take many variables into account, including the demographic data surrounding the bus stops, the metrics that the HDPT reports to the federal government, and the drastic change in population between when JMU is on or off session. These direct and data-driven models to quantify supply and demand and identify service gaps can generalize to other cities' bus systems.

We develop a new algorithm to compute a basis for $M_k(\Gamma_0(N))$, the space of weight $k$ holomorphic modular forms on $\Gamma_0(N)$, in the case when the graded algebra of modular forms over $\Gamma_0(N)$ is generated at weight two. Our tests show that this algorithm significantly outperforms a commonly used algorithm which relies more heavily on modular symbols.