Mathematics Department A collection of papers written by students and faculty of the Carthage College Mathematics Department http://hdl.handle.net/123456789/186 2019-06-25T08:06:24Z 2019-06-25T08:06:24Z The Double Exposure SI Model: A Senior Thesis Stapf, Kerry http://hdl.handle.net/123456789/5956 2017-12-21T06:44:30Z 2017-05-31T00:00:00Z The Double Exposure SI Model: A Senior Thesis Stapf, Kerry The SI model of disease spread is a fairly simple and well-known way to study disease. In the model, each person is either susceptible or infected, and the infection spreads when a susceptible person interacts with an infected person. A more realistic simulation on a network can be compared to this differential equation model in order to determine if the theoretical equation correctly predicts the behavior of the simulation. This research looks into a double-exposure model, where a person has to come into contact with at least two infected people before becoming infected. The theoretical equation will then be analyzed to see how it behaves, how it differs from the SI model, and if a simulated disease matches the theoretical curves. 2017-05-31T00:00:00Z Zero-Divisor Graphs of Direct Product Rings Salzman, Emily http://hdl.handle.net/123456789/5955 2017-12-21T06:44:28Z 2017-05-31T00:00:00Z Zero-Divisor Graphs of Direct Product Rings Salzman, Emily This research investigates zero-divisor graphs of direct product rings. In zero-divisor graphs, if elements are connected with an edge, they will multiply to the additive identity element of direct product rings, (0,0). We look at graphs of the form Z_2p×Z_2q, where p and q are prime numbers greater than 2. In order to generalize the structure of these graphs, many specific examples of Z_2p×Z_2q graphs are analyzed to better understand the common form. Upon investigating these rings, many “clumps”, or “families”, of zero-divisors appeared. Certain families will always connect to other families, and some families will never connect. This research investigates which families connect, making it possible to generalize the structure of these graphs. 2017-05-31T00:00:00Z Mathematical Representation of the Chlorine Dioxide-Iodine-Malonic Acid Reaction Rutter, Elisabeth http://hdl.handle.net/123456789/5954 2017-12-21T06:44:24Z 2017-05-31T00:00:00Z Mathematical Representation of the Chlorine Dioxide-Iodine-Malonic Acid Reaction Rutter, Elisabeth Mathematics is frequently used in chemistry to solve several different types of problems including thermodynamics, stoichiometry, and analyzing experimental data. However, there are more interesting ways to link these two disciplines using chemical kinetics. In this research, the oscillating chemical reaction, chlorine dioxide-iodine-malonic acid, was studied using differential equation modeling. The behavior of the reaction was investigated as the components of the reaction oscillated about its equilibrium point. The chemical reaction was modeled as a differential rate law equation and was simplified and non-dimensionalized in order to derive a system of differential equations that modeled the system. Numerical integration and linear stability analysis was used in order to observe how the chemical reaction behaved at different parameter values. 2017-05-31T00:00:00Z The Impact of Graph Density and Infection Probability on Disease Spread in Temporal Networks Northrup, Catherine http://hdl.handle.net/123456789/5953 2017-12-21T06:44:22Z 2017-05-31T00:00:00Z The Impact of Graph Density and Infection Probability on Disease Spread in Temporal Networks Northrup, Catherine Epidemiological processes, such as the spread of a disease through a population, can be represented in a simple way by applying SI dynamics to temporal networks. The spreading behavior is observed by running simulations and comparing the results against theoretical approximations using differential equations. Two types of graphs are used in the simulations: Bernoulli graphs, which are built based on probability, and Barabasi-Albert graphs, which are built using the concept of preferential attachment . For the first type, we use mean field theory to approximate the behavior, while for the second type we use a degree-based mean field approximation to account for the difference in structure. The main focus of this study is on the impact of two specific variables on spreading behavior, namely, graph density and infection probability. Here we research the effects of changes in these variables independently as well as in conjunction. Results show that changes in graph density and infection probability affect the overall percentage of the population that becomes infected and the time it takes the infection to fully spread. 2017-05-31T00:00:00Z