2018-2019 https://dspace.carthage.edu/handle/20.500.13007/10659 2021-12-01T09:28:43Z Infinite Snowman https://dspace.carthage.edu/handle/20.500.13007/9552 Infinite Snowman Montoya, Gladys In this research paper we explore how to create an infinite snowman within an isosceles triangle to find the sum of its total area. Our main objective was to find the sum of the areas of the circles inscribed within the isosceles triangle. 2019-05-17T00:00:00Z Minimizing Transpositions in Permutations with Indistinct Variables https://dspace.carthage.edu/handle/20.500.13007/9551 Minimizing Transpositions in Permutations with Indistinct Variables Hussey, Mary In any permutation within a symmetric group, the minimum number of transpositions required to return the permutation to the identity can be algorithmically determined. In fact, this algorithm determines this required number of transpositions based on the degree of the group and the cycle structure of the given permutation. This consistent property changes as the set acted upon by Sn becomes a multiset. In this paper, we explore how introducing these duplicates of some of the variables may reduce the required number of transpositions and determine the altered algorithm for this change in required number of transpositions. We find that this reduction is reliant on the number of indistinct variables for each unique variable, as well as the distribution of variables in the permutation and the cycles within the permutation. Since there are more intricacies to this algorithm than the base case of strictly unique variables, for any given cycle structure there are multiple possibilities of the required number of transpositions to return all variables to their original position. 2019-05-17T00:00:00Z Mathematically Modeling the Josephson Junction https://dspace.carthage.edu/handle/20.500.13007/9550 Mathematically Modeling the Josephson Junction Fiege, Nathan Josephson junctions are widely used in applications where a very precise voltage source is required, one example being metrology, where they are used to help define the values of several fundamental constants of nature. Modeling the Josephson junction furthers our understanding of the behavior of the junction which has had profound impacts in the metrological world. This junction and its voltage can be described by a nonlinear differential equation similar to that of the simple pendulum. Differential equations of this type are difficult to understand and very difficult if not impossible to solve explicitly. After deriving this nonlinear differential equation using quantum mechanical arguments, the solutions are found independently using both analytical and numerical approaches. The qualitative behavior of solutions change at a critical current, resulting in either no voltage across the junction or voltage asymptotically approaching the classical Ohm's law. Further modeling of this system could include modeling specific behaviors including the AC and DC Josephson effects. 2019-05-17T00:00:00Z Reaction-Diffusion Equations and Biochemical Pattern Formation https://dspace.carthage.edu/handle/20.500.13007/9549 Reaction-Diffusion Equations and Biochemical Pattern Formation Bresnahan, Brady Many biochemical reactions can be modeled using reaction-diffusion equations, some of which form patterns. Reaction-diffusion equations are partial differential equations that describe how the concentrations of two species change over time when subjected to diffusion and chemical reaction. Diffusion is represented in all cases by a first derivative with respect to time and a second derivative with respect to space. Chemical interactions are represented differently depending on the model, though they are typically described by nonlinear combinations of the dependent variables or concentrations. The two models explored here are the activator-substrate and the activator-inhibitor reaction-diffusion equations. We perform a stability analysis of each system and investigate solutions numerically under sinusoidal and random conditions with various parameters. The most interesting results come from the activator-inhibitor model under random conditions. 2019-05-17T00:00:00Z