Master's Theses (Mathematics)
http://hdl.handle.net/1957/16491
2016-10-22T15:25:52ZPersistence of Populations in Environments with an Interface
http://hdl.handle.net/1957/59861
Persistence of Populations in Environments with an Interface
Rekow, James A.
We model a fish population in a spatial region comprising a marine protected area and a fishing ground separated by an interface. The model assumes conservation of biomass density and takes the form of a reaction diffusion equation with a logistic reaction term. At the interface, in addition to continuity of biomass density flux, two possible matching conditions are considered: continuity of population density and continuity of biomass density. Neumann conditions are imposed at the physical boundaries.
The eigenvalues of the elliptic problem resulting from the linearization of the model are computed. Necessary and sufficient conditions for the largest eigenvalue to be positive are determined. We show that there exist positive eigenfunctions corresponding to this eigenvalue. If the largest eigenvalue is positive, the population persists, whereas if this eigenvalue is negative, the population goes extinct. A simple sufficient condition for persistence when biomass density is continuous is that the spatially averaged net growth rate is positive. Similarly, if the spatially averaged net body mass growth rate is positive, the population persists when population is continuous.
A brief introduction is given on connections between parabolic partial differential equations and stochastic processes. Questions relating branching stochastic processes and properties of population models that incorporate interfaces are identified.
Graduation date: 2017
2016-09-09T00:00:00ZInterpolation Schemes for Two Dimensional Flow with Applications
http://hdl.handle.net/1957/58804
Interpolation Schemes for Two Dimensional Flow with Applications
Umhoefer, Joseph G.
In this thesis we study a numerical analysis problem motivated by the need to simulate an event such as an oil spill in a deep water environment. Numerical simulation can help to mitigate the disastrous effects of such events by aiding the management of risk assessment and recovery efforts.
However, an accurate simulation of the physical processes involved in the oil spill requires highly sophisticated and accurate numerical models. Equally important is that such a simulation needs to have accurate hydrodynamics data which may either come from observations or from some other computation simulator which predicts the flow of water near the area involved in the spill. In this thesis we discuss a particular technical problem involved with proper interpretation and use of hydrodynamic data.
In numerical analysis, it is often necessary to approximate a given function or interpolate from discrete data. One may have discrete data from sampling or due to solving a partial differential equation at discrete points but require information between the nodes. The motivation for this investigation of interpolation on scattered data is to recreate a smooth function from hydrodynamic data. In other words, we will discuss algorithms that provide a smooth field from given discrete data. The Blowout and Spill Occurrence Model (BLOSOM) developed by the Department of Energy's National Energy Technology Laboratory models hydrocarbon release events from the sea floor to the final fate of the oil. The generated smooth field could be used in such a model, potentially improving the predicted outcomes.
The errors in prediction of the fate of oil arise, of course, from multiple sources. We study the errors due to the interpolation scheme applied. One particular aspect is also associated with whether the interpolated velocities have non-physical characteristics, specifically whether the interpolated velocities are conservative, given that the true velocities are. Ultimately, we achieve good results using radial basis function interpolation, but the scale of the problem needs to be considered further, as the large data sets in use may make the problem intractable.
Graduation date: 2016
2016-03-14T00:00:00ZActive Learning and the ALEKS Placement Test in College Algebra : An Observational Study
http://hdl.handle.net/1957/58518
Active Learning and the ALEKS Placement Test in College Algebra : An Observational Study
Dean, Raven (Raven E.)
In 2012, a university in the north western United States began offering a redesigned college algebra class that had a greater emphasis on active learning. Specifically, two out of four class periods were completely devoted to students working together in small groups on carefully designed worksheets. Two years later, the university began using a new mathematics placement system. Students were required to take the ALEKS placement test before enrolling in a course. ALEKS scores determined what classes a student was eligible to take. The purpose of this thesis is to examine how students in the redesign courses fared in comparison to students in the traditional courses, and to determine how ALEKS test score correlated with grade point value in college algebra. First, linear regression will be used to model the relationship between ALEKS score and grade point value. Next, the average multiple choice exam score of students in a traditional section of college algebra will be compared to the average score of students in a redesigned course, using data from one instructor during one term. Then, logistic regression will be used to model the relationship between course type and failure rates. Finally, there will be an examination of the flow of college algebra students to their next mathematics course. With this information, a recommendation will be made for the future of college algebra.
Graduation date: 2016
2016-03-09T00:00:00ZListing as a Potential Connection between Sets of Outcomes and Counting Processes
http://hdl.handle.net/1957/57952
Listing as a Potential Connection between Sets of Outcomes and Counting Processes
Erickson, Sarah A.
Counting problems are rich in opportunities for students to make meaningful mathematical connections and develop non-algorithmic thinking; their accessible nature and applications to computer science make counting problems a valuable part of mathematics curricula. However, students struggle in various ways with counting, and while previous studies have indicated that listing may be a useful way to address student difficulties, little work has been done toward understanding exactly how students may connect lists of outcomes to their solutions to counting problems. To begin to address this, I conducted twenty task-based interviews with undergraduate students to probe the ways in which students conceptualize the relationship between sets of outcomes and counting processes. In this thesis, I describe the ways that students listed outcomes using an elaboration of English's (1991) solution strategies, and I frame my findings about their understanding using Lockwood's (2013) model of students' combinatorial reasoning. I discover that students reason about the relationship between lists of outcomes and counting processes with varying levels of sophistication, and I suggest that teachers could help students by making connections between sets of outcomes and counting processes more explicit.
Graduation date: 2016
2015-12-04T00:00:00Z