Department of Mathematics
http://hdl.handle.net/1957/13737
Sat, 15 Apr 2017 00:53:32 GMT2017-04-15T00:53:32ZA New Algorithm for Computing the Veech Group of a Translation Surface
http://hdl.handle.net/1957/60633
A New Algorithm for Computing the Veech Group of a Translation Surface
Edwards, Brandon (Brandon Gary)
We give a new characterization of elements in the Veech group of a translation surface. This provides a computational test for Veech group membership. We use this computational test in an algorithm that detects when the Veech group is a lattice (has co-finite area), and in this case computes a fundamental polygon for the action of the Veech group on the hyperbolic plane. A standard result, essentially due to Poincaré, provides that a complete set of generators for the Veech group can then be obtained from the side pairings associated to this fundamental polygon.
Our approach introduces a new computational framework used to formulate a membership criterion for the Veech group of a compact translation surface (X,ω). We represent (X,ω) on a certain non-compact translation surface O that can be used to represent any translation surface within the SL(2,ℝ) orbit of the translation equivalence class of (X,ω). The surface O has an easily computed SL(2,ℝ)-action. When this action is restricted to the translation surface representations mentioned above, it corresponds to the usual SL(2,ℝ)-action on the set of equivalence classes of translation surfaces. The Veech group of a compact translation surface is therefore the stabilizer of its representation on O.
Graduation date: 2017
Fri, 17 Mar 2017 00:00:00 GMThttp://hdl.handle.net/1957/606332017-03-17T00:00:00ZQuantitative Study of Math Excel Calculus Courses
http://hdl.handle.net/1957/60588
Quantitative Study of Math Excel Calculus Courses
Watkins, Christopher H.
About twenty years ago, a large, rural, doctoral granting institution with an undergraduate population of approximately 24,000 in the pacific northwest of the United States established the Math Excel program. Students would attend lectures three times a week for 50 minutes like a traditional course, and they would also attend two workshops per week that are two hours each, in contrast to traditional courses with a 50 minute recitation once per week. For several years the university would offer a few sections of Math Excel for several 100- and 200-level mathematics courses each term. During the 2013-2014 academic year, the university dedicated all sections of Math Excel to a particular section of calculus and implemented a Math Excel section of a calculus every quarter with the sequence of courses consisting of differential calculus, integral calculus, and vector calculus. A Math Excel version of differential calculus, integral calculus, and vector calculus were offered during fall term, winter term, and spring term respectively. The purpose of this thesis is to investigate how students in the Math Excel calculus courses performed compared to students in traditional calculus courses. First, a logistic regression model will be used
to model the relationship between pass rates and enrollment in a Math Excel calculus course after controlling for predictor variables and a two-sample t-test will be performed to compare pass rates of students in a Math Excel calculus course and a traditional calculus course. Next, a linear regression model will be used to model the relationship between grade points and enrollment in a Math Excel calculus course and a two-sample t-test will be performed to compare grade points of students in a Math Excel calculus course and a traditional calculus course. Finally, a two-sample t-test will be performed to examine if there is a difference in average grade earned for students that took a Math Excel calculus course in the previous term and students that did not. In each of these cases I found that there is not significant evidence that the Math Excel program has a greater effect on student pass rates, grades, or future grades in calculus courses than traditional versions of the calculus courses. Based on these results, I suggest that more data is gathered to see if there is a change in the results, an in depth analysis of students’ demographics and activities outside the classroom, and looking at the instructor effect and execution in the classroom in order to understand how and whether the Math Excel program benefits students in different ways.
Graduation date: 2017
Tue, 28 Feb 2017 00:00:00 GMThttp://hdl.handle.net/1957/605882017-02-28T00:00:00ZVariations of Mathematics in College Algebra Instruction : An Investigation Through the Lenses
of Three Observation Protocols
http://hdl.handle.net/1957/60045
Variations of Mathematics in College Algebra Instruction : An Investigation Through the Lenses
of Three Observation Protocols
Gibbons, Claire J.
College Algebra is a prerequisite for calculus and is thus an important stepping stone in the careers of STEM-intending undergraduates. However, College Algebra has low pass rates across the United States, interrupting students’ pathways to success. To address this concern, a research-oriented university in the Northwest United States restructured its College Algebra course to increase student engagement and active learning practices. Despite a new common curriculum, wide variation in the mathematical content that is presented by instructors was observed. Through the lenses of three observation protocols applied to video recordings of College Algebra classrooms, this thesis investigates the mathematical content present in lessons covering two mathematical concepts, evaluates the protocols for their ability to capture the variation in mathematics, and synthesizes these results to offer ideas for future research in College Algebra instruction.
Graduation date: 2017
Tue, 06 Dec 2016 00:00:00 GMThttp://hdl.handle.net/1957/600452016-12-06T00:00:00ZNonuniform Sampling Of Band-limited Functions
http://hdl.handle.net/1957/59862
Nonuniform Sampling Of Band-limited Functions
Al-Hammali, Hussain Y.
In this thesis, we will study certain generalizations of the classical Shannon Sampling Theorem, which allows for the reconstruction of a pi-band-limited, square-integrable function from its samples on the integers. J. R. Higgins provided a generalization where the integers can be perturbed by less than 1/4, which includes nonuniform and nonperiodic sampling sets. We generalize Higgins’ theorem by allowing for sampling sets that are perturbations of the set of zeros of a π-sine-type function.
A second type of generalization allows for functions f that, while still band-limited, need not be square-integrable but may have polynomial growth when restricted to the real line. We investigate two ways to achieve this goal, again using nonuniform sampling sets. The first is an approximate method that uses the multiplication of f by a smooth and rapidly decaying auxiliary function. The second method is exact and uses oversampling by finitely many additional points. It is also shown that oversampling by finitely many points is not only economical and may lead to faster convergence of the series, but also enables the perturbed sampling points to go beyond a quarter from the integers. Furthermore, oversampling by finitely many points is applied to control the error stemming from a quantization of the sampled function values.
The final topic considered is the so-called peak value problem, where one seeks to find an upper bound for the infinity norm of a function from knowledge of the supremum of its sampled values. We generalize an existing approach by first proving and then applying a nonuniform version of the Valiron-Tschakaloff sampling theorem.
Graduation date: 2017
Mon, 22 Aug 2016 00:00:00 GMThttp://hdl.handle.net/1957/598622016-08-22T00:00:00Z