Using the finite element method as a numerical approximation in solving for eigenvalues of the hyperbolic Laplacian, this research investigates the estimates of the first two eigenvalues with Dirichlet and Neumann boundary conditions on bounded domains in the upper half plane. Examples of finite element code using Matlab are presented to illustrate how to obtain these approximations for hyperbolic polygons. These values can further be used to shed light on the Selberg and Fundamental Gap conjectures.

In the past, Personality Psychologists have commonly applied clustering techniques on questionnaire data to analyze personality structure. The purpose of this research is to determine if techniques from topological data analysis can provide a greater understanding of personality. Specifically, persistent homology was used to determine clusters topologically and analyze the ‘shape’ of the data. Previous work we have done focused on seeing how persistent homology can provide insight on identifying the key (most persistent) clusters in the data. New analysis looks to see how the Big 5 Personality Factors cluster together as a function of age.

An algebraic curve is a one-dimensional set defined by polynomial equations, such as a parabola in the plane (given by y-x^2=0) or the z-axis in the space (given by x=y=0). Let Y be an algebraic curve. Then a multiplicity structure on Y is another curve Z, which as a set has the same points as Y but with a higher and fixed multiplicity at each point. For example, the y-axis in the plane is given by the equation x=0 and if we intersect it with horizontal lines, say with y-b=0, we get the points (0,b) on the y-axis. Now if we take the line given by x^2=0 and intersect it with the horizontal lines as above we get the points (0,b) with multiplicity 2. Hence we call the later curve a double structure on the previous one. Similarly the equation x^3=0 gives a triple structure on the y-axis in the plane and so on. Structures like these might sound naive but they are crucial to understand the behaviors of families of curves. For example, the family of parabolas ty-x^2=0 deforms into the double line x^2=0 as t approaches 0. Although the notion of multiplicity is pretty geometric, we can use tools from abstract algebra to make it rigorous. This makes the subject challenging and yet very interesting at the same time. Classifying the multiplicity structures on a curve is still a wide open field in algebraic geometry. It is now well understood how the double and triple structures on a line look. A natural question then arises how do the double and triple structures look on conics? It turns out that the answers are much more complicated than for lines. In this poster I am going to show some of my research in that direction.

An algebraic integer is a complex number that is a root of a polynomial with integer coefficients and a leading coefficient of 1. This includes numbers like the square root of 2 and the cube root of 10, for example. A field is a set in which we can add, subtract, multiply, and divide (among other details). Consider the set of all algebraic integers in any given field containing the rational numbers. The index of an algebraic integer in this set is a natural number that measures how close the algebraic integer is to generating the set. For instance, the imaginary number i (the square root of -1) is an algebraic integer which generates the set of all complex numbers of the form a + bi where a and b are integers, and so has index 1. The closer the index is to 1, the closer the algebraic integer is to generating the set. We investigate these indices in cubic fields, determining not only which numbers occur as indices in given families, but also that the minimal index is unbounded as one traverses the set of all cubic fields in those families.

Background: To replace milk fats and eggs commonly found in ice cream, vegan varieties substitute with vegetable fats and/or pureed fruits. Vegan ingredient substitutions for ice cream must contain similar structural components to milk fat to preserve the expected texture/mouthfeel of the product. The purposes of this study were 1) to measure university students’ preferences and sensory ratings of vegan ice cream substitutions and 2) to identify which ingredients act as the best replacements.

Methods: 54 students enrolled in one of two TCU Nutritional Sciences (NTDT) courses participated in this single-blind, cross-sectional study. Subjects completed sensory evaluation of three homemade vegan ice creams containing different structural/flavor components. Sample A included coffee, cashews, and coconut cream; B used coconut cream and dates, and C contained coconut milk. Evaluations took place on two separate occasions in the NTDT Laboratory Kitchens. Flavor, sweetness, texture, mouthfeel, eye appeal, color, and overall rating of vegan ingredient substitutions for ice cream recipes were assessed. Sensory criteria responses were analyzed using SPSS XIX. Frequency distributions, ANOVAs, correlations, and descriptive statistics were determined to meet study objectives (p≤0.05). Protocol was approved by the TCU IRB.

Results: 53% of participants preferred the flavor of sample A, 42% of participants reported that sample B was the preferred flavor, and sample C received the lowest overall acceptability rating, with 73% of the participants disliking the flavor. More than 50% of participants stated sample A was most similar to traditionally-prepared non-vegan ice cream, and 78% of participants stated that they would consume these ice creams outside of the study.

Conclusions: Acceptable vegan ice cream fat substitutes are available. Cashew, coconut cream, dates, and coffee contributed to the rich flavors, creamy textures, and overall desirable sensory qualities in samples A and B. Coconut milk, utilized in sample C, contributed to an undesirable and unacceptable crystallized texture.