Abstracts: Contributed Talks

Section 5.2 Abstracts: Contributed Talks

If you or your students would like give a talk please email Hans Nordstrom (nordstro@up.edu) by June 11, 2021 with the title, short abstract using the abstract template.

Subsection 5.2.1 Contributed Talk Abstracts

Abstract 5.2.1. Dimensionality Reduction for Time-critical Support Vector Machine Applications.

Thomas Y. Chen: Academy for Mathematics, Science, and Engineering, presenter

Support vector machines constitute an important machine learning technique for prediction applications in a variety of interdisciplinary contexts. Dimensionality reduction is a technique that is utilized to reduce the time required to train and perform predictions on these algorithms. In this talk, we discuss how dimensionality reduction is conducted, in both a mathematical and programming context, as well as situations in which this approach is useful.

Abstract 5.2.2. Comparison of Change Point Detection Methods for Independent Data: Testing & Estimation.

Casey Christiansen: Western Washington University, presenter
Arick Grootveld
Jamie Shannon
Andrea Scolari
Ramadha Piyadi Gamage

Change point analysis (CPA) is a widely spread and valuable, research topic in many fields of applications. While there have been many proposed approaches for the detection of change points, only a small portion of the literature focuses on comparisons between such methods, let alone computational results to help distinguish between techniques. In this presentation, we discuss and compare between two prominent CPA methods, namely, the Cumulative Sum (CUSUM) and Likelihood Ratio (LR). The corresponding hypothesis test, test statistic, asymptotic distribution, and algorithm are summarized for each approach. Monte Carlo simulations are carried out to show the power of the test statistic using these two methods. We evaluate both methods on detecting change points in sequence of independent samples from different distributions. Further, we assess the accuracy of the estimated change location(s) relative to the actual change location(s) in order to illustrate the effectiveness of the methods at estimating the break locations.

Abstract 5.2.3. Using Euclidean Tools to Construct Elliptic Objects in the Klein Disk.

Genevieve Connolly: Oregon State University, presenter
Tevian Dray: Oregon State University

The Klein disk model of elliptic geometry can be realized as the image of (part of) the unit sphere under stereographic projection. We show how to construct elliptic lines, angles, and circles in the Klein disk using Euclidean tools but no measurement postulates. In other words, we construct an elliptic straightedge and compass using only a Euclidean straightedge and compass.

Abstract 5.2.4. Learning to Model Problems with GeoGebra.

Jose Contreras: Ball State University, presenter

In this presentation I will illustrate how learners can use GeoGebra as a tool to facilitate solving geometric problems. In particular, I will explore the picnic problem (a version of Viviani's problem): Three towns are the vertices of an equilateral triangle. The sides of the triangle are the roads that connect the towns. A picnic area will be constructed such that the sum of its distances to the roads is as small as possible. 1) What are all the possible locations for the picnic area? 2) For practical reasons, what is the best location for the picnic area? Justify your response.

Abstract 5.2.5. Specialization as a Problem-Posing Strategy: The Case of Varignon's Problem.

Jose Contreras: Ball State University, presenter

In this presentation I illustrate how we, instructors and students, can understand better the structure of a problem by investigating special cases of it. The use of Dynamic Geometry Software such as GeoGebra facilitates the formulation of conjectures, which can then be proved and reformulated as theorems. The processes of specialization and formulation of conjectures are illustrated with Varignon's problem: If E, F, G, and H are the midpoints of the consecutive sides of a quadrilateral ABCD, characterize quadrilateral EFGH.

Abstract 5.2.6. The Geometry of Numbers and the Four Squares Theorem.

Dibyajyoti Deb: Oregon Institute of Technology, presenter

We often associate proofs in number theory with abstractness, however, that is not always the case. Introducing ideas such as lattices and convex sets make these proofs more “visual”. In this talk we will look at a famous theorem of Minkowski on the geometry of numbers, and see how it can be used to prove a well-known result from number theory.

Abstract 5.2.7. Rethinking Assessment in the Age of Remote and Hybrid Instruction.

Michelle Ghrist: Gonzaga University, presenter

As we pivoted to online assessments last spring, I saw colleagues focused on how to give exams remotely in a secure fashion, including utilizing remote proctoring services; I also observed a sharp uptick in the number of academic integrity violations. I spent last summer rethinking my assessments, choosing to pivot to reduce dependence on exams that counted for a significant portion of students' grades, in an effort to find ways to do assessment that disincentivized students to cheat and focused on learning. This included rethinking my previously favored comprehensive final exams, as well as choosing to seek a “writing enriched” core designation for my Differential Equations course.

Abstract 5.2.8. Reconstructing images from derivative measurements by Discrete Laplacian Deconvolution.

David K. Hammond: Oregon Tech, presenter
Scott Prahl: Oregon Tech
Scott Breitenstein: Oregon Tech

We describe a novel method for recovering an unknown image from directional derivative measurements. Our method efficiently computes the ridge regression or the pseudoinverse estimate for the underlying image, using the fast fourier transform. We apply the method to empirical data from differential imaging contrast (DIC) microscopy. We show that under certain conditions our proposed method is equivalent to spiral phase integration, a commonly used method for this same image reconstruction problem. Through both simulation and application to empirical DIC data we show that the proposed method can give better reconstructions than the spiral phase method, through enabling use of multiple gradient directions and/or nonzero values for the ridge regression regularization constant.

Abstract 5.2.9. Generating Functions for Symmetric Polynomials.

Ethan Jensen: George Fox University, presenter

This talk will describe several generating functions for symmetric polynomials and use them to derive useful identities known as Newton's identities. Using these identities, a formula for solving nth order linear recurrence relations is proved. Finally, additional identities are derived for monomial symmetric polynomials and are used to compute values for a continuation of the Riemann zeta function, evaluated at a partition instead of at a complex number.

Abstract 5.2.10. Data analysis: Making use of large data sets.

Caroline Johnson: Davis Senior High School, Woodland Community College, presenter
Paul Johnson: Biostat Software Development

An investigator solves problems often by designing an experiment, collecting data and then testing hypotheses using the data. Models are developed. A procedure increasingly being used today is to download a data set from the internet and test if the investigator results based upon their data are consistent with those based upon a downloaded data set. There are many data sets available for download. Examples include daily air pollutant data from the EPA; water quality data from the USGS, census data, stock market data, COVID-19 case data, stream, river, lake, fish density data; climate and weather data from the National Oceanic and Atmospheric Administration. There is a wealth of information and often the data sets are so large they are in the form of zipped files. Before using such a large dataset a check should be made on their accuracy and validity. Statistical and graphical procedures looking for outliers, influential observations and errors can be used; but an initial investigative tool that may be used is Benford's law. Benford's law is that the leading significant nonzero digit (1,2, ... 9) obeys: P(first significant digit = d) = log10(1+1/d) for d = 1, 2, ...., 9. We used this law and examined several data sets, including census data from census.gov and water quality data from the United States Geological Survey (USGS). We compared the observed frequency counts of first significant digits to that expected. This resulted in questions being asked about anomalies seen, conflicts and discrepancies found. We catalogued these so that others can use the methods when using such large data sets for analysis. SAS® was used for the data analysis and for the production of graphs.

Abstract 5.2.11. Healthcare Loss Triangle Using Epidemiology Model.

Sooie-Hoe Loke: Central Washington University, presenter
Runhuan Feng: University of Illinois Urbana-Champaign
Longhao Jin: University of Illinois Urbana-Champaign

The impact of the COVID-19 pandemic in the health insurance industry have primarily been felt through increase in health spending and uncertainty in projecting future medical claims. One significant consideration is that health insurers pay different benefits to the infected individuals over their course of infection. However, traditional actuarial models lack the flexibility and robustness to describe the rapidly changing environment during a pandemic. To that end, we explore the epidemiology literature and combine some of the commonly used models with actuarial methodologies. We adopt the infection-age epidemic model and design epidemic insurance plans based on this model. Compared to classical life insurance, healthcare insurer's reserve function exhibit quite different patterns over the course of a pandemic and so we formulate conditions under which various reserve shapes arise. Inspired by well-known loss reserving techniques, we introduce a loss triangle, derived from the infection-age model, which is capable of predicting future unpaid healthcare liabilities during a pandemic.

Abstract 5.2.12. An Exploratory Data Analysis of Student Success in Mathematics Course Sequences.

Jen Nimtz: Western Washington University, presenter
Elias Bashir: Western Washington University

As mathematicians and mathematics educators, we see the logical progression of mathematics content in our course sequences, but how do we know if those sequences of courses are serving our students? What course sequences do students most commonly select? How might we analyze data to reveal student success in course sequences? In this presentation, we will outline the exploratory data analysis methods we are currently utilizing to evaluate student success in first year mathematics course sequences. The data set included 10 years of de-identified registrar data. We also will share future plans for expanding this analysis. Lastly, we welcome suggestions.

Abstract 5.2.13. A New Model for Rat-Flea—Driven Plague Transmission.

Andrew Oster: Eastern Washington University, presenter

Rats have long been thought to drive plague epidemics, specifically bubonic plague. However in a PNAS publication (2018), an alternative theory for plague transmission has been posited by Dean et al., where ectoparasites living on human hosts drive spread. In this talk, we will present a new mathematical model (developed with EWU undergraduate students: Ian Lynch and Luke Mattfeld) for the spread of the plague due to rat-flea interactions with the human population, and we will present a comparison of our results to existing models. Our results suggest that rat-flea transmission of the plague describes the data comparably to the Dean et al. model.

Abstract 5.2.14. A Square Deal For Squares: Lagrange's Recherches D'Arithmetique.

Erik Tou: University of Washington Tacoma, presenter

The history of modern number theory often begins with the work of Carl Friedrich Gauss. His 1801 work, Disquisitiones Arithmeticae, is a masterwork of mathematical writing and insight. Of course, Gauss relied on the work of those who preceded him, particularly Leonhard Euler and Joseph-Louis Lagrange. This talk explores some “rough draft” number theory produced by Lagrange in 1773 concerning quadratic forms, with the goal of telling a richer story about number theory during the late 18th century.

Abstract 5.2.15. Proving theorems in spherical geometry using the quaternions.

Marshall Whittlesey: California State University San Marcos, presenter

It is well known that the complex numbers can be used to do transformation geometry in the plane. In particular, rotation by angle \(\theta\) about the origin is accomplished via multiplication by the complex number \(e^{i\theta}=\cos(\theta)+i\sin(\theta)\text{.}\) It is less well known that the quaternion algebra (consisting of expressions of the form \(a+bi+cj+dk\) with \(i^2=j^2=k^2=-1\)) can be used to do similar transformations in three dimensional space. In this talk we show how to use quaternions to prove a significant classical theorem in spherical geometry. These methods are featured in the speaker's new book with CRC Press Spherical Geometry and its Applications, which the author hopes will be attractive for use in topics courses in geometry.

Abstract 5.2.16. Calculus Driven by Pandemic Data.

Aaron Wootton: University of Portland, presenter

Student engagement in math courses can be increased when driven by real life models of which they have direct experience. In this talk we shall see how many of the major topics in a first semester Calculus course, from the introduction of families of functions, through derivatives and integrals, can be motivated and explored using data collected from the Covid-19 pandemic.