Physical Sciences and Mathematics Commons^™

Calculations From On The Existence Of Periodic Traveling-Wave Solutions To Certain Systems Of Nonlinear, Dispersive Wave Equations, Jacob Daniels

Mathematics and Statistics Student Research and Class Projects

In the field of nonlinear waves, particular interest is given to periodic traveling-wave solutions of nonlinear, dispersive wave equations. This thesis aims to determine the existence of periodic traveling-wave solutions for several systems of water wave equations. These systems are the Schr¨odinger KdV-KdV, Schr¨odinger BBM-BBM, Schr¨odinger KdV-BBM, and Schr¨odinger BBM-KdV systems, and the abcd-system. In particular, it is shown that periodic traveling-wave solutions exist and are explicitly given in terms of cnoidal, the Jacobi elliptic function. Certain solitary-wave solutions are also established as a limiting case of the periodic traveling-wave solutions, that is, as the elliptic modulus approaches one.

Tasks For Learning Trigonometry, Sydnee Andreasen

All Graduate Reports and Creative Projects, Fall 2023 to Present

Many studies have been done using task-based learning within different mathematics courses. Within the field of trigonometry, task-based learning is lacking. The following research aimed to create engaging, mathematically rich tasks that meet the standards for the current trigonometry course at Utah State University and align with the State of Utah Core Standards for 7th through 12th grades. Four lessons were selected and developed based on the alignment of standards, the relevance to the remainder of the trigonometry course, and the relevance to courses beyond trigonometry. The four lessons that were chosen and developed were related to trigonometric ratios, graphing …

Vectors And Vector Borne Disease: Models For The Spread Of Curly Top Disease And Culex Mosquito Abundance, Rachel M. (Frantz) Georges

All Graduate Theses and Dissertations, Fall 2023 to Present

Mathematical models are useful tools in managing infectious disease. When designed appropriately, these models can provide insight into disease incidence patterns and transmission rates. In this work, we present several models that provide information that is useful in monitoring diseases spread by insects.

In the first part of this dissertation, we present two models that predict disease incidence patterns for Curly Top disease (CT) in tomato crops. CT affects a wide variety of plants and is spread through the bite of the Beet Leafhopper. This disease is particularly devastating to tomato crops. When infected, tomato plants present with stunted growth …

On The Existence Of Periodic Traveling-Wave Solutions To Certain Systems Of Nonlinear, Dispersive Wave Equations, Jacob Daniels

All Graduate Theses and Dissertations, Fall 2023 to Present

A variety of physical phenomena can be modeled by systems of nonlinear, dispersive wave equations. Such examples include the propagation of a wave through a canal, deep ocean waves with small amplitude and long wavelength, and even the propagation of long-crested waves on the surface of lakes. An important task in the study of water wave equations is to determine whether a solution exists. This thesis aims to determine whether there exists solutions that both travel at a constant speed and are periodic for several systems of water wave equations. The work done in this thesis contributes to the subfields …

A Comprehensive Uncertainty Quantification Methodology For Metrology Calibration And Method Comparison Problems Via Numeric Solutions To Maximum Likelihood Estimation And Parametric Bootstrapping, Aloka B. S. N. Dayarathne

All Graduate Theses and Dissertations, Fall 2023 to Present

In metrology, the science of measurements, straight line calibration models are frequently employed. These models help understand the instrumental response to an analyte, whose chemical constituents are unknown, and predict the analyte’s concentration in a sample. Techniques such as ordinary least squares and generalized least squares are commonly used to fit these calibration curves. However, these methods may yield biased estimates of slope and intercept when the calibrant, substance used to calibrate an analytical procedure with known chemical constituents (x-values), carries uncertainty. To address this, Ripley and Thompson (1987) proposed functional relationship estimation by maximum likelihood (FREML), which considers uncertainties …

Facilitating Mathematics And Computer Science Connections: A Cross-Curricular Approach, Kimberly E. Beck, Jessica F. Shumway, Umar Shehzad, Jody Clarke-Midura, Mimi Recker

Publications

In the United States, school curricula are often created and taught with distinct boundaries between disciplines. This division between curricular areas may serve as a hindrance to students' long-term learning and their ability to generalize. In contrast, cross-curricular pedagogy provides a way for students to think beyond the classroom walls and make important connections across disciplines. The purpose of this paper is a theoretical reflection on our use of Expansive Framing in our design of lessons across learning environments within the school. We provide a narrative account of our early work in using this theoretical framework to co-plan and enact …

Using Gamification To Foster Student Resilience And Motivation To Learn, And Using Games To Teach Significance Testing Concepts In The Statistics Classroom, Todd Partridge

All Graduate Theses and Dissertations, Fall 2023 to Present

Two studies are outlined in this dissertation.

In the first study, elements of Super Mario Bros. videos games were used to change the way college students in a beginners’ statistics course were graded on their work. This was part of an effort to help students remain optimistic in the face of challenging coursework and even failure on assignments and tests. The study shows that the changes made to the grading structure did help students to keep trying and to use the materials given to them by their professor until they achieved their desired grade in the course, and suggests ways …

Characterization Of The Long-Distance Dispersal Kernel Of White-Tailed Deer And Evaluating Its Impact On Chronic Wasting Disease Spread In Wisconsin, Mennatallah Gouda

All Graduate Theses and Dissertations, Fall 2023 to Present

Chronic Wasting Disease (CWD) is a fatal, untreatable neurodegenerative disease that infects deer and related species. It is highly contagious and caused by abnormal malfunction and assembly of normal cellular proteins into aggregation-prone proteins. The Centers for Disease Control and prevention report that the prevalence of CWD in free-ranging deer in the US is still relatively low. However, in several states the infection rates exceed 1 deer in 10. Deer may uptake CWD from direct interaction with infected individuals or from the environment. Infected individuals shed CWD into the environment through feces, urine, saliva or carcasses, and long-distance dispersal of …

Using Assessments To Promote Growth Mindset In College Algebra, Hannah M. Lewis, Kady Schneiter, David Lane Tait

Mathematics and Statistics Faculty Publications

Scientific evidence highlights the positive impact of a growth mindset on student achievement. Students with a growth mindset view errors and obstacles as opportunities for growth and welcome challenges and the opportunity to learn from their mistakes. Much has been written about promoting growth mindset through lectures and attitudes, however, assessments can also be an important avenue for encouraging a growth mindset in students. In this paper, we describe how we used assessments to promote growth mindset in a college algebra class. In the sections that follow, we discuss the need for these assessments and the principles that underly their …

A Novel Fuzzy Relative-Position-Coding Transformer For Breast Cancer Diagnosis Using Ultrasonography, Yanhui Guo, Ruquan Jiang, Xin Gu, Heng-Da Cheng, Harish Garg

Computer Science Faculty and Staff Publications

Breast cancer is a leading cause of death in women worldwide, and early detection is crucial for successful treatment. Computer-aided diagnosis (CAD) systems have been developed to assist doctors in identifying breast cancer on ultrasound images. In this paper, we propose a novel fuzzy relative-position-coding (FRPC) Transformer to classify breast ultrasound (BUS) images for breast cancer diagnosis. The proposed FRPC Transformer utilizes the self-attention mechanism of Transformer networks combined with fuzzy relative-position-coding to capture global and local features of the BUS images. The performance of the proposed method is evaluated on one benchmark dataset and compared with those obtained by …

An Interval-Valued Random Forests, Paul Gaona Partida

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

There is a growing demand for the development of new statistical models and the refinement of established methods to accommodate different data structures. This need arises from the recognition that traditional statistics often assume the value of each observation to be precise, which may not hold true in many real-world scenarios. Factors such as the collection process and technological advancements can introduce imprecision and uncertainty into the data.

For example, consider data collected over a long period of time, where newer measurement tools may offer greater accuracy and provide more information than previous methods. In such cases, it becomes crucial …

Stressor: An R Package For Benchmarking Machine Learning Models, Samuel A. Haycock

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Many discipline specific researchers need a way to quickly compare the accuracy of their predictive models to other alternatives. However, many of these researchers are not experienced with multiple programming languages. Python has recently been the leader in machine learning functionality, which includes the PyCaret library that allows users to develop high-performing machine learning models with only a few lines of code. The goal of the stressor package is to help users of the R programming language access the advantages of PyCaret without having to learn Python. This allows the user to leverage R’s powerful data analysis workflows, while simultaneously …

A Frobenius-Schur Extension For Real Projective Representation, Levi Gagnon‐Ririe

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Many problems in physics have explicit mathematical descriptions. This thesis aims to provide the mathematical tools for a particular problem in physics, that of Quantum Mechanical symmetries. In essence, we extend the known mathematics to a more general setting and provide a wider view of Real projective representation theory. The work done in this thesis contributes to the subfield of mathematics known as representation theory and to the subfield of physics concerned with time reversal symmetry.

Investigating The Effect Of Greediness On The Coordinate Exchange Algorithm For Generating Optimal Experimental Designs, William Thomas Gullion

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Design of Experiments (DoE) is the field of statistics concerned with helping researchers maximize the amount of information they gain from their experiments. Recently, researchers have been turning to optimal experimental designs instead of classical/catalog experimental designs. One of the most popular algorithms used today to generate optimal designs is the Coordinate Exchange (CEXCH) Algorithm. CEXCH is known to be a greedy algorithm, which means it tends to favor immediate, locally best designs instead of globally optimal designs. Previous research demonstrated that this tradeoff was efficacious in that it reduced the cost of a single run of CEXCH and allowed …

Examining Model Complexity's Effects When Predicting Continuous Measures From Ordinal Labels, Mckade S. Thomas

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Many real world problems require the prediction of ordinal variables where the values are a set of categories with an ordering to them. However, in many of these cases the categorical nature of the ordinal data is not a desirable outcome. As such, regression models treat ordinal variables as continuous and do not bind their predictions to discrete categories. Prior research has found that these models are capable of learning useful information between the discrete levels of the ordinal labels they are trained on, but complex models may learn ordinal labels too closely, missing the information between levels. In this …

I-Optimal Or G-Optimal: Do We Have To Choose?, Stephen J. Walsh, Lu Lu, Christine M. Anderson-Cook

Mathematics and Statistics Faculty Publications

When optimizing an experimental design for good prediction performance based on an assumed second order response surface model, it is common to focus on a single optimality criterion, either G-optimality, for best worst-case prediction precision, or I-optimality, for best average prediction precision. In this article, we illustrate how using particle swarm optimization to construct a Pareto front of non-dominated designs that balance these two criteria yields some highly desirable results. In most scenarios, there are designs that simultaneously perform well for both criteria. Seeing alternative designs that vary how they balance the performance of G- and I …

Geometry And Coding: Introducing An Interactive And Integrated Mathematics-Computer Science Unit, Kimberly Beck, Jessica F. Shumway

Publications

As part of a collaborative project between Utah State University, the Cache County School District, and Stanford, instructional units were designed for fifth-grade students. These units integrated math concepts of geometrical shapes and computer science concepts of sequences, conditionals, and loops. One component of the unit was implemented in math classrooms by math teachers, and the other component was implemented in computer labs. This presentation will focus on the math unit as presented at the National Council of Teachers of Mathematics (NCTM-V).

An Exploration Of Computational Text Analysis Of Co-Design Discourse In A Research-Practice Partnership, Mei Tan, Victor R. Lee

Publications

In combination with contextualized human interpretation, computational text analysis offers a quantitative approach to interrogating the nature of participation and social positioning in discourse. Using meeting transcript data from the development of a co-design research-practice partnership, we examine the roles and forms of participation that contribute to an effective collaboration between a multileveled school system and researcher partners. We apply computational methods to explore the language of co-design and multi-stakeholder perspectives in support of educational improvement science efforts and our theoretical understanding of partnership roles. Results indicate participation patterns align with documented roles in co- design partnerships and highlight the …

Epidemic Highs And Lows: A Stochastic Diffusion Model For Active Cases, Luis F. Gordillo, Priscilla E. Greenwood, Dana Strong

Mathematics and Statistics Faculty Publications

We derive a stochastic epidemic model for the evolving density of infective individuals in a large population. Data shows main features of a typical epidemic consist of low periods interspersed without breaks of various intensities and duration. In our stochastic differential model, a novel reproductive term combines a factor expressing the recent notion of ‘attenuated Allee effect’ and a capacity factor is controlling the size of the process. Simulation of this model produces sample paths of the stochastic density of infectives, which behave much like long-time Covid-19 case data of recent years. Writing the process as a stochastic diffusion allows …

Supplementary Files For "Adaptive Mapping Of Design Ground Snow Loads In The Conterminous United States", Jadon Wagstaff, Jesse Wheeler, Brennan Bean, Marc Maguire, Yan Sun

Browse all Datasets

Recent amendments to design ground snow load requirements in ASCE 7-22 have reduced the size of case study regions by 91% from what they were in ASCE 7-16, primarily in western states. This reduction is made possible through the development of highly accurate regional generalized additive regression models (RGAMs), stitched together with a novel smoothing scheme implemented in the R software package remap, to produce the continental- scale maps of reliability-targeted design ground snow loads available in ASCE 7-22. This approach allows for better characterizations of the changing relationship between temperature, elevation, and ground snow loads across the Conterminous United …

Co-Designing Elementary-Level Computer Science And Mathematics Lessons: An Expansive Framing Approach, Umar Shehzad, Jody Clarke-Midura, Kimberly Beck, Jessica Shumway, Mimi Recker

Publications

This study examines how a rural-serving school district aimed to provide elementary-level computer science (CS) by offering instruction during students’ computer lab time. As part of a research-practice partnership, cross-context mathematics and CS lessons were co-designed to expansively frame and highlight connections across – as opposed to integration within – the two subjects. Findings indicated that most students who engaged with the lessons across the lab and classroom contexts reported finding the lessons interesting, seeing connections to their mathematics classes, and understanding the programming. In contrast, a three-level logistic regression model showed that students who only learned about mathematics connections …

Data Visualization, Dimensionality Reduction, And Data Alignment Via Manifold Learning, Andrés Felipe Duque Correa

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The high dimensionality of modern data introduces significant challenges in descriptive and exploratory data analysis. These challenges gave rise to extensive work on dimensionality reduction and manifold learning aiming to provide low dimensional representations that preserve or uncover intrinsic patterns and structures in the data. In this thesis, we expand the current literature in manifold learning developing two methods called DIG (Dynamical Information Geometry) and GRAE (Geometry Regularized Autoencoders). DIG is a method capable of finding low-dimensional representations of high-frequency multivariate time series data, especially suited for visualization. GRAE is a general framework which splices the well-established machinery from kernel …

Symplectic Reduction Along A Submanifold, Peter Crooks, Maxence Mayrand

Mathematics and Statistics Faculty Publications

We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex algebraic varieties, and has an interpretation in terms of derived stacks in shifted symplectic geometry. It also encompasses Marsden-Weinstein-Meyer reduction, Mikami-Weinstein reduction, the pre-images of Poisson transversals under moment maps, symplectic cutting, symplectic implosion, and the Ginzburg-Kazhdan construction of Moore-Tachikawa varieties in topological quantum field theory. A key feature of our construction is a concrete and systematic association of a Hamiltonian G-space 𝔐_𝐺,𝑆 to …

Applying Expansive Framing To An Integrated Mathematics-Computer Science Unit, Kimberly Evagelatos Beck, Jessica F. Shumway

Publications

In this research report for the National Council of Teachers of Mathematics 2022 Research Conference, we discuss the theory of Expansive Framing and its application to an interdisciplinary mathematics-computer science curricular unit.

Leveraging The "Large" In Large Lecture Statistics Classes, Kady Schneiter, Kimberleigh Felix Hadfield, Jenny Lee Clements

Mathematics and Statistics Faculty Publications

Being a teacher or a student in a class with a large enrollment can be intimidating. Often, teachers view comforts that are common to small classes as unattainable in a larger class, including knowing students’ names, using active learning, employing group work, and creating group discussion. Students in large classes may find that the class size leads to isolation. At Utah State University, we offer introductory statistics classes for various audiences using a large lecture format. The authors have collectively led these large lectures dozens of times and found that, despite its shortcomings, the large lecture format can be an …

Recognizing And Reducing Ambiguity In Mathematics Curriculum And Relations Of Θ-Functions In Genus One And Two: A Geometric Perspective, Shantel Spatig

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Anxiety and mathematics come hand in hand for many individuals. This is due, in
part, to the fact that the only experience they have with mathematics is what some
mathematics educators refer to as "schoolmath," which uses a somewhat different
language than real mathematics. The language of schoolmath can cause individu-
als to have confusion and develop misconceptions related to several mathematical
concepts. One such concept is a fraction. In chapter one of this report, one possible
reason for this is discussed and a possible solution is purposed.
In chapter three of this report, genus-two curves admitting an elliptic involution …

Using The Reshetikhin-Turaev Link Invariant Approach With Non-Semisimple Categories, Adam Robertson

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Invariants of knots and links are useful because they give rise to invariants of 3-manifolds. In particular, combinatorial link invariants give rise to combinatorial invariants of 3-manifolds, which are hard to come by using traditional methods from classical topology. The Reshetikhin–Turaev approach, which is based in quantum topology, develops link invariants using semisimple ribbon categories. However, a large class of algebraically interesting ribbon categories are non-semisimple and so give trivial link invariants via the Reshetikhin–Turaev method. We modify the Reshetikhin–Turaev method to make it suitable for non-semisimple ribbon categories. We discuss explicitly the following three examples: semisimple modules for the …

Contributions To Random Forest Variable Importance With Applications In R, Kelvyn K. Bladen

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

A major focus in statistics is building and improving computational algorithms that can use data to predict a response. Two fundamental camps of research arise from such a goal. The first camp is researching ways to get more accurate predictions. Many sophisticated methods, collectively known as machine learning methods, have been developed for this very purpose. One such method that is widely used across industry and many other areas of investigation is called Random Forests.

The second camp of research is that of improving the interpretability of machine learning methods. This is worthy of attention when analysts desire to optimize …

Developing Confidence And Interest In Teaching Relevant Mathematical Modeling Lessons, Jacy Beck

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

What is mathematical modeling and how can inservice and pre-service teachers develop the skills and competencies necessary to increase confidence and interest in teaching relevant mathematical modeling lessons? Mathematical modeling is “the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions” (CSSM, 2010, p. 72). By providing students with an opportunity to engage in relevant mathematical modeling prompts, we provide them with transferable skills and knowledge. The aim of this paper will be to provide insight into the relevance of teaching mathematical modeling, provide resources for integrating modeling …

Dynamic System Discovery With Recursive Physics-Informed Neural Networks, Jarrod Mau

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

This thesis presents a novel method, recursive Physics informed neural network, to learn the right hand side of differential equations. The neural network takes in data, then trains, and then acts as a proxy for the differential equation which can be used for modeling. We show the theoretical superiority of the recursive approach. We also use computer simulations to demonstrate the proved properties.