Mathematics Commons^™

Collecting, Analyzing And Interpreting Bivariate Data From Leaky Buckets: A Project-Based Learning Unit, Florence Funmilayo Obielodan

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

Despite the significance and the emphasis placed on mathematics as a subject and field of study, achieving the right attitude to improve students‟ understanding and performance is still a challenge. Previous studies have shown that the problem cuts across nations around the world, both developing countries and developed alike. Teachers and educators of the subject have responsibilities to continuously develop innovative pedagogical approaches that will enhance students‟ interests and performance. Teaching approaches that emphasize real life applications of the subject have become imperative. It is believed that this will stimulate learners‟ interest in the subject as they will be able …

Solving Equations Applet Project, Kimberly Thatcher

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

The purpose of this paper is to summarize a Masters Project for the MMath Degree. The purpose of the project was to create and evaluate an applet that maintains the advantages of the existent manipulatives (Hands-On Equations® and the NLVM applet) while also overcoming the limitations of each. Another product of this project is accompanying lesson plans for teachers.

Ranking Score Vectors Of Tournaments, Sebrina Ruth Cropper

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

Given , a tournament on vertices, Landau derived a method to determine how close is to being transitive or regular. This comparison is based on the tournament’s hierarchy number, ̅, a value derived from its score vector ̅ ( ). Let be the set of all score vectors of tournaments on vertices with the entries listed in non-decreasing order. A partial order, poset, exists on the set using the following binary relation. Given ̅ ̅ such that ̅ ̅, let ̅ ̅ if Σ Σ for and Σ Σ . Let this poset be represented as ( ) ( ) …

B ̈Acklund Transformations For Darboux Integrable Equations, Ian M. Anderson

Mathematics and Statistics Faculty Presentations

Lie symmetry reduction is typically viewed as an integration method for differential systems of finite type, that is, systems of ordinary differential equations.

In this talk I shall present two new, recent applications of Lie symmetry reduction to the study of partial differential equations.

The first gives a remarkably simple method for constructing B ̈acklund transformations.

The second also gives a simple, very general method for constructing Darboux integrable equations.

The combination of these result in a new method for constructing B ̈acklund transformations for Darboux integrable equations.

The utility of this group theoretic approach will be illustrated by a …

On The Use Of Log-Transformation Vs. Nonlinear Regression For Analyzing Biological Power-Laws, Xiao Xiao

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

Power-law relationships are among the most well-studied functional relationships in biology . Recently the common practice of fitting power-laws using linear regression on log-transformed data (LR) has been criticized, calling into question the conclusions of hundreds of studies. It has been suggested that nonlinear regression (NLR) is preferable, but no rigorous comparison of these two methods has been conducted. Using Monte Carlo simulations we demonstrate that the error distribution determines which method performs better, with LR better characterizing data with multiplicative lognormal error and NLR better characterizing data with additive normal error. Analysis of 471 biological power-laws shows that both …

Rank 2 Distributions Of Monge Equations: Symmetries, Equivalences, Ex-Tensions, Ian M. Anderson, B. Kruglikov

Mathematics and Statistics Faculty Publications

By developing the Tanaka theory for rank 2 distributions, we completely classify classical Monge equations having maximal finite-dimensional symmetry algebras with fixed (albeit arbitrary) pair of its orders. Investigation of the corresponding Tanaka algebras leads to a new Lie-Backlund theorem. We prove that all flat Monge equations are successive integrable extensions of the Hilbert-Cartan equation. Many new examples are provided.

An Ethnographic Study Of In-Service And Pre-Service Teacher Collaboration In The Development And Execution Of Mathematics Lessons, Jessica Munns

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

While much is known about effective strategies for teaching mathematics, less is understood about how to lead new teachers to employ research-based practices in their work. To address this issue we followed a Methods of Teaching Secondary and Middle School Mathematics course at Utah State University where collaboration between inservice and pre-service teachers was emphasized. This report documents examples of the implementation of research-based teaching practices by collaborative groups during the course. The structure of collaboration followed a lesson study model involving research and planning, execution and observation, and reflection and revision of lessons. Preservice and in-service teachers met twice …

Virtual Manipulatives In The Classroom And Resulting Articles And Lesson Plans, Cheryl Juliana

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

Upon coming across mathematical manipulatives generated and produced by Utah State University, as a math teacher, I conducted a classroom teaching experiment in three pre-algebra classes with students of various achievement levels. After teaching the entire year using no manipulatives in the classroom, I tested my students with a general, end-of-year, core criterion, or cumulative test. Their scores were noted. The students in the study group were then given opportunities to try several manipulatives offered on the "National Library of Virtual Manipulatives," both as a class, and alone, and then retested. The following paper gives the parameters of the study, …

A Comparison Of Probe-Level And Probeset Models For Small-Sample Gene Expression Data, John R. Stevens, Jason L. Bell, Kenneth I. Aston, Kenneth L. White

Mathematics and Statistics Faculty Publications

Background: Statistical methods to tentatively identify differentially expressed genes in microarray studies typically assume larger sample sizes than are practical or even possible in some settings.

Results: The performance of several probe-level and probeset models was assessed graphically and numerically using three spike-in datasets. Based on the Affymetrix GeneChip, a novel nested factorial model was developed and found to perform competitively on small-sample spike-in experiments.

Conclusions: Statistical methods with test statistics related to the estimated log fold change tend to be more consistent in their performance on small-sample gene expression data. For such small-sample experiments, the nested factorial model can …

Development Of A Discrete Mathematics Textbook And Guide For High School Teachers, Rebecca A. Stokes

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

This project was to create the beginnings a textbook for teacher's to supplement instruction of a discrete mathematics course at the high school level. The development of the text was guided by past and current efforts to place discrete mathematics in high school curriculum. A review of the literature and experiences of instructors were viewed and analyzed to guide the construction of the textbook. The text was written with the goal to give teachers information about the topics of discrete mathematics, extra resources, lesson ideas , and optional worksheets for students. Several lessons were created and one was implemented in …

Classical Foundations For A Quantum Theory Of Time In A Two-Dimensional Spacetime, Nathan Thomas Carruth

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

We consider the set of all spacelike embeddings of the circle S1 into a spacetime R1 × S1 with a metric globally conformal to the Minkowski metric. We identify this set and the group of conformal isometries of this spacetime as quotients of semidirect products involving diffeomorphism groups and give a transitive action of the conformal group on the set of spacelike embeddings. We provide results showing that the group of conformal isometries is a topological group and that its action on the set of spacelike embeddings is continuous. Finally, we point out some directions for future research.

Dancing With Heretics: Essays On Orthodoxy, Questioning And Faith, Darren M. Edwards

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

While much has been written about the conflicts, supposed or actual, between logic and faith, science and religion, few accounts of the personal turmoil these conflicts can cause exist. Likewise, many of these nonfiction accounts are written from a distinctly polarized place leaning either to science or faith.

In this thesis, I mix research and history with memoir and a sense of poetry to explore my personal experience with this conflict. At its outset, I hoped for this project to capture my struggle as an orthodox member of The Church of Jesus Christ of Latter-day Saints (LDS) in dealing with …

The Effect Of Explicit Timing On Math Performance Using Interspersal Assignments With Students With Mild/Moderate Disabilities, Fangjuan Hou

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Explicit timing and interspersal assignments have been validated as effective methods to facilitate students' math practice. However, no researchers have explored the combinative effect of these two methods. In Study 1, we extended the literature by comparing the effect of explicit timing with interspersal assignments, and interspersal assignments without timing. Generally, participants' rate of digits correct on easy and hard addition problems was higher during the explicit timing condition than during the untimed condition. However, the participants' rate of digits correct decreased after initial implementation of the explicit timing condition.

Motivation plays a crucial role in maintaining performance levels and …

Simulating Power For One-Way Anova By Using Non-Normal Error Distributions, Caixia Xu

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

Usually we assume that the distribution of the additive errors in a one-way ANOVA linear model is normal. However, exceptions to this assumption about the error distribution may exist. In such cases, we might consider non-normal error distributions, but proceed with the "usual" ANOVA F-test analyses. This study focuses on simulating power for one-way ANOVA when using non-normal error distributions.

Improving Accuracy Of Large-Scale Prediction Of Forest Disease Incidence Through Bayesian Data Reconciliation, Ephraim M. Hanks

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

Increasing the accuracy of predictions made from ecological data typically involves replacing or replicating the data, but the cost of updating large-scale data sets can be prohibitive. Focusing resources on a small sample of locations from a large, less accurate data set can result in more reliable observations, though on a smaller scale. We present an approach for increasing the accuracy of predictions made from a large-scale eco logical data set through reconciliation with a small, highly accurate data set within a Bayesian hierarchical modeling framework. This approach is illustrated through a study of incidence of eastern spruce dwarf mistletoe …

Evolving Models: A Density-Based Approach To Modeling Sexual Dimorphism And Adaptive Speciation, Audrey Smith

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

In this paper, we begin by extending existing deterministic and individual-based ecological models for sexual dimorphism and adaptive speciation into density-based mathematical models describing the density or number of individuals with various trait values, or phenotypes. These density-based models describe the dynamics of a population of males and females using both clonal and sexual reproduction. Each generation, the populations are subject to mating, mutation, and ecological dynamics including infraspecific competition and carrying capacity of the environment. By avoiding individual-based models, we are able to avoid simulations and instead achieve repeatable results.

Implementing these models numerically, we are able to show …

Numerical Solution Of The Five-Moment Ideal Two-Fluid Equations In One Dimension, Marcus Scott

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

Plasmas are frequently treated as a single conducting fluid and modeled using the equations of magnetohydrodynamics. However, this regime works better for low-frequency plasmas. High-frequency plasmas may be modeled using the principles of kinetic theory. For plasmas with frequencies between these two extremes, a two-fluid approach can yield better results. In 2006, Ammar Hakim mathematically modeled a plasma with a set of equations called the five-moment ideal two-fluid equations. An attempt is made reproduce those results. A derivation of this set of equations by taking moments of the Boltzmann equation is presented. Electric and magnetic fields contribute to the source …

Assessment Of Utah Bankruptcies By Census Tracts: A Spatial Statistical Approach, Kenneth Pena

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

There are two questions raised when looking at the spatial pattern of the rate of bankruptcies in Utah: (i) are there similarities between the bankruptcy data in adjacent census tracts and (ii) can local cluster and outliers be identified within the data? Specifically, are there similar rates of bankruptcies in bordering census tracts and are there any localized areas of interest where we find extremely high or extremely low rates of bankruptcies? This study uses spatial statistics to perform tests for spatial autocorrelation to address these two questions. It also looks at commonalities in the clusters and differences in the …

Statistical Analysis Of Wastewater Remediation And Bio-Fuels Production Of Algae, Jay D. Jones

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

The Logan city wastewater treatment system consists of a series of seven large aerated ponds (460 acres) that biologically treats 15 million gallons per day of wastewater from Logan city and six other communities. Tighter regulations of allowed phosphorus levels in the effluent have recently been implemented due to environmental concerns of a downstream reservoir. The Biological Engineering program at Utah State University, the Bio-fuels Center, the Utah Water Research Laboratory (UWRL) and the city of Logan are working together to remediate the wastewater treatment system using microalgae. Algal growth requires the uptake of phosphorus. Thus, phosphorus in the effluent …

A Comparison Of Prediction Methods Of Functional Autoregressive Time Series, Devin Didericksen

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

Functional data analysis (FDA) is a relatively new branch of statistics that has seen a lot of expansion recently. With the advent of computer processing power and more efficient software packages we have entered the beginning stages of applying FDA methodology and techniques to data. Part of this undertaking should include an empirical assessment of the effectiveness of some of the tools of FDA, which are sound on theoretical grounds. In a small way, this project helps advance this objective.

This work begins by introducing FDA, scalar prediction techniques, and the functional autoregressive model of order one - FAR(1). Two …

Assessing The Precision And Accuracy In A Small Sample Of Actical Devices, Peter Sherick

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

Actigraphy is an increasingly popular approach in medicine to assess patient activity levels in a variety of scenarios. The devices are essentially accelerometers encased in a write-watch type assembly. This project sought to determine the device precision and accuracy for the Actical model. In a sample of four Acticals, it was found that intra-device variability was minimal. However, one device was found to be statistically biased in comparison to the other three. This bias could have adverse effects on aggregated or magnitude dependent data analysis. Also, inter-device comparisons may be problematic.

A Comparison Theorem For The Topological And Algebraic Classification Of Quaternionic Toric 8-Manifolds, Piotr Runge

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

In order to discuss topological properties of quaternionic toric 8-manifolds, we introduce the notion of an algebraic morphism in the category of toric spaces. We show that the classification of quaternionic toric 8-manifolds with respect to an algebraic isomorphism is finer than the oriented topological classification. We construct infinite families of quaternionic toric 8-manifolds in the same oriented homeomorphism type but algebraically distinct. To prove that the elements within each family are of the same oriented homeomorphism type, and that we have representatives of all such types of a quaternionic toric 8-manifold, we present and use a method of evaluating …

Leading Students To Investigate Diffusion As A Model Of Brine Shrimp Movement, Brynja R. Kohler, Rebecca J. Swank, James W. Haefner, James A. Powell

Mathematics and Statistics Faculty Publications

Integrating experimental biology laboratory exercises with mathematical modeling can be an effective tool to enhance mathematical relevance for biologists and to emphasize biological realism for mathematicians. This paper describes a lab project de-signed for and tested in an undergraduate biomathematics course. In the lab, students follow and track the paths of individual brine shrimp confined in shallow salt water in a Petri dish. Students investigate the question, “Is the movement well characterized as a 2-dimensional random walk?” Through open, but directed discussions, students derive the corresponding partial differential equation, gain an understanding of the solution behavior, and model brine shrimp …

Testing And Estimation For Functional Data With Applications To Magnetometer Records, Inga Maslova

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The functional linear model, Y_n = ψX_n + ε_n, with functional response and explanatory variables is considered. A simple test of the nullity of ψ based on the principal component decomposition is proposed. The test statistic has asymptotic chi-squared distribution, which is also an excellent approximation in finite samples. The methodology is applied to data from terrestrial magnetic observatories.

In recent years, the interaction of the auroral substorms with the equatorial and mid-latitude currents has been the subject of extensive research. We introduce a new statistical technique that allows us to test at a specified …

Simulation, Kriging, And Visualization Of Circular-Spatial Data, William James Morphet

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The circular dataimage is defined by displaying direction as the color at the same direction in a color wheel composed of a sequence of two-color gradients with color continuity between gradients. The resulting image of circular-spatial data is continuous with high resolution. Examples include ocean wind direction, Earth's main magnetic field, and rocket nozzle internal combustion flow. The cosineogram is defined as the mean cosine of the angle between random components of direction as a function of distance between observation locations. It expresses the spatial correlation of circular-spatial data. A circular kriging solution is developed based on a model fitted …

Modeling The Evolution Of Insect Phenology With Particular Reference To Mountain Pine Beetle, Brian P. Yurk

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Climate change is likely to disrupt the timing of developmental events (phenology) in insect populations in which development time is largely determined by temperature. Shifting phenology puts insects at risk of being exposed to seasonal weather extremes during sensitive life stages and losing synchrony with biotic resources. Additionally, warming may result in loss of developmental synchronization within a population, making it difficult to find mates or mount mass attacks against well-defended resources at low population densities. It is unknown whether genetic evolution of development time can occur rapidly enough to moderate these effects.

The work presented here is largely motivated …

Symmetry Analysis Of General Rank-3 Pfaffian Systems In Five Variables, Francesco Strazzullo

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

In this dissertation we applied geometric methods to study underdetermined second order scalar ordinary differential equations (called general Monge equations), nonlinear involutive systems of two scalar partial differential equations in two independent variables and one unknown and non-Monge-Ampere Goursat parabolic scalar PDE in the plane. These particular kinds of differential equations are related to general rank-3 Pfaffian systems in five variables. Cartan studied these objects in his 1910 paper. In this work Cartan provided normal forms only for some general rank-3 Pfaffian systems with 14-, 7-, and 6-dimensional symmetry algebra.

In this dissertation we provided normal forms of all general …

Teaching Probability Through Use Of An Applet, Melissa Gregory Jackson

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

The use of technology in the classroom is an ongoing debate by educators. Many teachers consider it to be a valuable teaching tool. Despite the many advantages, there are also drawbacks in using technology. A Java applet is a particular kind of multimedia technology proven to be useful in education. Because of students' struggles with learning basic probability in Statistics 1040, I have created a probability applet to reinforce the concept of probability. The applet was tested with two Statistics 1040 classes. The majority of students agreed that they learned more about probability from using the applet. Several of the …

Random Motion And The Diffusion Of Brine Shrimp: A Project Based Learning Unit, Rebecca Atkins

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

Over the years, there have been several trends, philosophies, and ideas about how mathematics should be taught. These have included progressive movements, essentialist movements, drill and practice, teacher-centered methodologies, student-centered methodologies, and others. However, despite these various instructional trends, research has found that mathematics teachers have basically taught the same way for years, with little or no variation with respect to the changing trends (Ward, 2001). For many students, the traditional teaching method employed in their mathematics classes is typically centered on the teacher and includes lecture in which the teacher provides the necessary definitions and notations and does several …

The Classification Of Simple Lie Algebras In Maple, D. Russell Sadler

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

Lie algebras are invaluable tools in mathematics and physics as they enable us to study certain geometric objects such as Lie groups and differentiable manifolds. The computer algebra system Maple has several tools in its Lie Algebras package to work with Lie algebras and Lie groups. The purpose of this paper is to supplement the existing software with tools that are essential for the classification of simple Lie algebras over C.

In particular, we use a method to find a Cartan subalgebra of a Lie algebra in polynomial time. From the Cartan subalgebra we can compute the corresponding root system. …