Mathematics Commons^™

Brun's 1920 Theorem On Goldbach's Conjecture, James A. Farrugia

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

One form of Goldbach’s Conjecture asserts that every even integer greater than 4is the sum of two odd primes. In 1920 Viggo Brun proved that every sufficiently large even number can be written as the sum of two numbers, each having at most nine prime factors. This thesis explains the overarching principles governing the intricate arguments Brun used to prove his result.

Though there do exist accounts of Brun’s methods, those accounts seem to miss the forest for the trees. In contrast, this thesis explains the relatively simple structure underlying Brun’s arguments, deliberately avoiding most of his elaborate machinery and …

Canonical Coordinates On Lie Groups And The Baker Campbell Hausdorff Formula, Nicholas Graner

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Lie Groups occur in math and physics as representations of continuous symmetries and are often described in terms of their Lie Algebra. This thesis is concerned with finding a concrete description of a Lie group given its associated Lie algebra. Several calculations toward this end are developed and then implemented in the Maple Differential Geometry package. Examples of the calculations are given.

Ensemble Estimation Of Information Divergence, Kevin R. Moon, Kumar Sricharan, Kristjan Greenewald, Alfred O. Hero Iii

Mathematics and Statistics Faculty Publications

Recent work has focused on the problem of nonparametric estimation of information divergence functionals between two continuous random variables. Many existing approaches require either restrictive assumptions about the density support set or difficult calculations at the support set boundary which must be known a priori. The mean squared error (MSE) convergence rate of a leave-one-out kernel density plug-in divergence functional estimator for general bounded density support sets is derived where knowledge of the support boundary, and therefore, the boundary correction is not required. The theory of optimally weighted ensemble estimation is generalized to derive a divergence estimator that achieves the …

Limit Behavior Of Mass Critical Hartree Minimization Problems With Steep Potential Wells, Yujin Guo, Yong Luo, Zhi-Qiang Wang

Mathematics and Statistics Faculty Publications

We consider minimizers of the following mass critical Hartree minimization problem: e_λ(N) ≔ inf{u∈H¹(ℝ^d),‖u‖²₂=N} E_λ(u), where d ≥ 3, λ > 0, and the Hartree energy functional E_λ(u) is defined by E_λ(u)≔∫_R^d|∇u(x)|²dx + λ∫_R^d g(x)u²(x)dx − 1/2∫_R^d∫_R^d {u²(x)u²(y)/|x−y|²} dxdy. Here the steep potential g(x) satisfies 0 = g(0) = inf_ℝ^dg(x) ≤ g(x) ≤ 1 and 1 …

The Effect Of Warmer Winters On The Demography Of An Outbreak Insect Is Hidden By Intraspecific Competition, Devin W. Goodsman, Guenchik Grosklos, Brian H. Aukema, Caroline Whitehouse, Katherine P. Bleiker, Nate G. Mcdowell, Richard S. Middleton, Chonggang Xu

Mathematics and Statistics Faculty Publications

Warmer climates are predicted to increase bark beetle outbreak frequency, severity, and range. Even in favorable climates, however, outbreaks can decelerate due to resource limitation, which necessitates the inclusion of competition for limited resources in analyses of climatic effects on populations. We evaluated several hypotheses of how climate impacts mountain pine beetle reproduction using an extensive 9‐year dataset, in which nearly 10,000 trees were sampled across a region of approximately 90,000 km², that was recently invaded by the mountain pine beetle in Alberta, Canada. Our analysis supports the hypothesis of a positive effect of warmer winter temperatures on …

Full Dyon Excitation Spectrum In Extended Levin-Wen Models, Yuting Hu, Alexandra Tebbs, Yong-Shi Wu

Mathematics and Statistics Faculty Publications

In Levin-Wen (LW) models, a wide class of exactly solvable discrete models, for two-dimensional topological phases, it is relatively easy to describe only single-fluxon excitations, but not the charge and dyonic as well as many-fluxon excitations. To incorporate charged and dyonic excitations in (doubled) topological phases, an extension of the LW models is proposed in this paper. We first enlarge the Hilbert space with adding a tail on one of the edges of each trivalent vertex to describe the internal charge degrees of freedom at the vertex. Then, we study the full dyon spectrum of the extended LW models, including …

Gaps In The Saturation Spectrum Of Trees, Ronald J. Gould, Paul Horn, Michael S. Jacobson, Brent J. Thomas

Mathematics and Statistics Faculty Publications

A graph G is H-saturated if H is not a subgraph of G but the addition of any edge from the complement of G to G results in a copy of H. The minimum number of edges (the size) of an H-saturated graph on n vertices is denoted sat(n, H), while the maximum size is the well studied extremal number, ex(n, H). The saturation spectrum for a graph H is the set of sizes of H-saturated graphs between sat(n, H) and ex(n, H). In …

Feasibility Of Predicting Vietnam’S Autumn Rainfall Regime Based On The Tree-Ring Record And Decadal Variability, Yan Sun, Shih-Yu (Simon) Wang, Rong Li, Brendan M. Buckley, Robert R. Gilies, Kyle G. Hansen

Mathematics and Statistics Faculty Publications

We investigate the feasibility of developing decadal prediction models for autumn rainfall ( R_A ) over Central Vietnam by utilizing a published tree-ring reconstruction of October–November (ON) rainfall derived from the earlywood width measurements from a type of Douglas-fir (Pseudotsuga sinensis). Autumn rainfall for this region accounts for a large percentage of the annual total, and is often the source of extreme flooding. Central Vietnam’s R_A along with its notable autocorrelation and significant cross-correlation with basin-wide Pacific sea surface temperature (SST) variability, to develop four discrete time-series models. The sparse autoregressive model, with Pacific SST as …

Possible Isolation Number Of A Matrix Over Nonnegative Integers, Leroy B. Beasley, Young Bae Jun, Seok-Zun Song

Mathematics and Statistics Faculty Publications

Let ℤ₊ be the semiring of all nonnegative integers and A an m × n matrix over ℤ₊. The rank of A is the smallest k such that A can be factored as an m × k matrix times a k×n matrix. The isolation number of A is the maximum number of nonzero entries in A such that no two are in any row or any column, and no two are in a 2 × 2 submatrix of all nonzero entries. We have that the isolation number of A is a lower bound of the rank of …

Relations Between Theta Functions Of Genus One And Two From Geometry, Thomas Hill

Undergraduate Honors Capstone Projects

Genus-two curves with special symmetries are related to pairs of genus-one curves by two and three-sheeted ramified coverings. This classical work dates back to early 20th century and is known as Jacobi and Hermite reduction. Jacobians of genus-two curves can be used to construct complex two-dimensional complex projective manifolds known as Kummer surfaces. On the other hand, the defining coordinates and parameters of both elliptic curves and Kummer surfaces can be related to Riemann Theta functions and Siegel Theta functions, respectively. This result goes back to the seminal work of Mumford in the 1980s. We use the geometric relation between …

Mindset, Attitudes, And Success In Statistics, Matthew Isaac

Undergraduate Honors Capstone Projects

Students in many disciplines are required to take an introductory statistics course while pursuing a college education. Despite the utility of statistical methods in future research and career pursuits, many students have negative views of statistics. We are interested in how students' mindsets and attitudes towards statistics impact their performance in an undergraduate statistics course. We administered a survey to students in several undergraduate statistics courses at Utah State University. This survey included questions addressing mathematics experience, attitudes towards statistics, mindset, and course performance. We observed that the majority of students indicated the presence of a growth mindset and positive …

A Series Of Papers On Detecting Examinees Who Used A Flawed Answer Key, Marcus W. Scott

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

One way that examinees can gain an unfair advantage on a test is by having prior access to the test questions and their answers, known as preknowledge. Determining which examinees had preknowledge can be a difficult task. Sometimes, the compromised test content that examinees use to get preknowledge has mistakes in the answer key. Examinees who had preknowledge can be identified by determining whether they used this flawed answer key. This research consisted of three papers aimed at helping testing programs detect examinees who used a flawed answer key.

The first paper developed three methods for detecting examinees who used …

Three Environmental Fluid Dynamics Papers, Eden Furtak-Cole

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Three papers are presented, applying computational fluid dynamics methods to fluid flows in the geosciences. In the first paper, a numerical method is developed for single phase potential flow in the subsurface. For a class of monotonically advancing flows, the method provides a computational savings as compared to classical methods and can be applied to problems such as forced groundwater recharge. The second paper investigates the shear stress reducing action of an erosion control roughness array. Incompressible Naiver-Stokes simulations are performed for multiple wind angles to understand the changing aerodynamics of individual and grouped roughness elements. In the third paper, …

Learning Logic: A Mixed Methods Study To Examine The Effects Of Context Ordering On Reasoning About Conditionals, Christina W. Lommatsch

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Logical statements are prevalent in mathematics, the sciences, law, and many areas of everyday life. The most common logical statements are conditionals, which have the form “If H..., then C...,” where “H” is a hypothesis (or condition) to be satisfied and “C” is a conclusion to follow. Reasoning about conditionals is a skill that is only superficially understood by most individuals and depends on four main conditional contexts (e.g., intuitive, abstract, symbolic, or counterintuitive). The purpose of this study was to test a theory about the effects of context ordering on reasoning about conditionals. To test the theory, the researcher …

Modeling The Spread Of Alfalfa Stem Nematodes: Insights Into Their Dynamics And Control, Scott G. Jordan

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Alfalfa is a major cash crop in the western United States, where fields that are infested with the alfalfa stem nematode (Ditylenchus dipsaci) can be found. With no nematicides available to control alfalfa stem nematode spread, growers can use nematode resistant varieties of alfalfa to manage nematode populations in a field. A deterministic, discrete-time, host-parasite model is presented that describes the spread of alfalfa stem nematodes on resistant hosts that was fit to experimental data obtained in Weber County, Utah. Numerical results obtained from simulations with the model are used to compare how varying levels of resistance can affect …

Jacobian Elliptic Kummer Surfaces And Special Function Identities, Elise Griffin, Andreas Malmendier

Mathematics and Statistics Faculty Publications

We derive formulas for the construction of all inequivalent Jacobian elliptic fibrations on the Kummer surface of two non-isogeneous elliptic curves from extremal rational elliptic surfaces by rational base transformations and quadratic twists. We then show that each such decomposition yields a description of the Picard-Fuchs system satisfied by the periods of the holomorphic two-form as either a tensor product of two Gauss' hypergeometric differential equations, an Appell hypergeometric system, or a GKZ differential system. As the answer must be independent of the fibration used, identities relating differential systems are obtained. They include a new identity relating Appell's hypergeometric system …

Genetic Variation Determines Which Feedbacks Drive And Alter Predator–Prey Eco-Evolutionary Cycles, Michael H. Cortez

Mathematics and Statistics Faculty Publications

Evolution can alter the ecological dynamics of communities, but the effects depend on the magnitudes of standing genetic variation in the evolving species. Using an eco‐coevolutionary predator–prey model, I identify how the magnitudes of prey and predator standing genetic variation determine when ecological, evolutionary, and eco‐evolutionary feedbacks influence system stability and the phase lags in predator–prey cycles. Here, feedbacks are defined by subsystems, i.e., the dynamics of a subset of the components of the whole system when the other components are held fixed; ecological (evolutionary) feedbacks involve the direct and indirect effects between population densities (species traits) and eco‐evolutionary feedbacks …

Second Order Fully Discrete Energy Stable Methods On Staggered Grids For Hydrodynamic Phase Field Models Of Binary Viscous Fluids, Yuezheng Gong, Jia Zhao, Qi Wang

Mathematics and Statistics Faculty Publications

We present second order, fully discrete, energy stable methods on spatially staggered grids for a hydrodynamic phase field model of binary viscous fluid mixtures in a confined geometry subject to both physical and periodic boundary conditions. We apply the energy quadratization strategy to develop a linear-implicit scheme. We then extend it to a decoupled, linear scheme by introducing an intermediate velocity term so that the phase variable, velocity field, and pressure can be solved sequentially. The two new, fully discrete linear schemes are then shown to be unconditionally energy stable, and the linear systems resulting from the schemes are proved …

Partitioning The Effects Of Eco-Evolutionary Feedbacks On Community Stability, Swati Patel, Michael H. Cortez, Sebastian J. Schreiber

Mathematics and Statistics Faculty Publications

A fundamental challenge in ecology continues to be identifying mechanisms that stabilize community dynamics. By altering the interactions within a community, eco-evolutionary feedbacks may play a role in community stability. Indeed, recent empirical and theoretical studies demonstrate that these feedbacks can stabilize or destabilize communities and, moreover, that this sometimes depends on the relative rate of ecological to evolutionary processes. So far, theory on how eco-evolutionary feedbacks impact stability exists only for a few special cases. In our work, we develop a general theory for determining the effects of eco-evolutionary feedbacks on stability in communities with an arbitrary number of …

Fully Discrete Second-Order Linear Schemes For Hydrodynamic Phase Field Models Of Binary Viscous Fluid Flows With Variable Densities, Yuezheng Gong, Jia Zhao, Xiaogang Yang, Qi Wang

Mathematics and Statistics Faculty Publications

We develop spatial-temporally second-order, energy stable numerical schemes for two classes of hydrodynamic phase field models of binary viscous fluid mixtures of different densities. One is quasi-incompressible while the other is incompressible. We introduce a novel energy quadratization technique to arrive at fully discrete linear schemes, where in each time step only a linear system needs to be solved. These schemes are then shown to be unconditionally energy stable rigorously subject to periodic boundary conditions so that a large time step is plausible. Both spatial and temporal mesh refinements are conducted to illustrate the second-order accuracy of the schemes. The …

Examples Of The Birkhoff Theorem And Its Generalizations, Charles G. Torre

Tutorials on... in 1 hour or less

In this worksheet I demonstrate three versions of Birkhoff's theorem, which is a characterization of spherically symmetric solutions of the Einstein equations. The three versions considered here correspond to taking the "Einstein equations" to be: (1) the vacuum Einstein equations; (2) the Einstein equations with a cosmological constant (3) the Einstein-Maxwell equations. I will restrict my attention to 4-dimensional spacetimes.

How To Make Tetrads, Charles G. Torre

How to... in 10 minutes or less

This is a worksheet which demonstrates tools for creating orthonormal and null tetrads for a given spacetime.

Symmetric Criticality In General Relativity, Charles G. Torre

Research Vignettes

In this worksheet I explore the local Lagrangian version of the Principle of Symmetric Criticality (PSC) due to Anderson, Fels, and Torre], which asserts the commutativity of the processes (i) of symmetry reduction (for finding group-invariant fields) and (ii) forming Euler-Lagrange equations. There are two obstructions to PSC, which I will call the Lie algebra obstruction and the isotropy obstruction. In this worksheet I will illustrate these obstructions in the General Theory of Relativity.

Mathematically Predicting The Aleut Tribe Population Using Archaeological Data, Jack Conrad Kiefer Ii, Paden Allen Thompson

Research on Capitol Hill

Sanak Island, located off the southern Alaska Peninsula, was home to the native Aleut peoples for thousands of years. Their hunter-gatherer society depended heavily on the arctic and marine ecosystem for food resources.

In 2015, a team of archaeologists from Idaho State and Utah State universities went to the island and collected data about the Aleut population size and their diet.

This study constructed a dynamical model to mathematically predict the Aleut population over time in order to gain insights into how food resources affected the Aleut people’s ability to survive.

Local And Global Dynamic Bifurcations Of Nonlinear Evolution Equations, Desheng Li, Zhi-Qiang Wang

Mathematics and Statistics Faculty Publications

We present new local and global dynamic bifurcation results for nonlinear evolution equations of the form ut + Au = fλ(u) on a Banach space X, where A is a sectorial operator, and λ ∈ R is the bifurcation parameter. Suppose the equation has a trivial solution branch {(0, λ) : λ ∈ R}. Denote Φλ the local semiflow generated by the initial value problem of the equation. It is shown that if the crossing number n at a bifurcation value λ = λ0 is nonzero and moreover, S0 = {0} is an isolated invariant set of Φλ0 , then …

Did They Learn Anything? - Learning Assessment In A General Business Analytics Course, Zsolt Ugray, Brynja Kohler

Mathematics and Statistics Faculty Publications

There is a projected need for 2 to 4 million of business translators who are not only experts in their business disciplines but also conversant and familiar with the use and limitations of business analytics. Preparing business students to step into the business translator role requires a general introductory analytics course. This paper presents a multi-component learning assessment to evaluate the depth of learning from such a course. This learning assessment is novel in a sense that it uses four distinct tools, both quantitative and qualitative ones, to measure students’ improvements in a comprehensive manner.

Expression Of Wnt-Signaling Pathway Genes And Their Associations With Mirnas In Colorectal Cancer, Martha L. Slattery, Lila E. Mullany, Lori C. Sakoda, Wade S. Samowitz, Roger K. Wolff, John R. Stevens, Jennifer S. Herrick

Mathematics and Statistics Faculty Publications

The Wnt-signaling pathway functions in regulating cell growth and thus is involved in the carcinogenic process of several cancers, including colorectal cancer. We tested the hypothesis that multiple genes in this signaling pathway are dysregulated and that miRNAs are associated with these dysregulated genes. We used data from 217 colorectal cancer (CRC) cases to evaluate differences in Wnt-signaling pathway gene expression between paired CRC and normal mucosa and identify miRNAs that are associated with these genes. Gene expression data from RNA-Seq and miRNA expression data from Agilent Human miRNA Microarray V19.0 were analyzed. We focused on genes most strongly associated …

Introduction To The Usu Library Of Solutions To The Einstein Field Equations, Ian M. Anderson, Charles G. Torre

Tutorials on... in 1 hour or less

This is a Maple worksheet providing an introduction to the USU Library of Solutions to the Einstein Field Equations. The library is part of the DifferentialGeometry software project and is a collection of symbolic data and metadata describing solutions to the Einstein equations.

Novel Statistical Models For Quantitative Shape-Gene Association Selection, Xiaotian Dai

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Other research reported that genetic mechanism plays a major role in the development process of biological shapes. The primary goal of this dissertation is to develop novel statistical models to investigate the quantitative relationships between biological shapes and genetic variants. However, these problems can be extremely challenging to traditional statistical models for a number of reasons: 1) the biological phenotypes cannot be effectively represented by single-valued traits, while traditional regression only handles one dependent variable; 2) in real-life genetic data, the number of candidate genes to be investigated is extremely large, and the signal-to-noise ratio of candidate genes is expected …

Extracting And Visualizing Data From Mobile And Static Eye Trackers In R And Matlab, Chunyang Li

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Eye tracking is the process of measuring where people are looking at with an eye tracker device. Eye tracking has been used in many scientific fields, such as education, usability research, sports, psychology, and marketing. Eye tracking data are often obtained from a static eye tracker or are manually extracted from a mobile eye tracker. Visualization usually plays an important role in the analysis of eye tracking data. So far, there existed no software package that contains a whole collection of eye tracking data processing and visualization tools. In this dissertation, we review the eye tracking technology, the eye tracking …