Mathematics Commons^™

Convergence Rates For Empirical Estimation Of Binary Classification Bounds, Salimeh Yasaei Sekeh, Morteza Noshad, Kevin R. Moon, Alfred O. Hero

Mathematics and Statistics Faculty Publications

Bounding the best achievable error probability for binary classification problems is relevant to many applications including machine learning, signal processing, and information theory. Many bounds on the Bayes binary classification error rate depend on information divergences between the pair of class distributions. Recently, the Henze–Penrose (HP) divergence has been proposed for bounding classification error probability. We consider the problem of empirically estimating the HP-divergence from random samples. We derive a bound on the convergence rate for the Friedman–Rafsky (FR) estimator of the HP-divergence, which is related to a multivariate runs statistic for testing between two distributions. The FR estimator is …

Feasibility Of Multi-Year Forecast For The Colorado River Water Supply: Time Series Modeling, Brian Plucinski, Yan Sun, Shih-Yu (Simon) Wang, Robert R. Gilies, James Eklund, Chih-Chia Wang

Mathematics and Statistics Faculty Publications

The future of the Colorado River water supply (WS) affects millions of people and the US economy. A recent study suggested a cross-basin correlation between the Colorado River and its neighboring Great Salt Lake (GSL). Following that study, the feasibility of using the previously developed multi-year prediction of the GSL water level to forecast the Colorado River WS was tested. Time-series models were developed to predict the changes in WS out to 10 years. Regressive methods and the GSL water level data were used for the depiction of decadal variability of the Colorado River WS. Various time-series models suggest a …

Nonlinear Reaction–Diffusion Process Models Improve Inference For Population Dynamics, Xinyi Lu, Perry J. Williams, Mevin B. Hooten, James A. Powell, Jamie N. Womble, Michael R. Bower

Mathematics and Statistics Faculty Publications

Partial differential equations (PDEs) are a useful tool for modeling spatiotemporal dynamics of ecological processes. However, as an ecological process evolves, we need statistical models that can adapt to changing dynamics as new data are collected. We developed a model that combines an ecological diffusion equation and logistic growth to characterize colonization processes of a population that establishes long‐term equilibrium over a heterogeneous environment. We also developed a homogenization strategy to statistically upscale the PDE for faster computation and adopted a hierarchical framework to accommodate multiple data sources collected at different spatial scales. We highlighted the advantages of using a …

(0,1)-Matrices, Discrepancy And Preservers, Leroy B. Beasley

Mathematics and Statistics Faculty Publications

Let m and n be positive integers, and let R = (r₁, . . . , r_m) and S = (s₁, . . . , s_n) be nonnegative integral vectors. Let A(R,S) be the set of all m × n (0, 1)-matrices with row sum vector R and column vector S. Let R and S be nonincreasing, and let F(R) be the m × n (0, 1)-matrix, where for each i, the ith row of F(R, …

From The Signature Theorem To Anomaly Cancellation, Andreas Malmendier, Michael T. Schultz

Mathematics and Statistics Faculty Publications

We survey the Hirzebruch signature theorem as a special case of the Atiyah–Singer index theorem. The family version of the Atiyah–Singer index theorem in the form of the Riemann–Roch–Grothendieck–Quillen (RRGQ) formula is then applied to the complexified signature operators varying along the universal family of elliptic curves. The RRGQ formula allows us to determine a generalized cohomology class on the base of the elliptic fibration that is known in physics as (a measure of) the local and global anomaly. Combining several anomalous operators allows us to cancel the local anomaly on a Jacobian elliptic surface, a construction that is based …

Machine Learning Techniques As Applied To Discrete And Combinatorial Structures, Samuel David Schwartz

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Machine Learning Techniques have been used on a wide array of input types: images, sound waves, text, and so forth. In articulating these input types to the almighty machine, there have been all sorts of amazing problems that have been solved for many practical purposes.

Nevertheless, there are some input types which don’t lend themselves nicely to the standard set of machine learning tools we have. Moreover, there are some provably difficult problems which are abysmally hard to solve within a reasonable time frame.

This thesis addresses several of these difficult problems. It frames these problems such that we can …

Interval-Valued Kriging Models With Applications In Design Ground Snow Load Prediction, Brennan L. Bean

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

One critical consideration in the design of buildings constructed in the western United States is the weight of settled snow on the roof of the structure. Engineers are tasked with selecting a design snow load that ensures that the building is safe and reliable, without making the construction overly expensive. Western states use historical snow records at weather stations scattered throughout the region to estimate appropriate design snow loads. Various mapping techniques are then used to predict design snow loads between the weather stations. Each state uses different mapping techniques to create their snow load requirements, yet these different techniques …

On Fractional Realizations Of Tournament Score Sequences, Kaitlin S. Murphy

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Contrary to popular belief, we can’t all be winners.

Suppose 6 people compete in a chess tournament in which all pairs of players compete directly and no ties are allowed; i.e., 6 people compete in a ‘round robin tournament’. Each player is assigned a ‘score’, namely the number of games they won, and the ‘score sequence’ of the tournament is a list of the players’ scores. Determining whether a given potential score sequence actually is a score sequence proves to be difficult. For instance, (0, 0, 3, 3, 3, 6) is not feasible because two players cannot both have …

Comparing Design Ground Snow Load Prediction In Utah And Idaho, Brennan L. Bean, Marc Maguire, Yan Sun

Mathematics and Statistics Faculty Publications

Snow loads in the western United States are largely undefined due to complex geography and climates, leaving the individual states to publish detailed studies for their region, usually through the local Structural Engineers Association (SEAs). These associations are typically made up of engineers not formally trained to develop or evaluate spatial statistical methods for their regions and there is little guidance from ASCE 7. Furthermore, little has been written to compare the independently developed design ground snow load prediction methods used by various western states. This paper addresses this topic by comparing the accuracy of a variety of spatial methods …

A 3d Numerical Study Of Interface Effects Influencing Viscous Gravity Currents In A Parabolic Fissure, With Implications For Modeling With 1d Nonlinear Diffusion Equations, Eden Furtak-Cole, Aleksey S. Telyakovskiy

Mathematics and Statistics Student Research and Class Projects

Although one-dimensional non-linear diffusion equations are commonly used to model flow dynamics in aquifers and fissures, they disregard multiple effects of real-life flows. Similarity analysis may allow further analytical reduction of these equations, but it is often difficult to provide applicable initial and boundary conditions in practice, or know the magnitude of effects neglected by the 1D model. Furthermore, when multiple simplifying assumptions are made, the sources of discrepancy between modeled and observed data are difficult to identify. We derive one such model of viscous flow in a parabolic fissure from first principals. The parabolic fissure is formed by extruding …

Methods For Meta–Analyses Of Rare Events, Sparse Data, And Heterogeneity, Brinley Zabriskie

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The vast and complex wealth of information available to researchers often leads to a systematic review, which involves a detailed and comprehensive plan and search strategy with the goal of identifying, appraising, and synthesizing all relevant studies on a particular topic. A meta–analysis, conducted ideally as part of a comprehensive systematic review, statistically synthesizes evidence from multiple independent studies to produce one overall conclusion. The increasingly widespread use of meta–analysis has led to growing interest in meta–analytic methods for rare events and sparse data. Conventional approaches tend to perform very poorly in such settings. Recent work in this area has …

Formation Of Radial Patterns Via Mixed Attractive And Repulsive Interactions For Schrödinger Systems, Jaeyoung Byeon, Youngae Lee, Zhi-Qiang Wang

Mathematics and Statistics Faculty Publications

This paper is concerned with the asymptotic behavior of least energy vector solutions for nonlinear Schrödinger systems with mixed couplings of attractive and repulsive forces. We focus here on the radially symmetric case while the general studies were already conducted in our earlier work [J. Byeon, Y. Sato, and Z.-Q. Wang, J. Math. Pures Appl. (9), 106 (2016), pp. 477--511], [J. Byeon, Y. Sato, and Z.-Q. Wang, J. Fixed Point Theory Appl., 19 (2017), pp. 559--583]. Though there is still the general phenomenon of component-wise pattern formation with co-existence of partial synchronization and segregation for positive least energy …

Association Of Rare Coding Mutations With Alzheimer Disease And Other Dementias Among Adults Of European Ancestry, Devanshi Patel, Jesse Mez, Badri N. Vardarajan, Lyndsay Staley, Jaeyoon Chung, Xiaoling Zhang, John J. Farrell, Michael J. Rynkiewicz, Lisa A. Cannon-Albright, Craig C. Teerlink, Jeffery Stevens, Chris Corcoran, Et Al.

Mathematics and Statistics Faculty Publications

IMPORTANCE Some of the unexplained heritability of Alzheimer disease (AD) may be due to rare variants whose effects are not captured in genome-wide association studies because very large samples are needed to observe statistically significant associations.

OBJECTIVE To identify genetic variants associated with AD risk using a nonstatistical approach.

DESIGN, SETTING, AND PARTICIPANTS Genetic association study in which rare variants were identified by whole-exome sequencing in unrelated individuals of European ancestry from the Alzheimer’s Disease Sequencing Project (ADSP). Data were analyzed between March 2017 and September 2018.

MAIN OUTCOMES AND MEASURES Minor alleles genome-wide and in 95 genes previously associated …

Linear Operators That Preserve The Genus Of A Graph, Leroy B. Beasley, Jeong Han Kim, Seok-Zun Song

Mathematics and Statistics Faculty Publications

A graph has genus k if it can be embedded without edge crossings on a smooth orientable surface of genus k and not on one of genus k−1. A mapping of the set of graphs on n vertices to itself is called a linear operator if the image of a union of graphs is the union of their images and if it maps the edgeless graph to the edgeless graph. We investigate linear operators on the set of graphs on n vertices that map graphs of genus k to graphs of genus k and graphs of genus k+1 to graphs …

Prediction Of Stress Increase At Ultimate In Unbonded Tendons Using Sparse Principal Component Analysis, Eric Mckinney, Minwoo Chang, Marc Maguire, Yan Sun

Mathematics and Statistics Faculty Publications

While internal and external unbonded tendons are widely utilized in concrete structures, an analytical solution for the increase in unbonded tendon stress at ultimate strength, Δ������, is challenging due to the lack of bond between strand and concrete. Moreover, most analysis methods do not provide high correlation due to the limited available test data. The aim of this paper is to use advanced statistical techniques to develop a solution to the unbonded strand stress increase problem, which phenomenological models by themselves have done poorly. In this paper, Principal Component Analysis (PCA), and Sparse Principal Component Analysis (SPCA) are employed on …

Drones And “Ghost Guns”: Unregulated Legal Space, Tori Bodine

Research on Capitol Hill

Law enforcement agencies are fighting a two - pronged battle when it comes to emerging technologies: keeping up with new ways criminals are using technology and developing effective ways to combat these innovations, while balancing these challenges against preserving the individual liberties of law - abiding citizens. This conflict is especially apparent with regard to criminal use of commercial drones and the developing fringe market surrounding homemade untraceable firearms (“ghost guns”).

Factors That Influence Mathematical Creativity, Joseph S. Kozlowski, Scott A. Chamberlin, Eric Mann

Mathematics and Statistics Faculty Publications

Creativity is a psychological construct that has gained research popularity (Akgul & Kaveci, 2016), however it remains a challenging one to define. The variety of definitions promulgated to understand creativity hints at the complexity of the mental process. Furthermore, as a subset of creativity, domain-specific mathematical creativity has also garnered a variety of definitions. The transdisciplinary research on creativity (Sriraman & Haavold, 2017) is seminal in this world of fast-paced innovation, invention, solution, and synthesis. Considering every human being with at least average cognitive abilities possesses the ability to think creatively (Baran, 2011), developing students’ creative talents and abilities must …

A Re-Emergent Analysis Of Early Algebraic Learning, Steven Boyce, Diana Moss

Extension Research

In this paper, we discuss a novel approach for collaborative retrospective analysis. One researcher was directly involved in a classroom teaching experiment, adopting an emergent perspective as an interpreter-witness of classroom interactions during a four-week algebra instructional unit with sixth-grade students. The other researcher experienced and analyzed the data in reverse chronological order. We describe how this re-emergent perspective revealed aspects of students’ early algebraic reasoning.

Exploring Mortality Rates For Major Causes Of Death In Korea, Hyo Jung Oh, Donng Min Yang, Chong Hyuck Kim, Jae Gyu Jeon, Nam Hyung Jung, Chan Young Kim, Jurgen Symanzik, Hyo Won Oh, Akugizibwe Edwin, Seong Ii Jo, Jeong Yong Ahn

Mathematics and Statistics Faculty Publications

Background: The trends and patterns of the mortality rates for causes of death are meaningful information. They can provide a basis for national demographic and health care policies by identifying the number, causes, and geographical distribution of deaths.

Objective: To explore and analyze the characteristics of the mortality rates for major causes of death in Korea.

Methods: Some common data analysis methods were used to describe the data. We also used some visualization techniques such as heat maps and line plots to present mortality rates by gender, age, and year.

Results: Our analysis shows the crude …

Upper Bounds For The Isolation Number Of A Matrix Over Semirings, Leroy B. Beasley, Seok-Zun Song

Mathematics and Statistics Faculty Publications

Let S be an antinegative semiring. The rank of an m×n matrix B over S is the minimal integer r such that B is a product of an m×r matrix and an r×n matrix. The isolation number of B is the maximal number of nonzero entries in the matrix such that no two entries are in the same column, in the same row, and in a submatrix of B of the form [b_i,j b_k,j

b_i,l b_k,l] with nonzero entries. We know that the isolation number of B is …

Spacetime Groups, Ian M. Anderson, Charles G. Torre

Publications

A spacetime group is a connected 4-dimensional Lie group G endowed with a left invariant Lorentz metric h and such that the connected component of the isometry group of h is G itself. The Newman-Penrose formalism is used to give an algebraic classification of spacetime groups, that is, we determine a complete list of inequivalent spacetime Lie algebras, which are pairs (g,η), with g being a 4-dimensional Lie algebra and η being a Lorentzian inner product on g. A full analysis of the equivalence problem for spacetime Lie algebras is given which leads to a completely algorithmic solution to the …

English Translation Of Einige Gesetze Ueber Die Theilung Der Ebene Und Des Raumes, Justin Heavilin

Mathematics and Statistics Faculty Publications

The article “Einige Gesetze ueber die Theilung der Ebene und des Raumes.” was published by J. Steiner in the very first volumn of the Journal fuer die reine und angewandte Mathematik in 1826. Journal fuer die reine und angewandte Mathematik is the oldest mathematics periodical in existence.† . My translation is meant to convey the ideas published by Steiner, and when presented with the choice between translating faithfully to the original text or clarity of his ideas, I admit to choosing the later. There are two footnotes original to the text, which appear with asterisks. Where helpful I included additional …

Power In Pairs: Assessing The Statistical Value Of Paired Samples In Tests For Differential Expression, John R. Stevens, Jennifer S. Herrick, Roger K. Wolff, Martha L. Slattery

Mathematics and Statistics Faculty Publications

Background: When genomics researchers design a high-throughput study to test for differential expression, some biological systems and research questions provide opportunities to use paired samples from subjects, and researchers can plan for a certain proportion of subjects to have paired samples. We consider the effect of this paired samples proportion on the statistical power of the study, using characteristics of both count (RNA-Seq) and continuous (microarray) expression data from a colorectal cancer study.

Results: We demonstrate that a higher proportion of subjects with paired samples yields higher statistical power, for various total numbers of samples, and for various strengths of …

Logarithmic Hennings Invariants For Restricted Quantum Sl (2), Anna Beliakova, Christian Blanchet, Alexandra Tebbs

Mathematics and Statistics Faculty Publications

We construct a Hennings-type logarithmic invariant for restricted quantum sl (2) at a 2p^th root of unity. This quantum group U is not quasitriangular and hence not ribbon, but factorizable. The invariant is defined for a pair: a 3–manifold M and a colored link L inside M. The link L is split into two parts colored by central elements and by trace classes, or elements in the 0^th Hochschild homology of U, respectively. The two main ingredients of our construction are the universal invariant of a string link with values in tensor powers of U, and the modified …

Multi-Resolution Analysis Using Wavelet Basis Conditioned On Homogenization, Abibat Adebisi Lasisi

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

This dissertation considers an approximation strategy using a wavelet reconstruction scheme for solving elliptic problems. The foci of the work are on (1) the approximate solution of differential equations using multiresolution analysis based on wavelet transforms and (2) the homogenization process for solving one and two-dimensional problems, to understand the solutions of second order elliptic problems. We employed homogenization to compute the average formula for permeability in a porous medium. The structure of the associated multiresolution analysis allows for the reconstruction of the approximate solution of the primary variable in the elliptic equation. Using a one-dimensional wavelet reconstruction algorithm proposed …

Calculating The Cohomology Of A Lie Algebra Using Maple And The Serre Hochschild Spectral Sequence, Jacob Kullberg

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

Lie algebra cohomology is an important tool in many branches of mathematics. It is used in the Topology of homogeneous spaces, Deformation theory, and Extension theory. There exists extensive theory for calculating the cohomology of semi simple Lie algebras, but more tools are needed for calculating the cohomology of general Lie algebras. To calculate the cohomology of general Lie algebras, I used the symbolic software program called Maple. I wrote software to calculate the cohomology in several different ways. I wrote several programs to calculate the cohomology directly. This proved to be computationally expensive as the number of differential forms …

Developmental Parameters Of A Southern Mountain Pine Beetle (Coleoptera: Curculionidae) Population Reveal Potential Source Of Latitudinal Differences In Generation Time, Anne E. Mcmanis, James A. Powell, Barbara J. Bentz

Mathematics and Statistics Faculty Publications

Mountain pine beetle (Dendroctonus ponderosae, Hopkins) is a major disturbance agent in pine ecosystems of western North America. Adaptation to local climates has resulted in primarily univoltine generation time across a thermally diverse latitudinal gradient. We hypothesized that voltinism patterns have been shaped by selection for slower developmental rates in southern populations inhabiting warmer climates. To investigate traits responsible for latitudinal differences we measured lifestage-specific development of southern mountain pine beetle eggs, larvae and pupae across a range of temperatures. Developmental rate curves were fit using maximum posterior likelihood estimation with a Bayesian prior to improve fit stability. …

How Design Features In Digital Math Games Support Learning And Mathematics Connections, Patricia S. Moyer-Packenham, Christina W. Lommatsch, Kristy Litster, Jill Ashby, Emma P. Bullock, Allison L. Roxburgh, Jessica F. Shumway, Emily Speed, Benjamin Covington, Christine Hartmann, Jody Clarke-Midura, Joel Skaria, Arla Westenskow, Beth L. Macdonald, Jurgen Symanzik, Kerry Jordan

Teacher Education and Leadership Faculty Publications

Current research shows that digital games can significantly enhance children’s learning. The purpose of this study was to examine how design features in 12 digital math games influenced children’s learning. The participants in this study were 193 children in Grades 2 through 6 (ages 8-12). During clinical interviews, children in the study completed pre-tests, interacted with digital math games, responded to questions about the digital math games, and completed post-tests. We recorded the interactions using two video perspectives that recorded children’s gameplay and responses to interviewers. We employed mixed methods to analyze the data and identify salient patterns in children’s …

Genome-Wide Association Study For Variants That Modulate Relationships Between Cerebrospinal Fluid Amyloid-Beta 42, Tau, And P-Tau Levels, Taylor J. Maxwell, Chris Corcoran, Jorge L. Del-Aguila, John P. Budde, Yuetiva Deming, Carlos Cruchaga, Alison M. Goate, John S. K. Kauwe, Alzheimer's Disease Neuroimaging Initiative

Mathematics and Statistics Faculty Publications

Background: A relationship quantitative trait locus exists when the correlation between multiple traits varies by genotype for that locus. Relationship quantitative trait loci (rQTL) are often involved in gene-by-gene (G×G) interactions or gene-by-environmental interactions, making them a powerful tool for detecting G×G.

Methods: We performed genome-wide association studies to identify rQTL between tau and Aβ42 and ptau and Aβ42 with over 3000 individuals using age, gender, series, APOE ε2, APOE ε4, and two principal components for population structure as covariates. Each significant rQTL was separately screened for interactions with other loci for each trait in the rQTL model. Parametric bootstrapping …

Remarks On Legendrian Self-Linking, Chris Beasley, Brendan Mclellan, Ruoran Zhang

Mathematics and Statistics Faculty Publications

The Thurston-Bennequin invariant provides one notion of self-linking for any homologically-trivial Legendrian curve in a contact three-manifold. Here we discuss related analytic notions of self-linking for Legendrian knots in R³. Our definition is based upon reformulation of the elementary Gauss linking integral and is motivated by ideas from supersymmetric gauge theory. We recover the Thurston-Bennequin invariant as a special case.