Theses and Dissertations at Montana State University (MSU)
Permanent URI for this collectionhttps://scholarworks.montana.edu/handle/1/733
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Item Quantifying robustness of the gap gene network(Montana State University - Bozeman, College of Letters & Science, 2024) Andreas, Elizabeth Anne; Chairperson, Graduate Committee: Tomas Gedeon; Bree Cummins (co-chair)Early development of Drosophila melanogaster (fruit fly) facilitated by the gap gene network has been shown to be incredibly robust, and the same patterns emerge even when the process is seriously disrupted. We investigate this robustness using a previously developed computational framework called DSGRN (Dynamic Signatures Generated by Regulatory Networks). Our mathematical innovations include the conceptual extension of this established modeling technique to enable modeling of spatially monotone environmental effects, as well as the development of a collection of graph theoretic robustness scores for network models. This allows us to rank order the robustness of network models of cellular systems where each cell contains the same genetic network topology but operates under a parameter regime that changes continuously from cell to cell. We demonstrate the power of this method by comparing the robustness of two previously introduced network models of gap gene expression along the anterior-posterior axis of the fruit fly embryo, both to each other and to a random sample of networks with same number of nodes and edges. We observe that there is a substantial difference in robustness scores between the two models. Our biological insight is that random network topologies are in general capable of reproducing complex patterns of expression, but that using measures of robustness to rank order networks permits a large reduction in hypothesis space for highly conserved systems such as developmental networks.Item Applications and diagnostics for dimension reduction of multivariate data(Montana State University - Bozeman, College of Letters & Science, 2022) Harmon, Paul Gary; Chairperson, Graduate Committee: Mark Greenwood; This is a manuscript style paper that includes co-authored chapters.Working with high-dimensional data involves various statistical challenges. This dissertation overviews a suite of tools and methods for dimension reduction, using latent- variable models, techniques for mapping high-dimensional data, clustering, and working with multivariate responses across a variety of use cases. First, we propose and develop a method for classifying institutions of higher education is and compare with the current standard for university classification: the Carnegie Classification. We present a classification tool based on Structural Equation Models that better allows for modeling of correlated indices than the PCA-based methodology that underlies the Carnegie Classification. Additionally, we create a Shiny-based web application that allows for assessment of sensitivity to changes in the underlying characteristics of each institution. Second, we develop a novel methodology that extends the Cook's Distance diagnostic for identifying influential points in regression to a new application on high-dimensional mapping tools. We highlight a PERMANOVA-based method for calculating the difference in the shape of resulting ordinations based on inclusion/exclusion of points, similar in style to the influence diagnostic Cook's Distance for regression. We present a set simulation studies with several mapping techniques and highlight where the method works well (Classical Multidimensional Scaling) and where the methods appear to work less effectively (t-distributed Stochastic Neighbor Embedding). Additionally, we examine several real data sets and assess the efficacy of the diagnostic on thsoe data sets. Finally, we introduce a new method for feature selection in a specific type of divisive clustering, called monothetic clustering. Utilizing a penalized matrix decomposition to re- weight the input data to the monothetic clustering algorithm allows for reduction in noise features allows this clustering method to better make splits based on single features at a time, leading to better cluster results. We present a method for tuning both the number of clusters, K, and the degree of sparsity, s, as well as simulation studies that highlight the efficacy of noise reduction in monothetic clustering solutions.Item Two-year community college students' understanding of rational expressions(Montana State University - Bozeman, College of Letters & Science, 2023) Kong, Chor Wan Amy; Chairperson, Graduate Committee: Jennifer Luebeck; Megan Wickstrom (co-chair)This research study investigated the gaps in knowledge held by two-year community college students in simplifying and operating with rational expressions and how these gaps affect their learning. The study employed multiple methods, including completion of a Diagnostic Problem Set, participating in collaborative and exploratory activities, and attending task-based interviews, to elicit and assess students' understanding of rational expressions. The study analyzed and categorized participants' responses based on the participants' different perspectives and learning processes. The research also explored how collaborative and exploratory learning, as well as the use of Knowledge-Eliciting Tasks, can help identify and address students' misconceptions. Qualitative analysis of the findings identified potential causes of the learning gaps and generated recommendations for instructional strategies that can bridge these gaps and improve students' understanding of rational expressions, which is crucial to student success in algebraic subjects and college academic achievement.Item Modeling saline fluid flow in subglacial ice-walled channels(Montana State University - Bozeman, College of Letters & Science, 2022) Jenson, Amy Jo; Chairperson, Graduate Committee: Scott McCallaSubglacial hydrological systems have impacts on ice dynamics as well as nutrient and sediment transport. There has been an extensive effort to understand the dynamics of subglacial drainage through numerical modeling, however these models have focused on freshwater, neglecting the consideration of brine. Saline fluid can exist in cold-based glacier systems where freshwater cannot. Therefore, there exist subglacial hydrological systems where the only fluid is brine. Understanding the routing of saline fluid is important for understanding geochemical and microbiological processes in these saline cryospheric habitats. In this thesis, I present a model of channelized drainage from a hypersaline subglacial lake and highlight the impact of saline fluid on melt rates in an ice-walled channel. The model results show that channel walls grow more quickly when fluid contains higher salt concentrations, which results in greater peak discharge and faster drainage for a fixed lake volume. This model provides a framework to assess the relative impact of brine on discharge and drainage duration.Item Analysis of dynamic biological systems imagery(Montana State University - Bozeman, College of Letters & Science, 2022) Dudiak, Cameron Drew; Chairperson, Graduate Committee: Scott McCallaBiological systems pose considerable challenges when attempting to isolate experimental variables of interest and obtain viable data. Developments in image analysis algorithms and techniques allow for further mathematical interpretation, model integration, and even model optimization ('training'). We formulate two distinct methods for obtaining robust quantitative data from time-series imagery of two biological systems: Paenibacillus dendritiformis bacterial colonies, and human gastric organoids. Boundary parameterizations of P. dendritiformis are extracted from timelapse image sequences displaying colony repulsion, and are subsequently used to 'train' a previously developed nonlocal PDE model through the means of error minimization between observation and simulation. Particle tracking is conducted for small colloidal beads embedded within human gastric organoids, and then used to perform particle tracking analysis. This information is analyzed to quantify the local complex viscoelastic properties of organoids' interior mucosal environment.Item Faithful sets of topological descriptors and the algebraic K-theory of multi-parameter zig-zag grid persistence modules(Montana State University - Bozeman, College of Letters & Science, 2023) Schenfisch, Anna Katherine; Chairperson, Graduate Committee: Tomas Gedeon and Brittany Fasy (co-chair)Given a geometric simplicial complex, the uncountable set of (augmented) persistence diagrams corresponding to lower-star filtrations taken with respect to all possible directions uniquely correspond to the simplicial complex, i.e., the set is faithful. While this hints towards interesting applications in shape comparison, the set of all possible directions is uncountably infinite, and so has no hope of computability. In practice, one might use a finite approximation, but faithfulness of this approximation is not guaranteed. Motivated by the need for both computability and provable faithfulness, we provide an explicit description of a finite faithful set of augmented persistence diagrams. We then show this construction applies to augmented Euler characteristic curves and augmented Betti curves, and is stable under particular perturbations. In the specific case where the underlying complex is a graph, we provide an improved construction that utilizes a radial binary search. We then shift focus to comparing the cardinalities of minimal faithful sets of descriptors as a way to define and order equivalence classes of topological descriptor types. Focusing on six topological descriptor types commonly used in practice, we give a partial order on their corresponding equivalence classes, as well as give bounds on the sizes of minimum faithful sets for each descriptor type. Next, we broaden our view to zig-zag grid persistence modules, functors whose domain categories are posets with grid-like structure. We begin by explicitly defining such persistence modules in terms of constructible cosheaves over stratified Euclidean space, including a careful treatment of augmented persistence modules, which are analogous to the aforementioned augmented descriptors who played a central role in discussion of faithful sets. Exodromy gives us an equivalence between persistence modules as a functor category and as constructible cosheaves; we furthermore show the equivalence of these categories with a category of constructible functors out of Rd with a fixed stratification and localized at weak equivalences, essentially "standardizing" modules so that the category has a clear monoid structure. We compute the algebraic K-theory of zig-zag grid persistence modules, using a double inductive argument to show the K-theory is additive over strata. Finally, we identify connections to related topics, such as the virtual diagrams of Bubenik and Elchensen, as well as Euler characteristic and Betti curves/surfaces/manifolds. We hope a study of K-groups will provide interesting insights into the nature of persistence modules, and we indicate ways in which the zeroth and first K-groups may be interpretedItem Rehumanizing college mathematics: centering the voices of Latin*, indigenous, LGBTQ+ and women STEM majors(Montana State University - Bozeman, College of Letters & Science, 2022) MacArthur, Kelly Ann; Chairperson, Graduate Committee: Derek A. WilliamsCalculus sequences are frequently experienced as gatekeeper courses for STEM-intending students, particularly for students from groups that have been historically marginalized in mathematics including Latin*, Indigenous, LGBTQ+ and women. I report here on research findings that explored attitudes of Calculus 2 students broadly, as well as more specifically from the above-listed groups regarding what practices, pedagogies, and structures feel humanizing to them. I used a transformative mixed methods design, built on a sociopolitical framework, namely the rehumanizing framework outlined by Gutiérrez (2018) that includes eight dimensions. The goal of this research is to answer a call from Gutiérrez in elevating and understanding the perspectives of students who are often ill-served and thereby impact future undergraduate teaching in positive and humanizing ways. The quantitative analysis of survey questions (n=153) showed that students generally find example scenarios that align with the eight rehumanizing dimensions to be humanizing, based on their ratings of feeling supported in their learning, feeling valued and a sense of belonging, and having connections between their mathematics class and their lives outside the classroom. From qualitative analysis of follow-up interviews with 20 students who self-identified as Latin*, Native American, LGBTQ+ and/or women, a student-driven definition of humanizing emerged. For these focal students, humanizing centers relationality and welcoming/caring/failure-tolerant classroom environment. Teaching actions that focal students described as humanizing were summed up as connections-connections to peers, teachers and to their lives outside the classroom. Blending the quantitative and qualitative analysis shed light on differences between dominant (white, heterosexual, cis-men) and focal group perceptions, especially regarding the Cultures & Theirstories rehumanizing dimension scenario. This was accompanied by cautions from focal students about how implementation of some scenarios matters in meeting a humanizing goal.Item Investigating newer statistics instructors' breakthroughs with and motivations for using active learning: a longitudinal case-study and a multi-phase approach towards nstrument development(Montana State University - Bozeman, College of Letters & Science, 2022) Meyer, Elijah Sterling; Co-Chairs, Graduate Committee: Stacey Hancock and Jennifer Green; This is a manuscript style paper that includes co-authored chapters.National recommendations call for a shift from using lecture-based approaches to using approaches that engage students in the learning process, primarily through active learning techniques. Despite these recommendations, the adoption of active learning techniques for newer statistics instructors remains limited. The goal of this research is to provide a more holistic understanding about statistics instruction, specifically as it relates to recommended active learning techniques and newer statistics instructors, including graduate student instructors (GSIs). In this research, I present two studies. In the first study, we investigated GSIs' breakthroughs in their knowledge about, emotions towards, and use of active learning over time by using a longitudinal collective case-study approach. Survey, interview, and observation data across four semesters revealed that the GSIs' breakthroughs in their use of active learning only occurred after their increased knowledge about active learning aligned with their emotions towards it. This study further revealed that the GSIs needed to feel confident in and be challenged by their course structure before implementing active learning techniques. The second study builds upon these findings by exploring statistics instructors' motivations or reasons for using active learning. Under the self-determination theory framework, we conducted a multi-phase study to develop an instrument that measures four different types of motivational constructs for using group work, a specific active learning approach. We constructed items using expert opinion and cognitive interviews, and then we conducted two pilot studies with newer statistics instructors. The resulting reliability and validity evidence suggest that this instrument may help support future studies' investigations of motivation, helping us to better understand newer statistics instructors' use of active learning. Together, these studies may help inform future recommendations on how to support newer statistics instructors' early adoption of such technique.Item Preservice teachers' construction of computational thinking practices through mathematical modeling activities(Montana State University - Bozeman, College of Letters & Science, 2022) Adeolu, Adewale Samson; Chairperson, Graduate Committee: Mary Alice Carlson and Elizabeth Burroughs (co-chair)The importance of learning computational thinking practices in K-12 settings is gaining momentum in the United States and worldwide. As a result, studies have been conducted on integrating these practices in mathematics teaching and learning. However, there is little study that focuses on how to prepare pre-service teachers who will teach the practices in K-12 settings. I investigated how pre-service teachers collaborated to develop computational thinking practices when working on modeling activities with computational tools. To carry out this research, I studied nine pre-service teachers working on modeling tasks for a semester. Five participants recorded their screens and were invited to participate in a stimulated recall interview. Using the interactional analysis procedures, findings showed that the presence of computational tools influenced the positioning (leadership and distributed authority) and collaborative processes (dividing and offloading labor, giving and receiving feedback, accommodation, and refining ideas) pre-service teachers used during modeling. This study found that pre-service teachers used ten computational thinking practices, which are sub-grouped into four broader practices -- data practices, mimicking and mathematizing, model exploration and extension, and model communication. This dissertation also found that pre-service teachers' mathematical knowledge and their ability to code were interdependent. From a research point of view, this study extends our knowledge of the social constructivist theory of doing research in the context of pre-service teachers engaging in modeling activities with computational tools. From the teacher education perspective, this study emphasizes the need to consider the impact of computational tools on the interactions of pre-service teachers during modeling. The study also reveals the need to structure the mathematical modeling curriculum to lead to a better learning experience for pre-service teachers.Item Critically fixed anti-rational maps, tischler graphs, and their applications(Montana State University - Bozeman, College of Letters & Science, 2022) McKay, Christopher Michael; Chairperson, Graduate Committee: Lukas GeyerWe are mainly concerned with maps which take the form of the complex conjugate of a rational map and where all critical points are fixed points, which are known as critically fixed anti-rational maps. These maps have a well understood combinatorial model by a planar graph. We make progress in answering three questions. Using this combinatorial model by planar graph, can we generate all critically fixed anti-rational maps from the most basic example, z -> z 2? Under repeated pullback by a critically fixed anti-rational map, does a simple closed curve off the critical set eventually land and stay in a finite set of homotopy classes of simple closed curves off the critical set? This is known as the global curve attractor problem and has been an area of interest since its introduction by Pilgrim in 2012. Lastly, anti-rational maps can be used to model the physical phenomenon of gravitational lensing, which is where the image of a far away light source is distorted and multiplied by large masses between the light source and the observer. Maximal lensing configurations are where n masses generate 5n - 5 lensed images of a single light source. There are very few known examples of maximal lensing configurations, all generated by Rhie in 2003. Can we use these combinatorial models to inspire new examples of maximal lensing configurations? In this dissertation we show one can generate all critically fixed anti-rational maps from the most basic example, z -> z 2 by a repeated 'blow-up' procedure. We also show that all critically fixed anti-rational maps with 4 or 5 critical points have a finite global curve attractor. Lastly we establish a connection between maximal lensing maps and Tischler graphs and generate new examples of maximal lensing maps.