Theses and Dissertations

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Open Access
A constrained optimization model for partitioning students into cooperative learning groups
(Colorado State University. Libraries, 2016) Heine, Matthew Alan, author; Kirby, Michael, advisor; Pinaud, Olivier, committee member; Henry, Kimberly, committee member
The problem of the constrained partitioning of a set using quantitative relationships amongst the elements is considered. An approach based on constrained integer programming is proposed that permits a group objective function to be optimized subject to group quality constraints. A motivation for this problem is the partitioning of students, e.g., in middle school, into groups that target educational objectives. The method is compared to another grouping algorithm in the literature on a data set collected in the Poudre School District.
Open Access
A direct D-bar reconstruction algorithm for complex admittivities in W2,∞(Ω) for the 2-D EIT problem
(Colorado State University. Libraries, 2012) Hamilton, Sarah Jane, author; Mueller, Jennifer L., advisor; Duchateau, Paul, committee member; Tavener, Simon, committee member; Lear, Kevin, committee member
Electrical Impedance Tomography (EIT) is a fairly new, portable, relatively inexpensive, imaging system that requires no ionizing radiation. Electrodes are placed at the surface of a body and low frequency, low amplitude current is applied on the electrodes, and the resulting voltage value on each electrode is measured. By applying a basis of current patterns, one can obtain sufficient information to recover the complex admittivity distribution of the region in the plane of the electrodes. In 2000, Elisa Francini presented a nearly constructive proof that was the first approach using D-bar methods to solve the full nonlinear problem for twice-differentiable conductivities and permittivities. In this thesis the necessary formulas to turn her proof into a direct D-bar reconstruction algorithm that solves the full nonlinear admittivity problem in 2-D are described. Reconstructions for simulated Finite Element data for circular and non-circular domains are presented.
Open Access
A geometric data analysis approach to dimension reduction in machine learning and data mining in medical and biological sensing
(Colorado State University. Libraries, 2017) Emerson, Tegan Halley, author; Kirby, Michael, advisor; Peterson, Chris, advisor; Nyborg, Jennifer, committee member; Chenney, Margaret, committee member
Geometric data analysis seeks to uncover and leverage structure in data for tasks in machine learning when data is visualized as points in some dimensional, abstract space. This dissertation considers data which is high dimensional with respect to varied notions of dimension. Algorithms developed herein seek to reduce or estimate dimension while preserving the ability to perform a specific task in detection, identification, or classification. In some of the applications the only property considered important to be preserved under dimension reduction is the ability to perform the indicated machine learning task while in others strictly geometric relationships between data points are required to be preserved or minimized. First presented is the development of a numerical representation of images of rare circulating cells in immunofluorescent images. This representation is paired with a support vector machine and is able to identify differentiating cell structure between cell populations under consideration. Moreover, this differentiating information can be visualized through inversion of the representation and was found to be consistent with classification criterion used by clinically trained pathologists. Considered second is the task of identification and tracking of aerosolized bioagents via a multispectral lidar system. A nonnegative matrix factorization problem arised out of this data mining task which can be solved in several ways including a ℓ1-norm regularized, convex but nondifferentiable optimization problem. Exisiting methodologies achieve excellent results when internal matrix factor dimension is known but fail or can be computationally prohibitive when this dimension is not known. A modified optimization problem is proposed that may help reveal the appropriate internal factoring dimension based on the sparsity of averages of nonnegative values. Third, we present an algorithmic framework for reducing dimension in the linear mixing model. The mean-squared error of a statistical estimator of a component of the linear mixing model can be considered as a function of the rank of different estimating matrices. We seek to minimize mean squared error as a function of the rank of the appropriate estimating matrix and yield interesting order determination rules and improved results, relative to full rank counterparts, in applications in matched subspace detection and generalized modal analysis. Finally, the culminating work of this dissertation explores the existence of nearly isometric, dimension reducing mappings between special manifolds characterized by different dimensions. Understanding the analogous problem between Euclidean spaces provides insights into potential challenges and pitfalls one could encounter in proving the existence of such mappings. Most significant of the contributions is the statement and proof of a theorem establishing a connection between packing problems on Grassmannian manifolds and nearly isometric mappings between Grassmannians. The frameworks and algorithms constructed and developed in this doctoral research consider multiple manifestations of the notion of dimension. Across applications arising from varied areas of medical and biological sensing we have shown there to be great benefits to taking a geometric perspective on challenges in machine learning and data mining.
Open Access
A local characterization of domino evacuation-shuffling
(Colorado State University. Libraries, 2024) McCann, Jacob, author; Gillespie, Maria, advisor; Peterson, Christopher, committee member; Huang, Dongzhou, committee member
We consider linear intersection problems in the Grassmanian (the space of k-dimensional subspaces of Cn), where the dimension of the intersection is 2. These spaces are called Schubert surfaces. We build of the previous work of Speyer [1] and Gillespie and Levinson [2]. Speyer showed there is a combinatorial interpretation for what happens to fibers of Schubert intersections above a "wall crossing", where marked points corresponding to the coordinates of partitions coincide. Building off Speyer's work, Levinson showed there is a combinatorial operation associated with the monodromy operator on Schubert curves, involving rectification, promotion, and shuffling of Littlewood-Richardson Young Tableaux, which overall is christened evacuation-shuffling. Gillespie and Levinson [2] further developed a localization of the evacuation-shuffling algorithm for Schubert curves. We fully develop a local description of the monodromy operator on certain classes of curves embedded inside Schubert surfaces [3].
Open Access
A posteriori error estimates for the Poisson problem on closed, two-dimensional surfaces
(Colorado State University. Libraries, 2011) Newton, William F., author; Estep, Donald J., 1959-, advisor; Holst, Michael J., committee member; Tavener, Simon, committee member; Zhou, Yongcheng, committee member; Breidt, F. Jay, committee member
The solution of partial differential equations on non-Euclidean Domains is an area of much research in recent years. The Poisson Problem is a partial differential equation that is useful on curved surfaces. On a curved surface, the Poisson Problem features the Laplace-Beltrami Operator, which is a generalization of the Laplacian and specific to the surface where the problem is being solved. A Finite Element Method for solving the Poisson Problem on a closed surface has been described and shown to converge with order h2. Here, we review this finite element method and the background material necessary for defining it. We then construct an adjoint-based a posteriori error estimate for the problem, discuss some computational issues that arise in solving the problem and show some numerical examples. The major sources of numerical error when solving the Poisson problem are geometric error, discretization error, quadrature error and measurement error. Geometric error occurs when distances, areas and angles are distorted by using a flat domain to parametrize a curved one. Discretization error is a result of using a finite-dimensional space of functions to approximate an infinite-dimensional space. Quadrature error arises when we use numerical quadrature to evaluate integrals necessary for the finite element method. Measurement error arises from error and uncertainty in our knowledge of the surface itself. We are able to estimate the amount of each of these types of error and show when each type of error will be significant.
Open Access
A quantum H*(T)-module via quasimap invariants
(Colorado State University. Libraries, 2024) Lee, Jae Hwang, author; Shoemaker, Mark, advisor; Cavalieri, Renzo, advisor; Gillespie, Maria, committee member; Peterson, Christopher, committee member; Hulpke, Alexander, committee member; Chen, Hua, committee member
For X a smooth projective variety, the quantum cohomology ring QH*(X) is a deformation of the usual cohomology ring H*(X), where the product structure is modified to incorporate quantum corrections. These correction terms are defined using Gromov-Witten invariants. When X is toric with geometric quotient description V//T, the cohomology ring H*(V//T) also has the structure of a H*(T)-module. In this paper, we introduce a new deformation of the cohomology of X using quasimap invariants with a light point. This defines a quantum H*(T)-module structure on H*(X) through a modified version of the WDVV equations. We explicitly compute this structure for the Hirzebruch surface of type 2. We conjecture that this new quantum module structure is isomorphic to the natural module structure of the Batyrev ring for a semipositive toric variety.
Open Access
A ratio ergodic theorem on Borel actions of Zd and Rd
(Colorado State University. Libraries, 2009) Holt, Eric Norman, author; Rudolph, Daniel, advisor
We prove a ratio ergodic theorem for free Borel actions of Zd and Rd on a standard Borel probability space. The proof employs an extension of the Besicovitch Covering Lemma, as well as a notion of coarse dimension that originates in an upcoming paper of Hochman. Due to possible singularity of the measure, we cannot use functional analytic arguments and therefore diffuse the measure onto the orbits of the action. This diffused measure is denoted μx, and our averages are of the form 1/μx(Bn) ∫ Bn f o T-v(x)dμx. A Følner condition on the orbits of the action is shown, which is the main tool used in the proof of the ergodic theorem. Also, an extension of a known example of divergence of a ratio average is presented for which the action is both conservative and free.
Open Access
A simplicial homotopy group model for K2 of a ring
(Colorado State University. Libraries, 2010) Whitfield, JaDon Saeed, author; Duflot, Jeanne, advisor; Miranda, Rick, committee member; Achter, Jeffrey D., committee member; Gelfand, Martin Paul, committee member
We construct an isomorphism between the group K2(R) from classical, algebraic K-Theory for a ring R and a simplicial homotopy group constructed using simplicial homotopy theory based on that same ring R. First I describe the basic aspects of simplicial homotopy theory. Special attention is paid to the use of category theory, which will be applied to the construction of a simplicial set. K-Theory for K0(R), K1(R) and K2(R) is then described before we set to work describing explicitly the nature of isomorphisms for K0(R) and K1(R) based on previous work. After introducing some theory related to K-Theory, some considerations and corrections on previous work motivate more new theory that helps the isomorphism with K2(R). Such theory is developed, mainly with regards to finitely generated projective modules over R and then elementary matrices with entries from R, culminating in the description of the Steinberg Relations that are central to the understanding of K2(R) in terms of homotopy classes. We then use new considerations on the previous work to show that a map whose image is constructed through this article is an isomorphism since it is the composition of isomorphisms.
Open Access
A two-field finite element solver for linear poroelasticity
(Colorado State University. Libraries, 2020) Wang, Zhuoran, author; Liu, Jiangguo, advisor; Tavener, Simon, advisor; Zhou, Yongcheng, committee member; Ma, Kaka, committee member
Poroelasticity models the interaction between an elastic porous medium and the fluid flowing in it. It has wide applications in biomechanics, geophysics, and soil mechanics. Due to difficulties of deriving analytical solutions for the poroelasticity equation system, finite element methods are powerful tools for obtaining numerical solutions. In this dissertation, we develop a two-field finite element solver for poroelasticity. The Darcy flow is discretized by a lowest order weak Galerkin (WG) finite element method for fluid pressure. The linear elasticity is discretized by enriched Lagrangian ($EQ_1$) elements for solid displacement. First order backward Euler time discretization is implemented to solve the coupled time-dependent system on quadrilateral meshes. This poroelasticity solver has some attractive features. There is no stabilization added to the system and it is free of Poisson locking and pressure oscillations. Poroelasticity locking is avoided through an appropriate coupling of finite element spaces for the displacement and pressure. In the equation governing the flow in pores, the dilation is calculated by taking the average over the element so that the dilation and the pressure are both approximated by constants. A rigorous error estimate is presented to show that our method has optimal convergence rates for the displacement and the fluid flow. Numerical experiments are presented to illustrate theoretical results. The implementation of this poroelasticity solver in deal.II couples the Darcy solver and the linear elasticity solver. We present the implementation of the Darcy solver and review the linear elasticity solver. Possible directions for future work are discussed.
Open Access
Abelian surfaces with real multiplication over finite fields
(Colorado State University. Libraries, 2014) Freese, Hilary, author; Achter, Jeffrey, advisor; Pries, Rachel, committee member; Peterson, Chris, committee member; Tavani, Daniele, committee member
Given a simple abelian surface A/Fq, the endomorphism algebra, End(A) ⊗ Q, contains a unique real quadratic subfield. We explore two different but related questions about when a particular real quadratic subfield K+ is the maximal real subfield of the endomorphism algebra. First, we compute the number of principally polarized abelian surfaces A/Fq such that K+ ⊂ End(A) ⊗ Q. Second, we consider an abelian surface A/Q, and its reduction Ap = A mod p, then ask for which primes p is K+ ⊂ End(A) ⊗ Q. The result from the first question leads to a heuristic for the second question, namely that the number of p < χ for which K+ ⊂ End(A) ⊗ Q grows like √χ/log(c).
Open Access
Abstract hyperovals, partial geometries, and transitive hyperovals
(Colorado State University. Libraries, 2015) Cooper, Benjamin C., author; Penttila, Timothy, advisor; Bohm, Wim, committee member; Cavalieri, Renzo, committee member; Duflot, Jeanne, committee member
A hyperoval is a (q+2)- arc of a projective plane π, of order q with q even. Let G denote the collineation group of π containing a hyperoval Ω. We say that Ω is transitive if for any pair of points x, y is an element of Ω, there exists a g is an element of G fixing Ω setwise such that xg = y. In1987, Billotti and Korchmaros proved that if 4
Open Access
Algorithms and geometric analysis of data sets that are invariant under a group action
(Colorado State University. Libraries, 2010) Smith, Elin Rose, author; Peterson, Christopher Scott, 1963-, advisor; Bates, Daniel J. (Daniel James), 1979-, committee member; Kirby, Michael, 1961-, committee member; McConnell, Ross M., committee member
We apply and develop pattern analysis techniques in the setting of data sets that are invariant under a group action. We apply Principal Component Analysis to data sets of images of a rotating object in Chapter 5 as a means of obtaining visual and low-dimensional representations of data. In Chapter 6, we propose an algorithm for finding distributions of points in a base space that are (locally) optimal in the sense that subspaces in the associated data bundle are distributed with locally maximal distance between neighbors. In Chapter 7, we define a distortion function that measures the quality of an approximation of a vector bundle by a set of points. We then use this function to compare the behavior of four standard distance metrics and one non-metric. Finally, in Chapter 8, we develop an algorithm to find the approximate intersection of two data sets.
Open Access
Algorithms for feature selection and pattern recognition on Grassmann manifolds
(Colorado State University. Libraries, 2015) Chepushtanova, Sofya, author; Kirby, Michael, advisor; Peterson, Chris, committee member; Bates, Dan, committee member; Ben-Hur, Asa, committee member
This dissertation presents three distinct application-driven research projects united by ideas and topics from geometric data analysis, optimization, computational topology, and machine learning. We first consider hyperspectral band selection problem solved by using sparse support vector machines (SSVMs). A supervised embedded approach is proposed using the property of SSVMs to exhibit a model structure that includes a clearly identifiable gap between zero and non-zero feature vector weights that permits important bands to be definitively selected in conjunction with the classification problem. An SSVM is trained using bootstrap aggregating to obtain a sample of SSVM models to reduce variability in the band selection process. This preliminary sample approach for band selection is followed by a secondary band selection which involves retraining the SSVM to further reduce the set of bands retained. We propose and compare three adaptations of the SSVM band selection algorithm for the multiclass problem. We illustrate the performance of these methods on two benchmark hyperspectral data sets. Second, we propose an approach for capturing the signal variability in data using the framework of the Grassmann manifold (Grassmannian). Labeled points from each class are sampled and used to form abstract points on the Grassmannian. The resulting points have representations as orthonormal matrices and as such do not reside in Euclidean space in the usual sense. There are a variety of metrics which allow us to determine distance matrices that can be used to realize the Grassmannian as an embedding in Euclidean space. Multidimensional scaling (MDS) determines a low dimensional Euclidean embedding of the manifold, preserving or approximating the Grassmannian geometry based on the distance measure. We illustrate that we can achieve an isometric embedding of the Grassmann manifold using the chordal metric while this is not the case with other distances. However, non-isometric embeddings generated by using the smallest principal angle pseudometric on the Grassmannian lead to the best classification results: we observe that as the dimension of the Grassmannian grows, the accuracy of the classification grows to 100% in binary classification experiments. To build a classification model, we use SSVMs to perform simultaneous dimension selection. The resulting classifier selects a subset of dimensions of the embedding without loss in classification performance. Lastly, we present an application of persistent homology to the detection of chemical plumes in hyperspectral movies. The pixels of the raw hyperspectral data cubes are mapped to the geometric framework of the Grassmann manifold where they are analyzed, contrasting our approach with the more standard framework in Euclidean space. An advantage of this approach is that it allows the time slices in a hyperspectral movie to be collapsed to a sequence of points in such a way that some of the key structure within and between the slices is encoded by the points on the Grassmannian. This motivates the search for topological structure, associated with the evolution of the frames of a hyperspectral movie, within the corresponding points on the manifold. The proposed framework affords the processing of large data sets, such as the hyperspectral movies explored in this investigation, while retaining valuable discriminative information. For a particular choice of a distance metric on the Grassmannian, it is possible to generate topological signals that capture changes in the scene after a chemical release.
Open Access
Algorithms in numerical algebraic geometry and applications
(Colorado State University. Libraries, 2015) Hanson, Eric M., author; Bates, Daniel J., advisor; Peterson, Chris, committee member; Cavalieri, Renzo, committee member; Maciejewski, Anthony, committee member
The topics in this dissertation, while independent, are unified under the field of numerical algebraic geometry. With ties to some of the oldest areas in mathematics, numerical algebraic geometry is relatively young as a field of study in its own right. The field is concerned with the numerical approximation of the solution sets of systems of polynomial equations and the manipulation of these sets. Given a polynomial system ƒ : CN → Cn, the methods of numerical algebraic geometry produce numerical approximations of the isolated solutions of ƒ(z) = 0, as well as points on any positive-dimensional components of the solution set, V(ƒ). In a short time, the work done in numerical algebraic geometry has significantly pushed the boundary of what is computable. This dissertation aims to further this work by contributing new algorithms to the field and using cutting edge techniques of the field to expand the scope of problems that can be addressed using numerical methods. We begin with an introduction to numerical algebraic geometry and subsequent chapters address independent topics: perturbed homotopies, exceptional sets and fiber products, and a numerical approach to finding unit distance embeddings of finite simple graphs. One of the most recent advances in numerical algebraic geometry is regeneration, an equation-by-equation homotopy method that is often more efficient than other approaches. However, the basic form of regeneration will not necessarily find all isolated singular solutions of a polynomial system without the additional cost of using deflation. In the second chapter, we present an alternative to deflation in the form of perturbed homotopies for solving polynomial systems. In particular, we propose first solving a perturbed version of the polynomial system, followed by a parameter homotopy to remove the perturbation. The aim of this chapter is two-fold. First, such perturbed homotopies are sometimes more efficient than regular homotopies, though they can also be less efficient. Second, a useful consequence is that the application of this perturbation to regeneration will yield all isolated solutions, including all singular isolated solutions. The third chapter considers families of polynomial systems which depend on parameters. There is a typical dimension for the variety defined by a system in the family; however, this dimension may jump for parameters in algebraic subsets of the parameter space. Sommese and Wampler exploited fiber products to give a numerical method for identifying these special parameter values. In this chapter, we propose a refined numerical approach to fiber products, which uses recent advancements in numerical algebraic geometry, such as regeneration extension. We show that this method is sometimes more efficient then known techniques. This gain in efficiency is due to the fact that regeneration extension allows the construction of the fiber product to be restricted to specified irreducible components. This work is motivated by applications in Kinematics - the study of mechanisms. As such we use an algebraic model of a two link arm to illustrate the algorithms developed in this chapter. The topic of the last chapter is the identification of unit distance embeddings of finite simple graphs. Given a graph G(V,E), a unit distance embedding is a map ɸ from the vertex set V into a metric space M such that if {vi,vj} is an element of E then the distance between ɸ (vi) and ɸ (vj) in M is one. Given G, we cast the question of the existence of a unit distance embedding in Rn as the question of the existence of a real solution to a system of polynomial equations. As a consequence, we are able to develop theoretic algorithms for determining the existence of a unit distance embedding and for determining the smallest dimension of Rn for which a unit distance embedding of G exists (that is, we determine the minimal embedding dimension of G). We put these algorithms into practice using the methods of numerical algebraic geometry. In particular, we consider unit distance embeddings of the Heawood Graph. This is the smallest example of a point-line incidence graph of a finite projective plan. In 1972, Chvátal conjectured that point-line incidence graphs of finite projective planes do not have unit-distance embeddings into R². In other words, Chvátal conjectured that the minimal embedding dimension of any point-line incidence graph of a finite projective plane is at least 3. We disprove this conjecture, adding hundreds of counterexamples to the 11 known counterexamples found by Gerbracht.
Open Access
An adaptation of K-means-type algorithms to the Grassmann manifold
(Colorado State University. Libraries, 2019) Stiverson, Shannon J., author; Kirby, Michael, advisor; Adams, Henry, committee member; Ben-Hur, Asa, committee member
The Grassmann manifold provides a robust framework for analysis of high-dimensional data through the use of subspaces. Treating data as subspaces allows for separability between data classes that is not otherwise achieved in Euclidean space, particularly with the use of the smallest principal angle pseudometric. Clustering algorithms focus on identifying similarities within data and highlighting the underlying structure. To exploit the properties of the Grassmannian for unsupervised data analysis, two variations of the popular K-means algorithm are adapted to perform clustering directly on the manifold. We provide the theoretical foundations needed for computations on the Grassmann manifold and detailed derivations of the key equations. Both algorithms are then thoroughly tested on toy data and two benchmark data sets from machine learning: the MNIST handwritten digit database and the AVIRIS Indian Pines hyperspectral data. Performance of algorithms is tested on manifolds of varying dimension. Unsupervised classification results on the benchmark data are compared to those currently found in the literature.
Open Access
An adaptive algorithm for an elliptic optimization problem, and stochastic-deterministic coupling: a mathematical framework
(Colorado State University. Libraries, 2008) Lee, Sheldon, author; Estep, Donald, advisor; Tavener, Simon, advisor
This dissertation consists of two parts. In the first part, we study optimization of a quantity of interest of a solution of an elliptic problem, with respect to parameters in the data using a gradient search algorithm. We use the generalized Green's function as an efficient way to compute the gradient. We analyze the effect of numerical error on a gradient search, and develop an efficient way to control these errors using a posteriori error analysis. Specifically, we devise an adaptive algorithm to refine and unrefine the finite element mesh at each step in the descent search algorithm. We give basic examples and apply this technique to a model of a healing wound. In the second part, we construct a mathematical framework for coupling atomistic models with continuum models. We first study the case of coupling two deterministic diffusive regions with a common interface. We construct a fixed point map by repeatedly solving the problems, while passing the flux in one direction and the concentration in the other direction. We examine criteria for the fixed point iteration to converge, and offer remedies such as reversing the direction of the coupling, or relaxation, for the case it does not. We then study the one dimensional case where the particles undergo a random walk on a lattice, next to a continuum region. As the atomistic region is random, this technique yields a fixed point iteration of distributions. We run numerical tests to study the long term behavior of such an iteration, and compare the results with the deterministic case. We also discuss a probability transition matrix approach, in which we assume that the boundary conditions at each iterations follow a Markov chain.
Open Access
An analysis of domain decomposition methods using deal.II
(Colorado State University. Libraries, 2021) Rigsby, Christina, author; Tavener, Simon, advisor; Bangerth, Wolfgang, committee member; Heyliger, Paul, committee member; Liu, Jiangguo, committee member
Iterative solvers have attracted significant attention since the mid-20th century as the computational problems of interest have grown to a size beyond which direct methods are viable. Projection methods, and the two classical iterative schemes, Jacobi and Gauss-Seidel, provide a framework in which many other methods may be understood. Parallel methods or Jacobi-like methods are particularly attractive as Moore's Law and computer architectures transition towards multiple cores on a chip. We implement and explore two such methods, the multiplicative and restricted additive Schwarz algorithms for overlapping domain decomposition. We implement these in deal.II software, which is written in C++ and uses the finite element method. Finally, we point out areas for potential improvement in the implementation and present a possible extension of this work to an agent-based modeling prototype currently being developed by the Air Force Research Laboratory's Autonomy Capability Team (ACT3).
Open Access
An analysis of factors affecting student success in Math 160 calculus for physical scientists I
(Colorado State University. Libraries, 2009) Reinholz, Daniel Lee, author; Klopfenstein, Kenneth F., advisor; Gloeckner, Gene William, 1950-, committee member; Hulpke, Alexander, committee member
The average success rate in MATH 160 Calculus for Physical Scientists I at Colorado State University has been near 60% for at least the past three years. Weak pre-calculus skills are often cited as one of the primary reasons students do not succeed in calculus. To investigate this conjecture we included the ALEKS Preparation for Calculus instructional software as a required component of MATH 160. Despite a perceived decrease in the number of algebra-related questions asked by students, we found no improvement in success rates. We also performed an analysis of other factors in relation to success, such as ACT scores and whether or not students had prior calculus experience. As a result of our investigations we conjecture that difficulty with conceptual thinking is a more significant factor than lack of mostly mechanical pre-calculus skills contributing to non-success in MATH 160.
Open Access
An implicit method for water wave problems
(Colorado State University. Libraries, 1983) Aston, Martha B., author; Thomas, J. W. (James William), 1941-, advisor; Zachmann, David W., committee member; Schubert, Wayne H., committee member
This paper presents an implicit scheme for numerically simulating fluid flow in the presence of a free surface. The scheme couples numerical generation of a boundary-fitted coordinate system with an efficient solution of the finite difference equations. The method solves the two dimensional Navier-Stokes equations by applying an implicit backward in time difference scheme which is linearized by Taylor series expansion about the known time level to produce a system of linear difference equations. The difference equations are solved by an Alternating-Direction-Implicit procedure which defines a sequence of one dimensional block tridiagonal matrix equations. A standard block elimination scheme solves the one dimensional equations. For each time step, solutions for all equations are calculated simultaneously and noniteratively. Preliminary solutions of free surface fluid flow in an open channel are presented. These solutions are examined to define initial stability criteria for the numerical scheme.
Open Access
An integrated mathematics/science activity for secondary students: development, implementation, and student feedback
(Colorado State University. Libraries, 2016) Gentry, Abigail Rose, author; Pilgrim, Mary, advisor; Shipman, Patrick, committee member; Gloeckner, Gene, committee member
Mathematics teachers are often challenged by their students to give reasoning for why learning mathematics is necessary. An approach to address this question is to show students the value in learning mathematics by enlightening them on the connections that mathematics has with other disciplines and the real-world applications of mathematics. Integration is a method of teaching that can be used to give students insight as to how mathematics is useful in a variety of different fields. In addition to engaging students with relevant curriculum, leading students to discover the connections between mathematics and science (among other fields) is helpful in showing students why learning mathematics is valuable. This thesis reports on my experiences in developing and implementing an integrated mathematics/science activity in a STEM Technology class at a local high school as well as discusses student feedback about the activity, about their interdisciplinary STEM Technology class, and about the integration of mathematics and science in the classroom.