Department of Mathematics
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Browsing Department of Mathematics by Author "Adams, Henry, committee member"
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Item 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 memberThe 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.Item Open Access Asymptotic enumeration of matrix groups(Colorado State University. Libraries, 2018) Tyburski, Brady A., author; Wilson, James B., advisor; Adams, Henry, committee member; Pries, Rachel, committee member; Wilson, Jesse W., committee memberWe prove that the general linear group GLd(pe) has between pd4e/64-O(d2) and pd4e2·log2p distinct isomorphism types of subgroups. The upper bound is obtained by elementary counting methods, where as the lower bound is found by counting the number of isomorphism types of subgroups of the generalized Heisenberg group. To count these subgroups, we use nuclei of a bilinear map alongside versor products - a division analog of the tensor product.Item Open Access Combinatorial structures of hyperelliptic Hodge integrals(Colorado State University. Libraries, 2021) Afandi, Adam, author; Cavalieri, Renzo, advisor; Shoemaker, Mark, advisor; Adams, Henry, committee member; Prasad, Ashok, committee memberThis dissertation explores the combinatorial structures that underlie hyperelliptic Hodge integrals. In order to compute hyperelliptic Hodge integrals, we use Atiyah-Bott (torus) localization on a stack of stable maps to [P1/Z2] = P1 × BZ2. The dissertation culminates in two results: a closed-form expression for hyperelliptic Hodge integrals with one λ-class insertion, and a structure theorem (polynomiality) for Hodge integrals with an arbitrary number of λ-class insertions.Item Open Access Commutative algebra in the graded category with applications to equivariant cohomology rings(Colorado State University. Libraries, 2018) Blumstein, Mark, author; Duflot, Jeanne, advisor; Adams, Henry, committee member; Bacon, Joel, committee member; Shonkwiler, Clayton, committee memberTo view the abstract, please see the full text of the document.Item Open Access Determining synchronization of certain classes of primitive groups of affine type(Colorado State University. Libraries, 2022) Story, Dustin, author; Hulpke, Alexander, advisor; Adams, Henry, committee member; Buchanan, Norm, committee member; Gillespie, Maria, committee memberThe class of permutation groups includes 2-homogeneous groups, synchronizing groups, and primitive groups. Moreover, 2-homogeneous implies synchronizing, and synchronizing in turn implies primitivity. A complete classification of synchronizing groups remains an open problem. Our search takes place amongst the primitive groups, looking for examples of synchronizing and non-synchronizing. Using a case distinction from Aschbacher classes, our main results are constructive proofs showing that three classes of primitive affine groups are nonsynchronizing.Item Open Access Generalizations of persistent homology(Colorado State University. Libraries, 2021) McCleary, Alexander J., author; Patel, Amit, advisor; Adams, Henry, committee member; Ben Hur, Asa, committee member; Peterson, Chris, committee memberPersistent homology typically starts with a filtered chain complex and produces an invariant called the persistence diagram. This invariant summarizes where holes are born and die in the filtration. In the traditional setting the filtered chain complex is a chain complex of vector spaces filtered over a totally ordered set. There are two natural directions to generalize the persistence diagram: we can consider filtrations of more general chain complexes and filtrations over more general partially ordered sets. In this dissertation we develop both of these generalizations by defining persistence diagrams for chain complexes in an essentially small abelian category filtered over any finite lattice.Item Open Access Hodge and Gelfand theory in Clifford analysis and tomography(Colorado State University. Libraries, 2022) Roberts, Colin, author; Shonkwiler, Clayton, advisor; Adams, Henry, committee member; Bangerth, Wolfgang, committee member; Roberts, Jacob, committee memberThere is an interesting inverse boundary value problem for Riemannian manifolds called the Calderón problem which asks if it is possible to determine a manifold and metric from the Dirichlet-to-Neumann (DN) operator. Work on this problem has been dominated by complex analysis and Hodge theory and Clifford analysis is a natural synthesis of the two. Clifford analysis analyzes multivector fields, their even-graded (spinor) components, and the vector-valued Hodge–Dirac operator whose square is the Laplace–Beltrami operator. Elements in the kernel of the Hodge–Dirac operator are called monogenic and since multivectors are multi-graded, we are able to capture the harmonic fields of Hodge theory and copies of complex holomorphic functions inside the space of monogenic fields simultaneously. We show that the space of multivector fields has a Hodge–Morrey-like decomposition into monogenic fields and the image of the Hodge–Dirac operator. Using the multivector formulation of electromagnetism, we generalize the electric and magnetic DN operators and find that they extract the absolute and relative cohomologies. Furthermore, those operators are the scalar components of the spinor DN operator whose kernel consists of the boundary traces of monogenic fields. We define a higher dimensional version of the Gelfand spectrum called the spinor spectrum which may be used in a higher dimensional version of the boundary control method. For compact regions of Euclidean space, the spinor spectrum is homeomorphic to the region itself. Lastly, we show that the monogenic fields form a sheaf that is locally homeomorphic to the underlying manifold which is a prime candidate for solving the Calderón problem using analytic continuation.Item Open Access Imprimitively generated designs(Colorado State University. Libraries, 2022) Lear, Aaron, author; Betten, Anton, advisor; Adams, Henry, committee member; Nielsen, Aaron, committee memberDesigns are a type of combinatorial object which uniformly cover all pairs in a base set V with subsets of V known as blocks. One important class of designs are those generated by a permutation group G acting on V and single initial block b subset of V. The most atomic examples of these designs would be generated by a primitive G. This thesis focuses on the less atomic case where G is imprimitive. Imprimitive permutation groups can be rearranged into a subset of easily understood groups which are derived from G and generate very symmetrical designs. This creates combinatorial restrictions on which group and block combinations can generate a design, turning a question about the existence of combinatorial objects into one more directly involving group theory. Specifically, the existence of imprimitively generated designs turns into a question about the existence of pair orbits of an appropriate size, for smaller permutation groups. This thesis introduces two restrictions on combinations of G and b which can generate designs, and discusses how they could be used to more efficiently enumerate imprimitively generated designs.Item Open Access Improved stick number upper bounds(Colorado State University. Libraries, 2019) Eddy, Thomas D., author; Shonkwiler, Clayton, advisor; Adams, Henry, committee member; Chitsaz, Hamid, committee memberA stick knot is a mathematical knot formed by a chain of straight line segments. For a knot K, define the stick number of K, denoted stick(K), to be the minimum number of straight edges necessary to form a stick knot which is equivalent to K. Stick number is a knot invariant whose precise value is unknown for the large majority of knots, although theoretical and observed bounds exist. There is a natural correspondence between stick knots and polygons in R3. Previous research has attempted to improve observed stick number upper bounds by computationally generating such polygons and identifying the knots that they form. This thesis presents a new variation on this method which generates equilateral polygons in tight confinement, thereby increasing the incidence of polygons forming complex knots. Our generation strategy is to sample from the space of confined polygons by leveraging the toric symplectic structure of this space. An efficient sampling algorithm based on this structure is described. This method was used to discover the precise stick number of knots 935, 939, 943, 945, and 948. In addition, the best-known stick number upper bounds were improved for 60 other knots with crossing number ten and below.Item Open Access Independence complexes of finite groups(Colorado State University. Libraries, 2021) Pinckney, Casey M., author; Hulpke, Alexander, advisor; Peterson, Chris, advisor; Adams, Henry, committee member; Neilson, James, committee memberUnderstanding generating sets for finite groups has been explored previously via the generating graph of a group, where vertices are group elements and edges are given by pairs of group elements that generate the group. We generalize this idea by considering minimal generating sets (with respect to inclusion) for subgroups of finite groups. These form a simplicial complex, which we call the independence complex. The vertices of the independence complex are nonidentity group elements and the faces of size k correspond to minimal generating sets of size k. We give a complete characterization via constructive algorithms, together with enumeration results, for the independence complexes of cyclic groups whose order is a squarefree product of primes, finite abelian groups whose order is a product of powers of distinct primes, and the nonabelian class of semidirect products Cp1p3…p2n-1 rtimes Cp2p4…p2n where p1,p2,…,p2n are distinct primes with p2i-1 > p2i for all 1 ≤ i ≤ n. In the latter case, we introduce a tool called a combinatorial diagram, which is a multipartite simplicial complex under certain numerical and minimal covering conditions. Combinatorial diagrams seem to be an interesting area of study on their own. We also include GAP and Polymake code which generates the facets of any (small enough) finite group, as well as visualize the independence complexes in small dimensions.Item Open Access k-simplex volume optimizing projection algorithms for high-dimensional data sets(Colorado State University. Libraries, 2021) Stiverson, Shannon J., author; Kirby, Michael, advisor; Peterson, Chris, advisor; Adams, Henry, committee member; Hess, Ann, committee memberMany applications produce data sets that contain hundreds or thousands of features, and consequently sit in very high dimensional space. It is desirable for purposes of analysis to reduce the dimension in a way that preserves certain important properties. Previous work has established conditions necessary for projecting data into lower dimensions while preserving pairwise distances up to some tolerance threshold, and algorithms have been developed to do so optimally. However, although similar criteria for projecting data into lower dimensions while preserving k-simplex volumes has been established, there are currently no algorithms seeking to optimally preserve such embedded volumes. In this work, two new algorithms are developed and tested: one which seeks to optimize the smallest projected k-simplex volume, and another which optimizes the average projected k-simplex volume.Item Open Access Laplacian Eigenmaps for time series analysis(Colorado State University. Libraries, 2020) Rosse, Patrick J., author; Kirby, Michael, advisor; Peterson, Chris, committee member; Adams, Henry, committee member; Anderson, Chuck, committee memberWith "Big Data" becoming more available in our day-to-day lives, it becomes necessary to make meaning of it. We seek to understand the structure of high-dimensional data that we are unable to easily plot. What shape is it? What points are "related" to each other? The primary goal is to simplify our understanding of the data both numerically and visually. First introduced by M. Belkin, and P. Niyogi in 2002, Laplacian Eigenmaps (LE) is a non-linear dimensional reduction tool that relies on the basic assumption that the raw data lies in a low-dimensional manifold in a high-dimensional space. Once constructed, the graph Laplacian is used to compute a low-dimensional representation of the data set that optimally preserves local neighborhood information. In this thesis, we present a detailed analysis of the method, the optimization problem it solves, and we put it to work on various time series data sets. We show that we are able to extract neighborhood features from a collection of time series, which allows us to cluster specific time series based on noticeable signatures within the raw data.Item Open Access Success in Calculus I: implications of students' precalculus content knowledge and their awareness of that knowledge(Colorado State University. Libraries, 2020) Sencindiver, Benjamin D., author; Hagman, Jess E., advisor; Pilgrim, Mary E., advisor; Adams, Henry, committee member; Gloeckner, Gene, committee member; Zarestky, Jill, committee memberHigh failure rates in Calculus I contribute to the course acting a filter, rather than a pump, for STEM disciplines. One often cited source of difficulty for students in Calculus I is their weak precalculus content knowledge. In this three-paper dissertation, I investigate Calculus I students' precalculus content knowledge and their awareness of that knowledge. In the first paper, I describe a methodology for collecting data about Calculus I students' tendency to regulate their precalculus content knowledge and analyze the utility of quantifying self-regulated learning as a means for identifying at-risk students. In the second paper, I focus on two factors (calibration and help-seeking) to investigate the how they correlate with Calculus I students' first exam performance. Results highlight the importance of calibration of precalculus content knowledge both directly on student success and how calibration accuracy mediates the benefits of help-seeking. Quantitative analyses of students' precalculus content knowledge highlight Calculus I students' difficulty with the concept of graph, despite students' high confidence in questions related to graph. In the third paper, I conduct interviews with Calculus I students to examine their conceptions of outputs and differences of outputs of a function in the graphical context to understand nuance in how students understand and reason with graphs. Results highlight that students' understandings of quantities and frames of references in graphs of functions can be varied and stable. Students' understanding of quantities also impacts their understanding of other concepts such as differences of outputs and difference quotient. Results of this dissertation have implications for educators, tutor center leaders, and researchers interested in students' understanding of graph, calibration, and help-seeking.Item Open Access Topological techniques for characterization of patterns in differential equations(Colorado State University. Libraries, 2017) Neville, Rachel A., author; Shipman, Patrick, advisor; Adams, Henry, committee member; Krummel, Amber, committee member; Shonkwiler, Clayton, committee memberComplex data can be challenging to untangle. Recent advances in computing capabilities has allowed for practical application of tools from algebraic topology, which have proven to be useful for qualitative and quantitative analysis of complex data. The primary tool in computational topology is persistent homology. It provides a valuable lens through which to study and characterize complex data arising as orbits of dynamical systems and solutions of PDEs. In some cases, this includes leveraging tools from machine learning to classify data based on topological characteristics. We see a unique pattern arising in the persistence diagram of a class of one-dimensional discrete dynamical systems--even in chaotic parameter regimes, and connect this to the dynamics of the system in Chapter 2. Geometric pattern structure tell us something about the parameters driving the dynamics in the system as is the case for anisotropic Kuramoto-Sivashinsky equation which displays chaotic bubbling. We will see this in Chapters 3 and 4. Defects in pattern-forming systems be detected and tracked and studied to characterize the degree of order in near-hexagonal nanodot structures formed by ion bombardment, which will be developed in Chapter 5.