Browsing by Author "Cavalieri, Renzo, committee member"
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Item 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 memberA 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 4Item 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 memberThe 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.Item Open Access Arithmetic properties of curves and Jacobians(Colorado State University. Libraries, 2020) Bisogno, Dean M., author; Pries, Rachel, advisor; Achter, Jeffrey, committee member; Cavalieri, Renzo, committee member; Tavani, Daniele, committee memberThis thesis is about algebraic curves and their Jacobians. The first chapter concerns Abhyankar's Inertia Conjecture which is about the existence of unramified covers of the affine line in positive characteristic with prescribed ramification behavior. The second chapter demonstrates the existence of a curve C for which a particular algebraic cycle, called the Ceresa cycle, is torsion in the Jacobian variety of C. The final chapter is a study of supersingular Hurwitz curves in positive characteristic.Item Open Access Covariance integral invariants of embedded Riemannian manifolds for manifold learning(Colorado State University. Libraries, 2018) Álvarez Vizoso, Javier, author; Peterson, Christopher, advisor; Kirby, Michael, advisor; Bates, Dan, committee member; Cavalieri, Renzo, committee member; Eykholt, Richard, committee memberThis thesis develops an effective theoretical foundation for the integral invariant approach to study submanifold geometry via the statistics of the underlying point-set, i.e., Manifold Learning from covariance analysis. We perform Principal Component Analysis over a domain determined by the intersection of an embedded Riemannian manifold with spheres or cylinders of varying scale in ambient space, in order to generalize to arbitrary dimension the relationship between curvature and the eigenvalue decomposition of covariance matrices. In the case of regular curves in general dimension, the covariance eigenvectors converge to the Frenet-Serret frame and the corresponding eigenvalues have ratios that asymptotically determine the generalized curvatures completely, up to a constant that we determine by proving a recursion relation for a certain sequence of Hankel determinants. For hypersurfaces, the eigenvalue decomposition has series expansion given in terms of the dimension and the principal curvatures, where the eigenvectors converge to the Darboux frame of principal and normal directions. In the most general case of embedded Riemannian manifolds, the eigenvalues and limit eigenvectors of the covariance matrices are found to have asymptotic behavior given in terms of the curvature information encoded by the third fundamental form of the manifold, a classical tensor that we generalize to arbitrary dimension, and which is related to the Weingarten map and Ricci operator. These results provide descriptors at scale for the principal curvatures and, in turn, for the second fundamental form and the Riemann curvature tensor of a submanifold, which can serve to perform multi-scale Geometry Processing and Manifold Learning, making use of the advantages of the integral invariant viewpoint when only a discrete sample of points is available.Item Open Access Finitely generated modules over Noetherian rings: interactions between algebra, geometry, and topology(Colorado State University. Libraries, 2020) Flores, Zachary J., author; Peterson, Christopher, advisor; Duflot, Jeanne, committee member; Cavalieri, Renzo, committee member; Ross, Kathryn, committee memberIn this dissertation, we aim to study finitely generated modules over several different Noetherian rings and from varying perspectives. This work is divided into four main parts: The first part is a study of algebraic K-theory for a certain class of local Noetherian rings; the second discusses extending well-known results on Lefschetz properties for graded complete intersection algebras to a class of graded finite length modules using geometric techniques; the third discusses the structure of various algebraic and geometric invariants attached to the finite length modules from the previous section; and lastly, we discuss the structure of annihilating ideals of classes of hyperplane arrangements in projective space.Item Open Access Group action on neighborhood complexes of Cayley graphs(Colorado State University. Libraries, 2014) Hughes, Justin, author; Hulpke, Alexander, advisor; Peterson, Chris, advisor; Berger, Bruce, committee member; Cavalieri, Renzo, committee member; Wilson, James, committee memberGiven G a group generated by S ≐ {g1, …, gn}, one can construct the Cayley Graph Cayley (G,S). Given a distance set D ⊂ Z≥0 and Cayley (G,S) one can construct a D-neighborhood complex. This neighborhood complex is a simplicial complex to which we can associate a chain complex. The group G acts on this chain complex and this leads to an action on the homology of the chain complex. These group actions decompose into several representations of G. This thesis uses tools from group theory, representation theory, homo-logical algebra, and topology to further our understanding of the interplay between generated groups (i.e. a group together with a set of generators), corresponding representations on their associated D-neighborhood complexes, and the homology of the D-neighborhood complexes.Item Open Access Modular group and modular forms(Colorado State University. Libraries, 2010) Schmidt, Eric, author; Hulpke, Alexander, advisor; Achter, Jeff, committee member; Cavalieri, Renzo, committee member; Harton, John, committee memberWe prove some results about the structure of SL2(Z) and related groups. We define modular forms for this group and develop the basic theory. We then use the theory of lattices to construct examples of modular forms.Item Open Access Number-theoretic properties of the binomial distribution with applications in arithmetic geometry(Colorado State University. Libraries, 2014) Schmidt, Eric, author; Achter, Jeffrey, advisor; Pries, Rachel, committee member; Cavalieri, Renzo, committee member; Bohm, Wim, committee memberAlina Bucur et al. showed that the distribution of the number of points on a smooth projective plane curve of degree d over a finite field of order q is approximated by a particular binomial distribution. We generalize their arguments to obtain a similar theorem concerning hypersurfaces in projective m-space. We briefly describe Bucur and Kedlaya's generalization to complete intersections. We then prove theorems concerning the probability that a binomial distribution yields an integer of various certain properties, such as being prime or being squarefree. Finally, we show how to apply such a theorem, concerning a property P, to yield results concerning the probability that the numbers of points on random complete intersections possess property P.Item Open Access Patterns in dynamics(Colorado State University. Libraries, 2012) Motta, Francis Charles, author; Shipman, Patrick D., advisor; Bradley, R. Mark, committee member; Cavalieri, Renzo, committee member; Dangelmayr, Gerhard, committee memberIn this paper we introduce and explore the idea of persistent homology (PH) and discuss several applications of this computational topology tool beyond its intended purpose. In particular we apply persistence to data generated by dynamical systems. The application of persistent homology to the circle map will lead us to rediscover the well-known result about the distribution of points in the orbit of this ergodic system called the Three Distance Theorem. We then apply PH to data extracted from several models of ion bombardment of a solid surface. This will present us with an opportunity to discuss new ways of interpreting PH data by introducing statistics on its output. Using these statistics we will begin to develop a technique to answer questions of interest to physicists about the degree of ordering present in the topography of a solid surface after ion bombardment. Finally we observe some inherent limitations in PH and, through simple examples, develop techniques to improve the technology. Specically we will implement algorithms to iteratively spread points on a real algebraic variety and demonstrate that the methodology works to improve the signals in the output of PH.Item Open Access Performance-computation tradeoffs in detection and estimation(Colorado State University. Libraries, 2023) Damale, Pranav U., author; Chong, Edwin K. P., advisor; Pezeshki, Ali, committee member; Tjalkens, Ronald B., committee member; Cavalieri, Renzo, committee memberDetection and estimation problems involve challenging tasks that often demand real-time, accurate results. Algorithms able to produce highly accurate results are often computationally expensive or inefficient. Naturally, we need to tailor algorithms to the specific needs of problems to optimally trade off between computation and accuracy. To explore this ever-present tradeoff, this dissertation describes three distinct problems in detection and estimation and our contribution to the decision-making process for choosing the best algorithms for solving these problems. First, we look at tradeoffs involved in designing a low-cost, camera-based autonomous gait acquisition and analysis system for inspecting gait impairments in mice. Specifically, we give a detailed description of our detection and classification algorithms for gait-event detection and gait-parameter extraction. Using the videos acquired in a live-animal study, we validate the performance of our system for assessing recovery in a mouse model of Parkinson's disease. Next, we analyze the tradeoffs involved in designing a modified data association algorithm for tracking multiple objects using measurements of uncertain origins, such as radar detection with false alarms and missed detection. Specifically, we explore the performance of the distance-weighting probabilistic data association approach in conjunction with the loopy-sum product algorithm and, using simulation data, we analyze its performance in terms of tracking accuracy and computation against other state-of-the-art data association methods for tracking multiple targets in clutter. Finally, to address the ill-conditioning of linear minimum mean square error estimation, we develop four approximate Wiener filter formulas that do not directly involve the inverse of the observation covariance matrix. Using real data, we evaluate the performance-complexity tradeoff for our approximated filters. The common underlying theme that connects our solutions to these distinct problems is that our decisions for selecting various parameters in each solution are based on the performance-computation tradeoff. Throughout this dissertation, we employ various methods to handle this tradeoff, such as receiver operating characteristics analysis and line-search procedure. Our analysis is beneficial for choosing the best algorithm to optimally trade off between performance and computation.Item Open Access Quantum Serre duality for quasimaps(Colorado State University. Libraries, 2022) Heath, Levi Nathaniel, author; Shoemaker, Mark, advisor; Cavalieri, Renzo, committee member; Gillespie, Maria, committee member; Gelfand, Martin, committee memberLet X be a smooth variety or orbifold and let Z ⊆ X be a complete intersection defined by a section of a vector bundle E → X. Originally proposed by Givental, quantum Serre duality refers to a precise relationship between the Gromov–Witten invariants of Z and those of the dual vector bundle E∨. In this paper we prove a quantum Serre duality statement for quasimap invariants. In shifting focus to quasimaps, we obtain a comparison which is simpler and which also holds for nonconvex complete intersections. By combining our results with the wall-crossing formula developed by Zhou, we recover a quantum Serre duality statement in Gromov-Witten theory without assuming convexity.Item Open Access Sparse matrix varieties, Daubechies spaces, and good compression regions of Grassmann manifolds(Colorado State University. Libraries, 2024) Collery, Brian, author; Peterson, Chris, advisor; Shonkwiler, Clayton, advisor; Cavalieri, Renzo, committee member; Kirby, Michael, committee member; Pouchet, Louis-Nöel, committee memberThe Grassmann manifold Gr(k, n) is a geometric object whose points parameterize k dimensional subspaces of Rn. The flag manifold is a generalization in that its points parameterize flags of vector spaces in Rn. This thesis concerns applications of the geometry of the Grassmann and flag manifolds, with an emphasis on image compression. As a motivating example, the discrete versions of Daubechies wavelets generate distinguished n-dimensional subspaces of R2n that can be considered as distinguished points on Gr(n, 2n). We show that geodesic paths between "Daubechies points" parameterize families of "good" image compression matrices. Furthermore, we show that these paths lie on a distinguished Schubert cell in the Grassmannian. Inspired by the structure of Daubechies wavelets, we define and explore sparse matrix varieties as a generalization. Keeping in that theme, we are interested in understanding geometric considerations that constrain the "good" compression region of a Grassmann manifold.Item Open Access Symmetric functions, shifted tableaux, and a class of distinct Schur Q-functions(Colorado State University. Libraries, 2022) Salois, Kyle, author; Gillespie, Maria, advisor; Cavalieri, Renzo, committee member; Hulpke, Alexander, committee member; Cooley, Daniel, committee memberThe Schur Q-functions form a basis of the algebra Ω of symmetric functions generated by the odd-degree power sum basis pd, and have ramifications in the projective representations of the symmetric group. So, as with ordinary Schur functions, it is relevant to consider the equality of skew Schur Q-functions Qλ/μ. This has been studied in 2008 by Barekat and van Willigenburg in the case when the shifted skew shape λ/μ is a ribbon. Building on this premise, we examine the case of near-ribbon shapes, formed by adding one box to a ribbon skew shape. We particularly consider frayed ribbons, that is, the near-ribbons whose shifted skew shape is not an ordinary skew shape. We conjecture with evidence that all Schur Q-functions for frayed ribbon shapes are distinct up to antipodal reflection. We prove this conjecture for several infinite families of frayed ribbons, using a new approach via the "lattice walks'' version of the shifted Littlewood-Richardson rule, discovered in 2018 by Gillespie, Levinson, and Purbhoo.Item Open Access The effects of maternal respiratory sinus arrhythmia and behavioral engagement on mother-child physiological coregulation(Colorado State University. Libraries, 2017) Skoranski, Amanda M., author; Lunkenheimer, Erika, advisor; Lucas-Thompson, Rachel, committee member; Cavalieri, Renzo, committee memberParent-child coregulation, observed as the active organization and coordination of parents' and children's behavioral and physiological states, is an important precursor for children's developing self-regulation, but we know little about how individual parent factors shape parent-child coregulation. We examined whether differences in maternal physiology and behavioral engagement were associated with coregulation of mothers' and their 3-year-old children's physiological states over time. We examined coregulation in real time by modeling maternal and child respiratory sinus arrhythmia (RSA) for 47 families across 18 minutes of dyadic interaction using multilevel coupled autoregressive models fitted in Mplus. Maternal basal RSA, maternal teaching, and maternal behavioral disengagement were each entered as between-subjects predictors to determine the extent to which mother-child coregulation was strengthened or weakened by maternal factors. Whereas greater maternal teaching during the mother-child interaction was associated with stronger coregulation in mother and child RSA over time, maternal disengagement was related to weaker coregulation: specifically, there were more-divergent parent and child RSA at higher levels of maternal disengagement. Coregulation in mother-child RSA was also weakened when mothers' basal RSA was higher. Findings contribute to the emerging knowledge base on real-time patterns of parent-child coregulation and suggest a role for parent-child physiological coregulation as a mechanism by which parent factors support or hinder children's developing self-regulation.Item Open Access Three projects in arithmetic geometry: torsion points and curves of low genus(Colorado State University. Libraries, 2019) Camacho-Navarro, Catalina, author; Pries, Rachel, advisor; Achter, Jeffrey, committee member; Cavalieri, Renzo, committee member; Peterson, Chris, committee member; Velasco, Marcela, committee memberThis paper is an exposition of three different projects in arithmetic geometry. All of them consider problems related to smooth curves with low genus and the torsion points of their Jacobians. The first project studies curves over finite fields and two invariants of the p-torsion part of their Jacobians: the a-number (a) and p-rank (f). There are many open questions in the literature about the existence of curves with a certain genus g and given values of a and f. In particular, not much is known when g = 4 and the curve is non-hyperelliptic. This is the case that we focus on here; we collect and analyze statistical data of curves over Fp for p = 3, 5, 7, 11 and their invariants. Then, we study the existence of Cartier points, which are also related to the structure of J[p]. For curves with 0 ≤ a < g, the number of Cartier points is bounded, and it depends on a and f. The second project addresses the problem of computing the endomorphism ring of a supersingular elliptic curve. This question has gained recent interest as the basis of alternative cryptosystems that hope to be resistant to quantum attacks. Our strategy is to generate these endomorphism rings by finding cycles in the l-isogeny graph which correspond to generators of the ring. We were able to find a condition for cycles to be linearly independent and an obstruction for two of them to be generators. Finally, the last chapter considers the Galois representations associated to the n-torsion points of elliptic curves over Q. In concrete, we construct models for the modular curves associated to applicable subgroups of GL₂(Z/nZ) and find the rational points on all of those which result in genus 0 or 1 curves, or prove that they have infinitely many. We also analyze the curves with a hyperelliptic genus 2 model and provably find the rational points on all but seven of them.Item Open Access Why prices matter: terms-of-trade, structural change, and development(Colorado State University. Libraries, 2018) Duvall-Pelham, Alexander, author; Tavani, Daniele, advisor; Vasudevan, Ramaa, advisor; Braunstein, Elissa, committee member; Cavalieri, Renzo, committee memberTo view the abstract, please see the full text of the document.