Mathematics Commons^™

Canonical Sequences Of Optimal Quantization For Condensation Measures, Doğan Çömez, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

We consider condensation measures of the form P:=13P∘S−11+13P∘S−12+13ν associated with the system (S,(13,13,13),ν), where S={Si}2i=1 are contractions and ν is a Borel probability measure on R with compact support. Let D(μ) denote the quantization dimension of a measure μ if it exists. In this paper, we study self-similar measures ν satisfying D(ν)>κ, D(ν)κ, the D(P)-dimensional lower and upper quantization coefficients are finite, positive and unequal; and for D(ν)≤κ, the D(P)-dimensional lower quantization coefficient is infinity.

We consider condensation measures of the form P:=13P∘S−11+13P∘S−12+13ν associated with the system (S,(13,13,13),ν), where S={Si}2i=1 are contractions and ν is a Borel probability measure …

Characterization Of A Family Of Rotationally Symmetric Spherical Quadrangulations, Lowell Abrams, Daniel Slilaty

Mathematics and Statistics Faculty Publications

A spherical quadrangulation is an embedding of a graph G in the sphere in which each facial boundary walk has length four. Vertices that are not of degree four in G are called curvature vertices. In this paper we classify all spherical quadrangulations with n-fold rotational symmetry (n ≥ 3) that have minimum degree 3 and the least possible number of curvature vertices, and describe all such spherical quadrangulations in terms of nets of quadrilaterals. The description reveals that such rotationally symmetric quadrangulations necessarily also have a pole-exchanging symmetry.

Mathematical Musings On The External Anatomy Of The Novel Coronavirus Part 2: Chasing After Quasi-Symmetry, Jyotirmoy Sarkar, Mamunur Rashid

Mathematics Faculty Publications

What is the shape of the novel Coronavirus which has turned our world upside down? Even though it looks dull, unattractive, and even disgusting under a microscope, creative artists have attributed to it bright colors, made it look pretty, and depicted it as a thing of beauty. What can a mathematician contribute to this effort? We take a purist’s point of view by imposing on it a quasi-symmetry and then deriving some consequences. In an idealistic world, far removed from reality but still constrained by the rules of mathematics, anyone can enjoy this ethereal beauty of the mind’s creation, beckoning …

Quantization For A Probability Distribution Generated By An Infinite Iterated Function System, Lakshmi Roychowdhury, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

Quantization for probability distributions concerns the best approximation of a d-dimensional probability distribution P by a discrete probability with a given number n of supporting points. In this paper, we have considered a probability measure generated by an infinite iterated function system associated with a probability vector on R. For such a probability measure P , an induction formula to determine the optimal sets of n-means and the nth quantization error for every natural number n is given. In addition, using the induction formula we give some results and observations about the optimal sets of n-means for all n ≥ …

The Adams Spectral Sequence For The Image-Of-J Spectrum, Robert R. Bruner, John Rognes

Mathematics Faculty Research Publications

We show that if we factor the long exact sequence in cohomology of a cofiber sequence of spectra into short exact sequences, then the d_2-differential in the Adams spectral sequence of any one term is related in a precise way to Yoneda composition with the 2-extension given by the complementary terms in the long exact sequence. We use this to give a complete analysis of the Adams spectral sequence for the connective image-of-J spectrum, finishing a calculation that was begun by D. Davis [Bol. Soc. Mat. Mexicana (2) 20 (1975), pp. 6–11].

Data Ethics: An Investigation Of Data, Algorithms, And Practice, Gabrialla S. Cockerell

Honors Projects

This paper encompasses an examination of defective data collection, algorithms, and practices that continue to be cycled through society under the illusion that all information is processed uniformly, and technological innovation consistently parallels societal betterment. However, vulnerable communities, typically the impoverished and racially discriminated, get ensnared in these harmful cycles due to their disadvantages. Their hindrances are reflected in their information due to the interconnectedness of data, such as race being highly correlated to wealth, education, and location. However, their information continues to be analyzed with the same measures as populations who are not significantly affected by racial bias. Not …

Universality And Synchronization In Complex Quadratic Networks (Cqns), Anca R. Radulescu, Danae Evans

Biology and Medicine Through Mathematics Conference

No abstract provided.

Implementation Of A Least Squares Method To A Navier-Stokes Solver, Jada P. Lytch, Taylor Boatwright, Ja'nya Breeden

Rose-Hulman Undergraduate Mathematics Journal

The Navier-Stokes equations are used to model fluid flow. Examples include fluid structure interactions in the heart, climate and weather modeling, and flow simulations in computer gaming and entertainment. The equations date back to the 1800s, but research and development of numerical approximation algorithms continues to be an active area. To numerically solve the Navier-Stokes equations we implement a least squares finite element algorithm based on work by Roland Glowinski and colleagues. We use the deal.II academic library , the C++ language, and the Linux operating system to implement the solver. We investigate convergence rates and apply the least squares …

Gene Drives And The Consequences Of Over-Suppression, Cole Butler

Biology and Medicine Through Mathematics Conference

No abstract provided.

An Even 2-Factor In The Line Graph Of A Cubic Graph, Seungjae Eom, Kenta Ozeki

Theory & Applications of Graphs

An even 2-factor is one such that each cycle is of even length. A 4- regular graph G is 4-edge-colorable if and only if G has two edge-disjoint even 2- factors whose union contains all edges in G. It is known that the line graph of a cubic graph without 3-edge-coloring is not 4-edge-colorable. Hence, we are interested in whether those graphs have an even 2-factor. Bonisoli and Bonvicini proved that the line graph of a connected cubic graph G with an even number of edges has an even 2-factor, if G has a perfect matching [Even cycles and …

Mathematical Model Of Immune-Inflammatory Response In Covid-19 Patients, Quiyana M. Murphy

Biology and Medicine Through Mathematics Conference

No abstract provided.

Estimating Glutamate Transporter Surface Density In Mouse Hippocampal Astrocytes, Anca R. Radulescu, Annalisa Scimemi

Biology and Medicine Through Mathematics Conference

No abstract provided.

Real-Time Interactive Simulation Of Large Bird Flocks: Toward Understanding Murmurations, Maxfield R. Comstock

Biology and Medicine Through Mathematics Conference

No abstract provided.

Optimal Time-Dependent Classification For Diagnostic Testing, Prajakta P. Bedekar, Paul Patrone, Anthony Kearsley

Biology and Medicine Through Mathematics Conference

No abstract provided.

Networks Of Necessity: Simulating Covid-19 Mitigation Strategies For Disabled People And Their Caregivers, Thomas E. Valles, Hannah Shoenhard, Joseph Zinski, Sarah Trick, Mason A. Porter, Michael R. Lindstrom

School of Mathematical & Statistical Sciences Faculty Publications

Abstract

A major strategy to prevent the spread of COVID-19 is the limiting of in-person contacts. However, limiting contacts is impractical or impossible for the many disabled people who do not live in care facilities but still require caregivers to assist them with activities of daily living. We seek to determine which interventions can best prevent infections of disabled people and their caregivers. To accomplish this, we simulate COVID-19 transmission with a compartmental model that includes susceptible, exposed, asymptomatic, symptomatically ill, hospitalized, and removed/recovered individuals. The networks on which we simulate disease spread incorporate heterogeneity in the risk levels of …

Will Reducing Drug Prices Slow Innovation?, Gregory Vaughan, Fred Ledley

Mathematical Sciences Faculty Publications

The pharmaceutical industry has long argued that high drug prices reflect the high cost of innovation and that reducing drug prices would necessarily slow the pipeline of new drugs. These arguments have been bolstered by studies of large pharmaceutical companies showing statistical associations between the projected market size or revenue for pharmaceutical products and research & development (R&D) activity. Our analysis recognizes the increasingly important role of small biopharmaceuticals in drug development , companies that typically have little revenue and negative earnings, but are now responsible for more than 40% of new drug approvals. We examine the relationship between changes …

Equidistant Sets In Spaces Of Bounded Curvature, Logan Scott Fox

Dissertations and Theses

Given a metric space (X,d), and two nonempty subsets A,B ⊆ X, we study the properties of the set of points of equal distance to A and B, which we call the equidistant set E(A,B). In general, the structure of the equidistant set is quite unpredictable, so we look for conditions on the ambient space, as well as the given subsets, which lead to some regularity of the properties of the equidistant set. At a minimum, we will always require that X is path connected (so that E( …

Matrix Factorizations With More Than Two Factors, Tim Tribone

Dissertations - ALL

Let $S$ be a regular local ring and $f$ a non-zero non-invertible element of $S$. In this thesis, we study the notion of a matrix factorization of $f$ with $d\ge 2$ factors, that is, we consider tuples of square matrices $(\phi_1,\phi_2,\dots,\phi_d)$, with entries in $S$, such that their product is $f$ times an identity matrix of the appropriate size. These objects have been studied thoroughly in the case $d=2$ and were originally introduced by Eisenbud in his study of free resolutions of modules over hypersurface rings. Many of the results given in this thesis are extensions of well-known results in …

Applications Of Powerset Operators, Especially To Matroids, Nathan Uricchio

Dissertations - ALL

Let \(\mathcal{V}\) denote a vector space over an arbitrary field with an inner product. For any collection \(\mathcal{S}\) of vectors from \(\mathcal{V}\) the collection of all vectors orthogonal to each vector in \(\mathcal{S}\) is a subspace, denoted as \(\mathcal{S}^{\perp_v}\) and called the \textit{orthogonal complement} of \(\mathcal{S}\). One of the fundamental theorems of vector space theory states that, \((\mathcal{S}^{\perp_v})^{\perp_v}\) is the subspace \textit{spanned} by \(\mathcal{S}\). Thus the ``spanning'' operator on the subsets of a vector space is the square of the ``orthogonal complement'' operator.

In matroid theory, the orthogonal complement of a matroid \(M\) is also well-defined and similarly results in …

Hodge Theoretic Compactification Of Period Maps, Haohua Deng

Arts & Sciences Graduate Student Theses and Dissertations

In this article, we review some aspects regarding Hodge-theoretic completion and boundarybehavior of period maps. First, we recall some classical results on compactification of classical period domains e.g. Baily-Borel, Ash-Mumford-Rapoport-Thai. The works produced by Kato-Usui aims at generalizing Mumford’s toroidal compactification to nonclassical period domain, which depends on construction of a strongly compatible fan. We prove such a fan can not exists universally, but for a single geometric variation of Hodge structure. We proceed such a construction on a 2-parameter geometric variation coming from Hosono-Takagi’s family of Calabi-Yau threefolds of type (1, 2, 2, 1). Moreover, we briefly review the …

Some Problems In Reproducing Kernel Spaces, Christopher Felder

Arts & Sciences Graduate Student Theses and Dissertations

The two chapters of this thesis are comprised of work in the setting of reproducing kernel (Hilbert) spaces. These are Banach (or Hilbert) spaces of functions defined on some set, with the special property that point evaluation, on the underlying set, is bounded. The first chapter deals with the study of inner functions. These functions have a rich history in function and operator theory in the Hardy spaces of the unit disk. The first section of this chapter studies the relationship between generalized inner functions and optimal polynomial approximants. The second section, which is joint work with Trieu Le, deals …

Properties Of Cyclic Functions, Jeet Sampat

Arts & Sciences Graduate Student Theses and Dissertations

For $1 \leq p < \infty$, consider the Hardy space $H^p(\mathbb{D}^n)$ on the unit polydisk. Beurling's theorem characterizes all shift cyclic functions in the Hardy spaces when $n = 1$. Such a theorem is not known to exist in most other analytic function spaces, even in the one variable case. Therefore, it becomes natural to ask what properties these functions satisfy in order to understand them better. The goal of this thesis is to showcase some important properties of cyclic functions in two different settings. 1. Fix $1 \leq p,q < \infty$ and $m, n \in \mathbb{N}$. Let $T : H^p(\mathbb{D}^n) \xrightarrow{} H^q(\mathbb{D}^m)$ be a bounded linear operator. Then $T$ preserves cyclic functions, i.e. $Tf$ is cyclic whenever $f$ is, if and only if $T$ is a weighted composition operator. 2. Let $\mathcal{H}$ be a normalized complete Nevanlinna-Pick space, and let $f, g \in \mathcal{H}$ be such that $fg \in \mathcal{H}$. Then $f$ and $g$ are multiplier cyclic if and only if their product $fg$ is. We also extend $(1)$ to a large class of analytic function spaces that includes the Dirichlet space, and the Drury-Arveson space on the unit ball $\mathbb{B}_n$ among others. Both of these properties generalize all previously known results of this type.

Quantum Curves And Asymptotic Hodge Theory, Soumya Sinha Babu

Arts & Sciences Graduate Student Theses and Dissertations

This dissertation explores a 2015 conjecture of Codesido-Grassi-Marino in topologicalstring theory that relates the enumerative invariants of toric CY 3-folds to the spectra of operators attached to their mirror curves. In the maximally supersymmetric case, our first theorem relates zeroes of the higher normal function associated to an integral K2-class on the mirror curve to the spectra of the operators for curves of genus one, and suggests a new link between analysis and arithmetic geometry. On the other hand in the ’t Hooft limit, [KM, MZ] deduced from the [CGM] conjecture that the limiting values of the local mirror map …

Classical And Quantum Markov Chains Derived From Billiard-Like Systems, Joshua Covey

Arts & Sciences Graduate Student Theses and Dissertations

Random billiards are a class of random dynamical systems related to dynamical billiards. We extend the study of random billiards and their associated Markov chains in two new directions. First, we introduce a new class of billiard-like systems called lensed billiards, which introduce a step potential to the usual billiard set-up, and conduct an exploratory study of random lensed billiards where we are mainly interested in how the newly-introduced potential parameter relates to the spectral gap and set of moments of the Markov operator associated to the random lensed system. Second, we recast the mathematical set-up of random billiards to …

A Nonconforming Finite Element Method For The 2d Vector Laplacian, Mary Barker

Arts & Sciences Graduate Student Theses and Dissertations

The vector Laplacian presents difficulties in finite element ap- proximation. It is well known that for nonconvex domains, H1- conforming approximation spaces form a closed subspace of the solution space H(div; Ω) ∩ H(curl; Ω). Hence H1-conforming approximations will fail to converge. This is problematic as it is highly difficult to construct more general finite dimensional ap- proximation spaces for this space. We will present an extension of a nonconforming method introduced by Brenner et al. The method was originally given for P1-nonconforming spaces in two dimensions. Our extension is given for degree r polynomials, but which agrees with the …

Weighted Estimates For The Bergman And Szegö Projections, Nathan Andrew Wagner

Arts & Sciences Graduate Student Theses and Dissertations

This thesis is a study of various weighted estimates for the Bergman and Szegö projections on domains in several complex variables. The starting point of our analysis is a bounded, pseudoconvex domain D ⊂ Cn. The Bergman and Szegö projections are both orthogonal projections onto spaces of holomorphic functions associated with D. While it is immediate that both of these operators are bounded on L2, it has been a topic of substantial interest to determine their mapping properties on Lp, where the boundary geometry of D plays amajor role. Given a linear operator T acting on measurable functions that is …

Evolution Of Eigenvalues Of A Geometric Operator Under Ricci Flow On A Riemannian Manifold, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

The behavior of the eigenvalues of a geometric operator closely related to the Laplacian under Ricci flow is investigated. These depend on a coupling parameter in the operator as well as an evolution parameter which gives a flow on a compact manifold of finite dimension. The main objective is to study the monotonicity properties of the eigenvalues.

Political Corruption And Corporate Risk-Taking, Hinh Khieu, Nam H. Nguyen, Hieu V. Phan, Jon A. Fulkerson

School of Mathematical & Statistical Sciences Faculty Publications

We use variation in corruption convictions across judicial districts in the US to examine the relationship between political corruption and risk-taking of public firms. Firms headquartered in regions with high levels of political corruption have lower total risk and lower idiosyncratic risk on average. Further analysis shows that corruption tends to encourage firms to pursue risk-decreasing investments, lower the riskiness of their operations, and decrease asset liquidity. While managerial ownership is intended to align the interests of managers and shareholders, the presence of corruption appears to encourage undiversified managers to decrease risk-taking. Our evidence is consistent with agency theory and …

On Two-Player Pebbling, Garth Isaak, Matthew Prudente, Andrea Potylycki, William Fagley, Joseph Marcinik

Communications on Number Theory and Combinatorial Theory

Graph pebbling can be extended to a two-player game on a graph G, called Two-Player Graph Pebbling, with players Mover and Defender. The players each use pebbling moves, the act of removing two pebbles from one vertex and placing one of the pebbles on an adjacent vertex, to win. Mover wins if they can place a pebble on a specified vertex. Defender wins if the specified vertex is pebble-free and there are no more pebbling moves on the vertices of G. The Two-Player Pebbling Number of a graph G, η(G), is the minimum m such that for every arrangement …

A Quantitative Study Of An Online Learning Platform’S Impact On High School Students' Engagement, Academic Achievement, And Student Satisfaction In A Mathematics Class, Mariah Minkkinen

Dissertations, Theses, and Projects

The present study investigated the impact using the online learning platform Pear Deck had on an online high school math class. The study measured student engagement, academic achievement, and students’ overall satisfaction with using the online learning platform. The participants in this study were online Algebra 2 students. The study was conducted during synchronous online lessons using an online learning system. Data was collected from two different live classes. One class used the online learning platform Pear Deck and the other did not. Engagement was measured by charting the number of student responses for each question posed. Students’ academic achievement …