Mathematics Commons^™

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 …

The Mathematical Foundation Of The Musical Scales And Overtones, Michaela Dubose-Schmitt

Theses and Dissertations

This thesis addresses the question of mathematical involvement in music, a topic long discussed going all the way back to Plato. It details the mathematical construction of the three main tuning systems (Pythagorean, just intonation, and equal temperament), the methods by which they were built and the mathematics that drives them through the lens of a historical perspective. It also briefly touches on the philosophical aspects of the tuning systems and whether their differences affect listeners. It further details the invention of the Fourier Series and their relation to the sound wave to explain the concept of overtones within the …

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 …

A Priori And A Posteriori Error Estimates For The Quad-Curl Eigenvalue Problem, Lixiu Wang, Qian Zhang, Jiguang Sun, Zhimin Zhang

Michigan Tech Publications, Part 1

In this paper, we consider a priori and a posteriori error estimates of the H(curl²)-conforming finite element when solving the quad-curl eigenvalue problem. An a priori estimate of eigenvalues with convergence order 2(s − 1) is obtained if the corresponding eigenvector u ∈ H^{s − 1}(Ω) and ∇ × u ∈ H^s(Ω). For the a posteriori estimate, by analyzing the associated source problem, we obtain lower and upper bounds for the errors of eigenvectors in the energy norm and upper bounds for the errors of eigenvalues. Numerical examples are presented for validation.

The Impact Of Academic Tracking And Mathematics Self-Concept On Mathematics Achievement., Kain M. Schow

Dissertations, Theses, and Projects

ABSTRACT

This study examines the effects of academic tracking, in high school math, on students’ mathematics self-concept (MSC) and how that correlates to students’ mathematics achievement. This study measured students’ MSC through a mathematics self-concept questionnaire and measured mathematics achievement by the students’ latest grade report. Participants included 60 students in grades 10-12 who had been or were currently enrolled in math courses in the researcher’s school district. The data collected will direct the researcher and school administration on the effects of academic tracking on students, allowing for further discussion about continuing tracking in the district.

Sheltered Math Curriculum For Middle School English Learners, Jasmine Ercink

Dissertations, Theses, and Projects

Language barriers have shown a need for differentiation and sheltered instruction in the classroom for English Learners (ELs) to be successful in the United States public school system. This project proposes a mathematics curriculum using SIOP so that both groups of students in the middle school level can increase their proficiency in the mathematics content area as well as experience opportunities for academic and social language development. The purpose of this report is to describe the processes, methods, data, and intent of the mathematics curriculum for these learners. The curriculum acts as an effective intervention to fill gaps in both …

Tiling Rectangles And 2-Deficient Rectangles With L-Pentominoes, Monica Kane

Rose-Hulman Undergraduate Mathematics Journal

We investigate tiling rectangles and 2-deficient rectangles with L-pentominoes. First, we determine exactly when a rectangle can be tiled with L-pentominoes. We then determine locations for pairs of unit squares that can always be removed from an m × n rectangle to produce a tileable 2-deficient rectangle when m ≡ 1 (mod 5), n ≡ 2 (mod 5) and when m ≡ 3 (mod 5), n ≡ 4 (mod 5).

A New Model For Predicting The Drag And Lift Forces Of Turbulent Newtonian Flow On Arbitrarily Shaped Shells On The Seafloor, Carley R. Walker, James V. Lambers, Julian Simeonov

Dissertations

Currently, all forecasts of currents, waves, and seafloor evolution are limited by a lack of fundamental knowledge and the parameterization of small-scale processes at the seafloor-ocean interface. Commonly used Euler-Lagrange models for sediment transport require parameterizations of the drag and lift forces acting on the particles. However, current parameterizations for these forces only work for spherical particles. In this dissertation we propose a new method for predicting the drag and lift forces on arbitrarily shaped objects at arbitrary orientations with respect to the direction of flow that will ultimately provide models for predicting the sediment sorting processes that lead to …

On Isomorphic K-Rational Groups Of Isogenous Elliptic Curves Over Finite Fields, Ben Kuehnert, Geneva Schlafly, Zecheng Yi

Rose-Hulman Undergraduate Mathematics Journal

It is well known that two elliptic curves are isogenous if and only if they have same number of rational points. In fact, isogenous curves can even have isomorphic groups of rational points in certain cases. In this paper, we consolidate all the current literature on this relationship and give a extensive classification of the conditions in which this relationship arises. First we prove two ordinary isogenous elliptic curves have isomorphic groups of rational points when they have the same $j$-invariant. Then, we extend this result to certain isogenous supersingular elliptic curves, namely those with equal $j$-invariant of either 0 …

Theoretical Aspects Of Calculation Of Photocurrent In Structures With Periodically Spaced Heterinclusions, R. Аyukhanov, A. Uteniyazov, S. Farmonov, N. Shernazarov

Karakalpak Scientific Journal

The processes of excitation of photoconductivity in structures with heteroinclusions with a band gap greater than the band gap of the base material of the matrix have been studied. Structures are studied when such macroinclusions are located periodically and are located at distances equal to the size of the inclusions. A technique has been developed and analytical expressions have been derived for calculating the photocurrent and specific photoconductivity in such structures.

Injection Depletion Effect In Semiconductors With Pair Impurity Complexes, A. Yu. Leyderman, A. Uteniyazov, R. Turmanova, M. Dauletbayev

Karakalpak Scientific Journal

The possibility of realizing the effect of injection depletion in a ⁺ -structure made on the basis of a semiconductor with paired donor-acceptor complexes is considered. It is shown that the appearance of this effect is possible under conditions of almost complete compensation of the shallow dopant donor impurity by the charge of the acceptor component of the pair complex. A necessary condition is also a strong asymmetry of the coefficients of capture of holes at the acceptor level in comparison with the coefficient of capture of electrons at the donor level.

Nonlinear Photoelasticity Processes Near Resonances At The Frequency Of Electronic Transitions From Deep Levels To The Conduction Zone, R. Аyukhanov, A. Uteniyazov, S. Farmonov, N. Shernazarov

Karakalpak Scientific Journal

The phenomenon of nonlinear photoelasticity, which consists of the occurrence of nonlinear photoelasticity coefficients comparable in magnitude to linear ones near the electronic resonance between the deep impurity levels and the bottom of the conduction zone, is considered theoretically. Analytical expressions for the Fourier decomposition coefficients of the dielectric permittivity of the crystal in the phase of the sound wave, which are proportional to the photoelasticity coefficients, are obtained. A general criterion for the nonlinear photoelasticity effect in the investigated frequency region is found numerically. It is shown that high deformation potentials of deep levels stimulate the appearance of this …

Combining Cardiac Monitoring With Actigraphy Aids Nocturnal Arousal Detection During Ambulatory Sleep Assessment In Insomnia, Lara Rösler, Glenn Van Der Lande, Jeanne Leerssen, Austin G. Vandegriffe, Oti Lakbila-Kamal, Jessica C. Foster-Dingley, Anne C.W. Albers, Eus J.W. Van Someren

Mathematics and Statistics Faculty Research & Creative Works

Study Objectives: The objective assessment of insomnia has remained difficult. Multisensory devices collecting heart rate (HR) and motion are regarded as the future of ambulatory sleep monitoring. Unfortunately, reports on altered average HR or heart rate variability (HRV) during sleep in insomnia are equivocal. Here, we evaluated whether the objective quantification of insomnia improves by assessing state-related changes in cardiac measures. Methods: We recorded electrocardiography, posture, and actigraphy in 33 people without sleep complaints and 158 patients with mild to severe insomnia over 4 d in their home environment. At the microscale, we investigated whether HR changed with proximity to …