Mathematics Commons^™

Birkhoff Summation Of Irrational Rotations: A Surprising Result For The Golden Mean, Heather Moore

University Honors Theses

This thesis presents a surprising result that the difference in a certain sums of constant rotations by the golden mean approaches exactly 1/5. Specifically, we focus on the Birkhoff sums of these rotations, with the number of terms equal to squared Fibonacci numbers. The proof relies on the properties of continued fraction approximants, Vajda's identity and the explicit formula for the Fibonacci numbers.

Understanding Waveguides In Resonance, Pieter Johannes Daniel Vandenberge

Dissertations and Theses

Several important classes of modern optical waveguides, including anti-resonant reflecting and photonic bandgap fibers, make use of geometries that guide energy in low refractive index material, a property that makes them of significant interest in numerous applications, notably including high-power delivery and guidance. These waveguides frequently exhibit resonance phenomena, in which their ability to propagate an input signal is sharply curtailed at particular operating frequencies. In this work we detail new advances in understanding these resonance effects and their implications for numerical modeling of these structures.

Part 1 focuses on the fields of slab waveguides, relatively simple structures for which …

The Negative Stigma Surrounding Mathematics, Marissa A. Greisen

PSU McNair Scholars Online Journal

There is a negative stigma that surrounds mathematics in our education system. It is important to bring notice to this for the benefit of future students. There is a lot of research claiming that math is looked down on, but they do not answer why, or what we can do to fix it. Why is there a greater negative stigma around math and not other subjects? What roles to teachers, parents, and peers play in this stigma? In this article, I created a survey for people to answer questions regarding their opinion on math, who they believe typically does well …

Forest Generating Functions Of Directed Graphs, Ewan Joaquin Kummel

Dissertations and Theses

A spanning forest polynomial is a multivariate generating function whose variables are indexed over both the vertex and edge sets of a given directed graph. In this thesis, we establish a general framework to study spanning forest polynomials, associating them with a generalized Laplacian matrix and studying its properties. We introduce a novel proof of the famous matrix-tree theorem and show how this extends to a parametric generalization of the all-minors matrix-forest theorem. As an application, we derive explicit formulas for the recently introduced class of directed threshold graphs.

We prove that multivariate forest polynomials are, in general, irreducible and …

Mathematics In The Woods: Exploring Low-Income Parents’ Perceptions Of And Involvement In Their Children’S Mathematical Learning, Lulu Sun

Northwest Journal of Teacher Education

This article features data from a three-day mathematics camping trip that offered parents and their children time and space to enjoy non-digital activities and mathematics-building tasks. Drawing upon data from a larger qualitative study of children and their parents, this article specifically focuses on 10 parents’ perceptions of their children’s mathematics learning, problem-solving, and wellbeing. Findings suggest that, although parents are interested in their children’s mathematics learning, they are most concerned with their children’s development of problem-solving abilities and social skills. Moreover, students’ own learning experience is important for their mathematics learning.

Introduction To Mathematical Analysis I - 3rd Edition, Beatriz Lafferriere, Gerardo Lafferriere, Mau Nam Nguyen

PDXOpen: Open Educational Resources

Video lectures explaining problem solving strategies are available

Our goal in this set of lecture notes is to provide students with a strong foundation in mathematical analysis. Such a foundation is crucial for future study of deeper topics of analysis. Students should be familiar with most of the concepts presented here after completing the calculus sequence. However, these concepts will be reinforced through rigorous proofs.

The lecture notes contain topics of real analysis usually covered in a 10-week course: the completeness axiom, sequences and convergence, continuity, and differentiation. In addition, the notes include many carefully selected exercises of various levels …

Policing By Proxy: Interrogating Big Tech's Role In Law Enforcement, Claire Elizabeth

University Honors Theses

Predictive policing, sometimes referred to as data-driven or actuarial policing, is a method of policing that uses a risk-based approach to law enforcement. For-profit technology companies market proprietary risk assessment algorithms to law enforcement organizations as tools meant to proactively mitigate crime. Using data collected from a vast array of sources, both personal and public, police are able to "predict" the likelihood of criminal activity in a given area using these algorithms. Proponents claim that risk assessment tools have the potential to fight crime with unbiased accuracy and speed by predicting when, where, and whom to police by relying on …

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( …

The Trace Of T2 Takes No Repeated Values, Liubomir Chiriac, Andrei Jorza

Mathematics and Statistics Faculty Publications and Presentations

We prove that the trace of the Hecke operator T₂" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">T2 acting on the vector space of cusp forms of level one takes no repeated values, except for 0, which only occurs when the space is trivial.

An Introduction To Number Theory, J. J. P. Veerman

PDXOpen: Open Educational Resources

These notes are intended for a graduate course in Number Theory. No prior familiarity with number theory is assumed.

Chapters 1-14 represent almost 3 trimesters of the course. Eventually we intend to publish a full year (3 trimesters) course on number theory. The current content represents courses the author taught in the academic years 2020-2021 and 2021-2022.

It is a work in progress. If you have questions or comments, please contact Peter Veerman (veerman@pdx.edu).

Minimality Of Integer Bar Visibility Graphs, Emily Dehoff

University Honors Theses

A visibility representation is an association between the set of vertices in a graph and a set of objects in the plane such that two objects have an unobstructed, positive-width line of sight between them if and only if their two associated vertices are adjacent. In this paper, we focus on integer bar visibility graphs (IBVGs), which use horizontal line segments with integer endpoints to represent the vertices of a given graph. We present results on the exact widths of IBVGs of paths, cycles, and stars, and lower bounds on trees and general graphs. In our main results, we find …

Linear Nearest Neighbor Flocks With All Distinct Agents, Robert G. Lyons

Dissertations and Theses

This dissertation analyzes the global dynamics of 1-dimensional agent arrays with nearest neighbor linear couplings. The equations of motion are second order linear ODE's with constant coefficients. The novel part of this research is that the couplings are different for each agent. We allow the forces to depend on the relative position and relative velocity (damping terms) of the agents, and the coupling magnitudes differ for each agent. Further, we do not assume that the forces are "Newtonian'" (i.e., the force due to A on B equals minus the force of B on A) as this assumption does not apply …

An Analysis Of Capillary Flow In Finite Length Interior Corners, Samuel Shaw Mohler

Dissertations and Theses

We analyze the mathematical robustness of slow massively parallel interior corner flows in low gravity environments. An interior corner provides a preferential orientation in low gravity environments. This is a luxury usually only found on earth. It also provides a passive pumping mechanism due to geometry of a conduit. The driving force for this flow is a pressure difference due to local surface curvature gradients. An alternative reasoning is that due to the geometrical constraints the interior corner surface energy is unbounded below. This results in the liquid wicking into corners indefinitely. Interior corner flow's main quantity of interest is …

Functional Role Of The N-Terminal Domain In Connexin 46/50 By In Silico Mutagenesis And Molecular Dynamics Simulation, Umair Khan

University Honors Theses

Connexins form intercellular channels known as gap junctions that facilitate diverse physiological roles, from long-range electrical and chemical coupling to nutrient exchange. Recent structural studies on Cx46 and Cx50 have defined a novel and stable open state and implicated the amino-terminal (NT) domain as a major contributor to functional differences between connexin isoforms. This thesis presents two studies which use molecular dynamics simulations with these new structures to provide mechanistic insight into the function and behavior of the NTH in Cx46 and Cx50. In the first, residues in the NTH that differ between Cx46 and Cx50 are swapped between the …

Numerical Techniques And Simulations For Studying Various High Power Optical Fiber Amplifiers, Particularly For Ytterbium (Yb+3), And Thulium (Tm+3) Doped Fibers, Tathagata Goswami

Dissertations and Theses

In this dissertation we present a simplified scalar numerical model, derived from Maxwell's field equations, for the fiber laser amplifier simulations. Maxwell's equations are reduced using a technique called Coupled Mode Theory (CMT).

The reduced model is made more efficient through a new scale model, referred to as an equivalent short fiber, which captures some of the essential characteristics of a longer fiber. The equivalent short fiber can be viewed as a fiber made using artificial (nonphysical) material properties that in some sense compensates for its reduced length. The computations can be accelerated by a factor approximately equal to the …

Spanning Trees Of Complete Graphs And Cycles, Minjin Enkhjargal

University Honors Theses

Spanning trees are typically used to solve least path problems for finding the minimal spanning tree of a graph. Given a number t ≥ 3 what is the least number n = α(t) such that there exists a graph on n vertices having precisely t spanning trees? Specifically, how will the factoring of t with the use of cycles connected by one vertex affect α(t)? Lower and upper bounds of α(t) are graphed by using properties of cycles and complete graphs. The upper bound of α(t) is then improved by constructing a graph of connected cycles {C_p1, C­ …

Group Theory Visualized Through The Rubik's Cube, Ashlyn Okamoto

University Honors Theses

In my thesis, I describe the work done to implement several Group Theory concepts in the context of the Rubik’s cube. A simulation of the cube was constructed using Processing-Java and with help from a YouTube series done by TheCodingTrain. I reflect on the struggles and difficulties that came with creating this program along with the inspiration behind the project. The concepts that are currently implemented at this time are: Identity, Associativity, Order, and Inverses. The functionality of the cube is described as it moves like a regular cube but has extra keypresses that demonstrate the concepts listed. Each concept …

Occam Manual, Martin Zwick

Systems Science Faculty Publications and Presentations

Occam is a Discrete Multivariate Modeling (DMM) tool based on the methodology of Reconstructability Analysis (RA). Its typical usage is for analysis of problems involving large numbers of discrete variables. Models are developed which consist of one or more components, which are then evaluated for their fit and statistical significance. Occam can search the lattice of all possible models, or can do detailed analysis on a specific model.

In Variable-Based Modeling (VBM), model components are collections of variables. In State-Based Modeling (SBM), components identify one or more specific states or substates.

Occam provides a web-based interface, which …

Nonlinear Multigrid Based On Local Spectral Coarsening For Heterogeneous Diffusion Problems, Chak Shing Lee, Francois Hamon, Nicola Castelletto, Panayot S. Vassilevski, Joshua A. White

Mathematics and Statistics Faculty Publications and Presentations

This work develops a nonlinear multigrid method for diffusion problems discretized by cell-centered finite volume methods on general unstructured grids. The multigrid hierarchy is constructed algebraically using aggregation of degrees of freedom and spectral decomposition of reference linear operators associated with the aggregates. For rapid convergence, it is important that the resulting coarse spaces have good approximation properties. In our approach, the approximation quality can be directly improved by including more spectral degrees of freedom in the coarsening process. Further, by exploiting local coarsening and a piecewise-constant approximation when evaluating the nonlinear component, the coarse level problems are assembled and …

A Posteriori Error Estimates For Maxwell's Equations Using Auxiliary Subspace Techniques, Ahmed El Sakori

Dissertations and Theses

The aim of our work is to construct provably efficient and reliable error estimates of discretization error for Nédélec (edge) element discretizations of Maxwell's equations on tetrahedral meshes. Our general approach for estimating the discretization error is to compute an approximate error function by solving an associated problem in an auxiliary space that is chosen so that:

-Efficiency and reliability results for the computed error estimates can be established under reasonable and verifiable assumptions.

-The linear system used to compute the approximate error function has condition number bounded independently of the discretization parameter.

In many applications, it is some functional …

A Posteriori Error Estimates For Elliptic Eigenvalue Problems Using Auxiliary Subspace Techniques, Stefano Giani, Luka Grubišić, Harri Hakula, Jeffrey S. Ovall

Mathematics and Statistics Faculty Publications and Presentations

We propose an a posteriori error estimator for high-order p- or hp-finite element discretizations of selfadjoint linear elliptic eigenvalue problems that is appropriate for estimating the error in the approximation of an eigenvalue cluster and the corresponding invariant subspace. The estimator is based on the computation of approximate error functions in a space that complements the one in which the approximate eigenvectors were computed. These error functions are used to construct estimates of collective measures of error, such as the Hausdorff distance between the true and approximate clusters of eigenvalues, and the subspace gap between the corresponding true and approximate …

Convex Analysis Of Minimal Time And Signed Minimal Time Functions, D. V. Cuong, B. S. Mordukhovich, Mau Nam Nguyen, M. L. Wells

Mathematics and Statistics Faculty Publications and Presentations

In this paper we first consider the class of minimal time functions in the general setting of locally convex topological vector (LCTV) spaces. The results obtained in this framework are based on a novel notion of closedness of target sets with respect to constant dynamics. Then we introduce and investigate a new class of signed minimal time functions, which are generalizations of the signed distance functions. Subdifferential formulas for the signed minimal time and distance functions are obtained under the convexity assumptions on the given data.

Analyzing Network Topology For Ddos Mitigation Using The Abelian Sandpile Model, Bhavana Panchumarthi, Monroe Ame Stephenson

altREU Projects

A Distributed Denial of Service (DDoS) is a cyber attack, which is capable of triggering a cascading failure in the victim network. While DDoS attacks come in different forms, their general goal is to make a network's service unavailable to its users. A common, but risky, countermeasure is to blackhole or null route the source, or the attacked destination. When a server becomes a blackhole, or referred to as the sink in the paper, the data that is assigned to it "disappears" or gets deleted. Our research shows how mathematical modeling can propose an alternative blackholing strategy that could improve …

Joint Lattice Of Reconstructability Analysis And Bayesian Network General Graphs, Marcus Harris, Martin Zwick

Systems Science Faculty Publications and Presentations

This paper integrates the structures considered in Reconstructability Analysis (RA) and those considered in Bayesian Networks (BN) into a joint lattice of probabilistic graphical models. This integration and associated lattice visualizations are done in this paper for four variables, but the approach can easily be expanded to more variables. The work builds on the RA work of Klir (1985), Krippendorff (1986), and Zwick (2001), and the BN work of Pearl (1985, 1987, 1988, 2000), Verma (1990), Heckerman (1994), Chickering (1995), Andersson (1997), and others. The RA four variable lattice and the BN four variable lattice partially overlap: there are ten …

Reconstructability Analysis & Its Occam Implementation, Martin Zwick

Systems Science Faculty Publications and Presentations

This talk will describe Reconstructability Analysis (RA), a probabilistic graphical modeling methodology deriving from the 1960s work of Ross Ashby and developed in the systems community in the 1980s and afterwards. RA, based on information theory and graph theory, resembles and partially overlaps Bayesian networks (BN) and log-linear techniques, but also has some unique capabilities. (A paper explaining the relationship between RA and BN will be given in this special session.) RA is designed for exploratory modeling although it can also be used for confirmatory hypothesis testing. In RA modeling, one either predicts some DV from a set of IVs …

Guided Reinvention As A Context For Investigating Students' Thinking About Mathematical Language And For Supporting Students In Gaining Fluency, Kristen Vroom

Dissertations and Theses

Fluency with mathematical language is important for students' engagement in many disciplinary practices such as defining, conjecturing, and proving; yet, there is growing evidence that mathematical language is challenging for undergraduate students. This dissertation study draws on two design experiments with pairs of students who were supported to encode their mathematical meanings with more formal language. I aimed to investigate the teaching and learning of mathematical language, and particularly the language in statements with multiple quantifiers, by engaging students in this type of activity. In the first paper, I investigated the complex ways in which the students in my study …

Structure Aware Runge–Kutta Time Stepping For Spacetime Tents, Jay Gopalakrishnan, Joachim Schöberl, Christoph Wintersteiger

Mathematics and Statistics Faculty Publications and Presentations

We introduce a new class of Runge–Kutta type methods suitable for time stepping to propagate hyperbolic solutions within tent-shaped spacetime regions. Unlike standard Runge–Kutta methods, the new methods yield expected convergence properties when standard high order spatial (discontinuous Galerkin) discretizations are used. After presenting a derivation of nonstandard order conditions for these methods, we show numerical examples of nonlinear hyperbolic systems to demonstrate the optimal convergence rates. We also report on the discrete stability properties of these methods applied to linear hyperbolic equations.

Convex And Nonconvex Optimization Techniques For Multifacility Location And Clustering, Tuyen Dang Thanh Tran

Dissertations and Theses

This thesis contains contributions in two main areas: calculus rules for generalized differentiation and optimization methods for solving nonsmooth nonconvex problems with applications to multifacility location and clustering. A variational geometric approach is used for developing calculus rules for subgradients and Fenchel conjugates of convex functions that are not necessarily differentiable in locally convex topological and Banach spaces. These calculus rules are useful for further applications to nonsmooth optimization from both theoretical and numerical aspects. Next, we consider optimization methods for solving nonsmooth optimization problems in which the objective functions are not necessarily convex. We particularly focus on the class …

On Dc And Local Dc Functions, Liam Jemison

University Honors Theses

In this project we investigate the class of functions which can be represented by a difference of convex functions, hereafter referred to simply as 'DC' functions. DC functions are of interest in optimization because they allow the use of convex optimization techniques in certain non-convex problems. We present known results about DC and locally DC functions, including detailed proofs of important theorems by Hartman and Vesely.

We also investigate the DCA algorithm for optimizing DC functions and implement it to solve the support vector machine problem.

Laurent Series Expansion And Its Applications, Anna Sobczyk

University Honors Theses

The Laurent expansion is a well-known topic in complex analysis for its application in obtaining residues of complex functions around their singularities. Computing the Laurent series of a function around its singularities turns out to be an efficient way to determine the residue of the function as well as to compute the integral of the function along any closed curves around its singularities. Based on the theory of the Laurent series, this paper provides several working examples where the Laurent series of a function is determined and then used to calculate the integral of the function along any closed curve …