Physical Sciences and Mathematics Commons^™

May Graduation, Samuel Coskey

Mathematics Faculty Publications and Presentations

Here I narrate the story of the last few days of my graduate program in mathematics. After the completion of the thesis and the delivery of the defense, several twists and turns await in the hours and even minutes before the last deadline.

New Jump Operators On Equivalence Relations, John D. Clemens, Samuel Coskey

Mathematics Faculty Publications and Presentations

We introduce a new family of jump operators on Borel equivalence relations; specifically, for each countable group Γ we introduce the Γ-jump. We study the elementary properties of the Γ-jumps and compare them with other previously studied jump operators. One of our main results is to establish that for many groups Γ, the Γ-jump is proper in the sense that for any Borel equivalence relation E the Γ-jump of E is strictly higher than E in the Borel reducibility hierarchy. On the other hand, there are examples of groups Γ for which the Γ-jump is not proper. To establish properness, …

Meshfree Methods For Pdes On Surfaces, Andrew Michael Jones

Boise State University Theses and Dissertations

This dissertation focuses on meshfree methods for solving surface partial differential equations (PDEs). These PDEs arise in many areas of science and engineering where they are used to model phenomena ranging from atmospheric dynamics on earth to chemical signaling on cell membranes. Meshfree methods have been shown to be effective for solving surface PDEs and are attractive alternatives to mesh-based methods such as finite differences/elements since they do not require a mesh and can be used for surfaces represented only by a point cloud. The dissertation is subdivided into two papers and software.

In the first paper, we examine the …

Implicit Surface Reconstruction With A Curl-Free Radial Basis Function Partition Of Unity Method, Kathryn P. Drake, Edward J. Fuselier, Grady B. Wright

Mathematics Faculty Publications and Presentations

Surface reconstruction from a set of scattered points, or a point cloud, has many applications ranging from computer graphics to remote sensing. We present a new method for this task that produces an implicit surface (zero-level set) approximation for an oriented point cloud using only information about (approximate) normals to the surface. The technique exploits the fundamental result from vector calculus that the normals to an implicit surface are curl-free. By using curl-free radial basis function (RBF) interpolation of the normals, we can extract a potential for the vector field whose zero-level surface approximates the point cloud. We use curl-free …

The Matrix Sortability Problem, Seth Cleaver

Boise State University Theses and Dissertations

Sorting is such a fundamental component of achieving efficiency that a significant body of mathematics is dedicated to the investigation of sorting. Any modern textbook on algorithms contains chapters on sorting.

One approach to arranging a disorganized list of items into an organized list is to successively identify two blocks of contiguous items, and swap the two blocks. In a fundamental paper D.A. Christie showed that a special version of block swapping, in recent times called context directed swapping and abbreviated cds, is the most efficient among block swapping strategies to achieve an organized list of items. The cds …

Waring Rank And Apolarity Of Some Symmetric Polynomials, Max Brian Sullivan

Boise State University Theses and Dissertations

We examine lower bounds for the Waring rank for certain types of symmetric polynomials. The first are Schur polynomials, a symmetric polynomial indexed by integer partitions. We prove some results about the Waring rank of certain types of Schur polynomials, based on their integer partition. We also make some observations about the Waring rank in general for Schur polynomials, based on the shape of their Semistandard Young Tableaux. The second type of polynomials we refer to as a Power of a Fermat-type polynomial, or a PFT polynomial. This is a Fermat type (or power sum) polynomial over n variables with …

Sinusoidal Projection For 360° Image Compression And Triangular Discrete Cosine Transform Impact In The Jpeg Pipeline, Iker Vazquez Lopez

Boise State University Theses and Dissertations

The equirectangular projection is commonly used to store and transmit 360' images. However, using the equirectangular projection to store and transmit 360° images is not efficient due to its natural topographic redundancy. To generate the 360° image, captured pixels that form a spherical point cloud in the 3D space are projected onto a 2D plane using the equirectangular projection; generating redundant pixels in the process. These extra pixels in the image add extra memory requirements that have low impact in the final image quality. This dissertation presents results of research into the compression of 360° spherical imagery. It examines and …

Tcr-L: An Analysis Tool For Evaluating The Association Between The T-Cell Receptor Repertoire And Clinical Phenotypes, Meiling Liu, Juna Goo, Yang Liu, Wei Sun, Michael C. Wu, Li Hsu, Qianchuan He

Mathematics Faculty Publications and Presentations

Background: T cell receptors (TCRs) play critical roles in adaptive immune responses, and recent advances in genome technology have made it possible to examine the T cell receptor (TCR) repertoire at the individual sequence level. The analysis of the TCR repertoire with respect to clinical phenotypes can yield novel insights into the etiology and progression of immune-mediated diseases. However, methods for association analysis of the TCR repertoire have not been well developed.

Methods: We introduce an analysis tool, TCR-L, for evaluating the association between the TCR repertoire and disease outcomes. Our approach is developed under a mixed effect modeling, where …

Relative Primeness And Borel Partition Properties For Equivalence Relations, John D. Clemens

Mathematics Faculty Publications and Presentations

We introduce a notion of relative primeness for equivalence relations, strengthening the notion of non-reducibility, and show for many standard benchmark equivalence relations that non-reducibility may be strengthened to relative primeness. We introduce several analogues of cardinal properties for Borel equivalence relations, including the notion of a prime equivalence relation and Borel partition properties on quotient spaces. In particular, we introduce a notion of Borel weak compactness, and characterize partition properties for the equivalence relations ��₂ and ��₁. We also discuss dichotomies related to primeness, and see that many natural questions related to Borel reducibility of equivalence …

Joint Full-Waveform Ground-Penetrating Radar And Electrical Resistivity Inversion Applied To Field Data Acquired On The Surface, Diego Domenzain, John Bradford, Jodi Mead

Mathematics Faculty Publications and Presentations

We exploit the different but complementary data sensitivities of ground-penetrating radar (GPR) and electrical resistivity (ER) by applying a multiphysics, multiparameter, simultaneous 2.5D joint inversion without invoking petrophysical relationships. Our method joins full-waveform inversion (FWI) GPR with adjoint derived ER sensitivities on the same computational domain. We incorporate a stable source estimation routine into the FWI-GPR. We apply our method in a controlled alluvial aquifer using only surface-acquired data. The site exhibits a shallow groundwater boundary and unconsolidated heterogeneous alluvial deposits. We compare our recovered parameters to individual FWI-GPR and ER results, and we compare them to log measurements of …

An Efficient High-Order Meshless Method For Advection-Diffusion Equations On Time-Varying Irregular Domains, Varun Shankar, Grady B. Wright, Aaron L. Fogelson

Mathematics Faculty Publications and Presentations

We present a high-order radial basis function finite difference (RBF-FD) framework for the solution of advection-diffusion equations on time-varying domains. Our framework is based on a generalization of the recently developed Overlapped RBF-FD method that utilizes a novel automatic procedure for computing RBF-FD weights on stencils in variable-sized regions around stencil centers. This procedure eliminates the overlap parameter δ, thereby enabling tuning-free assembly of RBF-FD differentiation matrices on moving domains. In addition, our framework utilizes a simple and efficient procedure for updating differentiation matrices on moving domains tiled by node sets of time-varying cardinality. Finally, advection-diffusion in time-varying domains …

Homflypt Skein Theory, String Topology And 2-Categories, Uwe Kaiser

Mathematics Faculty Publications and Presentations

We show that relations in Homflypt type skein theory of an oriented 3-manifold M are induced from a 2-groupoid defined from the fundamental 2-groupoid of a space of singular links M. The module relations are defined by homomorphisms related to string topology. They appear from a representation of the groupoid into free modules on a set of model objects. The construction on the fundamental 2-groupoid is defined by the singularity stratification and relates Vassiliev and skein theory. Several explicit properties are discussed, and some implications for skein modules are derived.

Lag Time Between State-Level Policy Interventions And Change Points In Covid-19 Outcomes In The United States, Jaechoul Lee

Mathematics Faculty Publications and Presentations

State-level policy interventions have been critical in managing the spread of the new coronavirus. Here, we study the lag time between policy interventions and change in COVID-19 outcome trajectory in the United States. We develop a stepwise drifts random walk model to account for non-stationarity and strong temporal correlation and subsequently apply a change-point detection algorithm to estimate the number and times of change points in the COVID-19 outcome data. Furthermore, we harmonize data on the estimated change points with non-pharmaceutical interventions adopted by each state of the United States, which provides us insights regarding the lag time between the …

Tukey Morphisms Between Finite Binary Relations, Rhett Barton

Boise State University Theses and Dissertations

Let A = (A₋, A₊, A) and B = (B₋, B₊, B) be relations. A morphism is a pair of maps φ₋ : B₋ → A₋ and φ₊ : A₊ → B₊ such that for all b ∈ B₋ and a ∈ A₊, φ₋(b)Aa ⟹ bBφ₊(a). We study the existence of morphisms between finite relations. The ultimate goal is to identify the conditions under which morphisms exist. In this thesis …

Efficient Inversion Of 2.5d Electrical Resistivity Data Using The Discrete Adjoint Method, Diego Domenzain, John Bradford, Jodi Mead

Mathematics Faculty Publications and Presentations

We have developed a memory and operation-count efficient 2.5D inversion algorithm of electrical resistivity (ER) data that can handle fine discretization domains imposed by other geophysical (e.g, ground penetrating radar or seismic) data. Due to numerical stability criteria and available computational memory, joint inversion of different types of geophysical data can impose different grid discretization constraints on the model parameters. Our algorithm enables the ER data sensitivities to be directly joined with other geophysical data without the need of interpolating or coarsening the discretization. We have used the adjoint method directly in the discretized Maxwell’s steady state equation to compute …

Exploring The Beginnings Of Algebraic K-Theory, Sarah Schott

Boise State University Theses and Dissertations

According to Atiyah, K-theory is that part of linear algebra that studies additive or abelian properties (e.g. the determinant). Because linear algebra, and its extensions to linear analysis, is ubiquitous in mathematics, K-theory has turned out to be useful and relevant in most branches of mathematics. Let R be a ring. One defines K₀(R) as the free abelian group whose basis are the finitely generated projective R-modules with the added relation P ⊕ Q = P + Q. The purpose of this thesis is to study simple settings of the K-theory for rings and …

Zariski Geometries And Quantum Mechanics, Milan Zanussi

Boise State University Theses and Dissertations

Model theory is the study of mathematical structures in terms of the logical relationships they define between their constituent objects. The logical relationships defined by these structures can be used to define topologies on the underlying sets. These topological structures will serve as a generalization of the notion of the Zariski topology from classical algebraic geometry. We will adapt properties and theorems from classical algebraic geometry to our topological structure setting. We will isolate a specific class of structures, called Zariski geometries, and demonstrate the main classification theorem of such structures. We will construct some Zariski structures where the classification …

A Partition Of Unity Method For Divergence-Free Or Curl-Free Radial Basis Function Approximation, Kathryn P. Drake, Edward J. Fuselier, Grady B. Wright

Mathematics Faculty Publications and Presentations

Divergence-free (div-free) and curl-free vector fields are pervasive in many areas of science and engineering, from fluid dynamics to electromagnetism. A common problem that arises in applications is that of constructing smooth approximants to these vector fields and/or their potentials based only on discrete samples. Additionally, it is often necessary that the vector approximants preserve the div-free or curl-free properties of the field to maintain certain physical constraints. Div/curl-free radial basis functions (RBFs) are a particularly good choice for this application as they are meshfree and analytically satisfy the div-free or curl-free property. However, this method can be computationally expensive …

Economics And Game Theory, Jeremiah Patrick Prenn

Mathematics Senior Showcase 2020

Game theory is one of the major fields of mathematics. Game theory is the study of how games, their players, and players’ strategies are defined, and how the games might play out. The outcomes of games are ultimately based on decisions, much like in the science of economics. Economics analyzes how scarce resources are to be allocated to suit unlimited needs. Every decision has an economic cost, and every decision has a utility value (utility being a quantitative measure of usefulness). Economics and game theory go hand in hand: Both analyze the effects of decisions and the rules imposed on …

The History And Application Of Benford's Law, Hunter Clark

Mathematics Senior Showcase 2020

My Poster is on the history and application of Benford’s law. This is a law that states that the leading digit of a set of numbers will be the number 1 approximately 30% of the time. This is a natural phenomenon and what I mean by that is that in order for this law to hold the numbers cannot be assigned. They must be random as in financial statements or logs. This law does not work on sets that are assigned such as time sheets and addresses. You will see in my poster that the original person to discover this …

Using The Chi-Square Test To Analyze Voter Behavior, Bailey Fadden

Mathematics Senior Showcase 2020

We explain the Chi-Square Test and how to use it to analyze voter behavior. Specifically we look at the behavior of U.S citizens and whether or not they voted in the 2016 U.S presidential election, and how this relates to income.

Morse-Code Encoded Eye Blinking As A Source Of Biometric Authentication Via Eeg, Ben Adams, Meghan Edgerton, Gabe Miles, Callum Young

Mathematics Senior Showcase 2020

Brain-Computer Interfaces (BCIs) have historically provided many uses in the medical field, including mobility for individuals with differing levels of paralysis. Present day research is focused around testing the efficacy of such devices on mental diseases such as Alzheimer's, Dementia, and Parkinson's. Leading companies that are spearheading the research of such devices, are looking at BCI's as a tool for solving many of the problems that these diseases produce, with the end goal of generalizing BCIs to appeal to the healthy layperson by providing an additional interface between them and the technological world. If such devices were present in society …

Internal Sorting Methods, Rebekah Marie Bitikofer

Mathematics Senior Showcase 2020

Internal sorting methods are possible when all of the items to be accessed fit in a computer's high-speed internal memory. There are quite a few (Knuth's third volume of The Art of Computer Programming covers 14 in total) but I will go over the four I found to be most versatile and useful. Each algorithm that I cover has a specific benefit that merits its' use in computer science. Some have faster run times (Heapsort), simpler code (Straight Insertion), run with a smaller memory space (Quicksort), or work well with large sets (Radix Sorting). Different sorting tasks lead users to …

Cybersecurity Of The Artificial Pancreas, D. J. Cooke, Andres Guzman, Robert Kinney, Christine Patterson, Josh Stone

Mathematics Senior Showcase 2020

We live in a world of cyber-enabled devices that enhance many aspects of life, including the treatment of diabetes. Type I Diabetes is a chronic autoimmune disorder characterized by destruction of pancreatic cells and subsequent deficiency of insulin - a crucial hormone in regulating blood glucose levels. The development of an Artificial Pancreas System is automating the maintenance of this disease by integrating wireless devices to continuously balance blood glucose levels without patient interaction. An integral part of this system is the Continuous Glucose Monitor (CGM) which wirelessly transmits blood glucose measurements every 5 minutes. CGMs and other Implantable Medical …

Construction Of A First Order Logic Theorem Prover, Luke Philip Tyler

Mathematics Senior Showcase 2020

There are many systems that have been researched in the past on automating the process of theorem proving in first-order logic. This research explores one of these systems, the tableau method. A point of interest within the tableau method is whether or not the method is sound and complete. This research was done in tandem with a computer implementation of the tableau method written in Haskell. The basic design of the implementation was to construct a fair rule for tableau expansion and expand the tableau until it was found to be closed, open, or infinite, thereby proving or disproving of …

Analytic Solutions For Diffusion On Path Graphs And Its Application To The Modeling Of The Evolution Of Electrically Indiscernible Conformational States Of Lysenin, K. Summer Ware

Boise State University Theses and Dissertations

Memory is traditionally thought of as a biological function of the brain. In recent years, however, researchers have found that some stimuli-responsive molecules exhibit memory-like behavior manifested as history-dependent hysteresis in response to external excitations. One example is lysenin, a pore-forming toxin found naturally in the coelomic fluid of the common earthworm Eisenia fetida. When reconstituted into a bilayer lipid membrane, this unassuming toxin undergoes conformational changes in response to applied voltages. However, lysenin is able to "remember" past history by adjusting its conformational state based not only on the amplitude of the stimulus but also on its previous …

A Fast And Accurate Algorithm For Spherical Harmonic Analysis On Healpix Grids With Applications To The Cosmic Microwave Background Radiation, Kathryn P. Drake, Grady B. Wright

Mathematics Faculty Publications and Presentations

The Hierarchical Equal Area isoLatitude Pixelation (HEALPix) scheme is used extensively in astrophysics for data collection and analysis on the sphere. The scheme was originally designed for studying the Cosmic Microwave Background (CMB) radiation, which represents the first light to travel during the early stages of the universe's development and gives the strongest evidence for the Big Bang theory to date. Refined analysis of the CMB angular power spectrum can lead to revolutionary developments in understanding the nature of dark matter and dark energy. In this paper, we present a new method for performing spherical harmonic analysis for HEALPix data, …

Meager Sets, Games And Singular Cardinals, Liljana Babinkostova, Marion Scheepers

Mathematics Faculty Publications and Presentations

We show that a statement concerning the existence of winning strategies of limited memory in an infinite two-person topological game is equivalent to a weak version of the Singular Cardinals Hypothesis.

The Classification Of Countable Models Of Set Theory, John Clemens, Samuel Coskey, Samuel Dworetzky

Mathematics Faculty Publications and Presentations

We study the complexity of the classification problem for countable models of set theory (ZFC). We prove that the classification of arbitrary countable models of ZFC is Borel complete, meaning that it is as complex as it can conceivably be. We then give partial results concerning the classification of countable well‐founded models of ZFC.

What Is The Derivative Of Music?, Thad B. Welch, Cameron H.G. Wright, Michael G. Morrow

Electrical and Computer Engineering Faculty Publications and Presentations

In our continuing effort to prove to students that Signals & Systems is not just another mathematics course taught by the ECE Department, we ask the question, “What is the Derivative of Music?”

The first-order difference (or first-difference) is an incredibly simple algorithm that very accurately approximates the numeric derivative operator, especially for oversampled signals. Its inverse also accurately approximates the numeric integration operator, but not without numeric difficulty.

Given a real-time demonstration using winDSK8, we can now show students that these mathematical operators provide powerful signal processing filtering tools for real-world signals.

During this ASEE session, we will include …