Mathematics Commons^™

Neural Networks For Threshold Dynamics Reconstruction, Elisa Negrini, Almanzo Jiahe Gao, Abigail Bowering, Wei Zhu, Luca Capogna

Mathematics Sciences: Faculty Publications

We introduce two convolutional neural network (CNN) architec- tures, inspired by the Merriman-Bence-Osher (MBO) algorithm and by cellular automatons, to model and learn threshold dynamics for front evolution from video data. The first model, termed the (single-dynamics) MBO network, learns a specific kernel and threshold for each input video without adapting to new dynamics, while the second, a meta-learning MBO network, generalizes across diverse threshold dynamics by adapting its parameters per input. Both models are evaluated on synthetic and real-world videos (ice melting and fire front propagation), with performance metrics indicating effective reconstruction and extrapolation of evolving boundaries, even under …

A Filtration On The Cohomology Rings Of Regular Nilpotent Hessenberg Varieties, Megumi Harada, Tatsuya Horiguchi, Satoshi Murai, Martha Precup, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Let n be a positive integer. The main result of this manuscript is a construction of a filtration on the cohomology ring of a regular nilpotent Hessenberg variety in GL(n, C) / B such that its associated graded ring has graded pieces (i.e., homogeneous components) isomorphic to rings which are related to the cohomology rings of Hessenberg varieties in GL(n- 1 , C) / B, showing the inductive nature of these rings. In previous work, the first two authors, together with Abe and Masuda, gave an explicit presentation of these cohomology rings in terms of generators and relations. We introduce …

Graphs Admitting Only Constant Splines, Katie Anders, Alissa S. Crans, Briana Foster-Greenwood, Blake Mellor, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

We study generalized graph splines, introduced by Gilbert, Tymoczko, and Viel (2016). For a large class of rings, we characterize the graphs that only admit constant splines. To do this, we prove that if a graph has a particular type of cutset (e.g., a bridge), then the space of splines naturally decomposes as a certain direct sum of submodules. As an application, we use these results to describe splines on a triangulation studied by Zhou and Lai, but over a different ring than they used.

Mathematics Cognitive And Content Abilities Across The Vincentian Student Population, Brendalee Rorena Cato

Walden Dissertations and Doctoral Studies

Mathematics achievement is a key component of student overall academic achievement. However, many students from Saint Vincent and the Grenadines (Vincentian students) continue to perform poorly on the regional Caribbean Secondary Examination Certificate (CSEC) mathematics examination. This poor mathematics performance is a concern for education stakeholders. The purpose of this quantitative, nonexperimental study was to explore the extent to which the CSEC mathematics scores of high-scoring Vincentian students versus low-scoring Vincentian students in the cognitive domains of knowledge, comprehension, and reasoning differ across the content domains of algebra; geometry; measurement; statistics; and relations, functions, and graphs (RFG). The theoretical foundation …

Positivity-Preserving Segregate-Flux Method For Infiltration Dynamics In Tumor Growth Models, Gilbert Danso Acheampong

Open Access Theses & Dissertations

We study the positivity preserving property and an incompressibility condition in a recently proposed tumor growth model as well as its numerical simulations. In this model, the biological process is described by a free-boundary problem of hyperbolic equations that govern the in-tumor motion of cancer cells and the infiltration of immune cells. Particularly, due to an assumption that cells take constant volume (the incompressibility condition), the tumor growth/shrinkage is closely correlated to the magnitude of infiltration of immune cells into the tumor.

Despite the fact that previous simulation results largely reproduced experimental data, there remain unanswered questions that are crucial …

Self-Similar Models: How Close The Diffusion Entropy Analysis And The Detrended Fluctuation Analysis Are From Other Models, William Kubin

Open Access Theses & Dissertations

Financial and seismic data, like many other high frequency data are known to exhibit memory effects. In this research, we apply the concepts of L ́evy processes, Diffusion Entropy Analysis (DEA) and the Detrended Fluctuation Analysis (DFA) to examine long-range persistence (long memory) behavior in time series data. L ́evy processes describe long memory effects. In other words, L ́evy process (where the increments are independent and follow the L ́evy distribution) is self-similar. We examine the relationship between the L ́evy parameter (α) characterizing the data and the scaling exponent of DEA (δ) and that of DFA (H) characterizing …

Free Semigroups And Identites For A Class Of Monoids, Enrique Salcido

Open Access Theses & Dissertations

The study of words as a mathematical object is a deep and rich field of study. Algebra, Combinatorics, Theoretical Computer Science etc., are major disciplines, which are fully using this study. Combinatorial properties (via Codes, Free Hulls, Infinite Words), and algebraic properties of words are presented in this Thesis. The free semigroup on a set (alphabet) X and finite presentation of semigroups have a central place in the algebraic study of words. The last part of the Thesis is devoted to the study of identities in the alphabet X = {x,y} for a class of monoids. The characterization of such …

Laplacian Spectra Of Kneser-Like Bipartite Graphs, Cesar Iram Vazquez

Open Access Theses & Dissertations

Given a,b ∈N such that a > b we define a Kneser-like bipartite graph G(a,b), whose two bipartite sets of vertices represent the a-subsets and b-subsets of S = {1,...,a + b + 1}, and whose edges are pairs of vertices X and Y such that X ∩Y = ∅. We prove that the eigenvalues of the Laplacian matrix of graphs G(a,1) are all nonnegative integers. In fact, we describe these eigenvalues, and their respective multiplicities.

Extrinsic Ricci Flow On Surfaces Of Revolution, Vincent E. Coll, Jeff Dodd, David L. Johnson

Research, Publications & Creative Work

An extrinsic representation of a Ricci flow on a differentiable n-manifold M is a family of submanifolds S(t), each smoothly embedded in Rn+k, evolving as a function of time t such that the metrics induced on the submanifolds S(t) by the ambient Euclidean metric yield the Ricci flow on M. When does such a representation exist? We formulate this question precisely and describe a new, comprehensive way of addressing it for surfaces of revolution in R3. Our approach is to build the desired embedded surfaces of revolution S(t) in R3 into the flow at the outset by rewriting the Ricci …

A Mathematical Analysis Of The Game Of Santorini, Carson Clyde Geissler

Senior Independent Study Theses

Santorini is a two player combinatorial board game. Santorini bears resemblance to the graph theory game of Geography, a game of moving and deleting vertices on a graph. We explore Santorini with game theory, complexity theory, and artificial intelligence. We present David Lichtenstein’s proof that Geography is PSPACE-hard and adapt the proof for generalized forms of Santorini. Last, we discuss the development of an AI built for a software implementation of Santorini and present a number of improvements to that AI.

Effects Of Aperiodicity And Frustration On The Magnetic Properties Of Artificial Quasicrystals, Barry Farmer

Theses and Dissertations--Physics and Astronomy

Quasicrystals have been shown to exhibit physical properties that are dramatically different from their periodic counterparts. A limited number of magnetic quasicrystals have been fabricated and measured, and they do not exhibit long-range magnetic order, which is in direct conflict with simulations that indicate such a state should be accessible. This dissertation adopts a metamaterials approach in which artificial quasicrystals are fabricated and studied with the specific goal of identifying how aperiodicity affects magnetic long-range order. Electron beam lithography techniques were used to pattern magnetic thin films into two types of aperiodic tilings, the Penrose P2, and Ammann-Beenker tilings. SQUID …

Statistical Analysis Of Demographic Effects On Insurance Coverage Of Perinatal And Neonatal Morbidity, Madeline Durbin

Undergraduate Honors Thesis Projects

In the United States of America, Ohio has one of the worst neonatal and perinatal death rates. Within Ohio, Montgomery County has an above average neonatal and perinatal death rate. This statistic can be lowered if more women in Montgomery County have health insurance. They would be more likely to seek out prenatal health care, since they would no longer have to pay as much money out-of-pocket. This would allow medical professionals to be able to diagnose and treat any potential issues in the mother or child earlier. Having health insurance would also prevent mothers-to-be from seeking out other potentially …

Invariance And Invertibility In Deep Neural Networks, Han Zhang

Theses and Dissertations

Machine learning is concerned with computer systems that learn from data instead of being explicitly programmed to solve a particular task. One of the main approaches behind recent advances in machine learning involves neural networks with a large number of layers, often referred to as deep learning. In this dissertation, we study how to equip deep neural networks with two useful properties: invariance and invertibility. The first part of our work is focused on constructing neural networks that are invariant to certain transformations in the input, that is, some outputs of the network stay the same even if the input …

Biasing Medial Axis Rapidly-Exploring Random Trees With Safe Hyperspheres, David Qin

Honors Theses

Motion planning is a challenging and widely researched problem in robotics. Motion planning algorithms aim to not only nd unobstructed paths, but also to construct paths with certain qualities, such as maximally avoiding obstacles to improve path safety. One such solution is a Rapidly-Exploring Random Tree (RRT) variant called Medial Axis RRT that generates the safest possible paths, but does so slowly. This paper introduces a RRT variant called Medial Axis Ball RRT (MABallRRT) that uses the concept of clearance -- a robot's distance from its nearest obstacle -- to efficiently construct a roadmap with safe paths. The safety of …

Fast Medial Axis Sampling For Use In Motion Planning, Hanglin Zhou

Honors Theses

Motion planning is a difficult but important problem in robotics. Research has tended toward approximations and randomized algorithms, like sampling-based planning. Probabilistic RoadMaps (PRMs) are one common sampling-based planning approach, but they lack safety guarantees. One main approach, Medial Axis PRM (MAPRM) addressed this deficiency by generating robot configurations as far away from the obstacles as possible, but it introduced an extensive computational burden. We present two techniques, Medial Axis Bridge and Medial Axis Spherical Step, to reduce the computational cost of sampling in MAPRM and additionally propose recycling previously computed clearance information to reduce the cost of connection in …

Estimating Value-At-Risk Of An Unconventional Portfolio, Elizabeth N. Mejía-Ricart

Honors Theses

Since the 2008 financial crisis, interest rates and bond yields have been low all through the recovery and expansion that followed, and they are still low. As a result, more investors have been attracted to US equities, a space of possibly higher returns. However, these returns come with a potential downside: risk of loss. One of the methods to assess this potential downside is value-at-risk (VaR), which gained momentum in the late 1990s. At the time, the market risk amendment to the 1988 Basle Capital Accord required commercial banks with significant trading activities to put aside capital to cover market …

Computer-Assisted Coloring-Graph Generation And Structural Analysis, Wesley Su

Honors Theses

Graphs are a well studied construction in discrete math, with one of the most common areas of study being graph coloring. The graph coloring problem asks for a color to be assigned to each vertex in a graph such that no two adjacent vertices share a color. An assignment of k colors that meets these criteria is called a k-coloring. The coloring graph Ck(G) is defined as the graph where every vertex represents a valid k-coloring of graph G and edges exist between colorings that di↵er by one vertex. We call graph G the base graph of the k-coloring graph …

Internal Migration Of Foreign-Born In Us: Impacts Of Population Concentration And Risk Aversion, Thin Yee Mon Su

Honors Theses

Internal migration in the US has been declining since the 1990s and research has mostly focused on labor market dynamics and aging population to explain the migration trends. This paper analyzes migration patterns of foreign-born groups in the US from 2000 to 2019. Along with the migration determinants such as education and employment, the paper focuses on population concentration as a factor that shapes foreign-born decisions to relocate in the US. Population concertation is defined to be a measure of how geographically concentrated each foreign-born group is across the US. I find that the likelihood of migrating to another state …

Identities And Inequalities For The Cosine And Sine Functions, Iosif Pinelis

Michigan Tech Publications, Part 1

Identities and inequalities for the cosine and sine functions are obtained.

Stability Of Anisotropic Parabolic Equations Without Boundary Conditions, Huashui Zhan, Zhaosheng Feng

School of Mathematical & Statistical Sciences Faculty Publications

No abstract provided.

Cartan’S Approach To Second Order Ordinary Differential Equations, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

In his work on projective connections, Cartan discusses his theory of second order differential equations. It is the aim here to look at how a normal projective connection can be constructed and how it relates to the geometry of a single second order differential equation. The calculations are presented in some detail in order to highlight the use of gauge conditions

Surviving Mathematics, Nathalie M. Luna

School of Mathematical & Statistical Sciences Faculty Publications

In this essay written in an informal voice, the author shares the ups and downs of her experience in academia. She shares her motivation to study mathematics, her undergraduate experience in Puerto Rico, and her graduate experience in South Texas.

Hiv-Associated Neurocognitive Disorder (Hand) Biomarker Identification: Significance Analysis Of Microarrays And Two Persuasive Approaches With Random Forest, Hansapani Rodrigo, Bryan Martinez, Roberto De La Garza, Upal Roy

School of Mathematical & Statistical Sciences Faculty Publications

Background: HIV Associated Neurological Disorders (HAND) is relatively common among people with HIV-1 infection, even those taking combined antiretroviral treatment (cART). Genome-wide screening of transcription regulation in brain tissue helps in identifying substantial abnormalities present in patients’ gene transcripts and to discover possible biomarkers for HAND. This study explores the possibility of identifying differentially expressed (DE) genes, which can serve as potential biomarkers to detect HAND. In this study, we have investigated the gene expression levels of three subject groups with different impairment levels of HAND along with a control group in three distinct brain sectors: white matter, frontal cortex, …

On Cohen-Macaulay Hopf Monoids In Species, Jacob A. White

School of Mathematical & Statistical Sciences Faculty Publications

We study Cohen-Macaulay Hopf monoids in the category of species. The goal is to apply techniques from topological combinatorics to the study of polynomial invariants arising from combinatorial Hopf algebras. Given a polynomial invariant arising from a linearized Hopf monoid, we show that under certain conditions it is the Hilbert polynomial of a relative simplicial complex. If the Hopf monoid is Cohen- Macaulay, we give necessary and sufficient conditions for the corresponding relative simplicial complex to be relatively Cohen-Macaulay, which implies that the polynomial has a nonnegative h-vector. We apply our results to the weak and strong chromatic polynomials of …

A Mathematical Model For Visco-Ferromagnetic Materials, Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden

Articles

Visco-ferromagnetic materials represented by non-local constitutive equation are considered in the paper. We use fractional derivatives in order to describe memory and spatial effects. Also, thermodynamic principles are formulated and studied.

Numerical Modeling Of Submicron Particles For Acoustic Concentration In Gaseous Flow, Jizhou Liu, Xiaodong Li, Fang Q. Hu

Mathematics & Statistics Faculty Publications

This paper intends to explore the rationality and feasibility of modeling dispersed submicron particles in air by a kinetic-based method called the unified gas-kinetic scheme (UGKS) and apply it to the simulation of particle concentration under a transverse standing wave. A gas-particle coupling scheme is proposed where the gas phase is modeled by the two-dimensional linearized Euler equations (LEE) and, through the analogous behavior between the rarefied gas molecules and the air-suspended particles, a modified UGKS is adopted to estimate the particle dynamics. The Stokes' drag force and the acoustic radiation force applied on particles are accounted for by introducing …

Classification Models Of Idiopathic Pulmonary Fibrosis Patients, Mohammed Alqawba, Luis R. Rodriguez, Norou Diawara, Rebecca T. Beuschel, Maryann Kaler, Amisha V. Barochia, Stewart J. Levine, Steven D. Nathan, Geraldine Grant

Mathematics & Statistics Faculty Publications

Idiopathic pulmonary fibrosis (IPF) is a chronic and fatal interstitial lung disease with no current cure. Progression of IPF is difficult to predict as the clinical course can be highly variable and range from a rapidly deteriorating state to a relatively stable state, or may be characterized by a slow progressive decline. Therefore, the need for an accurate diagnosis and improved tools for monitoring and managing IPF is of paramount importance, all for understanding the mitochondrial structure and the function played in the IPF. Mitochondrial DNA copy number (MtDCN) has been correlated with mortality in IPF patients and is a …

What's Your Sphericity Index? Rationalizing Surface Area And Volume, John A. Adam

Mathematics & Statistics Faculty Publications

Virginia Standards of Learning include mathematical content related to the surface area and the volume of various geometric objects. In the seventh grade, “Students... solve problems involving volume and surface area” In the eighth grade, “Proportional reasoning is expounded upon as students solve a variety of problems. Students find the volume and surface area of more complex three dimensional figures”. In high school geometry, “The student... use[s] surface area and volume of three-dimensional objects to solve practical problems” (Virginia Department of Education, 2016). The challenge is to find scenarios that are engaging to students and keep them interested in the …

Psu At Clef-2020 Arqmath Track: Unsupervised Re-Ranking Using Pretraining, Shaurya Rohatgi, Jian Wu, C. Lee Giles

Computer Science Faculty Publications

This paper elaborates on our submission to the ARQMath track at CLEF 2020. Our primary run for the main Task-1: Question Answering uses a two-stage retrieval technique in which the first stage is a fusion of traditional BM25 scoring and tf-idf with cosine similarity-based retrieval while the second stage is a finer re-ranking technique using contextualized embeddings. For the re-ranking we use a pre-trained robertabase model (110 million parameters) to make the language model more math-aware. Our approach achieves a higher NDCG0 score than the baseline, while our MAP and P@10 scores are competitive, performing better than the best submission …

Topological Pressure And Fractal Dimensions Of Cookie-Cutter-Like Sets, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

The cookie-cutter-like set is defined as the limit set of a sequence of classical cookie-cutter mappings. For this cookie-cutter set it is shown that the topological pressure function exists, and that the fractal dimensions such as the Hausdorff dimension, the packing dimension and the box-counting dimension are all equal to the unique zero h of the pressure function. Moreover, it is shown that the h-dimensional Hausdorff measure and the h-dimensional packing measure are finite and positive.