Applied Mathematics Commons^™

(Si10-123) Comparison Between The Homotopy Perturbation Method And Variational Iteration Method For Fuzzy Differential Equations, P. Chandru, B. Radhakrishnan

Applications and Applied Mathematics: An International Journal (AAM)

In this article, the authors discusses the numerical simulations of higher-order differential equations under a fuzzy environment by using Homotopy Perturbation Method and Variational Iteration Method. The fuzzy parameter and variables are represented by triangular fuzzy convex normalized sets. Comparison of the results are obtained by the homotopy perturbation method with those obtained by the variational iteration method. Examples are provided to demonstrate the theory.

A Data Driven Modeling Approach For Store Distributed Load And Trajectory Prediction, Nicholas Peters

Doctoral Dissertations and Master's Theses

The task of achieving successful store separation from aircraft and spacecraft has historically been and continues to be, a critical issue for the aerospace industry. Whether it be from store-on-store wake interactions, store-parent body interactions or free stream turbulence, a failed case of store separation poses a serious risk to aircraft operators. Cases of failed store separation do not simply imply missing an intended target, but also bring the risk of collision with, and destruction of, the parent body vehicle. Given this risk, numerous well-tested procedures have been developed to help analyze store separation within the safe confines of wind …

Approximation By Basis Pursuit: Background And Application To The Construction Of Efficient Spline Approximations, Babita Timalsina

Graduate Student Scholarship

Basis Pursuit was developed primarily as a tool in the field of signal processing, beginning in the mid 1990’s. The idea is to model the behavior of discrete signals using a wide range of functional behaviors and scales and to obtain an accurate and efficient representation of the signal using a minimal number of functions from a large “dictionary” of possible behaviors. The key observation is by formulating the representation as an ℓ1 optimization, the problem can be posed as a linear program so that the optimal solution uses no more than the number of constraints - it must be …

A Mode-Sum Prescription For The Renormalized Stress Energy Tensor On Black Hole Spacetimes, Peter Taylor, Cormac Breen, Adrian Ottewill

Articles

In this paper, we describe an extremely efficient method for computing the renormalized stress-energy tensor of a quantum scalar field in spherically symmetric black hole spacetimes. The method applies to a scalar field with arbitrary field parameters. We demonstrate the utility of the method by computing the renormalized stress-energy tensor for a scalar field in the Schwarzschild black hole spacetime, applying our results to discuss the null energy condition and the semiclassical backreaction.

Classification Of Pixel Tracks To Improve Track Reconstruction From Proton-Proton Collisions, Kebur Fantahun, Jobin Joseph, Halle Purdom, Nibhrat Lohia

SMU Data Science Review

In this paper, machine learning techniques are used to reconstruct particle collision pathways. CERN (Conseil européen pour la recherche nucléaire) uses a massive underground particle collider, called the Large Hadron Collider or LHC, to produce particle collisions at extremely high speeds. There are several layers of detectors in the collider that track the pathways of particles as they collide. The data produced from collisions contains an extraneous amount of background noise, i.e., decays from known particle collisions produce fake signal. Particularly, in the first layer of the detector, the pixel tracker, there is an overwhelming amount of background noise that …

Morton-Ordered Gpu Lattice Boltzmann Cfd Simulations With Application To Blood Flow, Gerald Gallagher, Fergal J. Boyle

Conference Papers

Computational fluid dynamics (CFD) is routinely used for numerically predicting cardiovascular-system medical device fluid flows. Most CFD simulations ignore the suspended cellular phases of blood due to computational constraints, which negatively affects simulation accuracy. A graphics processing unit (GPU) lattice Boltzmann-immersed boundary (LB-IB) CFD software package capable of accurately modelling blood flow is in development by the authors, focusing on the behaviour of plasma and stomatocyte, discocyte and echinocyte red blood cells during flow. Optimised memory ordering and layout schemes yield significant efficiency improvements for LB GPU simulations. In this work, comparisons of row-major-ordered Structure of Arrays (SoA) and Collected …

The Effect Of Habitat Fragmentation On Plant Communities In A Spatially-Implicit Grassland Model, Mika T. Cooney, Benjamin R. Hafner, Shelby E. Johnson, Sean Lee

Rose-Hulman Undergraduate Mathematics Journal

The spatially implicit Tilman-Levins ODE model helps to explain why so many plant species can coexist in grassland communities. This now-classic modeling framework assumes a trade-off between colonization and competition traits and predicts that habitat destruction can lead to long transient declines called ``extinction debts.'' Despite its strengths, the Tilman-Levins model does not explicitly account for landscape scale or the spatial configuration of viable habitat, two factors that may be decisive for population viability. We propose modifications to the model that explicitly capture habitat geometry and the spatial pattern of seed dispersal. The modified model retains implicit space and is …

Permitted Sets And Convex Coding In Nonthreshold Linear Networks, Steven Collazos, Duane Nykamp

Mathematics Publications

Hebbian theory proposes that ensembles of neurons form a basis for neural processing. It is possible to gain insight into the activity patterns of these neural ensembles through a binary analysis, regarding neurons as either active or inactive. The framework of permitted and forbidden sets, introduced by Hahnloser, Seung, and Slotine (2003), is a mathematical model of such a binary analysis: groups of coactive neurons can be permitted or forbidden depending on the network's structure.

In order to widen the applicability of the framework of permitted sets, we extend the permitted set analysis from the original threshold-linear regime. …

Abiogenesis: The Carter Argument Reconsidered, Daniel P. Whitmire

Mathematical Sciences Faculty Publications and Presentations

The observation of life on Earth is commonly believed to be uninformative regarding the probability of abiogenesis on other Earth-like planets. This belief is based on the selection effect of our existence. We necessarily had to find ourselves on a planet where abiogenesis occurred, thus nothing can be inferred about the probability of abiogenesis from this observation alone. This argument was first formalized in a Bayesian framework by Brandon Carter. Though we definitely had to find ourselves on a planet where abiogenesis occurred, I argue here that (1) the Carter conclusion is based on what is known as the ‘Old …

Entropy Analysis Of Sutterby Nanofluid Flow Over A Riga Sheet With Gyrotactic Microorganisms And Cattaneo–Christov Double Diffusion, M. Faizan, F. Ali, K. Loganathan, A. Zaib, C. A. Reddy, Sara I. Abdelsalam

Basic Science Engineering

In this article, a Riga plate is exhibited with an electric magnetization actuator consisting of permanent magnets and electrodes assembled alternatively. This exhibition produces electromagnetic hydrodynamic phenomena over a fluid flow. A new study model is formed with the Sutterby nanofluid flow through the Riga plate, which is crucial to the structure of several industrial and entering advancements, including thermal nuclear reactors, flow metres and nuclear reactor design. This article addresses the entropy analysis of Sutterby nanofluid flow over the Riga plate. The Cattaneo–Christov heat and mass flux were used to examine the behaviour of heat and mass relaxation time. …

Leveraging Subject Matter Expertise To Optimize Machine Learning Techniques For Air And Space Applications, Philip Y. Cho

Theses and Dissertations

We develop new machine learning and statistical methods that are tailored for Air and Space applications through the incorporation of subject matter expertise. In particular, we focus on three separate research thrusts that each represents a different type of subject matter knowledge, modeling approach, and application. In our first thrust, we incorporate knowledge of natural phenomena to design a neural network algorithm for localizing point defects in transmission electron microscopy (TEM) images of crystalline materials. In our second research thrust, we use Bayesian feature selection and regression to analyze the relationship between fighter pilot attributes and flight mishap rates. We …

Stochastic Modeling Of Flows In Membrane Pore Networks, Binan Gu

Dissertations

Membrane filters provide immediate solutions to many urgent problems such as water purification, and effective remedies to pressing environmental concerns such as waste and air treatment. The ubiquity of applications gives rise to a significant amount of research in membrane material selection and structural design to optimize filter efficiency. As physical experiments tend to be costly, numerical simulation and analysis of fluid flow, foulant transport and geometric evolution due to foulant deposition in complex geometries become particularly relevant. In this dissertation, several mathematical modeling and analytical aspects of the industrial membrane filtration process are investigated. A first-principles mathematical model for …

Computing Rational Powers Of Monomial Ideals, Pratik Dongre, Benjamin Drabkin, Josiah Lim, Ethan Partida, Ethan Roy, Dylan Ruff, Alexandra Seceleanu, Tingting Tang

Department of Mathematics: Faculty Publications

This paper concerns fractional powers of monomial ideals. Rational powers of a monomial ideal generalize the integral closure operation as well as recover the family of symbolic powers. They also highlight many interesting connections to the theory of convex polytopes. We provide multiple algorithms for computing the rational powers of a monomial ideal. We also introduce a mild generalization allowing real powers of monomial ideals. An important result is that given any monomial ideal I, the function taking a real number to the corresponding real power of I is a step function which is left continuous and has rational …

Maker Math: Exploring Mathematics Through Digitally Fabricated Tools With K–12 In-Service Teachers, Jason R. Harron, Yi Jin, Amy F. Hillen, Lindsey Mason, Lauren Siegel

Open Access Publishing Fund Collection

This paper reports on nine elementary, middle, and high school in-service teachers who participated in a series of workshops aimed at exploring the wonder, joy, and beauty of mathematics through the creation and application of digitally fabricated tools (i.e., laser-cut and 3D printed). Using the Technological Pedagogical and Content Knowledge (TPACK) framework to investigate technological, pedagogical, contextual, and content knowledge, researchers applied qualitative methods to uncover the affordances and constraints of teaching and learning math concepts with digitally fabricated tools and examined how the workshops supported broadening participation in mathematics by focusing on the connections between mathematical inquiry, nature, and …

Duality For Asymptotic Invariants Of Graded Families, Michael Dipasquale, Thái Thành Nguyễn, Alexandra Seceleanu

Department of Mathematics: Faculty Publications

The starting point of this paper is a duality for sequences of natural numbers which, under mild hypotheses, interchanges subadditive and superadditive sequences and inverts their asymptotic growth constants.

We are motivated to explore this sequence duality since it arises naturally in at least two important algebraic-geometric contexts. The first context is Macaulay- Matlis duality, where the sequence of initial degrees of the family of symbolic powers of a radical ideal is dual to the sequence of Castelnuovo-Mumford regularity values of a quotient by ideals generated by powers of linear forms. This philosophy is drawn from an influential paper of …

The Art Of Landslides: How Stochastic Mass Wasting Shapes Topography And Influences Landscape Dynamics, Benjamin Campforts, Charles Shobe, Irina Overeem, Gregory Tucker

Faculty & Staff Scholarship

Bedrock landslides shape topography and mobilize large volumes of sediment. Yet, interactions between landslide-produced sediment and fluvial systems that together govern large-scale landscape evolution are not well understood. To explain morphological patterns observed in steep, landslide-prone terrain, we explicitly model stochastic landsliding and associated sediment dynamics. The model accounts for several common landscape features such as slope frequency distributions, which include values in excess of regional stability limits, quasi-planar hillslopes decorated with straight, closely spaced channel-like features, and accumulation of sediment in valley networks rather than on hillslopes. Stochastic landsliding strongly affects the magnitude and timing of sediment supply to …

The Foundations Of Mathematics: Axiomatic Systems And Incredible Infinities, Catherine Ferris

Honors Program Theses and Projects

Often, people who study mathematics learn theorems to prove results in and about the vast array of branches of mathematics (Algebra, Analysis, Topology, Geometry, Combinatorics, etc.). This helps them move forward in their understanding; but few ever question the basis for these theorems or whether those foundations are sucient or even secure. Theorems come from our foundations of mathematics, Axioms, Logic and Set Theory. In the early20th century, mathematicians set out to formalize the methods, operations and techniques people were assuming. In other words, they were formulating axioms. The most common axiomatic system is known as the Zermelo-Fraenkel axioms with …

Data Engineering Techniques And Designs With Music Generating Neural Networks, Noah Solomon

Honors Program Theses and Projects

The generation of music artificially is an interesting concept to many and has received a lot of attention in recent years. The advancement of neural networks has allowed for the creation of models that can seemingly generate music creatively to mimic a specific genre or composer. This project delved deep into the many ways to construct music generating neural networks and compared different model architectures and data engineering techniques. Three main types of models were implemented and the resulting generated music was evaluated with respect to the melody, note agreeableness, and rhythm. These models used the Bach Chorales corpus as …

Vertex-Magic Graphs, Karissa Massud

Honors Program Theses and Projects

In this paper, we will study magic labelings. Magic labelings were first introduced by Sedláček in 1963 [3]. At this time, the labels on the graph were only assigned to the edges. In 1970, Kotzig and Rosa defined what are now known as edge-magic total labelings, where both the vertices and the edges of the graph are labeled. Following this in 1999, MacDougall, Miller, Slamin, and Wallis introduced the idea of vertex-magic total labelings. There are many different types of magic labelings. In this paper will focus on vertex-magictotal labelings.

The Role Of Surprise In Guessing Games, Justin Carpender

Honors Program Theses and Projects

In this thesis we will study the connection between game structure, surprise, and guessing strategies for these first two versions of a word guessing game. Our analysis will have three levels: one, a basic understanding of language and letter probabilities and the creation of programs that seek to use the structure of a game to efficiently guess words; two, an introduction of mathematical background and Information Theory; three, an analysis of the games and their corresponding guesses via a creative use of the key ideas of Information Theory, particularly, the concepts of surprise and entropy.

Vertex-Magic Total Labeling On G-Sun Graphs, Melissa Mejia

Honors Program Theses and Projects

Graph labeling is an immense area of research in mathematics, specifically graph theory. There are many types of graph labelings such as harmonious, magic, and lucky labelings. This paper will focus on magic labelings. Graph theorists are particularly interested in magic labelings because of a simple problem regarding tree graphs introduced in the 1990’s. The problem is still unsolved after almost thirty years. Researchers have studied magic labelings on other graphs in addition to tree graphs. In this paper we will consider vertex-magic labelings on G-sun graphs. We will give vertex-magic total labelings for ladder sun graphs and complete bipartite …

Spectral Analysis Of Multiscale Cultural Traits On Twitter, Chandler Squires, Nikhil Kunapuli, Yaneer Bar-Yam, Alfredo Morales

Northeast Journal of Complex Systems (NEJCS)

Understanding and mapping the emergence and boundaries of cultural areas is a challenge for social sciences. In this paper, we present a method for analyzing the cultural composition of regions via Twitter hashtags. Cultures can be described as distinct combination of traits which we capture via principal component analysis (PCA). We investigate the top 8 PCA components of an area including France, Spain, and Portugal, in terms of the geographic distribution of their hashtag composition. We also discuss relationships between components and the insights those relationships can provide into the structure of a cultural space. Finally, we compare the spatial …

Mathematical Models Yield Insights Into Cnns: Applications In Natural Image Restoration And Population Genetics, Ryan Cecil

Electronic Theses and Dissertations

Due to a rise in computational power, machine learning (ML) methods have become the state-of-the-art in a variety of fields. Known to be black-box approaches, however, these methods are oftentimes not well understood. In this work, we utilize our understanding of model-based approaches to derive insights into Convolutional Neural Networks (CNNs). In the field of Natural Image Restoration, we focus on the image denoising problem. Recent work have demonstrated the potential of mathematically motivated CNN architectures that learn both `geometric' and nonlinear higher order features and corresponding regularizers. We extend this work by showing that not only can geometric features …

Lorentzian Polynomials, Higher Hessians, And The Hodge-Riemann Property For Codimension Two Graded Artinian Gorenstein Algebras, Pedro Macias-Marques, Chris Mcdaniel, Alexandra Seceleanu, Junzo Watanabe

Department of Mathematics: Faculty Publications

We study the Hodge-Riemann property (HRP) for graded Artinian Gorenstein (AG) algebras. We classify AG algebras in codimension two that have HRP in terms of higher Hessian matrices and positivity of Schur functions associated to certain rectangular partitions.

In this paper we introduce the Hodge Riemann property (HRP) on an arbitrary graded oriented Artinian Gorenstein (AG) algebra defined over R, and we give a criterion on the higher Hessian matrix of its Macaulay dual generator (Theorem 3.1). AG algebras can be regarded as algebraic analogues of cohomology rings (in even degrees) of complex manifolds, and the HRP is analogous to …

Polynomial Growth Of Betti Sequences Over Local Rings, Luchezar L. Avramov, Alexandra Seceleanu, Zheng Yang

Department of Mathematics: Faculty Publications

We study sequences of Betti numbers (β^R_i (M)) of finite modules M over a complete intersection local ring, R. It is known that for every M the subsequence with even, respectively, odd indices i is eventually given by some polynomial in i. We prove that these polynomials agree for all R-modules if the ideal I^☐ generated by the quadratic relations of the associated graded ring of R satisfies height I^☐ ≥ codim R − 1, and that the converse holds when R is homogeneous and when codim R ≤ 4. Avramov, …

Low-Gain Integral Control For A Class Of Discrete-Time Lur’E Systems With Applications To Sampled-Data Control, Chris Guiver, Richard Rebarber, Stuart Townley

Department of Mathematics: Faculty Publications

We study low-gain (P)roportional (I)ntegral control of multivariate discrete-time, forced Lur’e systems to solve the output-tracking problem for constant reference signals. We formulate an incremental sector condition which is sufficient for a usual linear low-gain PI controller to achieve exponential disturbance-to-state and disturbance-to-tracking-error stability in closed-loop, for all sufficiently small integrator gains. Output tracking is achieved in the absence of exogenous disturbance (noise) terms. Our line of argument invokes a recent circle criterion for exponential incremental input-to-state stability. The discrete-time theory facilitates a similar result for a continuous-time forced Lur’e system in feedback with sampled-data low-gain integral control. The theory …

Debiasing Cyber Incidents – Correcting For Reporting Delays And Under-Reporting, Seema Sangari

Doctor of Data Science and Analytics Dissertations

This research addresses two key problems in the cyber insurance industry – reporting delays and under-reporting of cyber incidents. Both problems are important to understand the true picture of cyber incident rates. While reporting delays addresses the problem of delays in reporting due to delays in timely detection, under-reporting addresses the problem of cyber incidents frequently under-reported due to brand damage, reputation risk and eventual financial impacts.

The problem of reporting delays in cyber incidents is resolved by generating the distribution of reporting delays and fitting modeled parametric distributions on the given domain. The reporting delay distribution was found to …

Machine Learning Model Comparison And Arma Simulation Of Exhaled Breath Signals Classifying Covid-19 Patients, Aaron Christopher Segura

Mathematics & Statistics ETDs

This study compared the performance of machine learning models in classifying COVID-19 patients using exhaled breath signals and simulated datasets. Ground truth classification was determined by the gold standard Polymerase Chain Reaction (PCR) test results. A residual bootstrapped method generated the simulated datasets by fitting signal data to Autoregressive Moving Average (ARMA) models. Classification models included neural networks, k-nearest neighbors, naïve Bayes, random forest, and support vector machines. A Recursive Feature Elimination (RFE) study was performed to determine if reducing signal features would improve the classification models performance using Gini Importance scoring for the two classes. The top 25% of …

Stability And Differential Privacy Of Stochastic Gradient Methods, Zhenhuan Yang

Legacy Theses & Dissertations (2009 - 2024)

Recently there are a considerable amount of work devoted to the study of the algorithmic stability as well as differential privacy (DP) for stochastic gradient methods (SGM). However, most of the existing work focus on the empirical risk minimization (ERM) and the population risk minimization problems. In this paper, we study two types of optimization problems that enjoy wide applications in modern machine learning, namely the minimax problem and the pairwise learning problem.

Robust Uncertainty Quantification With Analysis Of Error In Standard And Non-Standard Quantities Of Interest, Zachary Stevens

Mathematics & Statistics ETDs

This thesis derives two Uncertainty Quantification (UQ) methods for differential equations that depend on random parameters: (\textbf{i}) error bounds for a computed cumulative distribution function (\textbf{ii}) a multi-level Monte Carlo (MLMC) algorithm with adaptively refined meshes and accurately computed stopping-criteria. Both UQ approaches utilize adjoint-based \textit{a posteriori} error analysis in order to accurately estimate the error in samples of numerically approximated quantities of interest. The adaptive MLMC algorithm developed in this thesis relies on the adjoint-based error analysis to adaptively create meshes and accurately monitor a stopping criteria. This is in contrast to classical MLMC algorithms which employ either a …