Analysis Commons^™

(R2151) Error Estimates Of Barycentric Lagrange Interpolation, Alvira Yawar, Swarnima Bahadur

Applications and Applied Mathematics: An International Journal (AAM)

Barycentric interpolation, which comes from Lagrange interpolation, is a useful method in numerical analysis. In this research paper, we explain how the barycentric interpolation formula is derived and discuss its features. We compare its stability and performance with the traditional Lagrange formula. First, we show how to get the barycentric formula from the Lagrange polynomial and present it as a rational function. We also provide an estimate of the error. Then, we use numerical examples to show that the barycentric formula is more stable and works better, especially when the degree of interpolation is high. Our results show that the …

Optimal Control Of Stochastic Systems: A Numerical Study Of The Stochastic Linear Quadratic Regulator Framework, Gülşen Orucova Büyüköz, Yaprak Güldoğan Dericioğlu, Tuğçem Partal

Mathematical Modelling and Numerical Simulation with Applications

Optimal control of stochastic linear systems is fundamental in control theory, with applications in robotics, finance, and engineering. The Stochastic Linear Quadratic Regulator (SLQR) derives optimal feedback laws via the Riccati equation but requires numerical discretization of the resulting stochastic dynamics. Despite extensive studies on numerical methods for stochastic differential equations, their performance within the SLQR framework remains insufficiently explored. This study compares two predictor–corrector schemes of different orders: the Order 1.0 Predictor-Corrector (PC) method and the Order 2.0 Weak PC method. A one-dimensional linear quadratic problem with a closed-form solution enables precise error evaluation against the analytical trajectory. Convergence …

Exploration Of Euclidean Distances In Five-Dimensional Color Space As Proof-Of-Concept For Identification Of Promising Alternatives To Platinum-Based Cancer Drug Candidates, Kiomi Tanakura, Elena G. Harvey

Student Scholar Symposium

Cisplatin, cis-Pt(NH₃)₂Cl₂, is one of the most successful anticancer drugs of all time. However, it has some negative aspects, including toxicity to healthy cells, side effects, and cost. Our goal in this project is to design metal-based anticancer drugs that would be less expensive, less toxic, and more efficient for cancer patients than cisplatin. To begin this research we calculated Euclidean distances - as used by computerized recognition systems - to compare the properties of colors within seasonal palettes that are proven and visible to the eye. These Euclidean distances were then used …

Data-Assimilation-Enabled Fast Convergence Of The Uzawa Scheme For Navier-Stokes Equations With Large Reynolds Numbers, Jessica C. Franklin

All Theses

This work studies efficient techniques utilized to solve the Navier-Stokes equations that model incompressible Newtonian flow in the setting where partial solution data is available. First, we analyze the Picard iteration and Picard with continuous data assimilation applied and test convergence. Then, we analyze the Arrow-Hurwicz (AH) and Uzawa iterative procedures and test convergence. Finally, we apply continuous data assimilation to Uzawa, and analytical and numerical results show that CDA improves Uzawa convergence for the NSE, similar to CDA-Picard but it is much more efficient.

On Sobolev Spaces And The Existence Of Weak Solutions To Boundary Value Problems, Skye X. Paul

Master's Theses

Many boundary value problems that arise in mathematical models have close connections to second order elliptic partial differential equations. This thesis introduces the idea of weak derivatives and Sobolev Spaces to generalize possible solutions. Using functional analysis centered around the Lax-Milgram theorem, we show the existence of these generalized solutions to boundary value problems including Laplace's Equation, 2nd order linear ODEs, and ultimately a general second order elliptic PDE. The work cumulates with recovering a number of central theorems of functional analysis in the context of Sobolev Spaces, creating a new perspective on the solvability of these boundary value problems.

Invitation To Polynomiography Via Chatgpt: For Teachers, Students And Artists, Bahman Kalantari

LASER Journal

Teachers, students, and artists in the United States and abroad who have encountered polynomiography, in lectures, demos, or software, consistently appreciate its educational value and artistic potential. While dedicated polynomiography programs require upkeep as systems evolve, AI chatbots now offer a practical, accessible alternative. This article invites readers to explore polynomiography with ChatGPT, broadening access beyond specialized tools. While the approach will not match the full range or polish of advanced software, it provides a powerful and flexible entry point with many possibilities.

At its core, polynomiography transforms polynomial equations, each encoding a finite set of points in the complex …

A Bdg Inequality For Stochastic Volterra Integrals, Alexandre Pannier

Journal of Stochastic Analysis

We establish Burkholder-Davis-Gundy-type inequalities for stochastic Volterra integrals with a completely monotone convolution kernel, which may exhibit singular behaviour at the origin. When the supremum is taken over a finite interval, the upper bound depends linearly on the Lγ-norm of the kernel, for any γ > 2. We demonstrate the utility of this inequality in quantifying the pathwise distance between two stochastic Volterra equations with distinct kernels, with a particular emphasis on the multifactor Markovian approximation. For kernels that decay sufficiently fast, we derive an alternative inequality valid over an infinite time interval, providing uniformin- time bounds for mean-reverting stochastic Volterra …

(Si15-140) Designing Bayesian Double Sampling Plans Based On Zero Inflated Poisson Distribution, Priyadharshini R., Shalini K., Hemalatha R., Sangeetha S.

Applications and Applied Mathematics: An International Journal (AAM)

The implementation of attribute-based sampling inspection serves as a quality control technique used across numerous industries to evaluate items or workflow processes. When the data exhibits a substantial number of zero counts, the zero-inflated Poisson (ZIP) distribution serves as an effective model for accommodating this zero-inflation. Double sampling plan (DSP) is a quality check method where the decision to approve or decline a batch comes after examining two samples, providing more conclusive information compared to a single sample plan (SSP). In practice, effective decision-making regarding submitted lots considers both within-lot and between-lot variations, which can be addressed through the use …

(Si15-142) Selection Of Single Sampling Plans Based On Zero Inflated Binomial Distribution Using Cost Optimization, Sangeetha S., Shalini K., Hemalatha R., Priyadharshini R.

Applications and Applied Mathematics: An International Journal (AAM)

Economic design of sampling plans involves creating sampling plans that minimize the total cost associated with the inspection process while ensuring quality. It aims to address the quality risk concerns of both producer and consumer, ensuring product quality while minimizing inspection costs. This article’s objective is to design single sampling plans by attributes based on Zero-inflated Binomial (ZIB) distribution, using cost optimization principles by developing an economic model aimed at achieving optimal total cost by considering the Average Total Inspection (ATI). Numerical illustration is provided to illustrate the selection of single sampling plans under ZIB distribution that minimizes producer’s total …

Matrices Induced By Scaled Hypercomplex Numbers Over The Real Field R, Daniel Alpay, Ilwoo Choo

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we construct, and study a certain type of definite, or indefinite inner product spaces over the real field R, induced by the scaled hypercomplex numbers H_t for a fixed scale t ∈ R, and some bounded operators acting on such vector spaces. In particular, we are interested in the vector spaces H^N_t consisting of all N-tuples of scaled hypercomplex numbers of H_t, and the (N x N)-matrices acting on H^N_t whose entries are from H_t, i.e., H_t-matrices, for all N ∈ N. For an arbitrarily fixed …

Assistance For The Calculation Of Sines (Translation Of E246), Julian Schennach

Euleriana

Paralleling his famous relation e^iφ = cos(φ) + i sin(φ), Euler establishes the equality (cos φ + i sin φ)ⁿ = (cos nφ + i sin nφ). He uses it to comprehensively derive trigonometric identities that convert arbitrary powers of sines and cosines of an angle (and products thereof) into sums of sines and cosines of multiples of that angle. Some negative and fractional powers are shown to yield infinite series. Euler further describes a general method to evaluate various infinite series involving weighted trigonometric functions. These results foreshadow Fourier series. As Euler points out, the scope of …

An Interpolating Regime For The Extreme Values Of A Random Zeta Model, Christine K. Chang

Dissertations, Theses, and Capstone Projects

We study the large values of a random model of the Riemann zeta function over short intervals. The extreme value statistics depend on the interval size: the log-correlated regime governs intervals of order one while the i.i.d. regime emerges over longer intervals. The main focus is to describe the transition between these two well-understood regimes as the interval varies in length. This thesis shows that there is an intermediate regime where the behavior of the zeta model’s maxima cannot be entirely captured by either extreme— i.i.d. or fully log-correlated. This suggests that the Riemann zeta function exhibits correlations around its …

Sugar: A Sequence Unfolding Based Transformer Model For Group Activity Recognition, Yash U. Gondkar

Graduate Masters Theses

Large Language Models have improved significantly in the past couple of years due to the adoption of transformers. However, transformers still find it challenging to process videos due to limited context size caused by their quadratic computing cost. Therefore, we studied a booming field in machine learning which powers applications like social scene analysis and video surveillance systems called Group Activity Recognition (GAR). We found that recent models were able to achieve more than 90% accuracy on popular datasets like the Volleyball dataset, however, it turned out that even they relied on transformers.

Therefore, in this work, we developed a …

Differentiation And Certain Operators On Scaled Sectional Hypercomplex Numbers, Daniel Alpay, Ilwoo Choo

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we study certain sectional structures of the t-scaled hypercomplex numbers H_t for a scale t ∈ R, including the quaternions H_-1, and the split quaternions H₁. For a fixed scale t ∈ R, by defining the collection S_t of certain pureimaginary t-scaled hypercomplex number in H_t , we sectionize H_t from the imaginaries of S_t. We concentrate on a section SH_It for an arbitrarily fixed imaginary I_t ∈ S_t , called the t-scaled section for I_t. Differentiation theory on the …

The Characteristic Function Of The Cube Of A Gaussian Random Variable, Andreas Boukas

Journal of Stochastic Analysis

Using the spectral resolution of the multiplication operator on the Schwartz class of L2(R,C), we compute the characteristic function of the cube of a Gaussian random variable.

A Study Of Complex Analysis After Whittaker And Watson, Crystal Steed

All Graduate Reports and Creative Projects, Fall 2023 to Present

The goal of this report is to provide solutions to the exercises found in chapter five of the book titled, A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions with an Account of the Principal Transcendental Functions by E.T. Whittaker and G.N. Watson. The fifth chapter is titled, "The Fundamental Properties of Analytic Functions; Taylor's, Laurent's and Liouville's Theorems." This report solves the end-of-chapter exercises in addition to providing details for some in-chapter exercises, which are left to the reader. Many of these exercises are results from famous mathematicians.

Traffic Prediction For Research And Education Networks: Anomaly-Aware Deep Learning And Benchmarking, Mohammad Arafath Uddin Shariff

School of Computing: Dissertations, Theses, and Student Research

Research and Education Networks (RENs) and High-Performance Computing (HPC) environments are critical infrastructures for modern scientific discovery, demanding sustained high-throughput and low-latency data transfers. Unlike commercial networks, RENs exhibit unique traffic characteristics, including predominant “elephant flows,” inherent burstiness, and complex temporal-spatial dynamics often decoupled from human-driven cycles. Traditional traffic forecasting methods, tailored for commercial Wide Area Networks (WANs), consistently fail to capture these distinct REN dynamics, leading to inefficient resource management and potential impediments to scientific progress.

This thesis addresses this critical gap by developing and validating a robust, scalable, and anomaly-aware traffic forecasting framework specifically tailored for REN/HPC networks. …

Mathematical Modeling Of Effects Of Tumor Location On Lung Function., Lamargaret Temukisa Johnson

Master of Engineering Theses

Lung cancer has the highest rates of incidence and mortality of all cancers. Most lung cancer tumors are Non-Small Cell Lung Cancer (NSCLC). NSCLC patients with lesions in the upper lobes are found to have better prognosis compared to those with lesions in the middle and lower lobes. Previous studies have suggested various causes for this discrepancy at both the organ-scale and tissue-scale. To model NSCLC growth in different locations within the lung, an organ scale lung model and tissue scale tumor model were coupled through the tissue pressure, and oxygen and carbon dioxide partial pressures. The coupling was used …

Analysis Of Popular Songs In The Us Market, Michael Myers

Theses, Dissertations and Culminating Projects

In today’s digitally driven music landscape, understanding what drive’s a song’s popularity requires insight not only into its acoustic and lyrical content, but also into patterns of listener engagement across platforms. This thesis explores the predictive and descriptive dimensions of song popularity by applying supervised and unsupervised machine learning models to a multi-source dataset integrating audio features, sentiment analysis, and temporal consumption behavior. Drawing from a novel, multi-platform dataset that includes Billboard Hot 100 rankings, Spotify acoustic features and popularity scores, streaming, airplay, and sales metrics as reported on Luminate’s Music Connect, and lyrics from AZLyrics, the study investigates the …

Dream Series, Jim Wolper

Journal of Humanistic Mathematics

Dream Series converge to numbers we cannot analyze. They exist, like dreams, but resist attempts to put them into order.

A Data-Driven Approach To Time Series Forecasting And Clustering Of U.S. Regional Drug Overdose Mortality, Koshali Hamy Muthunama Gonnage

Mathematics & Statistics ETDs

The increasing rate of drug overdose deaths in the United States poses a critical public health challenge, particularly due to the surge in synthetic opioids and other high-risk substances. This study presents a data-driven framework that integrates time series forecasting and clustering techniques. Monthly mortality data for five key drug types: cocaine, fentanyl, heroin, methamphetamine, and oxycodone were analyzed using four time series forecasting models: ARIMA, ETS, TBATS, and NNAR. These models were evaluated using standard accuracy metrics RMSE, MAPE, and MAE to assess predictive performance. Signal decomposition approach based on Singular Value Decomposition and subspace modeling was employed to …

Operator Information Quantities Of Semigroups Associated With Functions Of The Number Operator, Ryo Inayoshi, Kimiaki Saito

Journal of Stochastic Analysis

In this paper, we present recent developments on the operator information quantity acting on white noise functionals. In particular, we give a stochastic expression of the operator information quantity of a semigroup generated by some function of the number operator through a white noise delta distribution centered at an infinite dimensional Ornstein-Uhlenbeck process.

Quasistatic Peridynamics, Existence Of Unique Solution In The Presence Of Damage, Nuwanthi N. Samarawickrama

LSU Doctoral Dissertations

A mathematical model for damage propagation based on nonlocal potentials is developed within the framework of peridynamics. This model is applied to simulate damage evolution in cyclically loaded structures. By neglecting inertial effects, a well-posed quasistatic formulation for cyclic loading is obtained.\\ The resulting equation is expressed as a nonlocal and nonlinear integral operator that couples damage evolution to the deformation field.\\ This coupling occurs through the product of a damage factor and the derivative of a force potential. The damage factor ranges between zero and one, where one represents undamaged material and zero indicates complete damage.\\ It serves to …

Quantitative Boundary Doubling Estimates For Elliptic Equations, Jack Dalberg

LSU Doctoral Dissertations

We present an approach for obtaining quantitative boundary doubling inequalities for elliptic equations with Neumann boundary conditions. Carleman estimates are used to prove three-ball inequalities, which are then used to prove quantitative doubling inequalities, with bootstrapping from the interior to the boundary. This approach is illustrated by its application to the Laplace eigenvalue problem with homogeneous Neumann boundary conditions, where sharp doubling inequalities are recovered.

When then consider a equation with non homogeneous Neumann boundary conditions. By following the approach, we are able to obtain potentially sharp results. Finally, we are able to get an improvement on previously obtained results …

Strichartz Estimates For Many Particle Dispersive Equations, Tristan Reynoso

LSU Doctoral Dissertations

Dispersive equations are useful for describing a wide variety of phenomena in which solutions disperse through the space as time progresses. These equations show up frequently in physics, especially when studying quantum mechanical and fluid related systems. The single particle variants of equations such as the Schr\"{o}dinger and Wave equations have been studied at great length throughout modern history. Over the last couple decades progress has been made toward extending single particle dispersive equations to cover their many body counterparts. Space-time Strichartz estimates for the homogenous $N$-particle Schr\"{o}dinger equation with small interacting potentials was recently established on both $\mathbb{R}^d$ and …

Temporal Modeling And Forecasting Of Blood Glucose Dynamics In Individuals With Diabetes Mellitus, Mj Ruff

University Honors Theses

People living with Diabetes Mellitus face significant health risks, including an increased likelihood of heart disease, stroke, and fluctuations in blood glucose levels. The unpredictable nature of glucose levels can lead to dangerous conditions such as ketoacidosis and hypoglycemia. This study employs advanced time series analysis tools to forecast the glucose levels for an individual diagnosed with Type 1 Diabetes Mellitus.

On Excursions Associated With A Certain Local Time Of Simple Symmetric Random Walks, With Applications, Takahiko Fujita, Naohiro Yoshida

Journal of Stochastic Analysis

In this note, some applications of excursions associated with a certain local time of simple symmetric random walks are presented. Specifically, the excursions are applied to calculate some probability distributions of interest regarding the random walks. Furthermore, a solution of the Skorokhod embedding problem for random walks is obtained through the excursions.

Forecasting Influenza Rates Using Machine Learning: A Study Of Chatgpt's Predictive Accuracy, Sara Saleh

University Honors Theses

This study evaluates ChatGPT's ability to forecast influenza rates, such as the number of flu cases, hospitalizations, and death during peak season periods using CDC data, and comparing forecasts against actual results to calculate statistical accuracy and consistency. Influenza forecasting is essential for public health planning, but traditional methods may not always provide timely or accurate predictions. In this research study, ChatGPT was utilized to predict the influenza rates for the following week based on the previous week's data obtained from the FluView surveillance system. The predicted rates were compared to the actual influenza rates to assess the model's overall …

Banach Algebras And The Gelfand Theory Of Group Algebras On Locally Compact Abelian Groups, James Gabriel Bonvanie

Master's Theses

A Banach algebra is a complex algebra that is simultaneously a Banach space in which the norm is submultiplicative. Notably, $L^1(\mathbb{R})$ with the convolutional product is an Abelian, non-unital Banach algebra that admits an approximate identity. We rectify $L^1(\mathbb{R})$ lacking a unit via the unitization $L^1(\mathbb{R})\times\mathbb{C}$ with identity $(0,1)$. Unitization opens the discussion to the spectrum $\sigma(x)$ of a Banach algebra element, in which the spectrum is a nonempty, compact subset of the complex plane. The spectrum of an Abelian Banach algebra is fully characterized with multiplicative linear functionals, and we prove that the Fourier transform is the unique multiplicative …

Topiarism: The Kernel Embedding Of Distributions Applied To Modern Portfolio Theory, Stephen G. Cook

Master's Theses

The method of kernel embedding of distributions on a set $\Omega$ into a reproducing kernel Hilbert space is a method of studying the space of measures on a set $\Omega$ using Hilbert space geometry. Because positively weighted portfolios can be interpreted as nonnegative probability measures on the space of assets, we are able to apply this technique to portfolio theory. In this thesis, we discuss the theory of "topiarism", the study of positively weighted probability measures on compact sets under the kernel embedding of distributions. Given a specific payoff function $\psi$ on the set of assets $\Omega$, we optimize a …