Mathematics Commons^™

Toward Simulating 2d Cell Surfaces In A Disk, Myriam Allred

Mathematics & Statistics ETDs

Certain evolution models of cell surfaces (treated in two-dimensions) involve the solution of the Helmholtz equation with jump conditions enforced on an immersed closed curve. This thesis presents a sparse, modal spectral method for solving such Helmholtz problems. The solution is required to be continuous across the curve, but with a jump discontinuity in the normal derivative proportional to the planar curvature. The method relies on classical Fourier-Chebyshev basis functions, with the application of modal Chebyshev integration matrices to achieve sparse, banded approximations of the Helmholtz equation. The method achieves spectral convergence, despite the inherent low regularity of the relevant …

Derivation Of Adjoint Based Error Estimates For Nonlinear Ordinary Differential Equations With Application To Multistage Sir Models With Demographics, Daniel Alcala

Mathematics & Statistics ETDs

Ordinary Differential Equations (ODEs) are central to the mathematical modeling of various real-world phenomena, from mechanical systems governed by Newton’s laws to epidemic dynamics described by SIR-type ODEs. Since many ODEs do not admit closed-form analytic solutions, we approximate them numerically (e.g., with Euler’s, Runge–Kutta, or other such methods). This raises the key question: How accurate are these numerical solutions? In particular, reliably estimating the error in some quantity of interest (QoI) at time T without having an exact solution is of great scientific interest.

The first main contribution of this thesis is the development and analysis of adjoint-based error …

Logic Enriched Over A Quantale, Alexander Kurz

Engineering Faculty Articles and Research

Many-valued logics have a long history in mathematical logic as well as in applications to the semantics of programming languages and to engineering more generally. Typically these logics are rich with features motivated by the particular applications they stem from. In his 1973 article "Metric Spaces, Generalized Logic, and Closed Categories", Lawvere argued that any quantale Ω gives rise to a generalized Ω-valued logic that has as its models the categories enriched over the quantale. This suggests developing a uniform framework for many-valued logics parameterized in a quantale. In this talk we will review some previous and ongoing work in …

Asymptotic Behavior Of Evolution Equations Via Spectral Theory Of Functions On The Half Line, Sums Of Commuting Operators And Evolution Semigroups, Minh Nguyen, Vu Trong Luong

Faculty Scholarship

In this paper we give a survey of recent developments in the spectral theory of bounded functions on the half line, and its applications to study the asymptotic behavior of solutions of evolution equations of the form , where A(t) is periodic in t. For the applications we consider spectra of the evolution semigroups associated with the evolution equations in various spectral invariant function spaces and their generators. The main results presented in the paper are concerned with the existence and uniqueness within decaying functions with specific spectral properties. These asymptotic behaviors are related to the Katznelson–Tzafriri Theorem and Massera …

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.

Visions Of Kleinian Groups, Genevieve Nguyen

Master's Theses

Discrete groups of M¨obius transformations, called Kleinian groups, act on the Riemann sphere. We explore the connection between M¨obius transformations and PSL(2,C), its subgroups, and linear algebraic properties to understand the dynamics of various Kleinian groups, including Fuchsian groups and Schottky groups. In particular, we use Mathematica to program visualizations of Schottky groups and their limit sets.

Conditional Optimal Sets And The Quantization Coefficients For Some Uniform Distributions, Evans Nyanney, Megha Pandey, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

Bucklew and Wise (1982) showed that the quantization dimension of an absolutely continuous probability measure on a given Euclidean space is constant and equals the Euclidean dimension of the space, and the quantization coefficient exists as a finite positive number. By giving different examples, in this paper, we have shown that the quantization coefficients for absolutely continuous probability measures defined on the same Euclidean space can be different. We have taken uniform distribution as a prototype of an absolutely continuous probability measure. In addition, we have also calculated the conditional optimal sets of n-points and the nth conditional quantization errors …

Some Properties Of Rescaled Sasaki Type Metric On The Coframe Bundle, Habil Fattayev, Arif Salimov

Turkish Journal of Mathematics

In this paper we introduce the rescaled Sasaki type metric on the coframe bundle of a Riemannian manifold and investigate the Levi-Civita connection, curvature tensor, paracomplex structures of coframe bundle with this metric

Janowski Close-To-Convexity Harmonic Mappings, Bushra Arif, Jacek Dziok, Wasim Ul Haq, Mohsan Raza

Turkish Journal of Mathematics

In the present work we introduce harmonic univalent functions related to Janowski close-to-convexity and we provide their necessary and sufficient criteria. The extreme points and topological features of such harmonic mappings will also be investigated. By applying extreme points theory, we obtain integral mean inequalities, distortion theorems, and coefficient estimates for families of harmonic functions.

Approximate Controllability Of Fractional Backward Stochastic Differential Inclusions With Order 1/2 < P < 1, Müberaa Selah

Turkish Journal of Mathematics

In this paper, fractional backward stochastic differential equations are extended to multivalued forms. Since the problem of approximate controllability of the obtained fractional backward stochastic differential inclusions has not yet been addressed in the literature, we focus on the approximate controllability of these systems. The study’s main results, such as existence of the mild solution and approximate controllability for fractional backward stochastic differential inclusions, have been proven through fractional calculus, Bohnenblust-Karlin Theorem, and appropriate conditions.

The Circular Sum Of Points Of An Affine Segment, Mircea Crasmareanu, Marius Munteanu

Turkish Journal of Mathematics

This note introduces a sum for two interior points of an affine segment [AB] based on a trigonometric approach. Some properties of this operation, along with general examples (including squares of endpoints, the midpoint, and the centroid), are discussed. A main result is that the midpoint of a segment is the circular sum of the points dividing the segment with the ratios 4 and 9. In addition, we completely solve a Diophantine equation associated with the circular sum in order to identify all points with integer ratio whose circular sum is a point with integer ratio.

Weighted Inequalities For Discrete Bilinear Hardy-Type Operator With A Matrix, Nazerke Zhangabergenova, Ainur Temirkhanova

Turkish Journal of Mathematics

In this paper, we consider a new discrete bilinear inequality of Hardy-type involving a matrix operator. We establish weight characterizations of this inequality under some conditions on the matrix entries.

New Characterizations Of Weights In Dynamic Inequalities Of Hardy's Type On Time Scales, Mario Krnic, Mahmoud M. Osman, Samir H. Saker

Turkish Journal of Mathematics

The main aim of this paper is to establish some new characterizations of the weights in Hardy-type dynamic inequality, dual Hardy-type inequality, andreverse Hardy-type dynamic inequality on time scales. Some integral and discrete inequalities due to Copson, Leindler, and Hyun and Kim will bededuced as special cases. Moreover, new dynamic inequalities via convexity are proved.

Analytical Solution And Stability Of Ψ-Prabhakar Delayed Systems: Application To Antibiotic Production, Mustafa Aydin

Turkish Journal of Mathematics

The analytical solution of the linear ψ -Prabhakar delay differential equations is explored using the method of variation of parameters, in which a ψ -general delayed exponential matrix function is introduced. The ψ -Prabhakar calculus is enhanced in terms of the semi-group property, inversion results, and several simplified calculations and relations. The system's stability is analyzed within the framework of Ulam-Hyers stability. Finally, the theoretical findings are validated through an application to antibiotic production.

Integral Extensions On Growth And Higher Derivatives Of A Polynomial, Nirmal Kumar Singha, Fahreddi̇n G. Abdullayev, Barchad Chanam

Turkish Journal of Mathematics

A well-known theorem due to Ankeny and Rivlin states that if p(z) is a polynomial of degree n such that p(z) has no zero in |z| < 1, then
max_|z|=R≥1 |p(z)| ≤ (R^{n + 1} / 2) max_|z|=1 |p(z)|.
This research examines the polynomial p(z), ensuring that it has no zero in the disk |z| < k, where k ≥ 1. At the same time, we investigate the s^th derivative of this polynomial, where 0 ≤ s < n. In our effort to establish integral formulations of the inequalities related to the derivatives of this class of polynomials, we have successfully extended and generalized Ankeny and Rivlin’s inequality to integral settings. Additionally, part of our findings provides integral analogs of results by Mir [J. Anal., 27 (2019), 851−857]. Moreover, another aspect of our work leads to an improvement in the result of Jain [Turk. J. Math., 31 (2007), 89−94], which we have also verified using an example. We have also compared our results with a previously known result using this numerical example, where the bounds that are in terms of integral means are estimated numerically by numerical integration using Simpson’s 1/3^rd rule and illustrate graphically the obtained inequalities as regards sharpness.

On The Solutions Of Difference Equations Of The Volterra Type, Hakan Adigüzel

Turkish Journal of Mathematics

In this paper, we conduct a thorough analysis of the qualitative behavior of solutions to three different types of difference equations of the Volterra type. Utilizing the principles of discrete calculus and various well-known inequalities, we have presented our findings. To conclude our theoretical findings, we provide several examples that illustrate full alignment with the findings.

A Generalization Of Franklin’S Partition Identity And A Beck-Type Companion Identity, Gabriel Gray, David Hovey, Brandt Kronholm, Emily Payne, Holly Swisher, Ren Watson

School of Mathematical & Statistical Sciences Faculty Publications

Euler’s classic partition identity states that the number of partitions of n into odd parts equals the number of partitions of n into distinct parts. We develop a new generalization of this identity, which yields a previous generalization of Franklin as a special case, and prove an accompanying Beck-type companion identity.

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 …

How Multi-Scale Modeling Can Help Examine Social Determinants Of Health And Resulting Disparities, Kyoko Yoshida, Elsje Pienaar, Shalanda A. Bynum, Naomi Chesler, Mitchel J. Colebank, Jessie Heneghan, Nadra Tyus, Jasmine Miller-Kleinhenz, Bruce Y. Lee

Faculty Publications

Social determinants of health (SDOH) are the conditions in which people live, work, and play, and the wider set of factors (e.g., social and economic systems and policies) that shape a person’s daily life. SDOH can differ significantly across communities and populations, having positive impacts for some and negative impacts for others. Ultimately, this results in differences in health and disease distribution, that are known as health disparities. Despite the known impacts of SDOH and calls to characterize, address, reduce, and eliminate health disparities, they persist and, in some cases, have worsened. To address this challenge, a session at the …

Gain Threshold Optimization Using Fano Resonance, Alina Oktiabrskaia

LSU Doctoral Dissertations

The study of resonances in electromagnetics plays a critical role in the design of optical systems. This dissertation investigates the interaction between resonance and gain in optical structures to establish a universal principle for achieving ultra-low-threshold lasing. Through the analysis of geometric symmetries, material properties, and coupling mechanisms, this research develops prototype structures applicable to a wide range of optical and electromagnetic systems. A range of models is considered, starting from a simple onedimensional string-resonator system (based on the model of H. Lamb), then advancing to two- and three-dimensional waveguide models, and culminating with a realistic high-contrast model in open …