Mathematics Commons^™

The Boltzmann-Enskog Equation With Brownian Forcing, Aurora Wallace

LSU Doctoral Dissertations

Interacting particle systems (IPS) are used to rigorously derive partial differential equations modeling macroscopic behavior given by systems of stochastic differential equa- tions in physics, biology, and other sciences. These systems model the time evolution of n interacting particles with desired dynamics. In the n-system studied here, only two parti- cles may interact at a time (binary collisions). The particles are exchangeable; that is, the ordering of the particles does not play a role, and the interaction between the ith particle and the jth particle models an elastic collision. In this thesis, we use the method of inter- acting particle …

Component Order Edge Connectivity, Vertex Degrees, And Integer Partitions, Michael R. Yatauro

Theory & Applications of Graphs

Given a finite, simple graph G, the k-component order connectivity (resp. edge connectivity) of G is the minimum number of vertices (resp. edges) whose removal results in a subgraph in which every component has an order of at most k − 1. In general, determining the k-component order edge connectivity of a graph is NP-hard. We identify conditions on the vertex degrees of G that can be used to imply a lower bound on the k-component order edge connectivity of G. We will discuss the process for generating such conditions for a lower bound of 1 or 2, and we …

Conditional Quantization For Uniform Distributions On Line Segments And Regular Polygons, Pigar Biteng, Mathieu Caguiat, Tsianna Dominguez, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. If, in the quantization some of the elements in the support are preselected, then the quantization is called a conditional quantization. In this paper, we investigate the conditional quantization for the uniform distributions defined on the unit line segments and m-sided regular polygons, where 𝑚≥3, inscribed in a unit circle.

Dynamic Mean-Field Theory For Continuous Random Networks, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

This article studies the dynamics of the mean-field approximation of continuous random networks. These networks are stochastic integrodifferential equations driven by Gaussian noise. The kernels in the integral operators are realizations of generalized Gaussian random variables. The equation controls the time evolution of a macroscopic state interpreted as neural activity, which depends on position and time. Such a network corresponds to a statistical field theory (SFT) given by a momenta-generating functional. Discrete versions of the mentioned networks appeared in spin glasses and as models of artificial neural networks. Each of these discrete networks corresponds to a lattice SFT, where the …

An Operator Information Quantity Of A Semigroup And Associated Differential Equations, Kimiaki Saito, Ryo Inayoshi

Journal of Stochastic Analysis

No abstract provided.

Continuous Dependence Of Thermoelasticity Problem, Müge Meyvaci, Şevket Gür

Turkish Journal of Mathematics

The current study considers the initial boundary value problem for the thermoelastic model. The model in which heat conduction is given by Green and Naghdi's theory is called Type II. It is increasingly accepted that studying the structural stability of the initial boundary value problem is crucial. This is because even minor changes in the equation or its coefficients can lead to significant variations in the solution of the problem. From this point of view, we analyze the structural stability of the solutions, specifically examining the continuous dependence of the solutions on all the coefficients of the thermoelastic system, using …

Order Structure Of Order-To-Topological Continuous Operator, Gül Si̇nem Keleş, Bahri̇ Turan, Bi̇rol Altin

Turkish Journal of Mathematics

Let J be a vector lattice, and W be a topological vector space. An operator K: J → W is called an order-to-topological continuous operator if u_α → 0 in J implies K(u_α) → 0 in W for each net (u_α) in J. In this study, we examine the order structure of the space of order-to-topological continuous operators in general and the order structure of order-to-norm continuous operators in particular. We study the relationships between order-to-topology continuous operators and other classes of operators, such as order weakly compact and order continuous operators. Moreover, we give …

A Note On A Novel Type Of Alster Covering Property In Bitopological Setting, Necati̇ Can Açikgöz, Ceren Sultan Elmali

Turkish Journal of Mathematics

In this paper, we introduce a new type of Alster covering property in bitopological setting. We investigate its relations with other existing covering properties. By giving counterexamples, we indicate that our new property is different than other mentioned properties. We also construct some conditions for equivalences of those properties with the corresponding property. We also consider the preservation of this new form of bitopological Alster covering property. We investigate the productivity of bitopological nearly Lindelöfness and bitopological nearly Mengerness.

Improving Upper Bounds Of Berezin Number Inequalities Using Convex Function, Sana Khan, Mehmet Gürdal, Muhammad Saeed Akram

Turkish Journal of Mathematics

In this research article, new upper bounds are established for the Berezin number of bounded linear operators within the context of functional Hilbert space using the convex function. These bounds are achieved by exploiting an extension of Furuta's inequality and an extension of Kato's inequality. These Berezin number bounds are refinements of already existing bounds found in the literature.

Fractional Equivalent Of Fourth-Order Laguerre And Chebyshev Polynomials, Zahra Kavousi, Kazem Ghanbari

Turkish Journal of Mathematics

Orthogonal polynomials have very useful properties in the solution of mathematical and physical problems. In the last few years, there has been considerable interest in the properties of orthogonal polynomials satisfying differential equations (DE) of order greater than two, their connection to singular boundary value problems, their generalizations, and their classification as solutions of second-order DE. In this paper, we develop a fractional analog of the fourth-order Laguerre differential equation on the interval [0, ∞) and, in addition, we develop a fractional analog of the fourth-order Chebyshev differential equation on a symmetric interval [−α, α], where α is the order …

Biquandle Power Brackets Of Oriented Links, Nesli̇han Gügümcü, Sam Nelson

Turkish Journal of Mathematics

In this paper, we introduce biquandle power brackets, an infinite family of invariants of oriented links containing the classical skein invariants and the quandle and biquandle 2-cocycle invariants as special cases. Biquandle power brackets are generalizations of biquandle brackets in which the values of Kauffman states also depend on the biquandle colors they admit. We provide example computations and discuss the relationship between these new invariants and the previous cases.

Quantization Dimensions For Inhomogeneous Bi-Lipschitz Iterated Function Systems, Amit Priyadarshi, Mrinal Kanti Roychowdhury, Manuj Verma

School of Mathematical & Statistical Sciences Faculty Publications

Let ν be a Borel probability measure on a d-dimensional Euclidean space R d , d ≥ 1 , with a compact support, and let ( p 0 , p 1 , p 2 , … , p N ) be a probability vector with p j > 0 for 0 ≤ j ≤ N . Let { S j : 1 ≤ j ≤ N } be a set of contractive mappings on R d . Then, a Borel probability measure μ on R d such that μ = ∑ N j = 1 p j μ ∘ S − …

The Product Formula Of Multiple Stochastic Integrals With Respect To The Poisson Space Noise, Yuh-Jia Lee, Hsin-Hung Shih

Journal of Stochastic Analysis

No abstract provided.

American Option Pricing Using Generalised Stochastic Hybrid Systems, Evelyn Buckwar, Sascha Desmettre, Agnes Mallinger, Amira Meddah

Journal of Stochastic Analysis

No abstract provided.

About Fixed Points Of Quantum Channels, Raffaella Carbone

Journal of Stochastic Analysis

No abstract provided.

Stochastic Bundles, New Classes Of Gaussian Processes And White Noise-Space Analysis Indexed By Measures, Daniel Alpay, Palle Jorgensen

Mathematics, Physics, and Computer Science Faculty Articles and Research

Starting from a fixed measure space (X,F,μ), with μ a positive sigma-finite measure defined on the sigma-algebra F, we continue here our study of a generalization W(μ) of Brownian motion, and introduce a corresponding white-noise process. In detail, the generalized Brownian motion is a centered Gaussian process W^(μ), indexed by the elements A in F of finite μ measure, and with covariance function μ(A ∩ B). The purpose of our present paper is to make precise and study the corresponding whitenoise process, i.e., a point-wise process which is indexed by X, and which arises …

Subfitness In Distributive (Semi)Lattices, G. Bezhanishvili, J. Madden, M. A. Moshier, M. Tressl, Joanne Walters-Wayland

Mathematics, Physics, and Computer Science Faculty Articles and Research

We investigate whether the set of subfit elements of a distributive semilattice is an ideal. This question was raised by the second author at the BLAST conference in 2022. We show that in general it has a negative solution, however if the semilattice is a lattice, then the solution is positive. This is somewhat unexpected since, as we show, a semilattice is subfit if and only if so is its distributive lattice envelope.

Induced-Minor-Closed Classes Of Matroids, James Dylan Douthitt

LSU Doctoral Dissertations

A graph is chordal if every cycle of length at least four has a chord. In 1961, Dirac characterized chordal graphs as those graphs that can be built from complete graphs by repeated clique-sums. Generalizing this, we consider the class of simple GF(q)-representable matroids that can be built from projective geometries over GF(q) by repeated generalized parallel connections across projective geometries. We show that this class of matroids is closed under induced minors and characterize the class by its forbidden induced minors, noting that the case when q=2 is distinctive. Additionally, we show that the class of GF(2)-chordal matroids coincides …

The Intricacies Of Pairwise Modular Multiplicative Inverse In Lucas Numbers, Charles Liu

Rose-Hulman Undergraduate Mathematics Journal

Let (p,q) be a pair of relatively prime integers greater than 1. The pairwise modular multiplicative inverse (PMMI) of (p,q) is defined as the unique pair of positive integers (p′, q′) such that p p′ ≡ 1 (mod q), p′ < q, qq′ ≡ 1 (mod p), q′ < p. In this paper, we determine all pairs of Lucas numbers such that their PMMIs are pairs of Lucas numbers.

A Student Guide To Using Ai To Enhance Algebraic Understanding, Karan Puri

Open Educational Resources

This is a step-by-step guide that students can follow to use AI tools to check their understanding of concepts that have been tested in the introductory/college algebra classroom.

Time-Dependent Shear Flows And Their Applications In Parabolic–Parabolic Patlak–Keller–Segel Systems*, Siming He

Faculty Publications

In this study, we investigate the behavior of three-dimensional parabolic–parabolic Patlak–Keller–Segel systems in the presence of ambient shear flows. Our findings demonstrate that when the total mass of the cell density is below a specific threshold, the solution remains globally regular as long as the flow is sufficiently strong. The primary difficulty in our analysis stems from the fast creation of chemical gradients due to strong shear advection.

Counting Rotational Sets For Laminations Of The Unit Disk From First Principles, Michael J. Moorman, Gabriel B. Quijano, Matthew C. Williams Jr.

Rose-Hulman Undergraduate Mathematics Journal

By studying laminations of the unit disk, we can gain insight into the structure of Julia sets of polynomials and their dynamics in the complex plane. The polynomials of a given degree, d, have a parameter space. The hyperbolic components of such parameter spaces are in correspondence to rotational polygons, or classes of "rotational sets'', which we study in this paper. By studying the count of such rotational sets, and therefore the underlying structure of these rotational sets and polygons, we can gain insight into the interrelationship among hyperbolic components of the parameter space of these polynomials.

These rotational sets …

Finite Posets As Prime Spectra Of Commutative Noetherian Rings, David T. Alkass

Rose-Hulman Undergraduate Mathematics Journal

We study finite partially ordered sets of prime ideals as found in commutative Noetherian rings. In doing so, we establish that these posets have a bipartite structure and devise a construction for finding ring spectra that are order-isomorphic to many such posets. Specifically, we prove that any finite complete bipartite graph is order-isomorphic to the spectrum of a ring of essentially finite type over the field of rational numbers. Furthermore, we prove that prime spectra of such rings can also depict any finite path or even cycle.

Proportion Of P-Adic Polynomials Which Are Irreducible, Isaac Rajagopal

Rose-Hulman Undergraduate Mathematics Journal

We attempt to quantify the exact proportion of p-adic polynomials of degree n which are irreducible. We find an exact answer to this when n is prime and p != n, and also when n = 4 and p != 2. Our answers are rational functions in p. This relates to previous work done to find exact proportions of p-adic polynomials of degree n which have k roots.

Heart Disease Prediction Using Ensemble Tree Algorithms: A Supervised Learning Perspective, Enoch Sakyi-Yeboah, Edmund F. Agyemang, Vincent Agbenyeavu, Akua Osei- Nkwantabisa, Priscilla Kissi-Appiah, Lateef Moshood, Lawrence Agbota, Ezekiel N.N. Nortey

School of Mathematical & Statistical Sciences Faculty Publications

Heart disease stands as a leading cause of morbidity and mortality globally, presenting a significant public health challenge. Therefore, early prediction and detection are critical, leading to timely and appropriate interventions at early stages. Four ensemble tree-based algorithms were used in this study: adaptive boosting, extreme gradient boosting, random forest, and extremely randomized trees, investigating their ability to predict heart disease. Data related to heart disease clinical features was obtained from the open Kaggle Machine Learning Dataset repository. Adaptive Boosting stands out as the highest performer, achieving an average testing accuracy of 93.70%, precision of 93.71%, recall of 93.70%, and …

Extended High-Frequency Hearing And Suprathreshold Neural Synchrony In The Auditory Brainstem, Jithin Raj Balan, Sri Mishra, Hansapani Rodrigo

School of Mathematical & Statistical Sciences Faculty Publications

Elevated hearing thresholds in the extended high frequencies (EHFs) (>8 kHz) are often associated with poorer speech-in-noise recognition despite a clinically normal audiogram. However, whether EHF hearing loss is associated with disruptions in neural processing within the auditory brainstem remains uncertain. The objective of the present study was to investigate whether elevated EHF thresholds influence neural processing at lower frequencies in individuals with normal audiograms. Auditory brainstem responses (ABRs) were recorded at a suprathreshold level (80 dB normal hearing level) from 45 participants with clinically normal hearing. The recording protocol was optimized to obtain robust wave I of the …

Analysis Of Systematic Trade-Offs Between Military And Healthcare Expenditure Alongside Gdp Growth Of Select Asian And Western Exporting Economies In The 21st Century, Rahul Balamurugan, Carlos Gershenson, Preethi Nanjundan, Hiroki Sayama

Northeast Journal of Complex Systems (NEJCS)

This study explores the complexity in the trade-offs between military expenditure, healthcare expenditure, and GDP growth across select Asian nations and major weapon-exporting countries, examining how nations allocate finite resources between national security and human well-being over the past two decades. Using a systems science approach, the research integrates Granger causality testing to analyze temporal and directional relationships among GDP growth, military expenditure, and healthcare expenditure, uncovering their dynamic interdependencies. The methodology includes trend and slope analysis, Granger causality testing, outlier detection, and clustering to identify heterogeneity in resource allocation strategies. Developed, weapon-exporting nations exhibit complementary trends, with strong causality …

From Machine Learning To Human Learning: What Can Pedagogy Learn From Ai Successes, Victor L. Timchenko, Yury P. Kondratenko, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Many machine learning techniques -- including many techniques behind the current AI-based boom in machine learning -- come from the analysis of successful human learning strategies (and researchers expect that other human learning experiences can lead to even more effective AI-based systems). At this moment, so much experience have been accumulated in AI-based machine learning that it is time to start the analysis in the opposite direction -- to see what can human-based pedagogy learn from AI successes. In this chapter, we provide the first results of such an analysis -- some of which go somewhat against the current pedagogical …

Gurevich's Quizani Dialogs As An Example Of Explainable Mathematics, And How This Is Related To Quantum Space-Time Ideas That Can Speed Up Computations, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Everyone talks about the need for Explainable AI -- when, to supplement a long difficult-to-understand sequence of computational steps leading to AI's decision, we are looking for a shorter and understandable more-informal explanation for this decision. In this paper, we argue that this need is a particular case of what we call Explainable Mathematics -- when we want to supplement a long sequence of arguments and/or computations with a shorter and understandable more-informal explanation. Important instances of Explainable Mathematics are Yuri Gurevich's Quizani dialogs that help explain complex results from theoretical computer science and physicists' more-informal explanations of complex physical …

Unfortunately, The Universal Predictor Cannot Be Made Constructive, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A recent article in the Notices of the American Mathematical Society reminded the mathematics community that, under the Axiom of Choice, it is possible to have a universal predictor: if we input, into this predictor, the values of a function for all moments t < to for some to, then, for almost all to, this predictor correctly predicts the next values of this function on some interval [to, to + ε). This predictor cannot be used for actual predictions: it is based on the Axiom of Choice and is, therefore, not constructive. A natural question is: maybe it is possible to have another universal predictor, which is constructive? In this paper we show that, unfortunately, it is not possible to have a constructive universal predictor. In other words, the above universal predictor result cannot be used for actual predictions.