Physical Sciences and Mathematics Commons^™

Some New Dynamic Inequalities Involving The Dyanamic Hardy Operator With Kernels, Doaa M. Abdou, Haytham M. Rezk, Afaf S. Zaghrout, Samir H. Saker

Al-Azhar Bulletin of Science

In this article, by using dynamic Jensen’s inequality and its reverse, we prove some forms of dynamic inequalities involving the dynamic Hardy’s operator. As special cases of our results, we obtain refinements of some well-known Hardy-type inequalities and Hardy-Hilbert’s inequality for double integrals. Our findings are the generalization of some results in the literature.

Enumeration Of Lattice Paths With Restrictions, Vince White

Electronic Theses and Dissertations

Lattice path enumeration, through the lens of Catalan numbers, plays a crucial role in combinatorics. This thesis delves into enumerations of some of the most common lattice paths – north-east paths, up-down paths, and Dyck paths – with restrictions applied. The first restriction is counting north-east lattice paths that only cross the diagonal line, y=x, once. The second form of lattice paths with restrictions is up-down paths that cross the x-axis exactly once and fall to a fixed depth of k. While working through this module, a novel proof for a known integer sequence was used, then applied to generate …

Advanced Mathematical Graph-Based Machine Learning And Deep Learning Models For Drug Design, Farjana Tasnim Mukta

Theses and Dissertations--Mathematics

Drug discovery is a highly complicated and time-consuming process. One of the main challenges in drug development is predicting whether a drug-like molecule will interact with a specific target protein. This prediction accelerates target validation and drug development. Recent research in biomolecular sciences has shown significant interest in algebraic graph-based models for representing molecular complexes and predicting drug-target binding affinity. In this thesis, we present algebraic graph-based molecular representations to create data-driven scoring functions (SF) using extended atom types to capture wide-range interactions between targets and drug candidates. Our model employs multiscale weighted colored subgraphs for the protein-ligand complex, colored …

The Effect Of The Expanded Child And Dependent Care Tax Credit On Maternal Labor Supply, Abby Letocha

Honors Theses

Policies that subsidize childcare have many potential economic benefits such as mitigating the high cost of childcare, incentivizing families to have more children, increasing paid childcare participation, and increasing parental labor supply. In this paper, I focus on the effect of childcare subsidies on maternal labor supply through a tax policy expansion. The Child and Dependent Care Tax Credit (CDCTC) is the primary federal childcare subsidy in the United States, and it was temporarily expanded in 2021 under the American Rescue Plan Act. This expansion increased the generosity of the credit and made it fully refundable for the 2021 tax …

A Limit Order Book Model For High Frequency Trading With Rough Volatility, Yun S. Chen-Shue

Graduate Thesis and Dissertation 2023-2024

We introduce a financial model for limit order book with two main features: First, the limit orders and market orders for the given asset both appear and interact with each other. Second, the high frequency trading (HFT, for short) activities are allowed and described by the scaling limit of nearly-unstable multi-dimensional Hawkes processes with power law decay. The model eventually becomes a stochastic partial differential equation (SPDE, for short) with the diffusion coefficient determined by a Volterra integral equation governed by a Hawkes process, whose Hurst exponent is less than 1/2, which makes the volatility path of the stochastic PDE …

On A Fully Coupled Nonlocal Multipoint Boundary Value Problem For A Dual Hybrid System Of Nonlinear Q -Fractional Differential Equations, Ahmed Alsaedi, Martin Bohner, Bashir Ahmad, Boshra Alharbi

Mathematics and Statistics Faculty Research & Creative Works

A new class of nonlocal multipoint boundary value problems involving a dual hybrid system of nonlinear Riemann-Liouville-type q-fractional differential equations is studied in this paper. Existence and uniqueness results for the given problem are derived by applying the Leray-Schauder nonlinear alternative and the Banach contraction mapping principle. Examples are presented for illustrating the obtained results. The work established in this paper is a useful contribution to the existing literature on q-fractional differential equations. Some interesting special cases are also discussed.

Critical Point Approaches To Nonlinear Square Root Laplacian Equations, Martin Bohner, Giuseppe Caristi, Shapour Heidarkhani, Amjad Salari

Mathematics and Statistics Faculty Research & Creative Works

This work is devoted to the study of multiplicity results of solutions for a class of nonlinear equations involving the square root of the Laplacian. Indeed, we will use variational methods for smooth functionals, defined on reflexive Banach spaces, in order to achieve the existence of at least three solutions for the equations. Moreover, assuming that the nonlinear terms are nonnegative, we will prove that the solutions are nonnegative. Finally, by presenting an example, we will ensure the applicability of our results.

Solving Robert Wilson’S 𝑡 ≠ 2 Conjecture On Graham Sequences, Krishna Rajesh

HMC Senior Theses

Ron Graham's sequence is a surprising bijection from the natural numbers to the non-prime integers, which is constructed by looking at sequences whose product is square. In this thesis we will resolve a 22-year-old conjecture about this bijection, by construction of explicit sequences in a modified number theoretic context. Additionally, we will discuss the history of this problem, and give computational techniques for computing this bijection, levering ideas from linear algebra over the finite field of two elements.

The Dual Boundary Complex Of The Moduli Space Of Cyclic Compactifications, Toby Anderson

HMC Senior Theses

Moduli spaces provide a useful method for studying families of mathematical objects. We study certain moduli spaces of algebraic curves, which are generalizations of familiar lines and conics. This thesis focuses on, Δ^(r,n), the dual boundary complex of the moduli space of genus-zero cyclic curves. This complex is itself a moduli space of graphs and can be investigated with combinatorial methods. Remarkably, the combinatorics of this complex provides insight into the geometry and topology of the original moduli space. In this thesis, we investigate two topologically invariant properties of Δ^(r,n). We compute its Euler characteristic and …

Exploring Sigmoidal Bounded Confidence Models With Mean Field Methods, Tian Dong

HMC Senior Theses

Mathematicians use models of opinion dynamics to describe how opinions in a group of people change over time, which can yield insight into mechanisms behind phenomena like polarization and consensus. In these models, mathematicians represent the community as a graph, where nodes represent agents and edges represent possible interactions. Opinion updates are modeled with a system of differential equations (ODEs). Our work focuses on the sigmoidal bounded confidence model (SBCM), where agents update their opinion toward a weighted average of their neighbors' opinions by weighting similar opinions more heavily. Using tools developed in physics (mean-field theory), we derive a continuity …

A Combinatorial Model For Affine Demazure Crystals Of Levels Zero And One, Samuel Spellman

Electronic Theses & Dissertations (2024 - present)

The symmetric and non-symmetric Macdonald polynomials are special families of orthogonal polynomials with parameters q and t. They are indexed by dominant, (resp. arbitrary) weights associated to a root system and generalize several well-known polynomials such as the Schur polynomials, Jack polynomials, Hall-Littlewood polynomials, etc. There are two well-known combinatorial models for computing these polynomials: a tableau model in type A, due to Haglund, Haiman and Loehr, and a type-independent model due to Ram and Yip, based on alcove walks.

Crystals bases are an important construction encoding information about Lie algebra representations. It turns out that there is an interesting …

An Unsupervised Machine Learning Algorithm For Clustering Low Dimensional Data Points In Euclidean Grid Space, Josef Lazar

Senior Projects Spring 2024

Clustering algorithms provide a useful method for classifying data. The majority of well known clustering algorithms are designed to find globular clusters, however this is not always desirable. In this senior project I present a new clustering algorithm, GBCN (Grid Box Clustering with Noise), which applies a box grid to points in Euclidean space to identify areas of high point density. Points within the grid space that are in adjacent boxes are classified into the same cluster. Conversely, if a path from one point to another can only be completed by traversing an empty grid box, then they are classified …

Solid Angle Measure Approximation Methods For Polyhedral Cones, Allison Fitisone

Theses and Dissertations--Mathematics

Polyhedral cones are of interest in many fields, like geometry and optimization. A simple, yet fundamental question we may ask about a cone is how large it is. As cones are unbounded, we consider their solid angle measure: the proportion of space that they occupy. Beyond dimension three, definitive formulas for this measure are unknown. Consequently, devising methods to estimate this quantity is imperative. In this dissertation, we endeavor to enhance our understanding of solid angle measures and provide valuable insights into the efficacy of various approximation techniques.

Ribando and Aomoto independently discovered a Taylor series formula for solid angle …

Pairs Of Quadratic Forms Over P-Adic Fields, John Hall

Theses and Dissertations--Mathematics

Given two quadratic forms $Q_1, Q_2$ over a $p$-adic field $K$ in $n$ variables, we consider the pencil $\mathcal{P}_K(Q_1, Q_2)$, which contains all nontrivial $K$-linear combinations of $Q_1$ and $Q_2$. We define $D$ to be the maximal dimension of a subspace in $K^n$ on which $Q_1$ and $Q_2$ both vanish. We define $H$ to be the maximal number of hyperbolic planes that a form in $\mathcal{P}_K(Q_1, Q_2)$ splits off over $K$. We will determine which values for $(D, H)$ are possible for a nonsingular pair of quadratic forms over a $p$-adic field $K$.

Rado Numbers For Two Systems Of Linear Equations, Anthony Glackin

Electronic Theses and Dissertations

For any positive integer n and any equation E of either the form x1+x2+· · ·+xn = x0 or x1 + x2 + n = x0, the two-color Rado number R2(E) is the least integer such that any 2-coloring of the natural numbers 1 through R2(E) will contain a monochromatic solution to E. Let Ek be a system of k equations of the aforementioned form, where Ei represents the ith equation in Ek and the set I = {1, 2, . . . , k} is the set of indices of these equations. This thesis shows that the two-color Rado …

Contrastive Learning, With Application To Forensic Identification Of Source, Cole Ryan Patten

Electronic Theses and Dissertations

Forensic identification of source problems often fall under the category of verification problems, where recent advances in deep learning have been made by contrastive learning methods. Many forensic identification of source problems deal with a scarcity of data, an issue addressed by few-shot learning. In this work, we make specific what makes a neural network a contrastive network. We then consider the use of contrastive neural networks for few-shot learning classification problems and compare them to other statistical and deep learning methods. Our findings indicate similar performance between models trained by contrastive loss and models trained by cross-entropy loss. We …

Julia Limiting Directions Of Quasiregular Maps, Julie Marie Steranka

Graduate Research Theses & Dissertations

In this dissertation, we study the set of Julia limiting directions of quasiregular maps. The work combines the study of dynamics of quasiregular maps and applications of nonlinear potential theory to quasiregular maps. Our main result shows that the set of Julia limiting directions of a transcendental-type $K$-quasiregular map $f:\R^n\to \R^n$ must contain a component of a certain measure, depending on the dimension $n$, the maximal dilatation $K$, and the order of growth of $f$. In particular, we show that if the order of growth is small enough, then every direction is a Julia limiting direction. The main tool in …

Problems In Graph Theory With Applications To Topology And Modeling Rna, Rayan K. Ibrahim

Theses and Dissertations

In this thesis, we explore four projects. In the first project, we explore $r$-neighbor bootstrap percolation on a graph $G$. We establish upper bounds for the number of vertices required to percolate in the case that $r=2$ for particular classes of graphs. In the second project, we study the structure of graphs with independence number two. We prove a lower bound on the number of edges of such graphs, related to an upper bound on the number of edges in a triangle-saturated graph, and give a sufficient forbidden induced subgraph condition for independence number two graphs. In the third project, …

Extending Natural Mates In Euclidean 3-Space And Applications To Bertrand Pairs, Yun Myung Oh, Alexander Navarro

Faculty Publications

In Euclidean 3-space, a family of curves, the co-successor, is motivated and then introduced in relation to the natural mate. A complete characterization of co-successors is proved, followed by an application of the co-successor towards describing Bertrand curves and their mates.

Steklov Eigenvalue Problems On Nearly Spherical And Annular Domains, Nathan Philip Schroeder

CGU Theses & Dissertations

We consider Steklov eigenvalues on nearly spherical and nearly annular domains in d dimensions where d is any given positive integer. By using the Green-Beltrami identity for spherical harmonic functions, the derivatives of Steklov eigenvalues with respect to the domain perturbation parameter can be determined by the eigenvalues of a matrix involving the integral of the product of three spherical harmonic functions. By using the addition theorem for spherical harmonic functions, we determine conditions when the trace of this matrix becomes zero. These conditions can then be used to determine when spherical and annular regions are critical points while we …

Exploring Loss Functions In Machine Learning, Yujie Wang

CGU Theses & Dissertations

The loss function plays a critical role in machine learning. It is fundamental in training, evaluating, and optimizing machine learning models, directly impacting their effectiveness and efficiency in solving specific tasks. We explore three new loss functions and their applications. Softmax Cross-Entropy Loss, stands as a prevalent choice in neural network classification tasks. It treats all misclassifications uniformly. However, multi-class classification problems often have many semantically similar classes. We should expect that these semantically similar classes will have similar parameter vectors. We introduce a weighted loss function, the tree loss as a drop-in replacement for the cross entropy loss. The …

Multiple Imputation For Robust Cluster Analysis To Address Missingness In Medical Data, Arnold Harder, Gayla R. Olbricht, Godwin Ekuma, Daniel B. Hier, Tayo Obafemi-Ajayi

Mathematics and Statistics Faculty Research & Creative Works

Cluster Analysis Has Been Applied To A Wide Range Of Problems As An Exploratory Tool To Enhance Knowledge Discovery. Clustering Aids Disease Subtyping, I.e. Identifying Homogeneous Patient Subgroups, In Medical Data. Missing Data Is A Common Problem In Medical Research And Could Bias Clustering Results If Not Properly Handled. Yet, Multiple Imputation Has Been Under-Utilized To Address Missingness, When Clustering Medical Data. Its Limited Integration In Clustering Of Medical Data, Despite The Known Advantages And Benefits Of Multiple Imputation, Could Be Attributed To Many Factors. This Includes Methodological Complexity, Difficulties In Pooling Results To Obtain A Consensus Clustering, Uncertainty Regarding …

Modeling Inflation Using A Fast Fourier Transform (Fft), Blake Smith

Williams Honors College, Honors Research Projects

This paper utilizes a Fast Fourier Transform (FFT) algorithm to construct a trigonometric interpolant for the Consumer Price Index (CPI), which is then differentiated and used to obtain a continuous function for “instantaneous” (i.e., month-wise) inflation, as opposed to a 12-month percent-change. Fourier coefficients are analyzed to investigate underlying periodicities in the newly constructed function. This metric does not hold significant predictive value but it may prove helpful in retroactive analysis of inflation trends.

Neutrosophie : Un Cadre Interdisciplinaire Pour Une Meilleure Compréhension De L’Incertitude Et De L’Indétermination, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

La neutrosophie, un cadre philosophique et scientifique relativement récent introduit dans les années 1990,1 propose une approche robuste pour comprendre et modéliser l’indétermination, la contradiction et l’incertitude. Explorons les concepts fondamentaux de la neutrosophie et quelques diverses applications à travers plusieurs disciplines, notamment les mathématiques, la physique, la sociologie, la psychologie et la biologie.

Nuevos Tipos De Conjuntos Suaves: Conjunto Hiper Suave, Conjunto Suave Indeterminado, Conjunto Hiper Suave Indeterminado Y Conjunto Suave De Árbol, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This is an updated article, where we present the definitions and practical applications of the Soft Set and its extensions to the Hyper Soft Set, Indeterminate Soft Set, Indeterminate Hyper Soft Set, and Tree Soft Set.

Análisis Neutrosófico De Las Actitudes Hacia La Máquina De Experiencia De Nozick, Maikel Yelandi Leyva Vázquez, Jesús Estupiñán Ricardo, Noel Batista Hernández, Ricardo Sánchez Casanova, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This paper presents a neutrosophic study on attitudes towards virtual technology, with a special focus on the "Experience Machine". Employing a methodology that incorporates the theory of indecision and neutrality into response analysis, the perceptions of 50 university students were investigated. Opinions and feelings were transformed into neutrosophic numbers to calculate contradiction and conduct a detailed analysis. The results indicate a predisposition towards agreement with virtual technology, although notable indecision and a significant presence of neutral attitudes were detected. The correlations and clustering methods applied reveal a complex dynamic of attitudes, challenging binary interpretations and highlighting the need for more …

Two-Dimensional Steady Squeezing Flow Over A Vertical Porous Channel With Free Convective Heat/Mass Transfer And Invariable Suction, Zeeshan, Waris Khan, Taoufik Saidani, Florentin Smarandache, Muhammad Shahid Khan, Hamdi Ayed, M. Modather M. Abdou

Branch Mathematics and Statistics Faculty and Staff Publications

This research reports on the combined effects of heat and mass transfer (HMT) under the influences of the Soret and Dufour in natural convection steady 2D magnetohydrodynamic flow through the boundary layer in a porous vertical tube or duct. The current study is motivated by the significant applications of HMT in engineering processes such as casting and welding. The goal of this framework is to explore the assisting and opposing movements with HMT above a vertical porous channel under the influence of invariant suction and fluid dissipation which have not been reported in the earlier studies. The governing flow equations …

Plausible Photomolecular Effect And Microwave In Phase Transition Of Water As A New Dawn For Renewable Energy And Ensemble-Holistic Approach To Health Management, Victor Christianto, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Interaction among light and water molecules have baffled scientists for many decades, and even centuries. In this regards, photons in the visible spectrum, where bulk water normally doesn't absorb light, can surprisingly cleave off large water clusters from the water-vapor interface, according to a recent study by Tu and Chen (2023). This discovery, termed the "photomolecular effect," opens exciting possibilities for not only revolutionizing renewable energy but also paving the way for a more integrated-ensemble approach to health management (cf. Smarandache & Christianto, 2010; Tu & Chen, 2023; Tu et al., 2024). In a sense, other than with green or …

Neutrosophic Discrete Geometric Distribution, Rehan Ahmad Khan Sherwani, Sadia Iqbal, Shumaila Abbas, Muhammad Saleem, Muhammad Aslan, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Uncertainty, vagueness, and ambiguity surround us in many real-life problems and, therefore, always remain under consideration for researchers to quantify them. This study proposed neutrosophic discrete probability distribution as a generalization of classical or existing probability distributions, named neutrosophic geometric distribution. Case studies presented in the paper will help understand the concept and application of the proposed distribution. Several properties are derived, like the proposed distribution’s moment, characteristic, and probability-generating functions. Furthermore, the newly proposed distribution derives properties from the reliability analysis, such as survival function, hazard rate function, reversed hazard rate function, cumulative hazard rate function, mills ratio, and …

Finding A Basic Feasible Solution For Neutrosophic Linear Programming Models: Case Studies, Analysis, And Improvements, Maissam Ahmad Jdid, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Since the inception of operations research, linear programming has received the attention of researchers in this field due to the many areas of its use. The focus was on the methods used to find the optimal solution for linear models. The direct simplex method, with its three basic stages, begins by writing the linear model in standard form and then finding a basic solution that is improved according to the simplex steps until We get the optimal solution, but we encounter many linear models that do not give us a basic solution after we put it in a standard form, …