Mathematics Commons^™

Mathematics In Everyday Life: Exploring Practical Applications And Real-World Impact, Priyant Banerjee, Arshad Bhat

Himalayan Research Papers Archive

Mathematics is an essential part of daily life and influences decisions and problem-solving in various aspects of life. This study explores how mathematical concepts are embedded in daily activities such as financial management, cooking, travel planning, and technological interactions. We will show how arithmetic, algebra, geometry, and statistics are applied in real life to improve decision-making, efficiency, and productivity. Findings indicate that people with higher mathematical literacy solve problems more efficiently, especially in budgeting, as accurate calculations minimize financial mistakes and facilitate long-term financial planning. In cooking, proportional reasoning ensures the accuracy of recipes, thus providing consistent culinary results. Travel …

Long-Time Asymptotics For The Kadomtsev–Petviashvili I Equation With Small Initial Data, Samir Donmazov

Theses and Dissertations--Mathematics

We study the initial value problem for the Kadomtsev--Petviashvili I (KP I) equation (u_t + 6uu_x + u_xxx)_x = 3u_yy with small initial data belonging to a subspace of the energy space for the KP I equation. We establish the long-time asymptotics for solutions of the KP I equation using the inverse scattering transform formalism developed by Zhou. Within this framework, the inverse problem for the KP I equation is formulated as a nonlocal Riemann--Hilbert problem (RHP) in two spatial dimensions. As part of the asymptotic analysis, we determine the long-time behavior of the …

Multi-Valued Variational Inequalities For Variable Exponent Double Phase Problems: Comparison And Extremality Results, Siegfried Carl, Vy Khoi Le, Patrick Winkert

Mathematics and Statistics Faculty Research & Creative Works

We prove existence and comparison results for multi-valued variational inequalities in a bounded domain Ω of the form (Formula presented.) where A:W1,H(Ω)→W1,H(Ω)∗ given by (Formula presented.) for u∈W1,H(Ω), is the double phase operator with variable exponents and W1,H(Ω) is the associated Musielak–Orlicz Sobolev space. First, an existence result is proved under some weak coercivity condition. Our main focus aims at the treatment of the problem under consideration when coercivity fails. To this end we establish the method of sub–super-solution for the multi-valued variational inequality in the space W1, H(Ω) based on appropriately defined sub- and super-solutions, which yields the existence …

Line Graphs Of Directed Graphs I, Vaidy Sivaraman, Daniel Slilaty

Mathematics and Statistics Faculty Publications

We determine the forbidden induced subgraphs for the intersection of the classes of chordal bipartite graphs and line graphs of acyclic directed graphs. This is a first step towards finding the forbidden induced subgraphs for the class of line graphs of directed graphs.

Diophantine Avoidance, Number Fields, And Quadratic Forms, Sehun Jeong

CGU Theses & Dissertations

Diophantine avoidance has been studied by several authors in recent years. This term refers to effective results on existence of points of bounded size (where size is measured by norm or height, depending on the context) in a given algebraic set avoiding some specified subsets. The application of avoidance conditions allows to understand how ``well distributed" are points of bounded size in a given set. If it is possible to find them outside of some prescribed collection of subsets of the set in question, then it suggests that they are evenly distributed, in some appropriate sense. Our first result investigates …

Optimizing Decision-Making In A Cerebral Palsy Model Using Reinforcement Learning, Richard Ampah

Pitzer Senior Theses

This study presents an original interdisciplinary investigation into how reinforcement learning (RL) can model motor and cognitive defects and potentially improve motor and cognitive functions in individuals with cerebral palsy (CP), a non-progressive neurological disorder that impairs movement and adaptability. Integrating computational neuroscience and machine learning, the research applies policy gradient methods and Markov Decision Processes (MDPs) to simulate adaptive learning in agents with and without CP-related constraints.

The central aim is to compare the cumulative rewards of optimal policies, derived from value iteration, and human-like learning policies using the REINFORCE algorithm, both with and without the Bellman baseline. The …

The Anatomy Of A Reconstruction: From Fourier Space To Image Recovery In Computed Tomography, Charlotte P. Maurer

Honors Theses

This thesis develops the mathematical foundations of computed tomography (CT) reconstruction through the lens of harmonic analysis. Beginning with the Schwartz class, we introduce the Fourier transform and its role in expressing the Radon transform and its inversion via a fractional Laplacian. After constructing the Radon transform in general dimension R^d, we specialize to the cases d = 2 and d = 3, demonstrating explicit inversion formulas and the associated instability in lower dimensions. For its computational advantages, we study filtered back-projection using classical low-pass filters (Ram-Lak, Shepp–Logan, Cosine, Gaussian) and formulate a discrete reconstruction algorithm grounded in …

Integrating Sentiment Analysis In Predictive Models: A Comparative Study On Game Popularity On Steam, Khaleefa Alhemeiri

CMC Senior Theses

Over the past decades, the gaming industry has managed to evolve into a multi-billion-dollar enterprise. Gaming platforms such as Steam foster unprecedented amounts of engagement among players worldwide daily. In this thesis, we investigate the effect of incorporating sentiment-driven metrics, specifically YouTube view counts and positive reviews, into predictive models for game popularity. In addition, by comparing our linear regression sentiment-based approach to the Bayesian hierarchical folded normal model used by De Luisa et al. (2021), we can understand the many differences, strengths, and limitations of each methodology. In our thesis, we focus on three games. Each is of varying …

Mathematical Modeling Of Cancer Tumor Evolution, Belgacem Al-Azem

CGU Theses & Dissertations

Several mathematical models of cancer tumor evolution are presented. The aim of this endeavor is to write a thesis that fosters numerical and analytical understanding of certain mathematical models in the area of oncology as well as to offer potential predictive tools of therapeutic and clinical applicability. In this work, we give a panoramic view of the cancer phenotype, for to understand cancer, we need to "see" its biophysics. Indeed, the discovery of oncogenes led to the thinking that cancer tumor is perhaps a genetic disease. But the role of angiogenesis and other microenvironment-related discoveries (such as the role of …

Golden Spirals Everywhere?, John Adam

Mathematics & Statistics Faculty Publications

The article explores different types of spirals, including Archimedean, hyperbolic, and logarithmic spirals, with a focus on the golden ratio and golden spirals. It discusses the misconception that golden rectangles and spirals can be found in various natural and man-made objects, emphasizing the importance of understanding the properties of logarithmic spirals. The text provides mathematical equations for logarithmic spirals and poses questions for readers to explore the concept further. The author, John Adam, invites readers to engage in Fermi Questions and submit ideas for consideration.

Novel Generative And Language Model Architectures With Applications, Edison Mucllari

Theses and Dissertations--Mathematics

This dissertation investigates novel architectures to address fundamental challenges in machine learning, particularly focusing on transformer models, recurrent neural networks, GAN and continual learning and their applications in natural language processing and computer vision. We propose the Neumann-Cayley Gated Recurrent Unit (NC-GRU), which leverages a Neumann series-based Scaled Cayley transformation to maintain orthogonal weight matrices, effectively mitigating exploding gradients problems while improving long-term memory retention across prediction tasks. We demonstrate the practical applications of NC-GRU by implementing our proposed architecture into an autoencoder to derive neural molecular fingerprints. Building upon these advancements, we turn our attention to the transformer architecture, …

A New Simulation Framework For Analyzing Neutrosophic Data In Experimental Design, Muhammad Aslam, Nasrullah Khan, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

A recent simulation-based classical analysis has been developed for interval data. However, a review of the literature indicates that these existing simulations have notable limitations and fail to conform to the neutrosophic statistical framework. In this paper, we propose a novel simulation process designed to analyze neutrosophic data within an appropriate and rigorous neutrosophic framework. We demonstrate that the proposed simulation is more comprehensive and aligns closely with the principles of neutrosophic theory. The results will be obtained through simulation and compared with those of existing methods, with the expectation that the proposed approach provides substantial improvements and is better …

Weakly Reversible Deficiency Zero Realizations Of Polynomial Dynamical Systems, Neal Ryan Buxton

Graduate Theses, Dissertations, and Problem Reports (ETD)

Weakly reversible, deficiency zero (WR0) systems form a large class of polynomial ODEs modeling chemical reaction networks. Their behavior is exceptionally stable: they have unique positive steady states, which are locally asymptotically stable (they are also conjectured to be globally asymptotically stable). This powerful result (the Deficiency Zero Theorem) applies to a large class of high dimensional, nonlinear polynomial dynamics, and is independent of the choice of parameters in the model. In this dissertation we present two algorithms that expands the scope of the Deficiency Zero Theorem to:

1. Networks with WR0 realizations, i.e. networks that are not necessarily WR0 …

0th Order Solutions Of The Wavefunctions For The Quantum Elliptical Box And Microstrip Antenna, Nishtha Tikalal

Honors Undergraduate Theses

For a quantum particle confined to a two-dimensional elliptical box or electromagnetic wave in a microstrip antenna, geometrical and boundary condition interplay result in a spectrum of spatial patterns. Due to the asymmetrical nature of the ellipse, we are faced with continuous symmetry reductions, leaving both degenerate and nondegenerate solutions. Here, we present a complete derivation of an analytical solution and visualizations of the fundamental wavefunctions for both Dirichlet and Neumann boundary conditions respectively corresponding to the quantum elliptical box and the elliptical microstrip antenna.

We demonstrate that the eigenmodes, governed by eccentricity, directly correspond to the modal field distributions …

The Inverse Scattering Transform For The Nonlinear Schrödinger Equation, Ivan Casas-Rocha

Honors Undergraduate Theses

The Nonlinear Schrödinger (NLS) Equation, iψ_t + 1/2 ψ_xx ± |ψ|²ψ = 0, is a nonlinear partial differential equation which is used to model several physical phenomena including nonlinear effects inside optical fibers and the formation of rogue waves in shallow water. It is particu- larly difficult to study solutions to this equation due to the nonlinearity, and the nonlinearity leads to incredibly interesting solutions not found in linear PDEs such as solitons. In this thesis, we highlight two methods of obtaining solutions to the (NLS) equation: the Inverse Scattering Transform and the Dressing Method. Furthermore, …

Solvability Of Stochastic Linear-Quadratic Optimal Control Problems Under Partial Stabilizability Conditions, Al-Sadh Rahman Imadh

Honors Undergraduate Theses

Optimal Control Theory, a branch of Control Theory, is applicable in fields such as engineering, operations research, and economics. Stochastic Optimal Control deals with noisy systems and data using Ito’s formulation. Given a noisy system and a cost functional, the goal is to find a control that will minimize the cost. This thesis focuses on linear quadratic stochastic optimal control, and we explore state equations that are not stabilizable. We first address measurability concerns arising from the semigroup property of the state trajectory. The notions of partial stability and partial stabilizability are introduced, and we formulate their corresponding Lyapunov and …

Quantized Average Agreement Algorithms With Error Correction For Digraphs, Shuaib A. Mughal

Honors Undergraduate Theses

Multi-agent systems have become more and more prevalent as technology increasingly gets integrated into our daily lives. Some of these technological systems are large in size; for example, the smart grid where multiple devices are used to monitor and control different aspects of the energy grid. Another example is a team of autonomous systems deployed for a specific task. When these systems are spatially distributed, an important component of distributed algorithms is the ability for the agents to reach consensus on the global state of the system. Reaching agreement enables the spatially distributed agent make decisions or determine the next …

Centralized Deep Reinforcement Learning For Homogeneous Multi-Component Maintenance Optimization, Joseph W. Wittrock

Theses and Dissertations

This thesis explores an application of reinforcement learning (RL) in maintenance optimization. Recent advances in hardware-accelerated computation and deep learning have made RL a powerful tool for solving optimization problems which are too complex for traditional methods. Maintenance optimization involves improving the efficiency and effectiveness of maintenance activities through data-driven approaches, ultimately reducing costs and increasing asset availability. Making informed maintenance decisions is crucial to long-term sustainability.

A desirable maintenance policy maximizes a utility signal while minimizing the cost of maintenance. Techniques in sequential decision making such as dynamic programming (DP) and RL have found success in optimizing these maintenance …

Bayesian Networks For Safety-Critical Systems, Joseph Mietkiewicz

Theses

This thesis addresses a operational challenge in modern industrial operations: the increasing complexity of systems and the consequent cognitive burden on operators. As industrial technologies advance, the human-computer interface has become the primary conduit for information flow, playing a pivotal role in operational decision-making. However, the proliferation of data often leads to information overload, potentially compromising rather than enhancing operator performance. This research explores an approach to this pressing issue through the application of Bayesian networks as decision support systems in safety- critical scenarios. Our study employs a multi-faceted approach, combining theoretical modeling with empirical testing. Through collaboration with industry …

Facial Reduction Of Semidefinite Relaxations Of Binary Quadratic Problems, Ian Sugrue

Graduate Research Theses & Dissertations

A standard technique for bounding a combinatorial optimization problem is to form a semidefinite relaxation of the problem by "lifting" the problem into a higher dimensional space of symmetric matrices. For some combinatorial problems such as maximum k-cluster, the resulting semidefinite relaxation is not strictly feasible. In the context of the maximum k-cluster problem, we present an equivalent, strictly feasible semidefinite relaxation by way of the dimensionality reduction technique known as facial reduction. We provide a recipe to form this semidefinite relaxation directly, bypassing the lifting and facial reduction steps entirely. We then present our work on generalizing this result, …

Golden Spirals Everywhere? Solutions For Fermi Questions, January 2025, John Adam

Mathematics & Statistics Faculty Publications

The article discusses different types of spirals, including Archimedean, hyperbolic, and logarithmic spirals, with a focus on the golden ratio and golden spirals. It addresses the misconception that golden rectangles and spirals can be found in various natural and man-made structures, emphasizing the importance of understanding the properties of logarithmic spirals. The article provides mathematical explanations and solutions for questions related to pitch angles and self-similarity in logarithmic spirals, using examples like the nautilus shell and an ammonite-like stone. It concludes by referencing additional sources for further exploration of the golden ratio and golden spiral myths.

Exploring Concepts Of Hyperfuzzy, Hyperneutrosophic, And Hyperplithogenic Sets (I), Florentin Smarandache, Takaaki Fujita

Branch Mathematics and Statistics Faculty and Staff Publications

This work investigates the evolution of traditional set theory to address complex and ambiguous real-world phenomena. It introduces hierarchical hyperstructures and superhyperstructures, where superhyperstructures are formed by iteratively applying power sets to create nested abstractions. The focus is placed on three foundational set-based frameworks—Fuzzy Sets, Neutrosophic Sets, and Plithogenic Sets and their extensions into Hyperfuzzy Sets, HyperNeutrosophic Sets, and Hyperplithogenic Sets. These extensions are applied to various domains, including Statistics, TOPSIS, K-means Clustering, Evolutionary Theory, Topological Spaces, Decision Making, Probability, and Language Theory. By exploring these generalized forms, this paper seeks to guide and inspire further research and development in …

Exploring Concepts Of Hyperfuzzy, Hyperneutrosophic, And Hyperplithogenic Sets (Ii), Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This paper delves into the advancements of classical set theory to address the complexities and uncertainties inherent in real-world phenomena. It highlights three major extensions of traditional set theory - Fuzzy Sets [288], Neutrosophic Sets [237], and Plithogenic Sets [243] - and examines their further generalizations into Hyperfuzzy [106], HyperNeutrosophic [90], and Hyperplithogenic Sets [90]. Building on previous research [83], this study explores the potential applications of HyperNeutrosophic Sets and SuperHyperNeutrosophic Sets across various domains. Specifically, it extends f undamental c oncepts such as Neutrosophic Logic, Cognitive Maps, Graph Neural Networks, Classifiers, and Triplet Groups through these advanced set structures …

Forestfuzzy, Forestneutrosophic, Forestplithogenic, And Forestrough Set, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Concepts such as Fuzzy Sets [30, 72], Neutrosophic Sets [53, 55], Rough Sets [37], and Plithogenic Sets [59] have been extensively studied to address uncertainty, with diverse applications across various fields. Recently, TreeFuzzy, TreeNeutrosophic, TreePlithogenic, and TreeRough Sets have been defined [15]. This work examines their extensions: ForestFuzzy, ForestNeutrosophic, ForestPlithogenic, and ForestRough Sets.

Hyperplithogenic Cubic Set And Superhyperplithogenic Cubic Set, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Concepts such as Fuzzy Sets [23, 47], Neutrosophic Sets [32, 33], and Plithogenic Sets [35] have been extensively studied for addressing uncertainty, with diverse applications across numerous fields. Building on the Plithogenic Set, the HyperPlithogenic Set and SuperHyperPlithogenic Set have also gained recognition [15]. A Plithogenic Cubic Set integrates interval-valued and single-valued fuzzy memberships, augmented by multiattribute aggregation using plithogenic structures. This paper defines the HyperPlithogenic Cubic Set and SuperHyperPlithogenic Cubic Set and explores related concepts such as the HyperPlithogenic Fuzzy Cubic Set, HyperPlithogenic Intuitionistic Fuzzy Cubic Set, and HyperPlithogenic Neutrosophic Cubic Set.

Local-Neutrosophic Logic And Local-Neutrosophic Sets: Incorporating Locality With Applications, Florentin Smarandache, Takaaki Fujita

Branch Mathematics and Statistics Faculty and Staff Publications

The study of uncertainty has been a significant area of research, with concepts such as fuzzy sets [87], fuzzy graphs [51], and neutrosophic sets [58] receiving extensive attention. In Neutrosophic Logic, indeterminacy often arises from real-world complexities. This paper explores the concept of locality as a key factor in determining indeterminacy, building upon the framework introduced by F. Smarandache in [73]. Locality refers to processes constrained within a specific region, where an object or system is directly influenced by its immediate surroundings. In contrast, nonlocality involves effects that transcend spatial or temporal boundaries, where changes in one location have direct …

Artificial Intelligence, Society 5.0 And Smart City Adaptation Initiatives For Businesses: An Integrated Approach, Inês A.M. Gil, Fernando A.F. Ferreira, Neuza C.M.Q.F. Ferreira, Florentin Smarandache, Momtaj Khanam, Tugrul Daim

Branch Mathematics and Statistics Faculty and Staff Publications

The unprecedented migration of populations to urban areas has created major challenges for municipalities and service providers. To address these issues, decision-makers must embrace smart city and Society 5.0 paradigms, both of which focus on adaptability and sustainable development. Artificial intelligence (AI) plays a pivotal role by expanding service capacity, enabling automation, and processing vast data to align urban development with the UN’s Sustainable Development Goals (SDGs). This paper develops a multi-criteria analysis system designed to support decision-making in complex socio-technological contexts. Using cognitive mapping and the Decision-Making Trial and Evaluation Laboratory (DEMATEL) technique within a neutrosophic environment, the study …

Modelado Causal De La Violencia En El Área Metropolitana De Guayaquil: Evidencia Neutrosófica Y Análisis De Condiciones Necesarias, Maikel Y. Leyva Vázquez, Lorenzo Cevallos-Torres, Alfonso Guijarro Rodríguez, Douglas Iturburu-Salvador, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

La violencia urbana en el área metropolitana de Guayaquil plantea desafíos persistentes para el bienestar social y la gobernanza. Este estudio integra tres fuentes de evidencia para clarificar qué factores son necesarios y qué niveles mínimos habilitan altos niveles de violencia. Primero, se aplica stance detection con representación neutrosófica para construir un prior (T, I, F) de la literatura. Luego, evaluamos condiciones necesarias mediante fsQCA (inclusión difusa 𝑌⊆𝑋, umbral de consistencia ≥ 0.90) y estimamos umbrales (bottlenecks) con NCA (CE-FDH, CR-FDH, permutaciones). Usamos la misma matriz fuzzy en ambos métodos para asegurar coherencia. Con una encuesta a 179 jóvenes de …

Ppg-Based Sleep Stage Classification Using Pulse Wave Feature Fusion And Explainable Ai, Florentin Smarandache, Satyasri Akula, Saleh Alzahrani, Farrukh Arslan, Amir Ijaz

Branch Mathematics and Statistics Faculty and Staff Publications

Sleep monitoring plays a crucial role in understanding and managing various health conditions, including sleep disorders, cardiovascular diseases, and mental health. Traditional sleep monitoring methods rely on Electroencephalography (EEG) and Polysomnography (PSG) in clinical settings. However, these methods are expensive, difficult to administer, and unsuitable for home-based monitoring. In recent years, photoplethysmogram (PPG) has emerged as a promising noninvasive technology that is widely used in wearable devices and holds great potential for sleep assessment. Yet, most current sleep monitoring methods rely on deep learning models, which are inherently "black-box" and challenging in the clinical decision-making process. In this paper, we …

Soft Directed N-Superhypergraphs With Some Real-World Applications, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This paper introduces the Directed Soft Super Hyper Graph, a unified framework for modeling complex, multi-layered directed networks. It combines directionality, recursive hyperstructure, and soft-set parameterization to address the integration of Soft Super HyperGraphs and Directed SuperHyperGraphs, which remains largely unexplored. The paper provides formal definitions, core operations, and real-world examples, such as urban infrastructure and transportation networks, to demonstrate the framework's effectiveness in managing deep hierarchies and uncertain relationships simultaneously.