Applied Mathematics Commons^™

(R2165) Two Different Stages With Two Distinct Harvesting Processes, S. Vijaya, Krishika M

Applications and Applied Mathematics: An International Journal (AAM)

This study extends the classical Lotka-Volterra prey-predator model by incorporating a stage structured prey population consisting of juvenile and adult stages, while considering a single predator species governed by a Holling type II functional response. The growth of the juvenile prey depends on the adult prey population and since the juvenile prey has no reproduction capability. Two distinct harvesting strategies are introduced in the model. Constant-yield harvesting is applied to the juvenile stage of the prey population and represents a fixed amount of harvest regardless of population size. Constant effort harvesting is applied to the adult stage of the prey …

Toward Mapping Multiphase Multicomponent Mixtures With Neural Networks, Kristen L. Hallas, Melissa De Jesus, Christine J. Wu, Jianzhi Li, Jason Bernstein, Philip C. Myint

School of Mathematical & Statistical Sciences Faculty Publications

Equation of state (EOS) tables are commonly used in hydrodynamic simulations of high-pressure, high-temperature phenomena in fields like planetary science, astrophysics, and high-energy-density science. However, generating and storing EOS tables for multiphase, multicomponent mixtures over a wide range of pressures and temperatures is computationally infeasible due to their memory-intensive nature. To address this issue, we have developed a neural network-based machine learning model to predict new EOS tables for binary mixtures. In particular, a deep feedforward neural network trained on a set of ten EOS tables at particular mixture compositions is able to predict nine new (hold-out) EOS tables at …

Study Of A Nonlinear Delayed Parabolic Model For Prion Disease Dynamics With The Unfolded Protein Response, Gangadhara Boregowda, Laurent Pujo-Menjouet, Zhaosheng Feng, Michael R. Lindstrom

School of Mathematical & Statistical Sciences Faculty Publications

Prion diseases are neurodegenerative disorders characterized by the dynamic spread of misfolded toxic proteins in the brain. In this process, the normal cellular prion protein (PrPC) produced by neurons misfolds into a toxic form known as scrapie prion protein (PrPSc). These misfolded proteins propagate through the brain by converting healthy prions into their toxic form. This biological mechanism can be modeled by a system of nonlinear parabolic partial differential equations, accompanied by a nonlinear delayed integral boundary condition. Our primary objective is to establish the existence of nonnegative classical solutions to this system. Furthermore, we derive a priori estimates for …

Differential Equation Modeling For Sustainable Resource Management: A Steady-State Optimal Harvesting Approach, Iordanka N. Panayotova, Aleksei Talonov

CODEE Journal

Mathematical models based on differential equations provide a powerful framework for connecting real-world data to informed decision-making. In this work, we present a student-accessible project that uses an optimal-control framework to study the sustainable management of biological resources.

Motivated by fisheries management, we examine a predator--prey system in which harvesting decisions must balance ecological and economic considerations. The model is formulated as an optimal control problem that seeks to maximize the total discounted net revenue from harvesting. Rather than solving for the complete time-dependent harvesting trajectory, we restrict the analysis to positive controlled coexistence equilibria and characterize an interior stationary …

Taut And Dupin Submanifolds (Updated Version), Thomas E. Cecil

Mathematics and Computer Science Department Faculty Scholarship

This is an updated version of the paper [29] by the author which originally appeared in 1997. The original paper was a survey of the closely related fields of taut and Dupin submanifolds of Euclidean space, and this updated version includes many results in the field that have appeared since the publication of the original version. The emphasis is on stating results in their proper context and noting areas for future research, and relatively few proofs are given. The important class of isoparametric hypersurfaces is surveyed in detail, as is the relationship between the two concepts of taut and Dupin. …

Fast Computation And Model Order Reduction Of The Friction Stir Welding Process With Pod-Deim, Joshua Kay, Zilong Song

Mathematics and Statistics Student Research and Class Projects

Friction stir welding (FSW) is a solid-state manufacturing process widely used in joining aluminum and other metal workpieces. The FSW process can be modeled by a coupled system of non-Newtonian Navier–Stokes and heat-transfer equations. However, solving this non-linear system with high accuracy requires significant computational power. This work refines the system by introducing corrected coefficients and new treatments for boundary conditions near the tool. Then, model order reduction, including the Proper Orthogonal Decomposition (POD) and Discrete Empirical Interpolation Method (DEIM), is applied to efficiently solve the FSW system in a low-dimensional space. To enhance accuracy and effectiveness, two novel treatments …

Modeling The Clonal Rosette Composition Of A Bromeliaceae Genet: A Combinatorial Approach, Layla K. Lammers, Erin N. Bodine

Spora: A Journal of Biomathematics

Bromeliaceae, a neo-tropical plant family encompassing over 3,000 species, exhibit two modes of reproduction: sexual reproduction via flowers and seeds, and asexual reproduction via genetically identical clonal rosettes. The vegetative bodies of bromeliads form rosettes with new leaves emerging from the center and clonal rosettes emerging above a single leaf in the rosette, resulting in a genetic individual consisting of a seed-grown rosette and multiple iterations of clonal rosettes. This research develops a combinatorial model of probability that a single genetic individual will include at least n clonal rosettes when a single rosette can produce at most 1 or 2 …

The General Solution Analysis Of Homogeneous Linear Equations, Jacob Schwamb, Edward Whipple

Discovery Day - Daytona Beach

The general solution analysis of homogeneous linear equations are any systems of equations in which all constant terms are equal to zero is classified as a homogeneous linear equation. Some key characteristics of homogeneous linear equations are that there are “zero” solutions, where every system has at least a single solution where all variables are zero, also all solutions to any homogenous linear equation is linearly independent, along with having preserved homogeneity, where if any variable (x) may be added to the system, then any scalar multiple of the variable is also a solution. The General solution of any homogeneous …

Bridging Discrete And Continuous Systems: Fibonacci Sequences And Exponential Growth From Odes, Martyna Wojcik

Discovery Day - Daytona Beach

Bridging Discrete and Continuous Systems: Fibonacci Sequences and Exponential Growth from Ordinary Differential Equations       It has been observed that nature often exhibits specific patterns of growth and structure in biological systems and spiral formations. The Fibonacci sequence, defined as a discrete recursive sequence where each term is generated as the sum of the two preceding terms, “has been applied extensively to understand some natural phenomena” (Pakdemirli, 2023). In contrast, exponential growth describes a continuous process in which the rate of change of a quantity is proportional to its current value. Such behavior is modeled using differential equations that “produce solutions …

A Modeling Scenario For Cooling A Hot Vehicle In Florida, Jared Bunn, Bernadette Mullins, Elizabeth Hale, Jaeyoun Oh

CODEE Journal

This paper presents a group project assigned in a Calculus 2 course that has students work to develop, analyze, and draw conclusions about a modeling scenario for cooling a hot car. Using a modeling-first approach, instructors supported the students in class throughout the beginning of the project, enabling the groups to complete the remainder of the project on their own. Students used parameter estimation to tune their models to provided data: one for windows being up, and one for windows being down. This project provides an example of how modeling can be introduced early in a calculus course, rather than …

The Pi-Royal Tire, Erik Talvila

Journal of Humanistic Mathematics

In this humorous story, the half-wit proprietor of a tire manufacturing company thinks knowing pi to more digits will allow the production of rounder tires. An applied mathematician is recruited to fulfill a ridiculous industrial research agenda.

Waiting For The Magic: A Historical And Mathematical Study Of Queues In Disney Theme Parks, Nina E. Becket

Journal of Humanistic Mathematics

This paper, written as part of an honors research project in high school, provides an introduction to the line and queuing systems at Disney theme parks. We begin with a brief history of Disney World and its creators. We then introduce the basics of queuing theory and queuing notation. Finally, we examine modern line systems at Disney (as of 2023) and the future of its different queue systems.

Piracy, Terrorism, And The Law: Differential Equations In Hostage Situations, Gabriel Hallevy

Journal of Humanistic Mathematics

Pirates have taken the crew of an American ship hostage. They promise to release the hostages only if another pirate who is held in an American prison for commission of piracy crimes against American citizens, is released. Should the U.S. government enter into negotiations with them? Should they send armed forces and risk the hostages? Should they release the prisoner immediately and unconditionally? The article models and analyzes possible policies regarding sensitive situations involving hostages and other related risks using differential equations. The solutions are surprisingly simple, but not necessarily intuitive. Our analysis aims to demonstrate how powerful mathematics is …

A Method For Folding Origami Rectilinear Polygon Extrusions, Everett L. Jin

Journal of Humanistic Mathematics

We present a method for generating crease patterns for folding any rectilinear polygon extrusion with a specified uniform minimum edge-to-edge distance, using a rectangular sheet of paper. We demonstrate that the resulting models are efficient and watertight. In addition, they achieve optimal efficiency, meaning they attain the smallest possible scale factor from the original sheet of paper to the folded rectilinear polygon extrusion. Finally, we provide a code implementation of the method. Our study not only facilitates crease pattern design for origami artists but also offers an innovative, art-driven pedagogical tool for inspiring interest in mathematics.

The Uncertainty Principles, Lee Michael Felicetti

Mathematics & Statistics ETDs

The Heisenberg uncertainty principle is a central aspect of quantum mechanics, but also illustrates an essential quality of the Fourier transform. After Heisenberg, a variety of uncertainty inequalities emerged in the fields of physics and mathematics. In this thesis we will analyze the Heisenberg uncertainty principle in both the setting of quantum mechanics and Fourier analysis. We will then look at how the work of Heisenberg has been expanded upon in both physics and mathematics. Particularity, we will see how uncertainty principles can be applied to signal recovery and explore current research in this field.

Boundary Integral Method For A Modified Mullins–Sekerka System, Ly Le

Mathematics & Statistics ETDs

This thesis develops a boundary integral method for a modified Mullins–Sekerka system arising as the sharp-interface limit of a nonreciprocal Cahn–Hilliard model. Nonreciprocal coupling changes the classical Cahn–Hilliard structure by introducing an additional conserved field, leading to coupled elliptic and parabolic dynamics at the interface. Using matched asymptotic expansions, we formally derive the modified Mullins–Sekerka model and then apply the boundary integral method to rewrite it on the moving interface. The elliptic component is represented using the periodic Green’s function for the Laplace equation, while the parabolic component is represented using the periodic heat kernel. This method reduces the bulk …

Nerve Constructions And Mapper, Alexander Bram Fritschi

Mathematics & Statistics ETDs

Mapper is a data visualization tool commonly used in topological data analysis to study large, often high-dimensional datasets. Mapper operates through the selection of a lens function, a clustering algorithm, and a cover. The Mapper graph is constructed using the nerve of the cover after the clustering algorithm is performed; it is therefore useful to study nerves to better understand Mapper. In this thesis, we will utilize the properties of nerves to find the minimal point set that produces a given graph. We will then extend this to Mapper to determine what Mapper graphs may be constructed over a given …

Interpretable Case-Control Inference Through Log-Linear General Location Models, Zacharia Stuart

Mathematics & Statistics ETDs

This dissertation analyzes one of the few publicly available NFL injury datasets to study field type and non-contact lower-limb injuries. Field type is studied jointly with other risk factors to understand how these factors interact to affect injury risk. The data were gathered through a case-control sampling scheme, which limits direct inference on absolute injury probabilities. While not the most common approach for case-control data, this dissertation models the retrospective distribution directly through Log-Linear General Location Models (Log-Linear GLOMs). Through a log-linear structure placed on a log-odds-ratio reparameterization, the model provides directly interpretable marginal and interaction contributions to injury log-odds …

Fundamental Solutions To The Fractional Heat Operator, Jacob Flores

Math Theses

In this thesis, we are interested in showing the existence of a fundamental solution to the fractional heat operator. The fractional heat operator is a nonlocal linear operator used to model the time evolution of anomalous diffusion processes whose applicability arises in a wide variety of fields in the physical sciences, engineering, economics, and finance. Fundamental solutions to a partial differential operator are a class of generalized solutions formulated with rich mathematical analysis grounded in classes of well-behaved smooth functions referred to as test functions and their continuous linear functionals referred to as distributions. A primary tool that we use …

Cultural Traits May Replace Human Mobility Data In Forecasting Covid-19 Mortality: A Deep Learning Approach, Saif Abbas, Tamer Oraby, Michael G. Tyshenko, Samit Bhattacharyya

School of Mathematical & Statistical Sciences Faculty Publications

The COVID-19 pandemic highlighted the need for accurate epidemic forecasting to support public health decision-making. Most existing approaches depend heavily on human mobility data, while largely neglecting population behavior shaped by socio-cultural norms. In this study, we analyze daily COVID-19 mortality and Google mobility data from 72 countries during the first 130 d of the pandemic, a period characterized by high uncertainty and behavioral heterogeneity. In particular, we examine whether Hofstede’s country-level cultural dimensions can serve as latent behavioral forecasters of mortality in lieu of dynamic mobility indicators. Using 100 d for training and 30 d for forecasting, we employ …

A Bayesian-Optimized Ensemble Deep Learning Framework For Automated Detection And Classification Of Retinal Diseases In Ghana Using Oct Images, Gifty Duah, Eric Nyarko, Gideon Nana Amo, Theophilus Dwamena Frimpong, Anani Lotsi

School of Mathematical & Statistical Sciences Faculty Publications

Retinal diseases pose a significant global health challenge due to their potential to cause severe visual impairment and blindness. This study aimed to develop a robust deep learning ensemble framework for the automated detection and classification of retinal diseases from optical coherence tomography (OCT) images. This study used OCT images from WATBORG Eye Services in Ghana, including glaucoma, macular edema, posterior vitreous detachment (PVD), and healthy eyes. The data preprocessing steps included augmentation, resizing, and one-hot encoding. The dataset was divided into training (56%), validation (14%), and testing (30%) sets using stratified sampling. Six convolutional neural network (CNN) architectures, Visual …

Comparative Analysis Of Traditional And Deep Learning Time Series Architectures For Influenza A Infectious Disease Forecasting, Edmund Fosu Agyemang, Hansapani Rodrigo, Vincent Agbenyeavu

School of Mathematical & Statistical Sciences Faculty Publications

Influenza A remains a major cause of respiratory mortality worldwide, motivating accurate forecasting to support timely preparedness and resource allocation. This study presents a comparative evaluation of two traditional seasonal time series baselines, ARIMA and Holt–Winters exponential smoothing (ETS), and six deep learning (DL) architectures (Simple RNN, LSTM, GRU, BiLSTM, BiGRU, and a Transformer) for forecasting monthly Influenza A case counts in the United States. Data from January 2009 to December 2023 were analyzed, using January 2009 to December 2022 for training and January 2023 to December 2023 for out-of-sample testing. Models were tuned using a validation split and assessed …

Pseudodifferential Absorbing Boundary Conditions For Waves, Lauren Taylor

Mechanical Engineering Theses

Absorbing boundary conditions (ABCs) are required to truncate the computational domain when performing Finite Element Analyses of exterior acoustic problems where physical domain is unbounded. ABC is applied on a fictitious boundary containing the scatterer and ideally allows outgoing waves to leave without non-physical reflections. Preventing artificial reflections is essential to benefit from the accuracy of the numerical method used. Otherwise, ABC acts as a reflective surface, and artificial reflections distort the solution in the entire domain which is not recoverable by any type of refinement. Pseudodifferential ABCs were used to describe the Dirichlet-to-Neumann map which maps known boundary values …

A Categorical Framework For Modeling Genetic Drift, Taylor G. Mendes

Rose-Hulman Undergraduate Mathematics Journal

Genetic drift describes changes in allele frequencies that arise from chance sampling in finite populations. This paper develops a categorical framework for organizing the structural features of drift. Population states are modeled as objects, evolutionary transitions as morphisms, reversible transitions as groupoid morphisms, and structure-preserving comparisons between models as functors. Group actions are used to describe deterministic evolutionary operators such as mutation and selection, while orbits and fixed points identify reachable allele-frequency states and stable absorbing outcomes. Universal properties are then used to describe drift as a coherence condition connecting stochastic transitions with deterministic evolutionary maps. The resulting framework complements …

Multiplicity Of Laplacian Eigenvalues That Can Be Represented By Sum Of Two Squares Using Number Theory, Changfeng Zhou

Rose-Hulman Undergraduate Mathematics Journal

In this article, we use results of Number Theory to prove the conjecture on the eigenvalue problem of a 2D elliptic PDE proposed by P.Korman in his recent paper \cite{ref}: for any even integer $2k$, one can find an eigenvalue $N$ that can be represented as $N=a^{2}+b^{2}$, with integers $a\neq b$ with multiplicity $2k$, while for any odd integer $2k + 1$, one can find an integer $M$ that can be represented as $M=a^{2}+b^{2}$ with $a\neq b$ and multiplicity $2k+1$. In addition, the manuscript gives the formula to find those $N$'s.

A Differential Equation–Based Epidemiological Model Of Post-Operative Chronic Pain In Scoliosis Patients With Data-Driven Analysis, Paige Zhu, Padmanabhan Seshaiyer

CODEE Journal

Chronic post-surgical pain (CPSP) is a common and often overlooked complication following surgical correction of idiopathic scoliosis, impacting long-term patient wellbeing despite improvements in surgical outcomes. This project introduces a novel epidemiological framework to model the progression of CPSP using a compartmental structure. By applying a coupled system of nonlinear differential equations, we simulate pain trajectories over time and assess the effectiveness of surgical interventions. The model is implemented for a single-cohort population and extended to a two-cohort design to compare outcomes between Posterior Spinal Fusion (PSIF) and Vertebral Body Tethering (VBT) procedures. Further stratification by patient age enables us …

Cdt-1d Cnn Integration With Simpson-Sobolev Regularization For High-Frequency Options Trading: With Fem-Based Heston Option Pricing, Daniel M. Margolis, Johannes Tausch, Arthur K. Selender

Mathematics Theses and Dissertations

This dissertation presents a computational framework for high-frequency options trading that combines Cross-Data-Type 1-D Convolutional Neural Networks (CDT-1D CNN) with Simpson-Sobolev regularization for directional prediction, and finite element methods (FEM) for realistic option pricing during backtesting. The core innovation lies in developing a mathematically rigorous regularization approach that maintains the adaptability of modern deep learning while enabling accurate evaluation through stochastic volatility models. The primary contribution is the Simpson-Sobolev regularization scheme, which extends traditional Sobolev regularization by incorporating Simpson’s rule for numerical integration. This approach achieves higher-order accuracy in approximating the Sobolev norms that control function smoothness. Simpson’s rule attains …

(R2147) Effect Of Mass Variation With Log-Logistic Distribution In Perturbed Interacting Cr3bp, Abdullah Abdullah, Majhar Ali, S. K. Sahdev

Applications and Applied Mathematics: An International Journal (AAM)

This paper investigates the motion of the infinitesimal body in the perturbed restricted three-body problem where the primary is heterogeneous in shape and secondary is with modified Newtonian potential. With the use of log-logistic distribution, space-time transformation and the above-said perturbations, we determine the equations of motion and quasi-Jacobian integral. Further, we numerically perform the locations of equilibrium points, their stability, regions of motion, periodic orbits and Poincaré surfaces of section.

(R2159) A Novel Extension Of Picard’S Method For Fractional Initial Value Problems With Convergence Analysis On Finite And Infinite Intervals, Jag Mohan, Anju Sood

Applications and Applied Mathematics: An International Journal (AAM)

In recent years, fractional differential equations have emerged as powerful tools for modeling phenomena with memory and hereditary effects, owing to their non-local characteristics. These equations excel in tackling intricate problems across physics, engineering, and other fields. As analytical solutions are often infeasible, numerical methods play a vital role in their practical application. In this study, we have generalized Picard’s method to address fractional differential initial value problems with Caputo derivative, establishing an existence and uniqueness theorem applicable to both finite and infinite intervals. To substantiate our findings, we provide an example with graphical evidence demonstrating the convergence of the …

(R2175) Application Of Similarity Measures On Bipolar Complex Neutrosophic Matrices In United Nations’ Sdg-17 Using Python, T. Muthuraji, N. Krishnapraveen

Applications and Applied Mathematics: An International Journal (AAM)

The increasing complexity of decision-making environments demands mathematical frameworks capable of modeling bipolar, indeterminate, and phase-dependent uncertainty simultaneously. Bipolar Complex Neutrosophic theory provides such a structure, but the extension of similarity measures to matrix-based environments remains largely unexplored. In this study, we formally develop cosine, Dice, Jaccard, and hybrid vector similarity measures for Bipolar Complex Neutrosophic Matrices (BCNMs). Each matrix element is represented by a Bipolar Complex Neutrosophic Number (BCNN), enabling structured representation of multidimensional uncertainty within a matrix framework. We also design and implement efficient Python-based computational tools to automate similarity evaluation for BCNMs. The proposed algorithms reduce computational …