Applied Mathematics Commons^™

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 …

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 …

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 …

Analysis And Machine Learning Adaptation Of A Cognitive Model For Human Memory, Trevor Cross, Aihua W. Wood

Faculty Publications

In this paper, we use the Duolingo SLAM dataset to analyze several cognitive models of second language acquisition and develop new approaches for enhanced performance. In particular, we consider the Predictive Performance Equation and some of its underlying power laws. Leveraging insights from machine learning, we develop simple one-feature models as building blocks for combined models that match or in certain cases outperform the existing models at much reduced computational cost. In addition, a neural network with one fully connected hidden layer is constructed that outperforms all other models on sufficiently large datasets.

From Cork To Coasting: A Multi-Stage Ode Model Of Water-Rocket Flight, Viktoria Savatorova, Patryk Kustra, Ethan Dyer, Connor Carlson, Aleksei Talonov

CODEE Journal

Water rockets provide an affordable and engaging context for exploring applications of differential equations. Motivated by outreach activities conducted with undergraduate students, we develop a four-stage mathematical model of vertical water-rocket flight that is suitable for use in an ODE or mathematical modeling course. The model includes the cork-release phase, water-thrust propulsion, air-thrust propulsion with compressible and potentially choked flow, and the final ballistic stage with quadratic drag. While retaining key physical features, the model can be formulated as a system of ordinary differential equations that can be integrated numerically using tools familiar to students. We compare model predictions with …

Pinnlab: An Interactive Dashboard For Teaching Data-Driven Parameter Estimation In Differential Equations Using Physics-Informed Neural Networks, Mohan J. Parthasarathy, Padmanabhan Seshaiyer

CODEE Journal

Undergraduate instruction in ordinary differential equations (ODEs) is typically organized around the forward problem: finding solution trajectories when the governing equation and its parameters are known. In scientific practice, however, inverse problems are often more relevant, requiring unknown parameters to be inferred from noisy observations while assessing whether a proposed model is consistent with the data. We introduce PINNLab, an open-source MATLAB dashboard designed to help undergraduate students explore inverse modeling through physics-informed neural networks (PINNs). PINNLab presents PINNs as a complementary data-driven framework that connects differential equations, optimization, empirical data, and scientific machine learning. The instructional sequence is organized …

A Professional Development Course On Data-Driven Dynamical Systems At A Primarily Undergraduate Institution: Part A - Scientific Content, Alessandro M. Selvitella, Jeffrey R. Anderson

CODEE Journal

In the age of data-driven decision making, ordinary differential equations (ODEs) remain a powerful and interpretable framework for modeling dynamic processes, especially when integrated with modern tools from statistical learning and data-driven dynamical systems. Yet, general undergraduate and graduate curricula do not typically address key opportunities in data-driven dynamical systems.

This first paper in a series focuses on the mathematical and methodological core of a professional development course first developed in the academic year 2025-2026 at a Primarily Undergraduate Institution, Purdue University Fort Wayne. The curriculum developed in this course emphasized how regression, regularization, and sparse identification can be used …

Developing A Framework For Mooc Dropout: The Role Of Utilitarian And Hedonic Values In Student Retention, Bipllab Roy, Mave D'Souza, Madhura Laghane

Northeast Journal of Complex Systems (NEJCS)

Massive Open Online Courses (MOOCs) have expanded access to higher education but continue to face persistently high dropout rates, raising concerns about their long‑term effectiveness and sustainability. This study develops and empirically tests a structural framework that links utilitarian values (perceived usefulness, certificate value, time flexibility), hedonic values (enjoyment, variety and novelty, personal interest alignment), and individual characteristics (goal orientation, self‑efficacy, motivation type) to MOOC student retention. Data were collected through a structured questionnaire administered to 200 MOOC learners from Christ University, Lavasa Campus, and analyzed using Structural Equation Modelling (SEM) in AMOS. The results show that goal orientation, self‑efficacy …

Machine Learning For Modeling In An Elementary Differential Equations Class, Nathan Albin, Andrew G. Bennett, Abhinav Chand

CODEE Journal

Mixing machine learning with modeling is an area of increasing importance. This paper presents a lesson where students model a spring-mass system both using traditional analysis with linear damping and using machine learning to learn the damping from real data. The machine learning is implemented in a Jupyter notebook hosted on Google Colab, allowing students to train the neural network without requiring the students to carry out coding. Students get experience with how machine learning can fail, how it can work, and the time and data requirements for machine learning to succeed, and are asked to apply this knowledge to …

A New Approach To Generate Combinatorial Patterns In Logical Analysis Of Data And Its Application To Predict College Retention, Salihah Ahmed E. Jaafari

Theses and Dissertations

Student retention and degree completion remain central challenges for higher-education institutions, with significant implications for student success, institutional effectiveness, and public accountability. While advances in predictive analytics have enabled earlier identification of students at risk of withdrawal, many commonly used machine learning approaches suffer from limited interpretability, constraining their practical usefulness for advising, intervention, and policy decision making. This dissertation addresses the problem of predicting student persistence by developing and evaluating optimization based, interpretable classification models within the Logical Analysis of Data (LAD) framework. Building on existing LAD formulations, this research introduces two novel pattern generation models, the Best Term …

The Butterfly Effect In Economics: Exploring Chaos With A Simple Financial Model, Sandy Hs Herho

CODEE Journal

Wild swings in financial markets need not result from external shocks like earthquakes or wars—they can emerge from deterministic chaos. This article introduces kalimusada, an open-source Python library that lets students and instructors explore this phenomenon through a simple three- equation model of financial dynamics. The model couples interest rates, investment, and prices through nonlinear feedback, generating bounded but unpredictable oscillations characteristic of chaos. Tiny differences in starting conditions—smaller than any measurement could detect—grow exponentially until two initially identical economies follow completely different paths. The library provides ready-to-use tools for visualizing this “butterfly effect” in economics, computing divergence metrics, and …

Sparc Grade 2 Integrated Coding-Math Lessons, Spatial Activities And Robot Coding (Sparc)

Spatial Activities and Robot Coding (SPARC)

SPARC-Math Grade 2 Integrated Coding-Math Lessons with related Utah Core Standards and lessons for Spatial and Programming Language and Geometry.

Using Provided Guided Notes In Coordinated Introductory First-Year Mathematics Courses, Jennifer L. Huber

Mathematics Dissertations

The goal of this study is to investigate how standardized guided notes shape instructional practices and student engagement in coordinated introductory first-year college mathematics courses at a large public university. The researcher explored three multi-section introductory mathematics courses with overlapping learning objectives. Each course required students to purchase a student workbook as part of the instructional materials for the class. The instructors taught primarily from the workbook containing guided notes created by a former coordinator of the course. The researcher used a mixed-methods approach. Instructors and students participated in surveys, class observations and provided class meeting notes. Instructors shared additional …

(R2084) Clarification Of Mathematical Criteria For Conservative Forces And Vector Fields With A Point Singularity In Engineering Studies, Nelli Aleksandrova

Applications and Applied Mathematics: An International Journal (AAM)

The paper deals with a comprehensive analysis of existing mathematical criteria for conservative vector fields including algebraic and geometrical interpretation of Extended Green’s Theorem. Special attention is devoted to the fields containing a point singularity, which represents the primary challenge both in teaching Calculus and in carrying out scientific research in general in computer engineering. Examples of real-world physical fields such as gravitational, electric and magnetic ones are used to appreciate the proposed clarification while teaching various interchangeable disciplines such as mathematics, physics and engineering. Other examples from scientific research literature are provided as well to support the necessity of …

Code, Paper, Scissors: What To Cut In The Age Of Ai, Dmitry Kondrashov

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Mentoring First-Year Stem Students Through Collaborative Research In The Haynes Scholars Program, Alex Capaldi, Laura Tipton

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Expanding Ode Examples: Introducing Gene Regulation Dynamics Through Hill Functions, Maila Hallare, Jane M. Santamore

CODEE Journal

Gene regulation is a fundamental biological process that controls gene expression. It can explain phenomena such as cell differentiation, circadian rhythms, disease progression, and metabolic control, among many others. Despite their importance in mathematical biology, gene regulation models are rarely featured in traditional ODE textbooks, which focus mainly on examples from engineering, physics, chemistry, and population biology. This article introduces gene regulation dynamics as a valuable addition to ODE curricula, presenting the models from a mathematical perspective and building on properties of the Hill function. These models deepen the understanding of biology-inspired ODE applications, provide accessible research opportunities for students, …

Teaching Statistical Literacy Through An Excel Class Project, Patricia Berchiolli, Omar Babun Codorniu

Faculty and Staff Publications & Presentations

Lynn University’s core curriculum, The Dialogues, enables students to develop critical thinking, communication, and innovation skills. As part of this curriculum, Introductory Statistics introduces students to key statistical concepts while showing how they can be applied in real-world situations using Excel. The highlight of the course is the Statistics Excel Project, where students create their own dataset with a mix of quantitative and qualitative variables. To keep the focus on learning statistical techniques rather than data collection, students use Excel’s random number generator for quantitative data, which also avoids the need for IRB approval. From there, they calculate statistical measures, …

Potential Pitfalls In Visual Models Of Tipping Points - And How To Fix Them, Jonathan Dechert, Svetlana Gurevich, Stefan Heusler

Northeast Journal of Complex Systems (NEJCS)

Visual models play a crucial role in both science and science communication. However, the distinction between mere analogies and mathematically sound graphical representations is not easy and can be misunderstood not only by laypeople but also within academic literature itself. Moreover, even when the graphical representation exactly corresponds to the mathematical model, its interpretation is often far from obvious. In this paper we discuss the potential landscape visualization commonly used for tipping points in the context of nonlinear dynamics and reveal potential pitfalls, in particular when distinguishing bifurcation induced tipping (B-tipping) from noise-induced tipping (N-tipping).

We propose new visualization techniques …

Project-Based Learning With Odes: Modeling Straw Rocket Motion With Air Resistance, Viktoria Savatorova, Ethan Dyer, Aleksei Talonov

CODEE Journal

This paper presents a hands-on project that guides students through building and validating a mathematical model of projectile motion. The project starts with the idealized case of motion under gravity without air resistance and then introduces air drag : first as a linear force, and then as a nonlinear quadratic force, with the Reynolds number providing the justification for the quadratic model. Students perform experiments with vertical and angled launches, capturing and analyzing motion data using video analysis software. Vertical launch data allows parameter estimation via least squares fitting of the nonlinear drag model, yielding values for initial velocity and …

Stability Analysis And Extension Of A Discrete-Time Dynamical System For Scaffolded Learning, Kris H. Green, Bernard Ricca

Northeast Journal of Complex Systems (NEJCS)

Van Geert and Steenbeek [16] proposed a coupled, delayed, discrete-time deterministic dynamical system to model scaffolded learning. We provide a detailed analysis of their model, whose global dynamics are complicated by the presence of intersecting lines of non-isolated, nonhyperbolic fixed points. We also interpret some of the trajectories in the system that have interesting dynamics in the context of the teacher-student interactions, and propose an extension to the model that simultaneously collapses the lines of fixed points to single, isolated points and is easily interpreted. These results provide the foundation for guiding the collection and integration of experimental data to …

Leveraging Usage Of Ai In Education: Knowledge, Attitude And Behavioral Analysis On Students, Bipllab Roy, Purnendu Bikash Acharjee, Rohit Kumar Sharma, Ruptaheen Kramsapi, Shruti P

Northeast Journal of Complex Systems (NEJCS)

The paper explores the possible advantages and drawbacks of artificial intelligence (AI) on sustainability, with an emphasis on using AI to positively achieve SDGs. The study finds a significant vacuum in the literature on the association between knowledge, attitudes, and behaviors towards the use of AI tools and techniques in education and demographic characteristics (sex, age, education level, area of study, and city of origin). The purpose of this research is to close this knowledge gap and advance our understanding of how these demographic factors affect the integration of AI in educational environments. The study specifically aims to comprehend how …

The Other Side Of The Equation: De-Simplification, A Prerequisite For Calculus, Stephen L. Brown

ACMS Conference Proceedings 2005

No abstract provided.

Setting Up Students For Success: Analysis Of Effectiveness Of Mathematics Placement, Jeff Carvell, Jason Ho, David Klanderman, Sarah Klanderman

University Faculty Publications and Creative Works

How do we e!ectively and equitably place students into math classes in a way that provides them the best chance of success? As many higher education institutions veer away from placement based on standardized testing, many departments are seeking placement alternatives that will properly support students. Additionally, math placement determines not only a student’s mathematics courses but also influences their progress in related fields, including physics, chemistry, engineering, and more. This paper will describe three di!erent existing placement systems at each of our liberal arts institutions as well as the a!ordances and constraints of each approach. Further, we analyze data …

Fibonacci Product Cordial Labeling Of Generalized Triangular Snake And Other Graphs, Anthony S. Arial, Soumya Bhoumik

SACAD: Scholarly Activities

A Fibonacci Product Cordial labeling is an assignment of Fibonacci numbers to the vertices of a graph in a way that determines edge labels based on the product of vertex labels, modulo 2. The labeling is valid if the number of edges labeled 0 and 1 differ by at most 1. If a graph allows such a labeling, it is called a Fibonacci Cordial graph. We explore various snake graphs and Helm graphs that can have this type of labeling.

Predicting Battery Efficiency: A Theoretical Approach, Haiti Schafers

SACAD: Scholarly Activities

The Kansas ElectroRally Races continuously demand faster electric vehicles (EV). When designing a vehicle one of the most important components is the battery. This study was designed to find the following; What is the optimal throttle percentage to run a 36V 20Ah (768Wh) LiFePO4 Rechargeable Battery Pack[1].

The optimal throttle percentage was found with methods detailed in a different section. Based on the performance of the older cars this study choose to focus on 50% and 70%, while obtaining other relevant data to further that inference.

Ultrasonic Sensor-Based Sound Synthesis Using Raspberry Pi Pico W, Niraj Jaishwal

Williams Honors College, Honors Research Projects

At the intersection of Human Computer Interaction and digital art, this project transforms simple motion into musical expression. It explores an interactive real-time sound synthesis system using ultrasonic sensors to generate continuous audio. The objective is to design a system that maps physical distances into musical parameters such as pitch and amplitude, which will create a responsive audio environment. Two ultrasonic sensors are used in combination with the Raspberry Pi Pico W microcontroller running CircuitPython and Adafruit Audio Hat for real-time sound output. One sensor controls the pitch of the generated tone, while the other controls volume. This enables expressive …

Mathematics In Amusement Parks, Kacey Laumann

Honors Projects

This project focuses specifically on the Walt Disney World Park, Magic Kingdom. I started by collecting data through the MyDisneyExperience app. By recording the data, I was then able to create polynomial functions to the fifth the degree. Each attraction received a function which allowed me to predict the wait times for that attraction. Then, by graphing the functions and analyzing the graph using calculus the “best time” and “worst time” to go to the attraction were found. After the analysis the information is used to build a unique schedule for a guest. Then the guest receives this schedule after …

Shevtsov: Growing Pains And Growing Gains: The Arizona Experience, Tynan Lazarus, Jane Shevtsov

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

From Ecology To Modeling Apps - Newts, Crayfish, And Slopes, Timothy Lucas

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.