Applied Mathematics Commons^™

A Computational Approach To Periodic Orbits Of State-Dependent Delay Differential Equations, Noah Corbett

Electronic Theses and Dissertations 2020 - Present

The field of delay differential equations (DDEs) concerns the study of systems whose evolution depends on certain past states of the system. Of particular interest are the state-dependent DDEs, whose delay terms are non-constant and depend on the current state itself. In this thesis, we provide rigorous solution-finding techniques for a certain class of one-dimensional state-dependent DDEs, as well as a state-dependent delayed Van der Pol equation. This technique is inspired by the classical Picard-Lindelof theorem and is successful in proving the existence and uniqueness of orbits in such systems under certain reasonable restrictions. We then employ the Lagrange-Chebyshev interpolating …

The Digital Neuron: Neural Cellular Automata For Neural–Symbolic Translation, Nicole Assenza

SMU Data Science Review

A neural cellular automata (NCA) architecture, referred to as Pluto’s NCA, was developed to characterize bilateral communication and semantic reciprocity between symbolic representations and a spatially distributed update field. The architecture employs an encoder–automata–decoder pipeline that maps symbolic inputs into a multichannel state field and reconstructs them through agreement-driven attractor convergence within a stable semantic attractor landscape. System behavior was evaluated under controlled perturbations, including rhythmic desynchronization, graded ablations, correlated and independent noise, and percolation-based structural degradation. Quantities such as Agreement(t), internal coherence Aᵢ(t), the recovery time constant τ, and the critical percolation threshold pc were measured to assess stability, …

Emergent Dynamics In Multiplex Social Networks: Agent-Based Modeling Of Information Diffusion For Misinformation Control, Harshvardhan Prabhakar Ghongade, Anjali Ashokrao Bhadre, Shivani Agarwal, Harjitkumar Uttamrao Pawar, Harshal Subhash Rane

Northeast Journal of Complex Systems (NEJCS)

Information misrepresentation is widespread in multi-layered social networks which provide multiple avenues to communicate information. As such, it presents significant opportunities for both information integrity and public discourse to be undermined by disinformation. This paper outlines a new agent-based model, developed to capture emergent dynamics of multi-layered social networks and to help identify technical means to mitigate information misrepresentation in complex systems. A key component of this research includes a novel Multi-Layer Information Diffusion Model (MLIDM), integrating both cross-layer communication among agents, as well as heterogeneous agent behaviors and adaptive intervention strategies. Our methods employ a three-stage process to model …

Modeling Flood-Induced Cascading Disruptions In The Indian Electronics Supply Chain Using Influence Network Analysis, Surendra Orupalli, Hiroki Sayama

Northeast Journal of Complex Systems (NEJCS)

This study investigates flood induced disruptions in the Indian electronics supply chain using influence network analysis. Monsoon floods are recurring hazards that significantly impact economic activities, logistics, and industrial productivity. This study integrates district-level rainfall data (2020 to 2025) with supply chain network models to quantify cascading failures. The methodology applies rainfall thresholds (≥ 300 mm/month) to identify flood-prone districts and constructs a stochastic influence matrix representing inter-firm dependencies. Flood propagation dynamics are modeled iteratively with a propagation coefficient (α = 0.6) and convergence threshold (ε = 10⁻⁴). The resulting disruption profiles are mapped onto company-level revenues calibrated to India-specific …

Modeling Bitcoin Dynamics Using Differential Equations, Boone M. Fleenor

Departmental Honors & Graduate Capstone Projects

In this thesis, we develop and analyze two nonlinear systems of ordinary differential equations to model Bitcoin price dynamics. Analytical techniques are used to obtain exact or approximate solutions where possible. Then, numerical simulations using a fourth-order Runge–Kutta method are employed to explore system behavior beyond analytically tractable regimes. Finally, model outputs are compared to historical Bitcoin price data using normalized and resampled time series. These results suggest that deterministic models can provide meaningful insight into the structural behavior of Bitcoin markets, while highlighting the need for stochastic or time-dependent extensions for more realistic modeling.

Fractals Exploration Through Computer Visualization, Sihui Wei

SACAD: Scholarly Activities

This project explores the generation and visualization of fractals using computational methods. Several well-known fractal structures, including the Koch Snowflake, Sierpinski Triangle, Mandelbrot Set, and Julia Set, were implemented using C++ and the SFML graphics library.

The study focuses on how simple mathematical rules, when applied recursively or iteratively, can produce highly complex and self-similar structures. For geometric fractals, recursive algorithms were used to subdivide shapes and generate patterns. For complex-plane fractals, iterative formulas were applied pixel-by-pixel to determine set membership and visualize escape behavior.

The results demonstrate that small changes in parameters, such as recursion depth or iteration count, …

A Computational Study Of Taylor Approximation, Sihui Wei

SACAD: Scholarly Activities

This project investigates the accuracy of Taylor approximation using computational methods. Taylor polynomials of degree 1, 3, and 5 were applied to the functions e^x, sin x, and ln⁡(1+x), all centered at x=0.

Using C++, we generated both numerical data and graphical visualizations to analyze the absolute error∣f(x)−Tn(x)∣. The results show that higher-degree polynomials provide better approximation near the expansion point, while the error increases as the distance from the center grows.

In addition, the study reveals that the effectiveness of Taylor approximation depends not only on the polynomial degree but also on the structure of the function. In particular, …

A Dynamic Systems Framework For Customer Lifecycle Management: From Latent State Discovery To Robust Control Policy, Ali Nasirzonouzi

Northeast Journal of Complex Systems (NEJCS)

Traditional marketing often relies on static strategies that fail to capture dynamic customer behavior. This paper introduces an integrated framework to model and control the customer lifecycle, bridging the gap between empirical data and computational simulation. Using the Customer Personality Analysis dataset, we implemented a five-stage methodology. We first identified three distinct customer segments (At-Risk, Standard, High-Value) using Gaussian Mixture Models. To address the lack of longitudinal data, we calibrated a normative transition model based on customer inertia principles. Our analysis revealed that marketing effectiveness is highly state-dependent; notably, At-Risk customers exhibited a 33.5% lift when targeted with catalogs. Leveraging …

How To Effectively Trap Invasive Crayfish: A Discrete Life Stage Mathematical Model, Rini Pattison

Seaver College Research And Scholarly Achievement Symposium

The red swamp crayfish is an invasive species introduced into several streams within the Santa Monica Mountains (SMM). Crayfish predation decimates native aquatic species. The Mountains Restoration Trust (MRT) has worked to remove crayfish through regular trapping in Malibu Creek.

A prior student created a crayfish life cycle model with trapping, which we expand to better predict the efficacy of crayfish removal efforts in the SMM. We separate crayfish based upon life stages and sizes. We construct and parameterize this discrete crayfish population model with and without trapping. We use literature and crayfish removal data from MRT to fit the …

Irreversible K-Threshold Dynamics On Corona And Base-B Corona Product Graphs, Eric J. Moon, Soumya Bhoumik, Paul Flesher

SACAD: Scholarly Activities

This poster studies the irreversible k-threshold process on corona-type graph products, where a vertex becomes colored once at least k of its neighbors are colored and then remains colored permanently. We focus on corona, double corona, and base-b corona product graphs built from cycles and complete graphs, with particular attention to how graph structure affects complete activation from a minimum seed set.

A generalized reduction lemma is used to relate threshold dynamics on layered corona graphs to smaller residual graphs, yielding explicit formulas for the irreversible k-threshold conversion number on both corona and double corona families. The …

Transport Of Quantum Walks In Electric Fields, Yousef Mohammad Yousef Salah

Thesis/ Dissertation Defenses

This thesis presents an analysis of transport in one-dimensional discrete-time quantum walks (DTQWs) on the Hilbert space . Quantum walks serve as fundamental models of coherent quantum transport and exhibit ballistic spreading driven by superposition and interference. The primary focus of this work is the review and derivation of sharp maximal velocity bounds for several classes of quantum walk step operators, including the shift-coin walk, the split-step walk, and models with constant as well as position-dependent coin operators. We establish general a priori bounds that remain valid beyond the translation-invariant regime. For homogeneous models, Fourier and spectral analysis yield explicit …

Physics-Informed Neural Network Solution Of The 2d Helmholtz Equation With A Gaussian Source, Theodoros Panagiotakopoulos, Chris Velissaris, Aristotelis Nikolaos Rapsomanikis

Faculty Scholarship and Creative Works

We present a physics-informed neural network (PINN) framework for solving the complex-valued two-dimensional Helmholtz equation with a localized Gaussian source and spatially varying permittivity. Starting from Maxwell’s equations, the frequency-domain scalar Helmholtz formulation under transverse electric (TE) polarization is derived and enforced directly within the neural network loss function. The model employs a sinusoidal representation network (SIREN) architecture to capture the oscillatory nature of wave solutions and incorporates the Sommerfeld radiation condition to impose open boundary conditions. Training is performed using a hybrid collocation strategy combined with a two-stage optimization procedure consisting of Adam followed by L-BFGS. Numerical experiments in …

Digital Transformation And Market Microstructure: Analyzing The Impact Of Algorithmic Trading On National Stock Exchange Of India Price Discovery Mechanisms Through Complex Systems Theory., Mukesh Bhaskar Ahirrao, Harshal Anil Salunkhe, Vishal Sunil Rana

Northeast Journal of Complex Systems (NEJCS)

Abstract

This research examines the evolution of market microstructure at the National Stock Exchange of India (NSE) from 2020 to 2024, a period characterized by substantial growth in algorithmic trading from 35% to 44% of total trading volume. Using market microstructure data and analytical techniques grounded in complex systems perspectives, the study documents temporal patterns in price discovery, liquidity, volatility, and market efficiency associated with this digital transformation.

The analysis reveals several notable changes in market characteristics. Transaction costs improved significantly, with bid-ask spreads declining by 23.4% and market depth increasing by 18.1%. Price adjustment half-life decreased by 50%, indicating …

Analysis Of The Effects Of Magnetic Field, Heat Transfer, And Thermal Radiation On Blood Flow Through Bifurcated Artery To Enhance Tumor Treatments, Abdullahi Isah, Dauda Gulibur Yakubu, Ali Musa

Tanzania Journal of Science

The work presents the effects of some pertinent parameters on blood flow through bifurcated artery to enhance tumor treatments. Combining appropriately the basic equations, together with the fractionalized Maxwell fluid model allow us to determine the velocity, temperature and concentration of blood flow through bifurcated artery. The study adopted the Atangana-Baleanu fractional time derivative on fluid model to describe the non-Newtonian behavior of blood flow. The numerical simulations were performed using the combined Laplace transform and the method of undetermined coefficients and the results obtained with the aid of Mathcad software were simulated and presented graphically. From the graphical results, …

Complex Systems Mapping Of Fiscal Growth Dynamics At Strategic Maritime Chokepoints Using Time-Series Slopes, Rahul Balamurugan, Preethi Nanjundan, Avichal Sharma

Northeast Journal of Complex Systems (NEJCS)

This study examines how maritime and trading states allocate public resources between defence, health, and economic growth around three strategic chokepoints the Strait of Malacca, the Strait of Hormuz, and the Suez Canal. The analysis extends the classic “guns versus butter” framing by treating defence and health spending as co-evolving components of an interconnected fiscal-growth system. Using World Development Indicators data (1999-2024), trend slopes are estimated for military spending (% of GDP), healthcare spending (% of GDP), and GDP growth (annual %). Two derived indicators are computed, a defence-to-health slope ratio (military slope/health slope) and a fiscal-balance proxy (health slope …

Arum–D Nexus: Adaptive Reflexive Urban Metabolism For Complex Construction And Demolition Waste Governance In Bengaluru, Talat Naaz, Rosewine Joy

Northeast Journal of Complex Systems (NEJCS)

Urban material systems exhibit nonlinear dynamics governed by feedback, adaptation, and emergent coupling among institutions, markets, and behaviors. Construction and demolition (C&D) waste in Bengaluru is a great example of such complexity, where fragmented regulation, informal actors, and digital asymmetries coalesce into unstable waste flows and resource leakages. This study conceptualizes Bengaluru’s C&D waste system as a Complex Adaptive System (CAS), where institutional, market, behavioral, and metabolic subsystems co-evolve through nonlinear feedback interactions. A meta-analysis of secondary literature combined with benchmarking of government datasets is used to evaluate two key complexity indicators, i.e., response speed and feedback density. The advancement …

Solving Wordle Using Information Theory, Talal Aladaileh, Donald Stephens, Mallak Alqaisi, Congyu Wu

Northeast Journal of Complex Systems (NEJCS)

Wordle, a popular word-guessing game, challenges players to identify a five-letter secret word through iterative guesses and feedback on letter placement. The players must figure out the secret word within six guesses. After each guess, the letters will be color-coded based on different criteria. Optimizing the choice of guesses is critical for maximizing success within the limited attempts allowed. In this study, the application of Shannon entropy is explored as a strategy for selecting words that maximize information gain at each step of the game. By quantifying the uncertainty reduction achieved by potential guesses, this method prioritizes words that are …

A Multi-Layer Complex Adaptive System Framework For Ai-Driven Robo-Advisory Services, Jesty Mariam Philip, Mohit Boralkar, Alwin Joseph

Northeast Journal of Complex Systems (NEJCS)

The rapid integration of Artificial Intelligence (AI) into investment advisory services has changed financial decision-making, giving rise to adaptive robo-advisory systems capable of real-time analysis, personal recommendations, and autonomous portfolio optimization. Existing research evaluates these systems primarily through technological performance or investor adoption, overlooking the complex feedback-driven interactions that emerge when AI analytics, data environments, and human behavior operate together. This study addresses this gap by conceptualizing AI-enabled robo-advisors as a multi-layered Complex Adaptive System comprising historical data, real-time data, AI analytics, investor perception, and decision-making layers. A simulation model grounded in machine learning dynamics, behavioral finance, and complexity theory …

Regional Drought Modulation By Enso And Iod As Indicated By The Standardized Precipitation Index, Arpit Tiwari, Preethi Nanjundan, Tanu Sharma, Ravi Ranjan Kumar, Satyaban Bishoyi Ratna

Northeast Journal of Complex Systems (NEJCS)

Understanding the modulation of drought by large-scale ocean–atmosphere teleconnections is crucial for strengthening drought prediction and resilience in India. This study investigates the influence of the El Niño–Southern Oscillation (ENSO) and the Indian Ocean Dipole (IOD) on meteorological drought characteristics across India from 1950 to 2024 using the Standardized Precipitation Index (SPI) at a 12-month timescale. Drought events were quantified in terms of frequency, duration, severity, and intensity and linked to ENSO–IOD variability through composite, correlation, and mediation analyses. Results reveal that El Niño events consistently correspond to widespread and severe droughts, particularly over central and southern India, with drought …

Indexed Concatenation (I-Cat): A Method For Representing And Operating On Repeating Numbers*, Abigail Touma, Isaac Wolford

Campus Research Month

Our research presents Indexed Concatenation (I-Cat) notation as a structured way to represent numbers with repeating patterns, including both decimals and whole numbers. Instead of treating expressions like 0.333... or 735735735 as unstructured expansions, they are rewritten as compact repeating objects called I-Cats. The presentation demonstrates how arithmetic operations, including addition and multiplication, can be performed using hypothesized rules such as the "U = M/C" method, unpacking, and carry propagation. Examples progress from simple conversions and multiplication by integers to the multiplication of two I-Cats. A live visual demonstration will show how standard numerics transform into I-Cat form and how …