Applied Mathematics Commons^™

The Evolution Of Global Gold And Copper Trade Networks, Oleksandr Hulianskyi

Northeast Journal of Complex Systems (NEJCS)

Gold and copper have emerged as two of the most vital commodities in global trade. Despite serving distinct purposes, their international trade networks reveal interconnected patterns, critical to understanding the dynamics of global economics. This paper studies these attributes and their evolution during the last 36 years for both metals and finds correlations between them. The first part of the research is focused on the sustainability of networks through efficiency and robustness indexes; the second part is dedicated to interconnectedness – the Louvain and Bayesian SBM algorithms, partition, and modularity instruments are used. Community detection algorithms provide valuable insights into …

Exploring The Potential Of Large Language Models (Llms) To Simulate Social Group Dynamics: A Case Study Using The Board Game "Secret Hitler", Kaj Hansteen Izora, Christof Teuscher

Northeast Journal of Complex Systems (NEJCS)

This study explores the capacity of large language model-powered agents to simulate human-like behavior in multi-agent social systems. Using Secret Hitler — a hidden-role board game centered on trust, deception, and strategic communication — we evaluate how LLM agents navigate dynamic group interactions. Our findings show that agents exhibit human-like behaviors, including strategic temporal adaptation, contextual reasoning, and complex social cognition such as theory of mind and implicit coordination. Notably, 85% of agent decisions factored in at least two other players’ mental states, highlighting their capacity for multi-agent mental state inference. However, they struggled with key aspects of human gameplay, …

Coarse-Graining Spiking Reservoirs: Reducing Reservoir Size While Preserving Critical Dynamics, Tucker X. Mastin, Christof Teuscher

Northeast Journal of Complex Systems (NEJCS)

We propose an extension of renormalization into the domain of spiking neural networks, thereby providing a novel framework for coarse-graining neural networks without disrupting their critical properties. The proposed coarse-graining technique merges neurons and synaptic connections based on a graph-theoretic distance derived from synaptic weight strength and is configured to effectively prune the reservoir size while preserving the scale-free spiking dynamics indicative of criticality. Criticality in spiking neural networks may provide information-theoretic advantages by optimizing information processing and sensitivity to input. Using time-series prediction benchmarks, we demonstrate that networks operating at criticality exhibit up to 32% higher prediction accuracy before …

Evolutionary Dynamics Of Artificial Agents: Exploration And Learning In Games, Brian Mintz

Dartmouth College Ph.D Dissertations

The natural world abounds with examples of complex behavior in humans and many other species. Evolutionary game theory is a powerful mathematical framework to understand the origins of many such behaviors like cooperation. Since these behaviors are often selected against initially, understanding why they are so widespread has been a longstanding question. Rather than assuming agents' rationality, like in traditional game theory, this approach studies the mutation and selection of strategies themselves. However most behavior is neither perfectly rational nor entirely determined by genetics. This dissertation works to bridge the gap between these two perspectives by analyzing models where individuals …

Notes On The Invariance Of Tautness Under Lie Sphere Transformations, Thomas E. Cecil

Mathematics and Computer Science Department Faculty Scholarship

An embedding ϕ : V → Sⁿ of a compact, connected manifold V into the unit sphere Sⁿ ⊂ Rⁿ⁺¹ is said to be taut, if every nondegenerate spherical distance function d_p, p ∈ Sⁿ, is a perfect Morse function on V , i.e., it has the minimum number of critical points on V required by the Morse inequalities. In these notes, we give an exposition of the proof of the invariance of tautness under Lie sphere transformations due to ´Alvarez Paiva. First we extend the definition of tautness of submanifolds of S …

Multi-Level Differentiable Moving Particles With Partition Of Unity, Jinjin He

Dartmouth College Master’s Theses

Representing implicit geometry with intricate features has long been a challenge. Recent advances in Implicit Neural Representations (INRs) have shown great promise in applications such as 3D reconstruction, inverse rendering, and dynamic surface evolution. These methods leverage neural networks to model complex shapes continuously, offering advantages in resolution and flexibility over traditional discrete representations. Despite their success, efficiently handling fine geometric details and evolving dynamic scenes remains an open problem.

We introduce a differentiable moving particle representation based on the multi-level partition of unity (MPU) to model dynamic implicit geometries efficiently. Our approach employs two types of particles—feature particles and …

Orchestrating Complexity: The Art Of Virtual Leadership In System Modelling, Vijay Kumar Sonawane, Bipllab Roy, Purnendu Bikash Acharjee, Indu Pv

Northeast Journal of Complex Systems (NEJCS)

This paper explores the dynamics of virtual leadership within global remote work environments, focusing on the application of complex system modelling to understand and enhance leadership efficacy. The application of computational modelling has been a regular feature in economics, science and technology fields, however its application in virtual leadership with linkage to sport leadership appears to be a novel concept. Adopting a multidisciplinary approach, this paper incorporates Game Theory as a conceptual framework to make the leadership model more relevant and applicable that can offer simpler understanding of complex play of leadership drivers. The model incorporates five key leadership dimensional …

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.

Stability Insights From Modeling Chronic Myelogenous Leukemia, Giovani Thai

Master's Theses

This thesis centers around a model for chronic myelogenous leukemia (CML) as it behaves under imatinib treatment, a common medication for CML patients, and the anti-leukemia immune response. The dynamics are represented with a system of nonlinear delay-differential equations first constructed by Kim et al. in 2008, capturing population changes of T-cells and various CML growth stages. We investigate stability in both the clinical and mathematical sense. Through numerical simulations, we computationally incorporate a supplementary treatment plan to determine its effectiveness in aiding immune response and medication in achieving remission and full elimination. The primary goal is to conduct a …

Property Testing Ai: An Efficient Frontier, Paul Sopher Lintilhac

Dartmouth College Ph.D Dissertations

In this dissertation, we take a step towards addressing the major problem of a lack of standardized and rigorous approaches to testing and evaluation of AI systems. Taking inspiration from both the fields of Property Testing and Property Based Testing (for programs), we develop a novel taxonomy of partially overlapping classes of properties of AI systems, including simple properties, compound properties, higher order properties, data relation properties, and architecture-utility properties. We argue that this taxonomy categorizes a diverse set of AI traits -- including accuracy, fairness, robustness, monotonicity, point-wise and global privacy properties, sensitivity, and more -- according to the …

Filling Gaps In Scientific Data Sets Using Physics Informed Neural Networks: A Case Study In Velocity Fields, Ellen Saunders

Master's Theses

Gaps in scientific data sets are a persistent issue for researchers in a variety of fields, and while nothing makes up for missing out on real data, well-simulated synthetic data can be a useful tool. In the world of image processing, machine learning techniques have become quite sophisticated at taking an image with a missing component and filling in that space with something believable. The aim of this thesis is to take machine learning techniques similar to what gets used in image processing and repurpose them to infill gaps in scientific data sets in a realistic manner. This thesis compares …

Adaptive Volatility Forecasting Models, Jeffrey K. Tan

Master's Theses

In finance, risk is often quantified by volatility, and computing accurate volatility forecasts — while vital to financial decision making — remains one of the most challenging tasks in financial modeling. This thesis, motivated in part by the Black-Scholes-Merton Model and its limitations, adopts a statistical approach to volatility forecasting. The two main models of interest are the Exponential Weighted Moving Average (EWMA) model and the GARCH(1,1) model. Specifically, this work expands upon a potential adaptive lambda algorithm for EWMA models first proposed by Bernard Bollen (2014), and this work also utilizes the Momentum of Predictability (MoP) to generate adaptive …

Equiangularity From Compatible Orthobiangularity, Tyler J. Myers

Theses and Dissertations

An equiangular tight frame (ETF) is an equal norm sequence of vectors in a Hilbert space whose coherence achieves equality in the Welch bound. Such sequences necessarily have minimal coherence and thus are, in some sense, as "spread out" in space as possible. ETFs have a variety of applications, such as compressed sensing and waveform design. The main problem in the study of ETFs is determining the pairs (D, N) for which an ETF with N vectors in a D-dimensional space exists. Real ETFs are moreover equivalent to a special subset of a well-studied class of graphs known as strongly …

Machine Learning And Optimization For Intelligent Decision-Making, Elson Cibaku

Dissertations

This dissertation presents a series of innovative machine learning and optimization model designs that address complex operational challenges across logistics and power systems. By integrating advanced neural architectures with robust optimization techniques, the work delivers scalable solutions designed to improve efficiency, reliability, and decision-making in dynamic and real-world environments. The first study introduces a two-stage approach to effective vaccine distribution. This framework tackles the capacitated vehicle routing problem by combining adaptive clustering techniques with reinforcement learning and a simulated annealing pickup policy. Through extensive computational experiments, the approach demonstrates substantial improvements in routing efficiency, reducing both computational time and logistical …

From Neural Networks To Large Language Models: Innovations In Financial Ai, Mathematical Reasoning, And Structured Data Representation, Junyi Ye

Dissertations

This dissertation explores the evolution and application of artificial intelligence techniques across three critical domains: financial modeling, mathematical reasoning, and structured data analysis. The dissertation presents seven research projects that chart a progression from specialized neural architectures to sophisticated large language models (LLMs), contributing novel methodologies and frameworks at each stage.

In the financial domain, the research first introduces TS-Mixer, a MLP-based architecture for time-series forecasting that captures both feature relationships and temporal dependencies through a simple yet effective design, outperforming more complex models in S&P500 index prediction. The dissertation then presents DySTAGE, a dynamic graph representation learning framework that …

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 …

Nonlinear Phenomena Of Discrete Wave Systems With Nonlocal Coupling, Austin E. Marstaller

Mathematics Theses and Dissertations

There is a long and rich story of nonlinear dynamics in discrete lattices. A particular well studied model is the discrete Nonlinear Schr\"odinger Equation, with many applications, most notably in nonlinear optics.

Recent studies have explored dynamics in scenarios where coupling amongst the array elements is nonlocal. The interest in particular is that in the longwave approximation, it leads to fractional diffraction. In this contribution, we consider two long range interaction lattice wave models, the fractional discrete nonlinear Schr\"odinger equation and the Long Range Ablowitz-Ladik system. By use of perturbation methods, asymptotics and numerical simulations, we present results on modulational …

Constructing The Soliton Wave Structure And Stability Analysis To Generalized Calogero–Bogoyavlenskii–Schiff Equation Using Improved Simple Equation Method, Mina Fahim, Hamdy Mohamed Ahmed, Islam Samir Soliman, Mohamed Elsaid, Kamal Hassan Eldib

Basic Science Engineering

In this work, we investigated the (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation, which models long wave propagation in shallow water and plays a significant role in fluid mechanics and plasma physics. Using the improved simple equations method, we obtained various solutions, including dark, bright, and singular solitons, and combinations of singular periodic solutions and exponential rational solutions. Additionally, we performed a linear stability analysis to examine the stability properties of these wave solutions. To further illustrate their characteristics during propagation, we provided 3D and contour plots for some opted wave solutions.

Applications Of The Mathieu Groups And Information Theory In Dna Encoding Functions, Juan C. Nava Jr

Theses and Dissertations

A foundational idea in mathematics lies in breaking down existing components into their bare fundamentals. As evidenced by prime numbers and composites, we learn this idea at an early age. Categorizing these broken-down components into their simplest form allows mathematicians to construct proofs from emergent patterns. John Conway’s Atlas of Finite Groups in the 1990s was particularly concerned with the categorization of structures known as groups. There are certain axioms a group must adhere to, which amount to the retention of symmetry; ultimately a group helps us to better understand symmetric actions performed on a set with a binary operation. …