Applied Mathematics Commons^™

Knowledge-Enhanced Feature Store For Operational Ml And Llm Workflows, Saurabh Suman

Master's Theses

Modern machine learning (ML) organizations rely on feature stores to manage training and production data, yet as catalogs grow to thousands of features, semantic management becomes the bottleneck: discovery, governance, and metric selection remain largely manual. This thesis proposes and evaluates a knowledge-enhanced feature store that augments a dual-plane store with a hybrid knowledge layer—relational provenance, a lineage graph, semantic vector retrieval, and an online cache—and an LLM-driven multi-agent layer for profiling, matching, grounded metric recommendation, and pipeline generation. A working prototype was evaluated across three task families—semantic profiling, feature-matching retrieval, and discovery—using classification, ranking, and operational metrics computed by …

The Inverse Elasto-Acoustic Problem, Patrick Grice

Dissertations

A stable and numerically efficient boundary integral method formulation of the elasto-acoustic problem is presented, based on Fourier analysis. The method generalizes well to multiple scattering. The Frechet derivative of the elasto-acoustic problem with respect to shape perturbations is derived, and geometric flow theory is used to design stable numerical methods for the simulation of moving boundaries. The shape derivative is used to define a regularized Gauss-Newton algorithm for shape fitting of elasto-acoustic scatterers.

Geometric Convergence And State-Space Decompositions For Stochastic Gradient Descent Markov Chains, Philip Zaleski

Dissertations

No abstract provided.

Electronic Structure Discretization And Compression Using Diagonal Basis Sets, Casey Lee Dowdle

Dartmouth College Ph.D Dissertations

Numerically solving the electronic structure problem is a fundamentally difficult problem due to the exponential growth in the dimension of the Hilbert space as the system size increases. In order to solve problems at a chemically relevant accuracy, both the choice of basis set and numerical method are important factors that are intrinsically connected.

In this thesis, we study the discretization and resulting compression of electronic Hamiltonians using diagonal basis sets. A diagonal basis set approximately diagonalizes the matrix and tensor representations of the one- and two-body potentials. This can reduce storage, simplify matrix-vector products, and lower the complexity of …

Dynamics Of Microscale Droplets In Respiratory Airways, Md Shamser Ali Javed

Mathematics Theses and Dissertations

We investigate trajectories of microscale evaporating droplets in a stagnation point flow near a wall of a respiratory airway. The configuration is motivated by the problem of advection and deposition of microscale droplets of respiratory fluids in human airways during transmission of infectious diseases such as tuberculosis and COVID-19. Laminar boundary layer equations are solved to describe the air flow while the equations of motion of the droplet include contributions from gravity, aerodynamic drag, and Saffman force. Evaporation is accounted for at both the droplet surface and the wall of the respiratory airway and is shown to delay droplet deposition …

On A Classification Of Goy/Sabra-Type Shell Models Of Turbulence, Austin Ho

Theses and Dissertations

Since their inception in the 1970s, shell models have proved to be a fruitful testbed for studying theories of turbulence. This thesis considers variations of two classical shell models: the Gledzer–Ohkitani–Yamada (GOY) model and the Sabra model. Up to conjugacy classes, we classify all possible GOY-type and Sabra-type models. We numerically approximate the scaling exponents for the newly identified models and develop their mathematical well-posedness.

A Simulation Assessment Of The 'Law Of One Price', Caleb Wilkins

Computational and Data Sciences (MS) Theses

The ‘law of one price’ is an appealing notion regarding pricing of tradeable commodities that are priced in different currencies. It states that the prices of the same good in different markets should be equal after adjustment for exchange rates and that equality should persist through exchange rate fluctuations.

My research simulates the market conditions that should precipitate the ‘law of one price.’ Data was obtained from the simulated trade between algorithmic artificial intelligence agents that operated under induced boundedly rational market behaviors. Trade took place in two initially separate markets, a high-price market with a higher equilibrium price and …

Parameter Dependent Chen-Fliess Series And Their Nonrecursive Interconnections, Natalie T. Pham

Electrical & Computer Engineering Theses & Dissertations

In control theory, a Chen-Fliess functional series is a weighted sum of iterated integrals constructed from a given set of input functions. Such series can be used to represent nonlinear input-output systems. In applications, they have been employed to characterize interconnected nonlinear systems, to solve system inversion and tracking problems, and to design predictive and adaptive controllers.

Distributed parameter systems exhibit spatial dependence along with temporal dependence. Such systems are typically represented in terms of partial differential equations. In control theory, there appears to be no existing method for representing the input-output map of a distributed system via a Chen-Fliess …

Criticality In A Heterogeneous Neutron Transport Rod Model, Samuel Kaleb Crowford

All Graduate Reports and Creative Projects, Fall 2023 to Present

This work studies the stochastic behavior of neutron populations in a one-dimensional rod model using Monte Carlo simulation. The first part of this project reproduces the computational results of Dumonteil, Horton, Kyprianou, and Zoia (2025) by independently implementing the Monte Carlo algorithm described in their article, with the asymptotic behavior of the first moment analyzed in relation to the dominant eigenvalue and adjoint eigenfunction of the neutron transport operator. The model is then extended to a heterogeneous setting by introducing a central region where fission is suppressed. A global expectation over initial positions and directions is used to estimate the …

Convergence Theory For Deep And Multi-Grade Neural Architectures, Lei Huang

Mathematics & Statistics Theses & Dissertations

This dissertation studies two complementary notions of convergence arising in modern neural network models: the convergence of recursively constructed neural network architectures and the convergence of optimization algorithms used for training neural-network-based image restoration models. The first part develops a convergence theory for deep neural networks (DNNs) viewed as recursively generated sequences of functions. Within an Lp framework motivated by statistical learning, sufficient conditions are established under which increasing-depth neural network sequences converge to well-defined limiting functions. The analysis covers both bounded-width and unbounded-width architectures, establishes explicit convergence rates, and motivates a network initialization strategy derived from the convergence conditions. …

Timelines Over Tokens: Summarization, Prompting, And Explainable Fine-Tuning For User-Level Suicide Risk Detection, Aditya Tekale

Master's Theses

This research presents a two-stage pipeline for user-level suicide risk detection from Reddit: first, inference-only prompting with summarization; second, fine-tuned encoder classification with explainability and expert validation. The data are user-level: each of the 500 C-SSRS Reddit items is one user’s chronologically concatenated posts and comments (a user timeline), annotated by psychiatrists. Stage one: six prompting strategies zero-shot, few-shot, chain-of-thought, tree-of-thought, least-to-most, and self-consistency are evaluated across six LLMs on multi-class and binary formulations; simple zero-shot achieves the highest balanced accuracy (0.53 multi-class). Error analysis shows longer inputs associate with misclassification (p = 0.002); domain-specific summarization (timelines >2,000 tokens) reduces …

Information Theory Analysis Of Water Vapor Stable Isotopes From The Sail Campaign, Matthew John Rybecky

Earth and Planetary Sciences ETDs

Understanding the processes that control water vapor isotopic composition in mountain environ- ments is essential for interpreting isotope records and predicting water resource responses to cli- mate change. This thesis applies information theory to continuous, high-resolution water vapor stable isotope measurements from the Surface Atmosphere Integrated Field Laboratory (SAIL) campaign in the East River watershed of Colorado’s Upper Gunnison Basin, spanning the winter- to-spring transition of 2022–2023. The analysis employs Shannon entropy, mutual information, transfer entropy, and joint transfer en- tropy (JTE) to quantify how environmental variables, including surface meteorology, radiation, tur- bulent fluxes, and ERA5 reanalysis products, transfer information …

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 …

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.

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 …

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 …

Open Neighbourhood Edge Degree And Metric Descriptors Qspr And Multi Criteria Analysis Of Chemicals And Antiviral Drugs, Gayathri A Ms

Theses and Dissertations

Chemical Graph theory offers powerful computational methods for evaluating chemical structures, significantly impacting cheminformatics and drug analysis. It provides a robust mathematical framework to represent, model, and analyse molecular systems through topological indices and graph invariants. These indices serve as effective numerical descriptors, encapsulating significant structural information that enhances the prediction of physicochemical and biological properties. This thesis utilizes mathematical modeling, algorithmic computation, and decision making process to derive extensive insights to drug discovery and molecular analysis.

The thesis introduces a new class of topological indices ONE1, ONE2, ONE3, ONE4, ONE5,ONE6 and ONE7 based on open-neighbourhood-edge-degree and it evaluates the …

Numerical Range Of Rank-1 Partial Isometries And Elliptical Range Theorem, Lucas Saone

Theses

The Elliptical Range Theorem states that the numerical range of a 2x2 complex matrix A is an elliptical disc with foci at the eigenvalues of A and center at tr(A)/2. Many proofs have been offered for this theorem. In this paper, we provide another one. The strategy we will use is to decompose the matrix A into the sum of rank-1 operators of the form YxX*, where X and Y are in 2-D complex vectors. When analyzing the numerical range of a rank-1 operator of this form, three results can happen: i) the numerical range forms a disc at the …

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 …

Vibrations Of Tapered Beam Via The Exterior Matrix Method, Simranjit Kaur

Student Theses and Dissertations

Cell phone towers, utility poles, and traffic signal poles all use hollow, tapered beams as their main structural element. Because these structures are tall and exposed to wind forces, understanding their vibration behavior is important for ensuring stability and safety. We will use the Exterior Matrix Method to analyse a single beam, which can be used to analyse compound structures involving tapered beams. First, we find the system of four equations satisfied by the tapered beam, which can be converted to a 4 x 4 matrix. Then we find the exterior matrix, which is a 6 x 6 matrix, corresponding …

Bias, Structure, And Inference In Applied Network Analysis, Anna Vasenina

Dartmouth College Ph.D Dissertations

This dissertation develops mathematical and statistical methods for extracting reliable information from network data across biological applications, with an emphasis on understanding what observed network structure can and cannot resolve. The first study leverages protein–protein interaction network topology in the c-di-GMP signaling system of Pseudomonas fluorescens, showing that node centrality measures accurately classify protein domain types and that physical interaction structure contributes statistically significant predictive power for biofilm formation phenotypes across nearly 200 environments, while gene expression does not. The second study examines sampling bias in lemur-plant trophic interaction networks in Madagascar, demonstrating that differential detection of diurnal versus …

The Method Of Periodic Averaging Applied To Reduced Coupled Mode Theory Models For Fiber Laser Amplifiers, Rebecca Nicole Bryant

Dissertations and Theses

Fiber laser amplifier (FLA) models are often implemented without rigorous mathematical justification or thorough numerical validation. Without a proper theoretical basis for assumptions and approximations, or a technical analysis of model performance, there is significant uncertainty about the limitations of any given reduced model and its suitability for an application. This research aims to address the lack of comprehensive assessment of FLA models by directly comparing distinct models and recommending a mathematical alternative to replace heuristic model-reduction techniques. The work in this dissertation is divided into two projects: a comparative study that uses existing FLA models to assess the validity …

Mathematical Analysis Of Within-Host Models: Viral–Immune Dynamics, Bifurcations, And Disease Severity, Nazia Afrin

Doctoral Dissertations

My research develops and analyzes ODE-based within-host models at multiple scales. Using dynamical systems theory and numerical methods, I study host–pathogen interactions and immune responses, providing insights into disease dynamics and control. The first model describes the complex dynamics of Hepatitis B virus (HBV) infection and addresses the question: what mechanisms determine whether the infection is cleared during the acute phase or progresses to a chronic state? A key feature of this model is the assumption that all classes of liver cells (uninfected, infected, and protected from reinfection) proliferate at different rates. The findings provide insight into two important aspects …

A Bifurcation Theorem And Its Application To Discrete-Time Models In Ecology And Epidemiology, Jenita Jahangir

Doctoral Dissertations

Matrix models are useful for modeling populations or diseases that involve discrete developmental stages, multiple stages of infection, and interactions among species. To study the coexistence dynamics in matrix models, we extend a bifurcation theorem for resident-invader host-parasitoid type populations by allowing every block of the projection matrix, depending on the bifurcation parameter and the off-diagonal blocks, to be nonzero. As an application, in the first part of the dissertation, we propose a discrete-time host-parasitoid model with stage structure in both species. For this model, we establish conditions for the existence and global stability of the extinction and parasitoid-free equilibria. …

Strang-Type Exponential Integrators For Stiff Reaction-Diffusion Systems, Saburi Tolulope Rasheed

Doctoral Dissertations

Reaction-diffusion systems, as examples of semilinear parabolic partial differential equations, have played significant roles in the mathematical modeling of physical, chemical, and biological processes. Several reaction-diffusion systems typically do not have exact solutions in closed form, and numerically solving them also comes with challenges due to the presence of the nonlinear local interaction/chemical reaction dynamics representing the reaction term, coupling between components, multidimensionality of the diffusion operator, and stiffness of the diffusion and/or reaction terms. Wederive and analyze several second-order accurate exponential integrators of the Strang type for the time discretization of stiff reaction-diffusion systems. We utilize the finite difference …

Inverse Modeling The Geo-Spread Of Covid-19 In Brazil, Cameron Keith Mills

ETDs from 2020-2029

In 2021 Fitzgibbon, Morgan, Webb, and Wu used a modified SEIR (susceptible, exposed, infected, and recovered) model to predict how COVID-19 spread through Brazil [12]. For their model, six constant coefficients were used that were fitted, referenced, or assumed. In this thesis, we geo-spatially modify their SEIR model and formulate an inverse problem to recover the now spatial coefficients of the model. We first show there exists a unique solution to the modified model. To solve the inverse problem, we modify an inverse method [17] that focused on minimizing convex functionals. These recovered spatial coefficients can be used with Matlab’s …

Investigating An Energy-Preserving Spectral Finite Difference Numerical Method For A Wave Equation On The Metric Graph, Tue Minh Tran

University Honors Theses

To investigate the accuracy and long-term energy conservation of a spectral finite difference numerical method for a wave equation on metric graphs. In conservative systems, numerical methods should preserve total energy. However, explicit finite difference methods require impractically small space steps and exhibit energy drift at end points. To address these limitations, a spectral finite difference method is implemented using a Fourier transformation. This semi-spectral method improves stability at endpoints while maintaining second-order accuracy, achieving an overall error of O(∆t²). We implement the semi-spectral method on the IEEE14 metric graph and provide visuals showing the initial condition …

Traveling Wave Fronts For Sir Epidemic Models With Nonlocal Dispersal And Delayed Effects, Ashley Evette Embry

Theses and Dissertations

This dissertation investigates traveling wave solutions for two classes of delayed nonlocal dispersal susceptible--infected--recovered (SIR) epidemic models incorporating biologically realistic mechanisms such as delayed infectivity, delayed dispersal, nonlocal transmission, demographic turnover, and nonlinear incidence effects. These models extend classical spatial epidemic frameworks by allowing long-range population movement through nonlocal dispersal operators and incorporating temporal memory into both diffusion and transmission processes. The primary objective is to establish the existence of traveling wave solutions connecting disease-free equilibria to endemic states and to characterize threshold conditions governing epidemic propagation. The simultaneous presence of nonlocal dispersal, multiple delays, and non-monotone nonlinear incidence terms …