Mathematics Commons^™

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. …

A New Functional Setting For Term Structure Modeling Using The Health-Jarrow-Morton Framework, Michael Pokojovy, Ebenezer Nkum, Thomas M. Fullerton Jr.

Mathematics & Statistics Faculty Publications

The well-known Heath–Jarrow–Morton (HJM) framework provides a universal and efficacious instrument for modeling the stochastic evolution of an entire yield curve by explaining the interest rate dynamics in continuous time under no-arbitrage conditions. Existing implementations involve exponentially weighted function spaces as theoretical settings for the former stochastic evolution. While the choice of weight can have a drastic effect on model calibration and subsequent forecasting, it cannot be estimated from market data and does not allow for any objective interpretation. The proposed approach does not have this shortcoming as it adopts a suitably designed unweighted function space. The HJM equation is …

Identifying Relevant Covariates In Rna-Seq Analysis By Pseudo-Variable Augmentation, Yet Nguyen, Dan Nettleton

Mathematics & Statistics Faculty Publications

RNA-sequencing (RNA-seq) technology allows for the identification of differentially expressed genes, which are genes whose mean transcript abundance levels vary across conditions. In practice, RNA-seq datasets often include covariates that are of primary interest in addition to a set of covariates that are subject to selection. Some of these covariates may be relevant to gene expression levels, while others may be irrelevant. Ignoring relevant covariates or attempting to adjust for the effect of irrelevant covariates can compromise the identification of differentially expressed genes. To address this issue, we propose a variable selection method that uses pseudo-variables to control the expected …

An Association Test For Ordinal Outcomes In Clustered Data With Informative Cluster Size, Hasika K. Wickrama Senevirathne, Sandipan Dutta

Mathematics & Statistics Faculty Publications

In cluster-correlated data, the number of observations in a cluster can be associated with the outcome from that cluster. This phenomenon is known as informative cluster size which can occur in cluster-randomized clinical trial data. Several studies have found that ignoring the issue of informative cluster size can produce biased results in the analysis of clustered data. Most of the existing methods for addressing informative cluster size are suited to continuous outcomes. However, ordinal outcomes and covariates are often encountered in clustered data obtained from large clinical studies. The existing methods for ordinal association testing in clustered data can produce …

Logistic-T Multinomial Mixture Model For Clustering For Microbiome Data, Wenshu Dai, Yuan Fang, Sanjeena Subedi

Mathematics & Statistics Faculty Publications

The logistic-normal multinomial distribution has been used for modelling microbiome data obtained from high-throughput sequencing technologies, which are compositional in nature. A logistic-normal multinomial distribution is a hierarchical multinomial distribution that assumes the latent variable which are the additive log-ratio (ALR) transformed proportions in a multinomial distribution follows a Gaussian distribution. Model-based clustering algorithms have also been developed for clustering microbiome data based on the logistic-normal models. However, the Gaussian assumption may violated when the ALR transformed variable exhibit heavy-tailed distributions or has outliers. Our study introduces a novel mixture of logistic-t multinomial models that effectively address these challenges. Utilizing …

Integer-Valued Time Series Model Via Copula-Based Bivariate Skellam Distribution, Mohammed Alqawba, Norou Diawara, Mame Mor Sene

Mathematics & Statistics Faculty Publications

Time series analysis is crucial for modeling and forecasting diverse real-world phenomena. Traditional models typically assume continuous-valued data; however, many applications involve integer-valued series, often including negative integers. This paper introduces an approach that combines copula theory with the bivariate Skellam distribution to handle such integer-valued data effectively. Copulas are widely recognized for capturing complex dependencies among variables. By integrating copulas, our proposed method respects integer constraints while modeling positive, negative, and temporal dependencies accurately. Through simulation and an empirical study on a real-life example, we demonstrate that our class of models performs well. This approach has broad applicability in …

Mg-Spair: Multi-Grade Sparse-Guided Implicit Representation For Training-Data-Free Image Restoration, Jianmin Liao, Lei Huang, Ronglong Fang, Ashley Prater-Bennette, Lixin Shen, Yuesheng Xu

Mathematics & Statistics Faculty Publications

MG-SpaIR is a training-data-free framework for restoring a clean image from a single observation corrupted by a mixture of blur, downsampling, noise, and missing pixels. Building on implicit neural representations (INRs), we introduce a multi-grade residual hierarchy that progressively refines the reconstruction from low to high spatial frequencies across grades, improving representational fidelity and mitigating spectral limitations. To stabilize reconstruction optimization and suppress INR-induced artifacts, we further propose an explicit sparse proximal regularization (e.g., ℓ0 type) applied directly in the high-resolution image domain, which discourages spurious high-frequency patterns while preserving sharp structures. The resulting optimization is solved efficiently via a …

Adaptive Multi-Grade Deep Learning For Highly Oscillatory Fredholm Integral Equations Of The Second Kind, Jie Jiang, Yuesheng Xu

Mathematics & Statistics Faculty Publications

This paper studies the use of Multi-Grade Deep Learning (MGDL) for solving highly oscillatory Fredholm integral equations of the second kind. We provide rigorous error analyses of continuous and discrete MGDL models, showing that the discrete model retains the convergence and stability of its continuous counterpart under sufficiently small quadrature error. We identify the DNN training error as the primary source of approximation error, motivating a novel adaptive MGDL algorithm that selects the network grade based on training performance. Numerical experiments with highly oscillatory (including wavenumber 500) and singular solutions confirm the accuracy, effectiveness and robustness of the proposed approach.

A Composite Narxnn Approach To Photovoltaic Power Forecasting With Integrated Weather Inputs And Uncertainty Quantification, Denisse Urenda Castañeda, Sharmin Abdullah, Jackson Morgan, Honglun Xu, Michael Pokojovy, Tzu-Liang Tseng

Mathematics & Statistics Faculty Publications

Solar photovoltaics (PV) are a major source of sustainable energy. Yet, their power output is highly sensitive to environmental variability, particularly solar irradiance, cloud cover, wind, and temperature. Accurate forecasting of PV power is essential for efficient grid integration and energy planning, especially in applications requiring reliable longer-term forecasting rather than one-step-ahead predictions. This study presents a PV power forecasting approach using Nonlinear Autoregressive models with Exogenous Inputs (NARX), integrating large-scale numerical weather historical data as exogenous variables. Although NARX models effectively capture temporal dependencies, they can become overly dependent on historical power values, reducing responsiveness to real-time weather changes. …

A New Parallel-In-Time Direct Inverse Method For Nonlinear Differential Equations, Nail K. Yamaleev, Subhash Paudel

Mathematics & Statistics Faculty Publications

We propose a new method for parallelization of the first-order backward difference discretization (BDF1) of the first-order time derivative in nonlinear partial differential equations, such as conservation law equations. The time derivative term is discretized by using the method of lines based on the implicit BDF1 scheme, while the inviscid and viscous terms are approximated by conventional 2nd-order central discretizations of the 1st- and 2nd-order derivatives in each spatial direction. The global system of nonlinear discrete equations in the space-time domain is solved by the Newton method for all time levels simultaneously. For the BDF1 discretization, this all-at-once system at …

Near Real-Time Adaptive Isotropic And Anisotropic Image-To-Mesh Conversion For Cerebral Aneurysm Simulations, Kevin Garner, Chander Sadasivan, Nikos Chrisochoides

Computer Science Faculty Publications

This paper presents two performance optimization techniques for a mesh adaptation method that is designed to help streamline the discretization of complex vascular geometries within the numerical modeling process. This method is integrated into a pipeline with an image-to-mesh conversion tool to generate adaptive anisotropic meshes from segmented medical images. The pipeline is shown to satisfy quality, fidelity, smoothness, and robustness requirements while providing near real-time performance for medical image-to-mesh conversion. Tested with two brain aneurysm cases and utilizing up to 96 CPU cores within a single, multicore node on Purdue University’s Anvil supercomputer, the parallel adaptive anisotropic meshing method …

A Survey Of Ai Methods For Geometry Preparation And Mesh Generation In Engineering Simulation, Steven Owen, Nathan Brown, Nikos Chrisochoides, Rao Garimella, Xianfeng Gu, Franck Ledoux, Na Lei, Roshan Quadros, Navamita Ray, Nick Winovich, Yongjie Jessica Zhang

Computer Science Faculty Publications

Artificial intelligence is beginning to reduce the manual effort in the CAD-to-mesh pipeline. Written for meshing and geometry practitioners with limited AI background, this survey organizes recent work by workflow step. We cover part classification and segmentation, mesh quality prediction, and defeaturing. We review AI guidance for unstructured meshing, block-structured meshing in 2D and 3D, and volumetric parameterization, including reconstruction from implicit or sampled geometry. We also discuss parallel mesh generation and scripting automation via reinforcement learning and large language models. Across these topics, AI complements established geometry and meshing algorithms rather than replacing them. We conclude with practical lessons …

Analysis Of Multi Grade Deep Learning, Ronglong Fang

Mathematics & Statistics Theses & Dissertations

Multi-Grade Deep Learning (MGDL) is a training framework that incrementally builds deep neural networks. It does this by dividing the training process into multiple “grades,” where each grade sequentially trains a shallow neural network to learn the residue from the previous one, using the outputs of prior grades as input. This approach progresses from shallow to deep architectures. This dissertation offers a comprehensive theoretical and numerical analysis of the MGDL methodology.

We first demonstrate that MGDL can effectively learn target functions within the sum-composition learning format. In this context, MGDL approximates high-frequency components by composing multiple low-frequency functions. This unique …

Learning With Errors Parameter Analysis, Archana Parameswaran

Cybersecurity Undergraduate Research Showcase

We implement a systematic approach for generating, evaluating, and benchmarking Learning with Errors implementations in Sage Math by varying lattice dimensions, moduli, error standard deviations, and multiple error distributions to observe concrete security-efficiency tradeoffs. The security estimator maps parameter sets to concrete security levels and bits, while performance metrics measured computational efficiency and memory requirements. Results indicate that various distribution types do not significantly impact security, though binomial distributions require more computational overhead than discrete gaussian or uniform. Memory requirements increased when modulus q increased from 12289 to 65537. Larger dimensions have an exponentially growing requirement for memory, but this …

32 - Nested Two Level Decomposition For Quantum Computing, Andrew Maciejunes, John Stenger, Dan Gunlycke, Nikos Chrisochoides

Undergraduate Research Symposium

Abstract—We present a two-level decomposition strategy for solving the Vehicle Routing Problem (VRP) using the Quantum Approximate Optimization Algorithm (QAOA). A Problem-Level Decomposition (PLD) partitions a 9-node (72-qubit) VRP into smaller Traveling Salesman Problem (TSP) instances. Each TSP is then further simplified via Circuit-Level Decomposition (CLD), enabling execution on near-term quantum devices. Our approach achieves up to 90% reductions in circuit depth and qubit count. These results demonstrate the feasibility of solving VRPs previously too complex for quantum simulators and provide early evidence of potential quantum utility.

Stability Analysis In The Twist-Bend Nematic Liquid Crystal Model, Zhenqiang Li

Mathematics & Statistics Theses & Dissertations

The recently discovered twist-bend nematic liquid crystal (LC) phase is characterized by a nanoscale helical modulation of the nematic director n, forming a conical helix along the z-axis at an oblique angle θ. While many models assume a constant cone angle and equal elastic constants K11 = K22 = K33, this dissertation removes both assumptions by considering a fully anisotropic elastic energy with K11 ≠ K22 ≠ K33, and allowing θ to vary spatially. We analyze the stability of this system under frustrated and free boundary conditions using variational methods. …

Enhancing The Accuracy Of Image Classification For Degenerative Brain Diseases With Cnn Ensemble Models Using Mel-Spectrograms, Sang-Ha Sung, Michael Pokojovy, Do-Young Kang, Woo-Yong Bae, Yeon-Jae Hong, Sangjin Kim

Mathematics & Statistics Faculty Publications

Alzheimer’s disease (AD) and Parkinson’s disease (PD) are prevalent neurodegenerative disorders among the elderly, leading to cognitive decline and motor impairments. As the population ages, the prevalence of these neurodegenerative disorders is increasing, providing motivation for active research in this area. However, most studies are conducted using brain imaging, with relatively few studies utilizing voice data. Using voice data offers advantages in accessibility compared to brain imaging analysis. This study introduces a novel ensemble-based classification model that utilizes Mel spectrograms and Convolutional Neural Networks (CNNs) to distinguish between healthy individuals (NM), AD, and PD patients. A total of 700 voice …

Optimal Control And Structurally-Informed Gradient Optimization Of A Custom 4-Dof Rigid-Body, Brock Marcinczyk, Logan E. Beaver

Mechanical & Aerospace Engineering Faculty Publications

This work develops a control-centric framework for a custom 4-DOF rigid-body manipulator by coupling a reduced-order Pontryagin’s Maximum Principle (PMP) controller with a physics-informed Gradient Descent stage. The reduced PMP model provides a closed-form optimal control law for the joint accelerations, while the Gradient Descent module determines the corresponding time horizons by minimizing a cost functional built directly from the full Rigid-Body Dynamics. Structural-mechanics reaction analysis is used only to initialize feasible joint velocities—most critically the azimuthal component—ensuring that the optimizer begins in a physically admissible region. The resulting kinematic trajectories and dynamically consistent time horizons are then supplied to …

Multipliers On Weighted Sequence Spaces, Gilbert Acheampong, Raymond Cheng

Mathematics & Statistics Faculty Publications

The space ℓp,α of complex sequences a = (a0, a1,a2,...) for which

[[formula omitted]]

is studied. Each such sequence can be identified with the analytic function with power series

[[formula omitted]]

In this setting, the point evaluation and the difference quotient mappings are shown to be bounded; the cases are identified in which ℓp,α is boundedly contained in ℓr,β. Conditions on the parameters are derived for the analytic functions of ℓp,α to have radial limits almost everywhere on the boundary, and for ℓp,α to be an algebra. Smoothness properties of the boundary function are investigated. Basic properties of multipliers on …

Golden Spirals Everywhere?, John Adam

Mathematics & Statistics Faculty Publications

The article explores different types of spirals, including Archimedean, hyperbolic, and logarithmic spirals, with a focus on the golden ratio and golden spirals. It discusses the misconception that golden rectangles and spirals can be found in various natural and man-made objects, emphasizing the importance of understanding the properties of logarithmic spirals. The text provides mathematical equations for logarithmic spirals and poses questions for readers to explore the concept further. The author, John Adam, invites readers to engage in Fermi Questions and submit ideas for consideration.

Model-Free Organization Of Patient Reported Outcomes Data: Geometrical Rep-Resentation Of The Modified Compartmen-Talization Method, Manasi Sheth, N. Rao Chaganty

Mathematics & Statistics Faculty Publications

There is a recent advancement in the field of mathematics and statistics to understand the geometry or connectedness of the data due to the massive amounts of data being generated. The data provided for analyses are usually very large and need to be organized and minimized in order to make it more useful and meaningful. In biostatistics or medical field, it is important for patients to have access to high-quality, safe and effective and/ or efficacious medical products. It is quite necessary to ascertain that the patients and their care-partners stay at the center of the regulatory decision-making process. In …

A Decision-Space Model Explains Context-Specific Decision Making, Dirk W. Beck, Cory N. Heaton, Luis D. Davila, Lara I. Rakocevic, Sabrina M. Drammis, Danil Tyulmankov, Atanu Giri, Shreeya Umashankar Beck, Qingyang Zhang, Michael Pokojovy, Kenichiro Negishi, Alexis A. Salcido, Neftali F. Reyes, Andrea Y. Macias, Serina A. Batson, Paulina Vara, Raquel J. Ibáñez Alcalá, Safa B. Hossain, Graham L. Waller, Laura E. O'Dell, Travis M. Moschak, Ki A. Goosens, Alexander Friedman

Mathematics & Statistics Faculty Publications

Optimal decision-making requires consideration of internal and external contexts. Biased decision-making is a transdiagnostic symptom of neuropsychiatric disorders. We created a computational model demonstrating how the striosome compartment of the striatum constructs a context-dependent mathematical space for decision-making computations, and how the matrix compartment uses this space to define action value. The model explains multiple experimental results and unifies other theories like reward prediction error, roles of the direct versus indirect pathways, and roles of the striosome versus matrix, under one framework. We also found, through new analyses, that striosome and matrix neurons increase their synchrony during difficult tasks, caused …

A Bridge Too Low: Solutions For Fermi Questions, May 2025, John Adam

Mathematics & Statistics Faculty Publications

The article discusses a low bridge in Keswick, England, near the River Greta, with an arch shaped like a semiellipse. It presents Fermi questions related to the bridge's dimensions, such as the maximum distance a person of a certain height can walk under it without hitting their head. The solutions involve mathematical calculations and approximations, including the use of formulas provided by the Indian mathematician Srinivasa Ramanujan.

The Effect Of Electrode Geometry On Excited Species Production In Atmospheric Pressure Air-Hydrogen Streamer Discharge, Shirshak Kumar Dhali, Stuart Reyes

Electrical & Computer Engineering Faculty Publications

When a gas is overvolted at or near atmospheric pressure, it results in a streamer discharge formation. Electrode geometries exert significant impact on the electrical breakdown of gases by altering the spatial profile of the electric field. In many applications the efficient generation of radicals is critical and is determined by the characteristics of the streamer discharge. We examine the effect of electrode geometry on the streamer characteristics and the production of radicals. This is performed for three different electrode geometries: plane–plane, pin–plane, and pin–pin. A two-dimensional rotationally symmetric fluid model is used for the streamer discharge simulation in the …

Isochronous And Period-Doubling Diagrams For Symplectic Maps Of The Plane, T. Zolkin, S. Nagaitsev, I. Morozov, S. Kladov, Y. -K. Kim

Physics Faculty Publications

Symplectic mappings of the plane serve as key models for exploring the fundamental nature of complex behavior in nonlinear systems. Central to this exploration is the effective visualization of stability regimes, which enables the interpretation of how systems evolve under varying conditions. While the area-preserving quadratic Hénon map has received significant theoretical attention, a comprehensive description of its mixed parameter-space dynamics remain lacking. This limitation arises from early attempts to reduce the full two-dimensional phase space to a one-dimensional projection, a simplification that resulted in the loss of important dynamical features. Consequently, there is a clear need for a more …

Golden Spirals Everywhere? Solutions For Fermi Questions, January 2025, John Adam

Mathematics & Statistics Faculty Publications

The article discusses different types of spirals, including Archimedean, hyperbolic, and logarithmic spirals, with a focus on the golden ratio and golden spirals. It addresses the misconception that golden rectangles and spirals can be found in various natural and man-made structures, emphasizing the importance of understanding the properties of logarithmic spirals. The article provides mathematical explanations and solutions for questions related to pitch angles and self-similarity in logarithmic spirals, using examples like the nautilus shell and an ammonite-like stone. It concludes by referencing additional sources for further exploration of the golden ratio and golden spiral myths.

Multipliers Between ℓᴾ Spaces, Raymond Cheng

Mathematics & Statistics Faculty Publications

For 0 < p ⩽ ∞ and 0 < r ⩽ ∞, the space 𝔐_p,r of (coefficient) multipliers from ℓ^p and ℓ^r is completely characterized. This is elementary in most instances. The interesting case 0 < r < p < ∞ requires more effort, and it is shown that a sequence of complex numbers belongs to 𝔐_p,r if and only if the sequence of their absolute values has a non increasing rearrangement (h₀,h₁,h₂,...) satisfying

(∞

Σ (k +1)^(p-r)/p (h^r_k - h^r_k+1)^1/r) < ∞

k = 0

In that case, the expression on the left is the norm of the multiplier, and it is a compact operator. Further upper and lower bounds are given for the multiplier norm.

A Bridge Too Low, John Adam

Mathematics & Statistics Faculty Publications

The article "A bridge too low" in the Physics Teacher journal discusses a low bridge near the River Greta in Keswick, England, with an arch shaped like a semiellipse. It presents questions about the maximum height a person of a certain height can walk under the bridge without hitting their head, the cross-sectional area of the arch, its eccentricity, and perimeter. The article also mentions the approximation by Indian mathematician Srinivasa Ramanujan for the perimeter of an ellipse and invites readers to find the answers online.

A Question Of Transparency, John Adam

Mathematics & Statistics Faculty Publications

Question 1: Why is it easier to see through rain than fog?

Start thinking about this by imagining a fixed volume (V) of water being dispersed into, say, N identical droplets of diameter d. Surface area and volume considerations should lead to the answer in terms of V and d.

Question 2: (a) How "long" (in meters) might such a rain shower or fog bank be?

Hint: Suppose you are looking along a linear stack of S cubes with 1-m sides (through the rain or fog). Each cube contains N drops. If p is the visibility (i.e., the fraction of …

A Question Of Transparency: Solutions For Fermi Questions, March 2025, John Adam

Mathematics & Statistics Faculty Publications

Question 1: Why is it easier to see through rain than fog?

Start thinking about this by imagining a fixed volume (V) of water being dispersed into, say, N identical droplets of diameter d. Surface area and volume considerations should lead to the answer in terms of V and d.

Solution to Question 1: N = VI(πd³/6) = 6V/πd³ ≈ 2V/d³.

The cross-sectional area A of each drop is πd²/4 ≈ 3d²/4, so the total area blocked off (assuming no overlapping drops—so this is an upper bound) is NA ≈ 1.5V/d, so the area blocked off is inversely proportional to …