Mathematics Commons^™

Is Energy Local? Counterintuitive Non-Locality Of Energy In General Relativity Can Be Naturally Explained On The Newtonian Level, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

From the physics viewpoint, energy is the ability to perform work. To estimate how much work we can perform, physicists developed several formalisms. For example, for the fields, once we know the Lagrangian, we can find the energy density and, by integrating it, estimate the overall energy of the field. Usually, this adequately describe how much work this field can perform. However, there is an exception -- gravitational field in General Relativity. The known formalism to compute its energy density leads to 0 -- and by integrating this 0, we get a counterintuitive conclusion that the overall energy of the …

Logarithmic Number System Is Optimal For Ai Computations: Theoretical Explanation Of Empirical Success, Olga Kosheleva, Vladik Kreinovich, Christoph Lauter, Kristalys Ruiz-Rohena

Departmental Technical Reports (CS)

Everyone knows the success story of machine-learning AI. However, the current AI tools are not perfect. We know how to make them better: every time we increase the amount of computations by the order of magnitude, we get a drastic improvement in the performance of the resulting machine learning tools. Training modern AI system requires a tremendous amount of computations -- that already take a lot of time. So, to increase the number of computations, we need to make each computation step faster. One way to do that is to use low-precision arithmetic operations, e.g., with 1 byte per real …

Algotric: Symmetric And Asymmetric Encryption Algorithms For Cryptography – A Comparative Analysis In Ai Era, Naresh Kshetri, Mir Mehedi Rahman, Md Masud Rana, Omar Faruq Osama, James Hutson

Faculty Scholarship

The increasing integration of artificial intelligence (AI) within cybersecurity has necessitated stronger encryption methods to ensure data security. This paper presents a comparative analysis of symmetric (SE) and asymmetric encryption (AE) algorithms, focusing on their role in securing sensitive information in AI-driven environments. Through an in-depth study of various encryption algorithms such as AES, RSA, and others, this research evaluates the efficiency, complexity, and security of these algorithms within modern cybersecurity frameworks. Utilizing both qualitative and quantitative analysis, this research explores the historical evolution of encryption algorithms and their growing relevance in AI applications. The comparison of SE and AE …

Machine Learning Methods For Quantification Of Glacier Variations Through Satellite Imagery, Robert D. Breininger

Theses and Dissertations

Glaciers around the world have experienced a trend of recession within the past century. Quantification of glacier variations using satellite imagery is of great interest due to the importance of glaciers as freshwater resources and as indicators of climate change. The potential methods to quantify glacier variations with increasing complexity include detecting the terminus location, quantifying the glacier surface area, and measuring glacier volume. Although there are methods in literature designed purposefully for glacier area segmentation that have achieved acceptable results, they are often localized to the region where their training data were acquired and further rely on training sets …

Latent Characterization Of The Complete Batse Gamma-Ray Bursts Catalogue Using Gaussian Mixture Of Factor Analysers And Model-Estimated Overlap-Based Syncytial Clustering, Fan Dai, Ranjan Maitra

Michigan Tech Publications

Characterizing and distinguishing gamma-ray bursts (GRBs) has interested astronomers for many decades. While some authors have found two or three groups of GRBs by analysing only a few parameters, recent work identified five ellipsoidally shaped groups upon considering nine parameters T50, T90, F1, F2, F3, F4, P64, P256, P1024. Yet others suggest subclasses within the two or three groups found earlier. Using a mixture model of Gaussian factor analysers, we analysed 1150 GRBs, that had nine parameters observed, from the current Burst and Transient Source Experiment (BATSE) catalogue, and again established five ellipsoidal-shaped groups to describe the GRBs. These five …

An Examination Of Hilbert Spaces, Fnu Sumaiya

Theses and Dissertations

This independent study focuses on the exploration of Hilbert spaces, a cornerstone of modern analysis and a fundamental construct in mathematics. It starts by introducing vector space, metric and metric spaces and then Hilbert spaces and its properties.

This study includes good amount of mathematical proofs, theorems and examples. Through extensive research and analysis, this independent study aims to offer a comprehensive examination of Hilbert spaces, detailing their properties and their pivotal role in modern analysis.

An Analysis Of The Properties Of Polar Codes, Luke Szramowski

All Theses

Polar Codes have risen to the forefront of practical coding theory, due to their incredible efficiency and ease of construction. Originally introduced by Arikan in his 2009 paper, they are the first code defined with an explicit construction that achieved channel capacity. Moreover, polar codes possess some physically practical properties that make their implementation alluring. In the same paper as mentioned above, Arikan elaborated on his construction and noted that the construction given was one specific instance of a polar code and that there is a family of polar codes that can be produced by the same method. Since this …

Analysis Of Impulsive Differential Equation Models Of Cell Populations Undergoing Radiation Therapy, Abigail D'Ovidio Long

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

Radiation therapy is a mode of treatment which is implemented for approximately 50% of cancer patients. Treatment needs to be able to kill cancer cells, but also do minimal damage to surrounding healthy tissue. We propose two main impulsive differential equation models of radiation therapy to capture the periodic nature of the treatment. These models build off of previous studies using clinical data to ensure biological relevance. The first model incorporates only cancer cell populations, and we provide parameter relationships which theoretically ensure treatment outcomes of cancer eradication, cancer approaching a carrying capacity, and cancer approaching a periodic solution. We …

Nestings Of Bibds With Block Size Four, Marco Buratti, Donald L. Kreher, Douglas R. Stinson

Michigan Tech Publications

In a nesting of a balanced incomplete block design (or BIBD), we wish to add a point (the nested point) to every block of a (Formula presented.) -BIBD in such a way that we end up with a partial (Formula presented.) -BIBD. In the case where the partial (Formula presented.) -BIBD is in fact a (Formula presented.) -BIBD, we have a perfect nesting. We show that a nesting is perfect if and only if (Formula presented.). Perfect nestings were previously known to exist in the case of Steiner triple systems (i.e., (Formula presented.) -BIBDs) when (Formula presented.), as well as …

Surgical Suturing Skill Assessment Using Estimated Hand Roll Angle From A Deep-Learning Computer Vision Algorithm, Jianxin Gao, Amir Mehdi Shayan, Simar P. Singh, Joe Bible, Ravikiran Singapogu, Richard E. Groff

Publications

This paper proposes a deep-learning computer vision algorithm to estimate hand roll angles for metric-based assessment of surgical suturing skills. The number of rolls metric, previously calculated directly from IMU data, counts the number of hand roll reversals during a single suture. To calculate this metric using computer vision, we apply a deep-learning algorithm that can reliably estimate hand roll angles after training on suturing videos collected on the SutureCoach simulator. Results show that the estimation accuracy of the deep-learning algorithm is robust to different video backgrounds. The number of rolls metrics were used to analyze suturing performance in the …

Computational Representation, Analysis And Verification Of Requirements In Engineering Design And Systems Engineering, Chandan Kumar Sahu

All Dissertations

Systems are developed to satisfy a set of requirements derived from stakeholders’ needs, defining the problem space for which the system is created as a feasible solution. The system design process begins with eliciting these requirements and concludes with validating whether the created system meets them. Requirements engineering (RE) encompasses elicitation, representation, analysis, documentation, verification, and validation. However, challenges in RE, such as imprecision in natural language (NL), proprietary restrictions, and a lack of standardized quality metrics, hinder the creation of well-formed and comprehensive requirements. These challenges complicate formalization and analysis of requirements.

This dissertation addresses these challenges by proposing …

Andre-Quillen Homology And Special Classes Of Ring Homomorphisms, Hossein Faridian

All Dissertations

This thesis is comprised of three chapters. The first chapter deals with a purely algebraic proof of a deep result of Quillen stating that the category of simplicial commutative algebras over a commutative ring is a model category. The central focus of our approach is on the study of shuffle product of connective chain complexes that provides a bridge to translate the constructions in the simplicial algebra world to the chain complex world.

The second chapter delves into Quillen's fundamental spectral sequences that relate Andre-Quillen homology and cohomology to Tor and Ext functors. Our comprehensive treatment develops and streamlines the …

(R2093) Fixed Point Of Hybrid Jaggi-Meir-Keeler Type Multivalued Contraction, Sirajo Yahaya, Mohammed Shehu Shagari

Applications and Applied Mathematics: An International Journal (AAM)

One of the most applicable results in metric fixed point theory is based on the contractive inequalities, including both rational and non-rational types. In this manuscript, a general idea under the name Jaggi-Meir-Keeler hybrid type multivalued contraction is introduced. We investigate the existence of fixed points for such operators in the setting of a complete metric space. The presented concept herein unifies the above-mentioned contractions and the corresponding invariant point results. A comparative nontrivial example is constructed to show the connection between the main idea in this paper and the related literature.

Decoding Neural Networks: An Information-Theoretic Guide To Interpretability, Error Analysis And Efficiency, Mackenzie J. Meni

Theses and Dissertations

This dissertation addresses critical challenges in neural network design by leveraging entropy-based techniques to improve model efficiency, interpretability, and bias reduction. Focusing on the unique demands of computer vision applications, particularly object detection and classification for real-time systems, this work introduces a series of innovative methods centered on information theory. At the core of these methods is the Probabilistic Explanations of Entropic Knowledge (PEEK) framework, a tool developed to analyze and visualize entropy distributions across feature maps. PEEK offers insights into information flow within neural networks, making it possible to pinpoint layers that contribute meaningfully to decision-making or identify those …

On The Lagrange And Hermite Quadrature Formula, Shuxia Li, Yonghong Chen

School of Mathematical & Statistical Sciences Faculty Publications

This paper builds on the error analysis method for Newton-Cotes quadrature formulas developed by D. R. Hayes and L. Rubin in 1970, which utilizes Lagrange interpolation polynomials. By adopting and extending their approach, this work derives the error estimate for Hermite interpolation quadrature. Specifically, we construct a polynomial P(x) analogous to the scaling function A(x) used by Hayes and Rubin, and prove its non-negativity over the interval. This allows us to establish a precise error formula for Hermite interpolation quadrature. The results provide a novel application of Hayes and Rubin's methodology, offering new insights …

Enhancing Mathematical Models For Covid-19 Pandemic Response: A Philippine Study, Timothy Robin Teng, Elvira De Lara-Tuprio, Ma. Regina Justina Estuar, Christian Pulmano, Lu Christian S. Ong, Zachary Pangan, Lenard Paulo V. Tamayo, Jasper John V. Segismundo, Mark Anthony C. Tolentino, Alyssa Nicole N. Ty

Mathematics Faculty Publications

Mathematical models supported by a robust automated data pipeline proved to be useful tools for a data-driven and science-based response and policy-making during the COVID-19 pandemic in the Philippines. In the first year of the pandemic, FASSSTER (Feasibility Analysis on Syndromic Surveillance using Spatio-Temporal Epidemiological modeleR) used a compartmental model to generate scenario-based projections of COVID-19 cases. The emergence of the Delta variant, however, and the administration of vaccines over the second half of 2021 caused significant changes in the Philippine pandemic landscape. This necessitated making adjustments to the model to better capture the local disease transmission dynamics and address …

A Machine Learning Approach For Survival Analysis Of Transplanted Kidneys Based On Donors’ And Recipients’ Factors., Alain Edward Despeignes

Theses and Dissertations

Over seven thousand people on average die each year in the United States waiting for an organ transplant due to the shortage of donated organs. With this alarming concern, efforts from the health organizations like the United Network Organ Sharing (UNOS) and government officials have considered avenues to remedy this distress, one of which is to investigate the characteristics among donors and recipients that affects the longevity of donated organs. The goal of this project is to investigate the survival time of transplanted kidneys from 1987 to 2018 with regards to the donors’ and the recipients’ characteristics including gender, ethnicity, …

Teachers’ Perceptions On The Impact And Effectiveness Of Certification Programs In Preparing And Retaining Educators, Teresa De Jesus Padilla

Theses and Dissertations

Mathematics is integral to STEM fields, making math critically important for the stability and development of the nation. As a result, mathematics teachers have a crucial role in our society. A role whose importance needs the necessary support to accomplish its numerous responsibilities. However, research indicates that certification routes—traditional and alternative—often fail to adequately prepare math teachers for the challenges they face, leading to high turnover rates. This study explores the impact of these certification routes on teachers’ abilities to support student achievement, address diverse learning needs, and manage additional duties. Surveying secondary math teachers in the Rio Grande Valley, …

Mathematical And Statistical Methods To Harness Limited Data In Models For Ecological Space Use Under Global Change, Sarah C. Bogen

All Graduate Theses and Dissertations, Fall 2023 to Present

The dynamics of how plants and animals use space in their habitats has important implications for the fields of ecology and conservation. However, understanding and responding to these spatial and temporal dynamics is often limited by data availability, financial resources and biases. As average global temperatures increase, suitable habitats shift poleward and require local populations to move with suitable habitat, adapt to the changing environment, or risk extinction. Capacity to persist without movement may be estimated by considering changes to a combination of habitat characteristics. Capacity to track suitable habitat may be modeled through synthesizing information on species demographic mechanisms …

Bounding The Convex Hull Relaxation Of The Unit Commitment Problem With The Shapley-Folkman Theorem, Lauren Henderson

All Theses

The Unit Commitment (UC) problem finds an optimal schedule for a set of generators by minimizing the total operation cost subject to demand and operational constraints. The UC problem is often modeled with a mixed-integer linear program (MILP). We employ the Shapley-Folkman Theorem to provide a bound on the size of fractional solutions of its convex hull relaxation. This result is used to obtain a bound on the optimality gap between the MILP and the convex hull relaxation, which is further tightened using several problem-specific properties of UC. We conduct extensive numerical experiments to study the tightness of this threshold, …

On Representations Of The Super-Yangian Of The Queer Lie Superalgebra, Elena Poletaeva

School of Mathematical & Statistical Sciences Faculty Publications

Let Q(n) be the queer Lie superalgebra. We determine conditions under which two 1-dimensional modules over the super-Yangian of Q(n) can be extended nontrivially. We describe the dual modules of the simple finite-dimensional modules over YQ(1) . We use these results to describe blocks in the subcategory of finite-dimensional YQ(1) -modules admitting the zero generalized central character.

An Adaptive And Parallel Direct Solver For Elliptic Partial Differential Equations, Damyn Chipman

Boise State University Theses and Dissertations

We introduce the quadtree-adaptive Hierarchical Poincaré-Steklov (QAHPS) method, an adaptive direct method for solving elliptic partial differential equations on a hierarchy of adaptively refined finite volume meshes. The QAHPS method builds up a solution operator set with O(N^3/2) complexity that acts as the factorization of the system matrix, with linear O(N) complexity for the application of the solution operator set to any number of right-hand side vectors. As the solution operator set is built up by merging local subdomains, it can be adapted as the mesh is refined and coarsened. The method is an …

A Comparative Analysis Of Early Algebraic Thinking Activities From U.S. And Singapore Primary Textbooks, Christian Joel Hernandez

Theses and Dissertations

Many studies highlight the challenges students face when transitioning to algebra at the secondary level. Introducing algebraic concepts and fostering algebraic thinking at the primary level can help mitigate these difficulties. Prior to formal algebra instruction, early algebra can be cultivated as a mode of thinking known as algebraic thinking. Several international curricula, such as Singapore Math, incorporate early algebraic thinking into the early stages of schooling. Singapore Math, renowned for its high performance in international assessments, has been widely adopted by schools seeking to replicate its success.

This study compares two primary-level mathematics curricula—CCSSM-aligned textbooks and Singapore Math—specifically focusing …

Modelling Tinnitus Functional Index Reduction Using Supervised Machine Learning Algorithms, Edmund F. Agyemang

School of Mathematical & Statistical Sciences Faculty Publications

This study aims to model the reduction in the Tinnitus Functional Index (TFI) utilizing supervised machine learning algorithms, focusing primarily on Ordinary Least Squares (OLS), K-Nearest Neighbor (KNN), Ridge, and Lasso regressions. Our analysis highlighted Group, ISI, and SWLS as significant predictors of TFI reduction, identified through the best subset selection and confirmed by both forward and backward selection criteria in the OLS regression. Notably, the shrinkage methods, Ridge and Lasso regressions, demonstrated superior performance compared to OLS and KNN, with the Ridge regression presenting the smallest test mean square error (MSE) of 318.30. This finding establishes the Ridge regression …

A Novel Phenotype Imputation Method With Copula Model, Jianjun Zhang, Jane Zizhen Zhao, Samantha Gonzales, Xuexia Wang, Qiuying Sha

Michigan Tech Publications

BACKGROUND: Jointly analyzing multiple phenotype/traits may increase power in genetic association studies by aggregating weak genetic effects. The chance that at least one phenotype is missing increases exponentially as the number of phenotype increases especially for a real dataset. It is a common practice to discard individuals with missing phenotype or phenotype with a large proportion of missing values. Such a discarding method may lead to a loss of power or even an insufficient sample size for analysis. To our knowledge, many existing phenotype imputing methods are built on multivariate normal assumptions for analysis. Violation of these assumptions may lead …

Bioconvection Dynamics In Rotating And Stationary Cone-Disk Systems, Puneet Rana, Mahanthesh Basavarajappa

School of Mathematical & Statistical Sciences Faculty Publications

This work focuses on the study of bioconvection in a conical region of rotating and stationary cone-disk systems utilizing nanofluids involving gyrotactic micro-organisms. The flow geometry encompasses two different configurations, namely, rotating cone-disk system (RCDS) and stationary cone-disk system (SCDS). For RCDS, four unique configurations are considered: rotating cone static disk (Model-I), static cone rotating disk (Model-II), co-rotating cone-disk (Model-III), and counter-rotating cone-disk (Model-IV), while SCDS includes both swirling and non-swirling flow scenarios. A total of six different physical configurations that differ in boundary conditions are investigated. The mathematical model comprises Navier–Stokes, energy, nanoparticle volume fraction (NVF), and micro-organism density …

On The Total Perimeter Of Pairwise Disjoint Convex Bodies, Arseniy Akopyan, Alexey Glazyrin

School of Mathematical & Statistical Sciences Faculty Publications

In this note we introduce a pseudometric on closed convex planar curves based on distances between normal lines and show its basic properties. Then we use this pseudometric to give a shorter proof of the theorem by Pinchasi that the sum of perimeters of 𝑘 convex planar bodies with disjoint interiors contained in a convex body of perimeter 𝑝 and diameter 𝑑 is not greater than 𝑝 + 2(𝑘 − 1)𝑑.

Quantum Markov Chains Related To Certain Lattice Models, Ali Alalaai

Thesis/ Dissertation Defenses

A central open problem in quantum field theory is the construction of a general theory of quantum field, this thesis introduces quantum probability and applies it via the construction of quantum Markov chains on different hierarchical lattices (Cayley trees). Furthermore, these trees correspond to the Ising-XY-Model which then the existence of a unique Markov chain can be utilized to detect phase transitions.

A Dynamical Systems Approach For Modeling Malware Propagating Through A Network And Potential Solutions Towards Mitigating Spread, James Johnson

Cybersecurity Undergraduate Research Showcase

Many people draw close parallels between malware propagating through a network and an epidemic spreading through a population. Epidemics are often modeled by a Susceptible-Infected-Recovered (SIR) model, in which a similar system of equations can model the spread of a virus through a computer network, and can be simplified when making assumptions about the network itself and its fixed number of nodes and edges. In this instance, malware propagating in a network also should reflect the network it is propagating through, in which the dynamical system will factor in the nodes of the network and their properties. The system itself …

Implementing Bernstein Operational Matrices To Solve A Fractional‐Order Smoking Epidemic Model, Jalal Al Hallak, Mohammed Alshbool, Ishak Hashim

All Works

This paper leverages the Bernstein operational matrices method for the first time in order to resolve the nonlinear fractional smoking epidemic model presented in terms of Caputo’s fractional derivative. An approximate solution is derived using Bernstein’s operational matrices and strategically chosen collocation points. This is followed by the validation of the proposed method’s accuracy and reliability against the established Runge–Kutta fourth‐order method. Furthermore, a comprehensive comparative analysis is conducted against two prominent techniques: the fractional differential transform method (FDTM) and the q‐homotopy analysis transform method (q‐HATM). The results show a superior and significant performance regarding accuracy as well as approximation. …