Mathematics Commons^™

The Impact Of Loss Function Topology On Gradient Descent, Robert B. Skudnig Jr.

Theses and Dissertations

Gradient descent is a popular optimization method that utilizes a model’s prediction error to iteratively improve its parameters for a given task. The functions that measure this error can be defined to align with the user’s goals and sometimes satisfy metric or norm properties. It is common for these functions to measure over Rn, but any differentiable space allows for gradient descent to occur. There has been some research investigating the influence of topological spaces on optimization methods, but it is a limited field of study. This thesis further explores this phenomenon by applying a transformation prediction model to multiple …

The Combinatorics Of Integer Partitions Enumerated By Some Exotic Weights, Hunter Waldron

Dissertations, Master's Theses and Master's Reports

Identities of integer partitions generally state that two dissimilar appearing families of partitions are in fact equinumerous when both are restricted to any fixed size. Euler's theorem is a classic example of such an identity, which equates the number of partitions with odd parts to the number of partitions with distinct parts. Lately, analogs of known partition identities involving weights other than size have begun to attract research interest. This dissertation is an investigation of two such weights. In Chapter 2, we study Schmidt weights, which count only parts with indices belonging to some given subset of the positive integers. …

Accelerated Optimization Of Implicit Neural Representations For Ct Reconstruction, Mahrokh Najaf, Gregory Ongie

Mathematical and Statistical Science Faculty Research and Publications

Inspired by their success in solving challenging inverse prob-lems in computer vision, implicit neural representations (INRs) have been recently proposed for reconstruction in low-dose/sparse-view X-ray computed tomography (CT). An INR represents a CT image as a small-scale neural network that takes spatial coordinates as inputs and outputs attenuation values. Fitting an INR to sinogram data is similar to classical model-based iterative reconstruction methods. However, training INRs with losses and gradient-based algorithms can be prohibitively slow, taking many thousands of iterations to converge. This paper investigates strategies to accelerate the optimization of INRs for CT reconstruction. In particular, we propose two …

Abstractions Of The Game “Set”, Martin Grant

Master's Projects

Each card in the game of SET can be represented as a point in Z⁴₃, where Z₃ is the f ield of 3 elements; an in-game position without any SETs can be represented as a cap set. We find the largest cap sets in Zⁿ ₃ for n ≤ 4 and prove their uniqueness. Then, we provide more insight into Ellenberg and Gijswijt’s proof of upper bound for the maximum size of cap set in Fⁿ_q, where F_q is the field of q elements, which they find to be o(c^{n …}

Students’ Concept Maps Profile On Environmental Pollution Material During The Covid-19 Pandemic, Sunarni Sunarni, Sri Widoretno, Puguh Karyanto, Bowo Sugiharto, Sri Dwiastuti, Chandra Adi Prabowo

Jurnal Pendidikan Sains

The study investigates the performance of students in constructing Concept Maps (CM) related to environmental pollution material during the COVID-19 pandemic. The data reveals that the overall achievement in CM construction was suboptimal, with no component reaching more than half of its maximum potential. The highest achievement was recorded in the valid relationship component, which attained 9.75% of its maximum possible score. This was followed by the hierarchy component, achieving 7.34%, and the branching component at 3.42%. The pattern component ranked fourth with 1.31%, closely followed by the example component at 1.30%. The lowest performance was observed in the crosslink …

Analysis Of Students’ Problem-Solving Skills In Newton’S Law With 5e Learning Cycle, Nidya Desyana, Sutopo Sutopo, Nandang Mufti

Jurnal Pendidikan Sains

The aim of this study is to identify students’ problem-solving skills in Newton’s laws through the implementation of the 5E learning cycle in the classroom. A pre-experimental design with a one-group pretest-posttest approach was employed. Data on problem-solving skills were collected from students’ pre-test and post-test responses to two essay questions. The indicator with the highest percentage of problem-solving skills was useful description (83% of students), while the indicator with the lowest percentage was specific application of physics (33% of students). Due to the non-normal distribution of the data, the Wilcoxon test was utilized. The results indicated a significant difference …

Development Of Conceptual Scaffolding-Assisted Formative Assessment For Enhancing Understanding Of Temperature And Heat, Ana Lailatul Fitriyah, Sentot Kusairi, Markus Diantoro

Jurnal Pendidikan Sains

This research aims to develop formative assessment media assisted by conceptual scaffolding to improve students’ understanding of temperature and heat, test its validity and practicality, and gauge student responses. Utilizing the ADDIE model, the research follows five stages: Analysis, Design, Development, Implementation, and Evaluation. In the Analysis stage, a literature review was conducted to identify critical components, including formative assessment, conceptual scaffolding, concept mastery, and previous research. The Design stage involved identifying users, defining competencies and indicators, creating a storyboard, integrating assessment components, and determining mastery levels. During the Development stage, the assessment media were developed into a functional product. …

Composite Volume Learning Media Based On Augmented Reality To Enhance Students’ Visual-Spatial Intelligence, Ria Kurniawati, Susiswo Susiswo, Syamsul Hadi

Jurnal Pendidikan Sains

Augmented Reality (AR) technology, a three-dimensional visual tool that integrates virtual objects or environments into real-world displays in real time (Azuma et al., 2001), is particularly useful for grasping complex concepts such as the composite volume of solids, which often confuses students when they need to decompose solids into their individual components. While previous research has predominantly focused on the introduction of solids, spatial nets, and the volume of singular shapes, this study aims to develop composite volume learning media based on AR to enhance students’ visual-spatial intelligence in a valid, practical, and effective manner, aligning with their concrete operational …

Design Of Motivational-Realistic-Pancasila Learning In Mathematics, Marup Marup, Suradi Tahmir, Ahmad Talib, Nurzatulshima Kamaruddin

Jurnal Pendidikan Sains

Learning mathematics in class greatly influences students' learning success both in terms of knowledge and attitudes. In carrying out learning, a learning design is needed that can foster student motivation so that they provide learning outcomes to the learning objectives. Motivational-Realistic-Pancasila learning design is considered to be an alternative solution. Morela is a design that combines motivational, realistic indicators and Pancasila values. The teaching of mathematics in the classroom significantly influences students' academic success, both in terms of knowledge and attitude. To effectively carry out the teaching process, a well-designed instructional approach is necessary to foster students' motivation to achieve …

A Novel Format For Math Circles: Circles With No Center, Frederick A. Peck, Ian P. Renga, Nick Grener, David R. Erickson, Matt B. Roscoe, Ke Wu

Journal of Math Circles

This paper shares a novel format for Math Circles, which we call “circles with no center.” Circles with no center do not have a traditional facilitator. The problem or activity is novel to everyone in the room. All members engage in mathematical inquiry, and everyone takes responsibility for running the gathering. The “no center” format has benefits for participants and organizers. For participants, the format offers more freedom and more responsibility. For organizers, the format is easier to organize, more fun, and involves less pressure. Our assessment of a 3-day retreat for teachers suggests participants may prefer a circle with …

Understanding Women, Infant And Children Feeding Decisions In Emergencies: The Case Of Covid-19 And The Formula Shortage, Davida Mensah, Edmund F. Agyemang, Constance Gewa

School of Mathematical & Statistical Sciences Faculty Publications

Background: The advantages of breastfeeding and its role in emergencies are undeniable. Studies show that many women in developed countries choose formula over breastmilk, even in emergencies. Nevertheless, no one knows if postpartum women under the Women, Infants and Children (WIC) program turned out to breastfeed or re-lactate their infants during these crises. This study aims to better understand WIC participants' perceptions about breastfeeding their babies in emergencies.

Methods: A chi-square test of independence was used to assess the association between breastfeeding choice and demographic characteristics and potential covariates of respondents. A post hoc analysis using Dunn's multiple comparisons test …

Effects Of Missing Data Imputation Methods On Univariate Blood Pressure Time Series Data Analysis And Forecasting With Arima And Lstm, Nicholas Niako, Jesus D. Melgarejo, Gladys E. Maestre, Kristina Vatcheva

School of Mathematical & Statistical Sciences Faculty Publications

Background: Missing observations within the univariate time series are common in real-life and cause analytical problems in the flow of the analysis. Imputation of missing values is an inevitable step in every incomplete univariate time series. Most of the existing studies focus on comparing the distributions of imputed data. There is a gap of knowledge on how different imputation methods for univariate time series affect the forecasting performance of time series models. We evaluated the prediction performance of autoregressive integrated moving average (ARIMA) and long short-term memory (LSTM) network models on imputed time series data using ten different imputation techniques. …

Apex Graphs And Cographs, Jagdeep Singh, Vaidy Sivaraman, Thomas Zaslavsky

Theory & Applications of Graphs

A class G of graphs is called hereditary if it is closed under taking induced subgraphs. We denote by G^{apex} the class of graphs G that contain a vertex v such that G − v is in G. Borowiecki, Drgas-Burchardt, and Sidorowicz proved that if a hereditary class G has finitely many forbidden induced subgraphs, then so does G^{apex}. We provide an elementary proof of this result.

The hereditary class of cographs consists of all graphs G that can be generated from K_1 using complementation and disjoint union. A graph is an apex cograph if it contains a …

Parallel Circles: A Math Club As A University Math Circle, Sarah C. Cobb, Marcos Lopez

Journal of Math Circles

This paper will discuss a monthly social event for the University Math Club. Over the past several years, the event has been shaped by the philosophy of a Student Math Circle. We will discuss the event's organization and the ways it has benefited from the influence of the Math Circle community.

On The Power Series Method For Solitons And Peakons Of Nonlinear Pde, Erika Olivares, Vesselin Vatchev

School of Mathematical & Statistical Sciences Faculty Publications

In this paper we discuss the power series method to obtain traveling wave solutions of nonlinear PDE in one space variable. The method, including a recursive procedure for the coefficients and error estimate for the partial sums, is presented for the Camassa-Holm equation. Numerical examples of approximate single soliton and single peakon solutions are included.

Existence And Uniqueness Of Solutions Of Fourth Order Iterative Boundary Value Problems, Zach Whaley

Theses and Dissertations

In this dissertation, we study the fourth-order iterative differential equation \begin{displaymath} x^{(4)}(t) = \f(t, x(t), x^{[2]}(t), \dots, x^{[m]}(t)) \end{displaymath} where $x^{[2]}(t) = x(x(t))$ and $x^{[j]}(t) = x(x^{[j - 1]}(t))$ for $j > 2$. We consider the above equation with multiple sets of boundary conditions, and we state results on the existence and uniqueness of solutions for each set of boundary conditions. In Chapter 2, the boundary conditions are conjugate boundary conditions, \begin{align*} &x(-a) = -a, \ x'(-a) = b, \ x''(-a) = c, \ x(a) = a \\ &x(-a) = -a, \ x(a) = a, \ x'(a) = b, \ x''(a) …

Mathematical Modelling Of Disease Outbreak, Favour Christian, Matthew Molloy

SURE Journal: Science Undergraduate Research Experience Journal

Establishing a model framework for more research necessitates a thorough understanding of the causes, distribution, prevalence, and evolution of infectious illnesses. The main mathematical concept used in this modelling simulation is ordinary differential equations (ODEs). The purpose of this study was to investigate the significance of the many criteria linked to a zombie virus spread. The zombie framework provides an accessible and relatively simple representation of the nature of infectious disease spread, allowing for tractable assumptions and the development of more complex situations.

The models are designed around a zombie outbreak in which the zombie virus is spread through a …

Theory And Algorithms To Learn, Propagate, And Exploit Uncertainty For Stochastic Optimal Control Of Dynamical Systems, Vignesh Sivaramakrishnan

Electrical and Computer Engineering ETDs

Non-Gaussian uncertainty frequently arises in learning and control problems involving stochastic dynamical systems, particularly in autonomous vehicles, UAVs, satellites, and robotics. In this dissertation, we propose a new framework that leverages characteristic functions that provides a frequency-domain representation of random variables. The dissertation is structured into three key areas. First, we address model-based stochastic optimal control for linear systems with non-Gaussian noise, demonstrating that characteristic functions can be used to enforce chance constraints and control systems toward desired distributions. Second, we explore data-driven stochastic control, utilizing empirical characteristic functions to handle systems with unknown disturbances. In addition, we derive several …

Operations On Submodules With The Multiplicative And Quotient Properties, Jiekai Pang

Mathematics & Statistics ETDs

Inspired by the works of Petro, Epstein, Vassilev, and Morre, this thesis aims to study the generalized definitions of the semiprime operation, weakly prime operation, and standard closure operation on rings, that is, the multiplicative operation, weakly multiplicative operation, and standardly multiplicative operation on submodules. Then, we will use Matlis duality to induce the dual notions of these definitions on submodules of Matlis-dualizable Artinian modules. In order to understand the dual notion of the standardly multiplicative operation, that is, the standardly quotient operation, we will classify the operations on the injective hull of residue field of the ring K[[x,y]]/(xy) which …

Calculation And Statistical Analysis Of Wins Above Replacement, Joshua Taylor

Departmental Honors & Graduate Capstone Projects

The Wins Above Replacement (WAR) statistic in Major League Baseball is a prominent metric used to estimate player value by quantifying all aspects of play in terms of wins added to a baseball team. We will use R to calculate WAR for all players from 1871 to 2012 and use data from those years to construct multivariate predictive models to attempt to estimate WAR for players from 2013 to 2024. We find strong correlations between predicted and actual WAR values for most models, with the exception of the polynomial predictive model for non-qualified pitchers.