Mathematics Commons^™

An Elementary Approach To The Probability Distribution Of The Product Of Multiple Random Variables, Suwen Tian

Rose-Hulman Undergraduate Mathematics Journal

Abstract There arenindependent identically distributed (i.i.d.) uniform random variables defined as ξ1,ξ2,…,ξn, valued in interval [0,k](k>0), and there is a constantmvalued in interval (0,kn). We study the distribution of the product of these random numbers and prove a formula calculating Pr(∏i=1nξi≤m). Interestingly, we find that the result is exactly the sum of the firstnterms in the Taylor series expansion of the function exp(x) with x=nlnk-lnm. Through considering the corresponding probability density function, we make an extension of the formula calculating Pr(∏i=1nξi≤m) to any positive realn, and the extended formula can be written in a …

Exploring System Identification Of Non-Linear Dynamics Using The Weighted Composition Operator And The Liouville Operator, Chukwuebuka Amagwula

USF Tampa Graduate Theses and Dissertations

System identification is the process of determining mathematical models that describe the dynamics of a system from data. Dynamic Mode Decomposition (DMD) and Sparse Identification of Nonlinear Dynamical Systems (SINDy) are two distinct approaches used for this purpose.

DMD identifies dominant spatiotemporal modes and eigenvalues that describe the evo lution of a system. It assumes a near-linear representation of dynamics and is closely linked to the Koopman operator, making it ideal for analyzing fluid flows, oscillatory systems, and modal structures. The DMD method uses time series data where each data point is referred to as a snapshot and represents the …

On The Hamilton-Waterloo Problem With A Single Factor Of 6-Cycles, Zazil Santizo Huerta, Melissa S. Keranen

Michigan Tech Publications

The uniform Hamilton-Waterloo Problem (HWP) asks for a resolvable (CM, CN)decomposition of Kv into α CM-factors and β CN-factors. We denote a solution to the uniform Hamilton-Waterloo problem by HWP(v, M, N, α, β). Our research concentrates on addressing some of the remaining unresolved cases, which pose a significant challenge to generalize. We place a particular emphasis on instances where the gcd(M, N) = {2, 3}, with a specific focus on the parameter M = 6. We introduce modifications to some known structures, and develop new approaches to resolving these outstanding challenges in the construction of uniform 2-factorizations. This innovative …

Functional Equivariance And Backward Error Analysis, Sanah Suri

Arts & Sciences Graduate Student Theses and Dissertations

Geometric, or structure-preserving, numerical integration has long been used as a framework for studying integrators that preserve a systems invariants. In backward error analysis, the approximate numerical flow of a system is viewed as the exact flow of a modified problem, allowing us to gain qualitative insights into the behavior of the numerical solution. The preservation of conservation laws by a numerical integrator can be generalized to $F$-functionally equivariant integrators, where $F$ represents an observable of the system in consideration. This thesis describes the behavior of geometric integrators through the lens of backward error analysis. First, we extend the idea …

Analysis For Idempotent States On Quantum Permutation Groups, J.P. Mccarthy

Department of Mathematics Publications

Woronowicz proved the existence of the Haar state for compact quantum groups under a separability assumption later removed by Van Daele in a new existence proof. A minor adaptation of Van Daele’s proof yields an idempotent state in any non-empty weak-* compact convolution-closed convex subset of the state space. Such subsets, and their associated idempotent states, are studied in the case of quantum permutation groups.

Dice Math And Probability, Warren Campbell

SEAS Faculty Publications

The manufacture of dice for tabletop games is a billion-dollar industry. In gaming circles and online forums, the concept of “cursed dice” is a popular topic. Dice are inherently unfair due to the difficulty of manufacturing them with perfect geometric tolerances and uniform material densities. A common method for testing dice fairness is the chi-square statistic, which is typically assumed to follow the chi-square distribution. However, this assumption is only asymptotically valid.

Exact distributions of the statistic can be computed for dice with few sides and a limited number of rolls—such as 2-sided (D2) and 4-sided (D4) dice—but the computational …

Let’S Teach More Accurate And Inclusive History! The Case Of Islamic Contributions To Mathematics And Science, Noah (Nuh) Aydin

Faculty Publications

Schools commonly teach a history of mathematics and science that is Eurocentric. This selective version of the history of science, called Classical Narrative by some, is rooted in colonial times and mindset, and has reinforced certain negative opinions about other cultures. It ignores or downplays contributions to mathematics and sciences from non-western civilizations, and falsely attributes many scientific discoveries to European scholars. Medieval Islamic Civilization, which had strong connections to Renaissance Europe, serves as a clear example of how this narrative is distorted. Based on research on primary sources since the middle of the 20th century, we now know that …

Exploring The Exceptional Extreme Rays Of The Copositive Cone, Trent Holmgren

All NMU Master's Theses

In this paper we will look at the exceptional extreme rays of COP^5 and COP^6 and identify if they are exposed or nonexposed. We start with some necessary background material on copositive matrices and cones. Then we construct two algorithms to see if a matrix is exposed or nonexposed.

Bicomplex Hardy Classes Of Solutions To Beltrami Equations And The Schwarz Boundary Value Problem, William L. Blair

Math Faculty Publications and Presentations

We define Hardy classes of bicomplex-valued functions on the complex unit disk which solve bicomplex versions of the Beltrami and related equations. Using representations in terms of their complex-valued counterparts, we show these bicomplex-valued functions recover the boundary behavior associated with the classic holomorphic Hardy spaces. This work generalizes known results for complex-valued functions and continues recent work in the setting of bicomplex analogues of Hardy spaces of both holomorphic and generalized analytic functions. Also, we show Schwarz and Dirichlet boundary value problems associated with the bicomplex Beltrami equation are solvable and provide solution formulas.

Torus Probabilistic Principal Component Analysis, Anahita Nodehi, Mousa Golalizadeh, Mehdi Maadooliat, Claudio Agostinelli

Mathematical and Statistical Science Faculty Research and Publications

Analyzing data in non-Euclidean spaces, such as bioinformatics, biology, and geology, where variables represent directions or angles, poses unique challenges. This type of data is known as circular data in univariate cases and can be termed spherical or toroidal in multivariate contexts. In this paper, we introduce a novel extension of probabilistic principal component analysis (PPCA) designed for toroidal (or torus) data, termed torus probabilistic PCA (TPPCA). We provide detailed algorithms for implementing TPPCA and demonstrate its applicability to torus data. To assess the efficacy of TPPCA, we perform comparative analyses using a simulation study and three real datasets. Our …

Graph Convolutional Networks Enable Fast Hemorrhagic Stroke Monitoring With Electrical Impedance Tomography, J. Toivanen, V. Kolehmainen, A. Paldanius, A. Hänninen, A. Hauptmann, Sarah J. Hamilton

Mathematical and Statistical Science Faculty Research and Publications

Objective: To develop a fast image reconstruction method for stroke monitoring with electrical impedance tomography with image quality comparable to computationally expensive nonlinear model-based methods. Methods: A post-processing approach with graph convolutional networks is employed. Utilizing the flexibility of the graph setting, a graph U-net is trained on linear difference reconstructions from 2D simulated stroke data and applied to fully 3D images from realistic simulated and experimental data. An additional network, trained on 3D vs. 2D images, is also considered for comparison. Results: Post-processing the linear difference reconstructions through the graph U-net significantly improved the image quality, resulting in images …

Exploring Communication In Multi-Agent Cooperative Reinforcement Learning, Matthew Kalarickal

Math and Computer Science Honors Theses

This work focuses on communication strategies within a cooperative multi-agent reinforcement learning system. The goal is to explore how communication can be used among agents to potentially improve performance. The research operates within the scope of “learning tasks with communication,” where the primary aim is to solve domain-specific tasks through information exchange using explicit communication protocols. Three distinct communication strategies were implemented and explored: Combinatorial Ghost, Feature Sharing Ghost, and Move Sharing Ghost. Fully Centralized Training and Execution and Centralized Training with Decentralized Execution training approaches were based on the different communication strategy used. Performance metrics were recorded for these …

P-Adic Cellular Neural Networks With Delay, Baboucarr Dibba

Theses and Dissertations

This dissertation presents a novel framework for p-adic reaction-diffusion cellular neural networks (CNNs) with delay, offering new insights into the stability and dynamic behavior of these networks. Through numerical simulations, we explore their response to various conditions, highlighting their capability to model complex systems. Additionally, this work reviews cutting-edge developments in p-adic CNNs, particularly their application to advanced image processing tasks such as edge detection and noise filtering, demonstrating their effectiveness in preserving critical image features while filtering out noise. This dissertation is written in collaboration with my Ph.D supervisor and Dr. Zambrano-Luna, Brian.

A Study Of Quasigroups, A Computational Approach On Isotopism And Isomorphism Hierarchical Structure, Runaldo Montrose

Theses and Dissertations

Quasigroups morphism classification can present a real computational challenge. As of today, the number of quasigroups that can be generated from a finite set S of cardinality n is known for very small value of n. Though we know the number of quasigroups for small n, generating them in real time is a whole other issue that leads to a bigger challenge of building their isomorphy classes, because the algorithm used required large computing memory resources. In this paper we will expose the computational complexity of constructing their hierarchical morphism structure from isotopism to isomorphism. As an improvement, …

Explorations In Extremal Combinatorics: Graph Spectra, Codegree Squared Sum Of Hypergraphs And Lin-Lu-Yau Ricci Curvature, George Henry Brooks

Theses and Dissertations

This dissertation explores a variety of extremal problems in spectral graph theory, hypergraph Tur\'an theory in the $\ell_2$-norm, and Lin-Lu-Yau Ricci curvature. We begin by studying the maximization of linear combinations of eigenvalues in relation to the adjacency matrix of a graph. For a graph $G$ on $n$ vertices, we order its eigenvalues $\lambda_1 \geq \dots \geq \lambda_n$. First, we study a generalization of the maximum spread. Specifically, we consider the $(i,j)$-spread, defined as $\lambda_{i+1}-\lambda_{n-j}$, and aim to maximize that quantity over all simple graphs on $n$ vertices. We establish asymptotic bounds for all $i,j\geq 0$ and identify infinitely many …

On A Conjecture On Covering Systems And An Irreducibility Question On Sparse 0,1-Polynomials, Alexandros Kalogirou

Theses and Dissertations

In 1952, H. Davenport posed the problem of determining a condition on the minimum modulus $m_0$ in a finite distinct covering system that would imply that the sum of the reciprocals of the moduli in the covering system is bounded away from 1. In 1973, P.~Erd\H os and J.~Selfridge indicated that they believed that $m_0$ > 4 would suffice. We provide a proof that this is the case in Chapter 2. Chapters 3 and 4 are dedicated to showing that $0,1$-polynomials of high degree and few terms are irreducible with high probability. Formally, let $k\in\mathbb{N}$ and $F(x)=1+\sum_{i=1}^kx^{n_i}$, where $ 0

On Berge Pancyclicity For Uniform Hypergraphs, Teegan Cole Bailey

Theses and Dissertations

An $n$-vertex graph $G$ is \textit{hamiltonian} if it contains a length $n$ cycle as a subgraph. A stronger notion of hamiltoncity is pancyclicity. A graph $G$ is \textit{pancyclic} if $G$ contains a cycle of length $k$ for every $3\leq k \leq n$. Dirac's classical result on hamiltonicity states that if an $n$-vertex graph $G$ has minimum degree $\delta(G)\geq \frac{n}{2}$, then $G$ is hamiltonian. Under the same minimum degree condition as Dirac, J. A. Bondy showed that $G$ must also be pancyclic or the exceptional graph $K_{\frac{n}{2}, \frac{n}{2}}$. In 1971, Bondy proposed a so called meta-conjecture claiming that \textit{``almost any nontrivial …

Dissections Of Lacunary Eta Quotients And Identically Vanishing Coefficients, Timothy Huber, James Mclaughlin, Dongxi Ye

School of Mathematical & Statistical Sciences Faculty Publications

For any function A(q)=∑∞n=0anqn defineA(0):={n∈N:an=0}.Now suppose C(q) and D(q) are two functions whose m-dissections are given byC(q)=c0G0(qm)+c1qG1(qm)+…+cm−1qm−1Gm−1(qm),D(q)=d0G0(qm)+d1qG1(qm)+…+dm−1qm−1Gm−1(qm).If it is the case that ci=0⟺di=0, i=0,1,…,m−1, then we say that C(q) and D(q) have similar m-dissections, and then it is also clear that C(0)=D(0), in which case we say that C(q) and D(q) have identically vanishing coefficients. In the present paper some new 4-dissections of particular eta quotients are developed. These are used in conjunction with known 2- and 3-dissections to prove many results on the identical vanishing of coefficients of various eta quotients, results which were found experimentally …

Stability Analysis Of The Eulerian-Lagrangian Finite Volume Methods For Nonlinear Hyperbolic Equations In One Space Dimension, Yang Yang, Jiajie Chen, Jing Mei Qiu

Michigan Tech Publications

In this paper, we construct a novel Eulerian-Lagrangian finite volume (ELFV) method for nonlinear scalar hyperbolic equations in one space dimension. It is well known that the exact solutions to such problems may contain shocks though the initial conditions are smooth, and direct numerical methods may suffer from restricted time step sizes. To relieve the restriction, we propose an ELFV method, where the space-time domain was separated by the partition lines originated from the cell interfaces whose slopes are obtained following the Rakine-Hugoniot junmp condition. Unfortunately, to avoid the intersection of the partition lines, the time step sizes are still …

The Effort To Improve Students’ Problem-Solving Ability: Physics E-Modules For High School Students, Hidayatullah Hana Putra, Edi Supriana, Nasikhuddin Nasikhuddin, Annisa U. Yana, Suhailee Sohnui

Jurnal Pendidikan Sains

Physics teachers in Indonesia still face obstacles during learning, such as students who are less active and have difficulty in solving physics problems. The physics learning media used are also still not very interesting and are not based on authentic problem solving. Therefore, this study focuses on the development of PBL-based e-modules on work and energy to improve the physics problem-solving abilities of high school students. This research and development used the ADDIE design and was tested on 90 high school students in High School 1 Sumber Pucung Malang in the 2022/2023 academic year. The questionnaire instrument was used in …

Ethnoscience-Based Science Learning Through Composite Board Making Utilizing Eggshell Waste And Water Hyacinth Fiber (Eichornia Crassipes), Asep Khaidir Arpan, Yulia Sukmawardani, Pina Pitriana, Ade Yeti Nuryantini

Jurnal Pendidikan Sains

The demand for boards continues to increase every year, while overexploitation of wood leads to environmental degradation and forest loss. On the other hand, eggshell and water hyacinth waste continues to grow without optimal utilization. This research aims to make composite boards based on eggshell waste and water hyacinth fibers and analyze the potential of ethnoscience in science learning. This research used a qualitative descriptive method with a literature and documentation study approach. The materials used in this experiment are eggshell waste, water hyacinth fiber, and polyester resin. The composite board was made by mixing water hyacinth fiber and eggshell …

Contextual Approach To Improve Problem Solving Skills In Ipas Subject At Elementary School: A Study Development Of Teaching Materials, Nur Hidayah, Sumarno Sumarno, Joko Siswanto

Jurnal Pendidikan Sains

The objective of this study is to assess the validity, practicality, and efficacy of instructional materials. We refer to this research as research and development, with a focus on the creation of teaching materials. The research design proposed by Borg and Gall consists of six steps: identification of potential and difficulties, data collecting, product design, design validation, design revision, and product trial. The utilized instruments comprise expert and practitioner validity sheets, teacher and student response surveys, and validation sheets for teaching materials and modules, as well as pretest and posttest questions. Experts performed the validity assessment, attaining a score of …

Teaching Materials Containing Kse To Support Character Education In Mathematics Subjects In Elementary Schools, Kunaenah Kunaenah, Rasiman Rasiman, Bagus Ardi Saputro

Jurnal Pendidikan Sains

The purpose of the study was to determine the validity, practicality, and effectiveness of teaching materials containing KSE to support character education in mathematics lesson content. This study employs a development research approach based on Borg and Gall's research design. The subjects of the study were in class V at public elementary school Kaliboyo. The data collection instruments used were validity sheets, response questionnaires, validation sheets, and observation sheets. The findings show that experts carried out the validity test, achieving a value of 93.30% in the very valid category. The practicality test results were derived from various sources: students' questionnaires, …

Development Of Differentiated Teaching Modules With A Scientific Approach To Improve Elementary School Students’ Science Literacy, Supriyanto Supriyanto, Joko Sulianto, Achmad Buchori

Jurnal Pendidikan Sains

The study aim is to determine the validity, practicality, and effectiveness of teaching materials. This type of research involves the development and creation of teaching module products. The research design used is ADDIE (Analysis, Design, Development, Implementation, Evaluation). The instruments used are validity sheets for experts and practitioners, teacher response questionnaires, teaching module validation sheets, and pretest and posttest questions. The findings show that experts conducted the validity test, achieving a score of 92.22 in the very valid category. The practicality test was obtained from the teacher questionnaire, which was stated as practical with an average value of 91.25, and …

Differentiated Teaching Module Using Problem Solving Model: Numerical Ability Of Elementary School Students, Risa Alifatin, Agus Sutono, Ida Dwijayanti

Jurnal Pendidikan Sains

This study aims to develop a differentiated teaching module using the problem-solving model that is valid, practical, and effective to improve students' numerical abilities. This study uses the ADDIE Model research and development approach. The sample in this study was class III of elementary school Tenggulangharjo as an experimental class totaling 22 students and class III of elementary school Karangtengah as a control class totaling 19 students. Data analysis used the t-test and N-Gain test. The study found that (1) the average validity score of the three parts of the Differentiated Teaching Module was 3.74, which is 93.62%, placing it …

Analysis Of Students' Creative Thinking Skills In Learning Hydrocarbons By Using Daily Life-Based Practicum Method, Yenni Kurniawati, Laras Dyaz Handayani, Dea Ramadani, Nurhikmah Sari

Jurnal Pendidikan Sains

Creative thinking skill is a competence of the 21st century that is needed very much and needs to be increased by utilizing chemistry in life. This research aimed at analyzing student creative thinking skills in experimental and control groups and finding out the effect of the daily life-based practical work learning method on student creative thinking skills. This research was conducted in the eleventh grade at High School 1 Pekanbaru on the hydrocarbon lesson. The quasi-experimental method was used in this research with pretest-posttest and non-equivalent control group designs. The instruments used in this research were essay tests, observation, and …

Exploring Students’ Perceptions Of Science Laboratory Practices: A Comparative Study Between Rural And Urban Schools, Safrida Dwiningsih, Ina Lestari, Roma Ismi, Sri Wahyu Rahmadina, Masdalena Sinaga, M. Rahmad

Jurnal Pendidikan Sains

This study aims to analyze students' perceptions of the implementation of science laboratory practicums at public middle school 020 Pekanbaru and public middle school 7 Pangkalan Kuras, Riau Province. Using a quantitative survey method, this study involved 75 student respondents. The instrument used was a questionnaire consisting of 26 statements, analyzed descriptively. The results show that students' perceptions of the preparation, implementation, and conclusion stages of the practicum fall into the "good" category, with urban schools having a higher percentage. In conclusion, the implementation of practicums in urban schools is better than in rural schools, bringing attention to the need …

Digital Comic Learning Media (E-Comic) To Increase Student Learning Interest In Reaction Rate Material At High School, Berliana Afsohin Nabila, Achmad Lutfi

Jurnal Pendidikan Sains

This study aims to develop digital comic learning media that is effective in increasing students' interest in learning about reaction rate material, focusing on the media's validity, practicality, and overall effectiveness. This research utilizes the research and development method with the ADDIE model. Research instruments were used, including validity sheets to assess the media's validity, student activity observations, and student response questionnaires to evaluate the media's practicality, as well as student interest questionnaires and learning outcomes to measure the effectiveness of the learning media. The findings show that validity results showed a mode score of 4, indicating a very valid …

Analysis Of Misconceptions In Learning Biology Mutation Material At High School, Fitri Hayati Kurnia, Tri Wahyu Agustina, Wahyuni Handayani

Jurnal Pendidikan Sains

This research aims to analyze student misconceptions regarding mutation material in class XII. The research method uses a descriptive approach with mixed methods, which combines qualitative and quantitative methods. Qualitative data was obtained through interviews with biology teachers, while quantitative data was obtained through questionnaires and essay tests on students. The research results indicate that the material that students like is growth and development, while the material they don't like is gene mutation and genetic material. Students' misconceptions occur due to preconception aspects, associative thinking, and a lack of understanding of genetic concepts. It was found that students' understanding of …

Regularity, Uniqueness And The Relative Size Of Small And Large Scales In Sqg Flows, Z. Akridge, Z. Bradshaw

Mathematical Sciences Faculty Publications and Presentations

The problems of regularity and uniqueness are open for the supercritically dissipative surface quasi-geostrophic equations in certain classes. In this note we examine the extent to which small or large scales are necessarily active both for the temperature in a hypothetical blow-up scenario and for the error in hypothetical non-uniqueness scenarios, the latter understood within the class of Marchand's solutions. This extends prior work for the 3D Navier-Stokes equations. The extension is complicated by the fact that mild solution techniques are unavailable for supercritical SQG. This forces us to develop a new approach using energy methods and Littlewood-Paley theory.