Mathematics Commons^™

Why Resilient Modulus Is Proportional To The Square Root Of Unconfined Compressive Strength (Ucs): A Qualitative Explanation, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

The strength of the pavement is determine by its resilient modulus, i.e., by its ability to withstand (practically) instantaneous stresses caused by the passing traffic. However, the resilient modulus is not easy to measure: its measurement requires a special expensive equipment that many labs do not have. So, instead of measuring it, practitioners often measure easier-to-measure Unconfined Compressive Strength (UCS) -- that describes the effect of a continuously applied force -- and estimate the resilient modulus based on the result of this measurement. An empirical formula shows that the resilient modulus is proportional to the square root of the Unconfined …

How To Estimate Unknown Unknowns: From Cosmic Light To Election Polls, Talha Azfar, Vignesh Ponraj, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

In two different areas of research -- in the study of space light and in the study of voting -- the observed value of the quantity of interest is twice larger than what we would expect. That the observed value is larger makes perfect sense: there are phenomena that we do not take into account in our estimations. However, the fact that the observed value is exactly twice larger deserves explanation. In this paper, we show that Laplace Indeterminacy Principle leads to such an explanation.

Lagrange’S Proof Of Wilson’S Theorem—And More!, Carl Lienert

Number Theory

No abstract provided.

Lagrange’S Proof Of The Converse Of Wilson’S Theorem, Carl Lienert

Number Theory

No abstract provided.

Lagrange’S Alternate Proof Of Wilson’S Theorem, Carl Lienert

Number Theory

No abstract provided.

May Graduation, Samuel Coskey

Mathematics Faculty Publications and Presentations

Here I narrate the story of the last few days of my graduate program in mathematics. After the completion of the thesis and the delivery of the defense, several twists and turns await in the hours and even minutes before the last deadline.

How To Combine Probabilistic And Fuzzy Uncertainty: Theoretical Explanation Of Clustering-Related Empirical Result, Lázló Szilágyi, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In contrast to crisp clustering techniques that assign each object to a class, fuzzy clustering algorithms assign, to each object and to each class, a degree to which this object belongs to this class. In the most widely used fuzzy clustering algorithm -- fuzzy c-means -- for each object, degrees corresponding to different classes add up to 1. From this viewpoint, these degrees act as probabilities. There exist alternative fuzzy-based clustering techniques in which, in line with the general idea of the fuzzy set, the largest of the degrees is equal to 1. In some practical situations, the probability-type fuzzy …

Fuzzy Techniques Explain The Effectiveness Of Relu Activation Function In Deep Learning, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the last decades, deep learning has led to spectacular successes. One of the reasons for these successes was the fact that deep neural networks use a special Rectified Linear Unit (ReLU) activation function s(x) = max(0,x). Why this activation function is so successful is largely a mystery. In this paper, we show that common sense ideas -- as formalized by fuzzy logic -- can explain this mysterious effectiveness.

Why 6-Labels Uncertainty Scale In Geosciences: Probability-Based Explanation, Aaron Velasco, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To describe uncertainty in geosciences, several researchers have recently proposed a 6-labels uncertainty scale, in which one the labels corresponds to full certainty, one label to the absence of any knowledge, and the remaining four labels correspond to the degrees of confidence from the intervals [0,0.25], [0.25,0.5], [0.5,0.75], and [0.75,1]. Tests of this 6-labels scale indicate that it indeed conveys uncertainty information to geoscientists much more effectively than previously proposed uncertainty schemes. In this paper, we use probability-related techniques to explain this effectiveness.

Complex Numbers Explain Why In Chinese Tradition, 4 Is Bad But 8 Is Good, Luc Longpre, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the traditional Chinese culture, 4 is considered to be an unlucky number, while the number 8 is considered to be very lucky. In this paper, we show that both "badness" and "goodness" can be explained if we take into account the role of complex numbers in the analysis of general dynamical systems.

Why Bump Reward Function Works Well In Training Insulin Delivery Systems, Lehel Dénes-Fazakas, Lásló Szilágyi, Gyorgy Eigner, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

Diabetes is a disease when the body can no longer properly regulate blood glucose level, which can lead to life-threatening situations. To avoid such situations and regulate blood glucose level, patients with severe form of diabetes need insulin injections. Ideally, the system should automatically decide when best to inject insulin and how much to inject. To find the optimal control, researchers applied machine learning with different reward functions. It turns out that the most effective learning occurred when they used the so-called bump function. In this paper, we provide a possible explanation for this empirical result.

We Can Always Reduce A Non-Linear Dynamical System To Linear -- At Least Locally -- But Does It Help?, Orsolya Csiszar, Gábor Csiszar, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

Many real-life phenomena are described by dynamical systems. Sometimes, these dynamical systems are linear. For such systems, solutions are well known. In some cases, it is possible to transform a nonlinear system into a linear one by appropriately transforming its variables, and this helps to solve the original nonlinear system. For other nonlinear systems -- even for the simplest ones -- such transformation is not known. A natural question is: which nonlinear systems allow such transformations? In this paper, we show that we can always reduce a nonlinear system to a linear one -- but, in general, it does not …

What Was More Frequently Used -- "And" Or "Or": Based On Analysis Of European Languages, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditional logic has two main connectives: "and" and "or". A natural question is: which of the two is more frequently used? This question is easy to answer for the current usage of these connectives -- we can simply analyze all the texts, but what can we say about the past usage? To answer this question, we use the known linguistics fact that, in general, notions that are more frequently used are described by shorter words. It turns out that in most European languages, the word for "and" is shorter -- or of the same length -- as the word for …

How To Propagate Interval (And Fuzzy) Uncertainty: Optimism-Pessimism Approach, Vinícius F. Wasques, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, inputs to a data processing algorithm are known with interval uncertainty, and we need to propagate this uncertainty through the algorithm, i.e., estimate the uncertainty of the result of data processing. Traditional interval computation techniques provide guaranteed estimates, but from the practical viewpoint, these bounds are too pessimistic: they take into account highly improbable worst-case situations when all the measurement and estimation errors happen to be strongly correlated. In this paper, we show that a natural idea of having more realistic estimates leads to the use of so-called interactive addition of intervals, techniques that has already …

Incidence And Laplacian Matrices Of Wheel Graphs And Their Inverses, Jerad Ipsen, Sudipta Mallik

Mathematics Faculty Research

It has been an open problem to find the Moore-Penrose inverses of the incidence, Laplacian, and signless Laplacian matrices of families of graphs except trees and unicyclic graphs. Since the inverse formulas for an odd unicyclic graph and an even unicyclic graph are quite different, we consider wheel graphs as they are formed from odd or even cycles. In this article we solve the open problem for wheel graphs. This work has an interesting connection to inverses of circulant matrices.

Interpolation Problems And The Characterization Of The Hilbert Function, Bryant Xie

Mathematical Sciences Undergraduate Honors Theses

In mathematics, it is often useful to approximate the values of functions that are either too awkward and difficult to evaluate or not readily differentiable or integrable. To approximate its values, we attempt to replace such functions with more well-behaving examples such as polynomials or trigonometric functions. Over the algebraically closed field C, a polynomial passing through r distinct points with multiplicities m1, ..., mr on the affine complex line in one variable is determined by its zeros and the vanishing conditions up to its mi − 1 derivative for each point. A natural question would then be to consider …

Edge Covers Of Joined Cycle And Path Graphs, Rowan Kennedy

Mathematics Undergraduate Research

A graph is a mathematical structure consisting of vertices, representing objects, and edges that connect pairs of vertices, representing relationships between objects. When a specific graph structure can be extended in a consistent pattern we get a graph family, such as the families of path and cycle graphs. An edge cover of a graph is a subset of the graph's edges chosen so that each vertex is an endpoint of at least one edge in this subset. The edge cover counts of certain graph families, such as the path and cycle graphs, correspond to known sequences, the Fibonacci and Lucas …

Generation Of Cardiac Chamber Models Using Interpretable Generative Neural Networks For Electrophysiology Studies, Sunil Mathew

Dissertations (1934 -)

An Electrophysiology study is conducted to diagnose and treat heart rhythm disorders, such as arrhythmias (abnormal heartbeat) like atrial fibrillation. A catheter is inserted into the chamber of interest to acquire 3D location and electrical information to create an electroanatomical map. This dissertation explores the design of a mapping system based on interpretable generative neural networks for generating patient specific cardiac models. Chapter 1 provides an introduction to electroanatomical mapping, the need for interpretability in neural networks and other relevant topics that are discussed in detail in the chapters that follow. Neural networks are often very large models with millions …

Deep Learning Methods For Some Problems In Scientific Computing, Yuankai Teng

Theses and Dissertations

Deep learning has emerged as a powerful approach for solving complex problems in scientific computing due to the increasing availability of large-scale data and computational resources. This thesis explores the potential of deep learning methods for three specific problems in scientific computing: (i) reducing the dimensions of variables in function approximation, (ii) solving linear reaction-diffusion equations, and (iii) finding the parametric representations of parameters in the numerical schemes for solving time-dependent partial differential equations.

For the first problem, a novel deep learning architecture is developed for reducing the dimensions of variables in function approximation. The proposed method achieves state-of-the-art performance …

Widely Digitally Delicate Brier Primes And Irreducibility Results For Some Classes Of Polynomials, Thomas David Luckner

Theses and Dissertations

This dissertation considers three different sections of results. In the first part of the dissertation, a result on consecutive primes which are widely digitally delicate and Brier numbers is discussed. Making use of covering systems and a theorem of D. Shiu, M. Filaseta and J. Juillerat showed that for every positive integer k, there exist k consecutive widely digitally delicate primes. They also noted that for every positive integer k, there exist k consecutive primes which are Brier numbers. We show that for every positive integer k, there exist k consecutive primes that are both widely digitally …

Kinetic Particle Simulations Of Plasma Charging At Lunar Craters Under Severe Conditions, David Lund, Xiaoming He, Daoru Frank Han

Mathematics and Statistics Faculty Research & Creative Works

This paper presents fully kinetic particle simulations of plasma charging at lunar craters with the presence of lunar lander modules using the recently developed Parallel Immersed-Finite-Element Particle-in-Cell (PIFE-PIC) code. The computation model explicitly includes the lunar regolith layer on top of the lunar bedrock, taking into account the regolith layer thickness and permittivity as well as the lunar lander module in the simulation domain, resolving a nontrivial surface terrain or lunar lander configuration. Simulations were carried out to study the lunar surface and lunar lander module charging near craters at the lunar terminator region under mean and severe plasma environments. …

Modified Geometries, Clifford Algebras And Graphs: Their Impact On Discreteness, Locality And Symmetr, Roman Sverdlov

Mathematics & Statistics ETDs

In this dissertation I will explore the question whether various entities commonly used in quantum field theory can be “constructed". In particular, can spacetime be “constructed" out of building blocks, and can Berezin integral be “constructed" in terms of Riemann integrals.

As far as “constructing" spacetime out of building blocks, it has been attempted by multiple scientific communities and various models were proposed. But the common downfall is they break the principles of relativity. I will explore the ways of doing so in such a way that principles of relativity are respected. One of my approaches is to replace points …

Covering By Planks And Avoiding Zeros Of Polynomials, Alexey Glazyrin, Roman Karasev, Alexandr Polyanskii

School of Mathematical & Statistical Sciences Faculty Publications

We note that the recent polynomial proofs of the spherical and complex plank covering problems by Zhao and Ortega-Moreno give some general information on zeros of real and complex polynomials restricted to the unit sphere. As a corollary of these results, we establish several generalizations of the celebrated Bang plank covering theorem. We prove a tight polynomial analog of the Bang theorem for the Euclidean ball and an even stronger polynomial version for the complex projective space. Specifically, for the ball, we show that for every real nonzero d-variate polynomial P of degree n⁠, there exists a point in the …

Integrable Discretizations For A Generalized Sine-Gordon Equation And The Reductions To The Sine-Gordon Equation And The Short Pulse Equation, Han-Han Sheng, Bao-Feng Feng, Guo-Fu Yu

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we propose fully discrete analogues of a generalized sine-Gordon (gsG) equation utx=(1+ν∂2x)sinu. The bilinear equations of the discrete KP hierarchy and the proper definition of discrete hodograph transformations are the keys to the construction. Then we derive semi-discrete analogues of the gsG equation from the fully discrete gsG equation by taking the temporal parameter b→0. Especially, one full-discrete gsG equation is reduced to a semi-discrete gsG equation in the case of ν=−1 (Feng {\it et al. Numer. Algorithms} 2023). Furthermore, N-soliton solutions to the semi- and fully discrete analogues of the gsG equation in the determinant form …

Contractive Symmetric Matrix Completion Problems Related To Graphs, Sangmin Chun, In Hyoun Kim, Jaewoong Kim, Jasang Yoon

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we consider the contractive real symmetric matrix completion problems motivated in part by studies on sparse (or dense) matrices for weighted sparse recovery problems and rating matrices with rating density in recommender systems. We completely characterize symmetric patterns P with the property (C) that every partially contractive real symmetric matrix with pattern P has a contractive real symmetric completion using graphs.

Special Case Of Partial Fraction Expansion With Laplace Transform Application, Laurie A. Florio, Ryan D. Hanc

CODEE Journal

Partial fraction expansion is often used with the Laplace Transforms to formulate algebraic expressions for which the inverse Laplace Transform can be easily found. This paper demonstrates a special case for which a linear, constant coefficient, second order ordinary differential equation with no damping term and a harmonic function non-homogeneous term leads to a simplified partial fraction expansion due to the decoupling of the partial fraction expansion coefficients of s and the constant coefficients. Recognizing this special form can allow for quicker calculations and automation of the solution to the differential equation form which is commonly used to model physical …

Fucik Spectrum With Weights And Existence Of Solutions For Nonlinear Elliptic Equations With Nonlinear Boundary Conditions, Nsoki Mavinga, Q. A. Morris, S. B. Robinson

Mathematics & Statistics Faculty Works

We consider the boundary value problem −Δu + c(x)u = αm(x)u⁺ − βm(x)u⁻ + f(x,u), x∈Ω, (∂u)/(∂η) + σ(x)u = αρ(x)u⁺ − βρ(x)u⁻ + g(x,u), x∈∂Ω, where (α,β) ∈R2, c, m ∈ L^∞(Ω), σ, ρ ∈ L^∞(∂Ω), and the nonlinearities f and g are bounded …

Exploring High School Students' Concept Mastery Of Work And Energy Topics: A Quantitative Descriptive Study With Insights For Stem Learning, Maharani Karunia Putri, Parno Parno, Nuril Munfaridah, Marlina Ali

Jurnal Pendidikan Sains

This quantitative descriptive study investigates the concept mastery of work and energy among high school students, specifically focusing on 105 students from the XI MIPA Laboratory UM High School. The research employs a concept mastery test, adapting Bloom's taxonomy indicators from C2 to C6, comprising 15 reasoned multiple-choice questions with a reliability value of 0.704. Student responses, categorized into No Response (NR), No Understanding (NU), Incorrect Understanding (IU), Partial Understanding (PU), and Sound Understanding (SU), reveal a prevalent mastery level in the category of Incorrect Understanding or partial comprehension of misconceptions. Specifically, NR, NU, IU, PU, and SU categories are …

Exploring The Influence Of Online Inquiry Learning With E-Scaffolding Prompting Question On Problem-Solving Skills In The Work And Energy Topic For Islamic Senior High School Students, Arrika Wifqotu Lailin Nafisah, Supriyono Koes Handayanto, Sunaryono Sunaryono, Chokchai Yuenyong

Jurnal Pendidikan Sains

This comprehensive investigation delves into the impact of integrating e-scaffolding prompting questions into online inquiry learning on students' problem-solving skills (PSS). With an explanatory design and a mixed-methods approach, the study focuses on XI-MIPA class students at MAN 1 Kediri City during the academic year 2022-2023. Utilizing meticulously crafted learning instruments, five problem-solving skills questions, and precise interview guidelines, the research employs triangulation through pre-tests, post-tests, and in- depth interviews. Quantitative data undergoes rigorous scrutiny via Analysis of Covariance (ANCOVA). The primary objective is to unveil the discernible influence of online inquiry learning complemented by e-scaffolding on the nuanced domain …

Study Of Scientific Reasoning And Concept Mastery Of Students Through Guided Inquiry Learning Model Assisted By Formative Assessment, Selvina Visbad Saiya, Sentot Kusairi, Sunaryono Sunaryono

Jurnal Pendidikan Sains

This study endeavors to assess the impact of guided inquiry learning, complemented by formative assessment, on students’ scientific reasoning (SR) and concept mastery (CM). The research involved 20 students from class XI A at SMA Muhammadiyah 02 Sumberpucung, utilizing questions adapted from the modified Lawson classroom test of scientific reasoning (MLCTSR) to measure SR, and the force concept inventory (FCI) to gauge CM. The instructional focus was Newton’s law of motion. Both qualitative and quantitative analyses were employed for data interpretation. The findings revealed a substantial influence of the guided inquiry learning model on SR, with a d-effect size value …