Physical Sciences and Mathematics Commons^™

Statistics Pedagogy: Modernizing Introductory College Statistics Courses For Students With Non-Stem Majors, Samantha Hazan

Honors Projects in Mathematics and Economics

With the growing role of technology in society, statistics and statistical literacy have become ever more important. In reaction to this, many colleges now require students, regardless of major, to take an introductory statistics course. The intention behind this is to create a workforce of people who are capable of reasoning about statistics. While good intentioned, preliminary research suggests that current statistics curriculum is not effective in instilling the necessary skills to achieve statistical literacy, particularly for students not majoring in a STEM field. For this reason, this thesis looks to empirically evaluate how the disjuncture between teaching statistics and …

Visualizing The Standard Deviation Via Revolution Using R/Rstudio, Hieu Nguyen '25, Trung Pham '25, Mamunur Rashid, Jyotirmoy Sarkar

Student Research

The standard deviation is a commonly used statistical measure to quantify the level of variation present in a set of numbers or in a random variable. Sarkar and Rashid (2016) introduced an interpretation of the population standard deviation as the radius of a cylinder with a volume equivalent to that of the solid of revolution when the 2-D graph of the empirical cumulative distribution function is revolved about the vertical line through the mean. This article demonstrates step-by-step how to use the RevSD package in R/RStudio to visualize the standard deviation of data using this innovative technique. The RevSD package …

How Difficult Is It To Comprehend A Program That Has Significant Repetitions: Fuzzy-Related Explanations Of Empirical Results, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In teaching computing and in gauging the programmers' productivity, it is important to property estimate how much time it will take to comprehend a program. There are techniques for estimating this time, but these techniques do not take into account that some program segments are similar, and this similarity decreases the time needed to comprehend the second segment. Recently, experiments were performed to describe this decrease. These experiments found an empirical formula for the corresponding decrease. In this paper, we use fuzzy-related ideas to provide commonsense-based theoretical explanation for this empirical formula.

Mcfadden's Discrete Choice And Softmax Under Interval (And Other) Uncertainty: Revisited, Bartlomiej Jacek Kubica, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Studies of how people actually make decisions have led to an empirical formula that predicts the probability of different decisions based on the utilities of different alternatives. This formula is known as McFadden's formula, after a Nobel prize winning economist who discovered it. A similar formula -- known as softmax -- describes the probability that the classification predicted by a deep neural network is correct, based on the neural network's degrees of confidence in the object belonging to each class. In practice, we usually do not know the exact values of the utilities -- or of the degrees of confidence. …

Why Bernstein Polynomials: Yet Another Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many computational situations -- in particular, in computations under interval or fuzzy uncertainty -- it is convenient to approximate a function by a polynomial. Usually, a polynomial is represented by coefficients at its monomials. However, in many cases, it turns out more efficient to represent a general polynomial by using a different basis -- of so-called Bernstein polynomials. In this paper, we provide a new explanation for the computational efficiency of this basis.

Somewhat Surprisingly, (Subjective) Fuzzy Technique Can Help To Better Combine Measurement Results And Expert Estimates Into A Model With Guaranteed Accuracy: Digital Twins And Beyond, Niklas Winnewisser, Michael Beer, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To understand how different factors and different control strategies will affect a system -- be it a plant, an airplane, etc. -- it is desirable to form an accurate digital model of this system. Such models are known as digital twins. To make a digital twin as accurate as possible, it is desirable to incorporate all available knowledge of the system into this model. In many cases, a significant part of this knowledge comes in terms of expert statements, statements that are often formulated by using imprecise ("fuzzy") words from natural language such as "small", "very possible", etc. To translate …

How To Gauge Inequality And Fairness: A Complete Description Of All Decomposable Versions Of Theil Index, Saeid Tizpaz-Niari, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In general, in statistics, the most widely used way to describe the difference between different elements of a sample if by using standard deviation. This characteristic has a nice property of being decomposable: e.g., to compute the mean and standard deviation of the income overall the whole US, it is sufficient to compute the number of people, mean, and standard deviation over each state; this state-by-state information is sufficient to uniquely reconstruct the overall standard deviation. However, e.g., for gauging income inequality, standard deviation is not very adequate: it provides too much weight to outliers like billionaires, and thus, does …

Update From Aristotle To Newton, From Sets To Fuzzy Sets, And From Sigmoid To Relu: What Do All These Transitions Have In Common?, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show that there is a -- somewhat unexpected -- common trend behind several seemingly unrelated historic transitions: from Aristotelian physics to modern (Newton's) approach, from crisp sets (such as intervals) to fuzzy sets, and from traditional neural networks, with close-to-step-function sigmoid activation functions to modern successful deep neural networks that use a completely different ReLU activation function. In all these cases, the main idea of the corresponding transition can be explained, in mathematical terms, as going from the first order to second order differential equations.

How To Make A Decision Under Interval Uncertainty If We Do Not Know The Utility Function, Jeffrey Escamilla, Vladik Kreinovich

Departmental Technical Reports (CS)

Decision theory describes how to make decisions, in particular, how to make decisions under interval uncertainty. However, this theory's recommendations assume that we know the utility function -- a function that describes the decision maker's preferences. Sometimes, we can make a recommendation even when we do not know the utility function. In this paper, we provide a complete description of all such cases.

Paradox Of Causality And Paradoxes Of Set Theory, Alondra Baquier, Bradley Beltran, Gabriel Miki-Silva, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Logical paradoxes show that human reasoning is not always fully captured by the traditional 2-valued logic, that this logic's extensions -- such as multi-valued logics -- are needed. Because of this, the study of paradoxes is important for research on multi-valued logics. In this paper, we focus on paradoxes of set theory. Specifically, we show their analogy with the known paradox of causality, and we use this analogy to come up with similar set-theoretic paradoxes.

Number Representation With Varying Number Of Bits, Anuradha Choudhury, Md Ahsanul Haque, Saeefa Rubaiyet Nowmi, Ahmed Ann Noor Ryen, Sabrina Saika, Vladik Kreinovich

Departmental Technical Reports (CS)

In a computer, usually, all real numbers are stored by using the same number of bits: usually, 8 bytes, i.e., 64 bits. This amount of bits enables us to represent numbers with high accuracy -- up to 19 decimal digits. However, in most cases -- whether we process measurement results or whether we process expert-generated membership degrees -- we do not need that accuracy, so most bits are wasted. To save space, it is therefore reasonable to consider representations with varying number of bits. This would save space used for representing numbers themselves, but we would also need to store …

How To Fairly Allocate Safety Benefits Of Self-Driving Cars, Fernando Munoz, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we describe how to fairly allocated safety benefits of self-driving cars between drivers and pedestrians -- so as to minimize the overall harm.

Using Known Relation Between Quantities To Make Measurements More Accurate And More Reliable, Niklas Winnewisser, Felix Mett, Michael Beer, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Most of our knowledge comes, ultimately, from measurements and from processing measurement results. In this, metrology is very valuable: it teaches us how to gauge the accuracy of the measurement results and of the results of data processing, and how to calibrate the measuring instruments so as to reach the maximum accuracy. However, traditional metrology mostly concentrates on individual measurements. In practice, often, there are also relations between the current values of different quantities. For example, there is usually an known upper bound on the difference between the values of the same quantity at close moments of time or at …

Exploring Quaternion Neural Network Loss Surfaces, Jeremiah Bill, Bruce A. Cox

Faculty Publications

This paper explores the superior performance of quaternion multi-layer perceptron (QMLP) neural networks over real-valued multi-layer perceptron (MLP) neural networks, a phenomenon that has been empirically observed but not thoroughly investigated. The study utilizes loss surface visualization and projection techniques to examine quaternion-based optimization loss surfaces for the first time. The primary contribution of this research is the statistical evidence that QMLP models yield smoother loss surfaces than real-valued neural networks, which are measured and compared using a robust quantitative measure of loss surface “goodness” based on estimates of surface curvature. Extensive computational testing validates the effectiveness of these surface …

Data Fusion Is More Complex Than Data Processing: A Proof, Robert Alvarez, Salvador Ruiz, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

Empirical data shows that, in general, data fusion takes more computation time than data processing. In this paper, we provide a proof that data fusion is indeed more complex than data processing.

Towards An Optimal Design: What Can We Recommend To Elon Musk?, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich, Hung T. Nguyen

Departmental Technical Reports (CS)

Elon Musk's successful "move fast and break things" strategy is based on the fact that in many cases, we do not need to satisfy all usual constraints to be successful. By sequentially trying smaller number of constraints, he finds the smallest number of constraints that are still needed to succeed -- and using this smaller number of constrains leads to a much cheaper (and thus, more practical) design. In this strategy, Musk relies on his intuition -- which, as all intuitions, sometimes works and sometimes doesn't. To replace this intuition, we propose an algorithm that minimizes the worst-case cost of …

On Generating Bijections For Permutations And Inversion Sequences, Melanie J. Ferreri

Dartmouth College Ph.D Dissertations

Given an algebraic proof of a combinatorial identity, we use recursive methods to construct a bijection demonstrating the identity.

Our first application centers around derangements and nonderangements. A derangement is a permutation with no fixed point, and a nonderangement is a permutation with at least one fixed point. There is a one-term recurrence for the number of derangements of n elements, and we describe a bijective proof of this recurrence which can be found using a recursive map. We then show the combinatorial interpretation of this bijection and how it compares with other known bijections, and show how this extends …

The Impacts Of Covid-19 On The U.S. Life Insurance Industry: A Study Of Pandemic Mortality Rates, Allison Wheaton

Honors Projects in Mathematics and Economics

The COVID-19 pandemic has had substantial and widespread impacts around the world that are not yet completely understood but will likely be experienced for years to come. This research seeks to analyzed and quantify the pandemic’s effects on the life insurance industry in the United States through the use of mortality models. Because the life insurance industry relies on the ability to accurately predict an individual’s mortality, it is necessary to investigate the impacts of the pandemic on mortality predictions and accuracy. Impairment factors were developed using pandemic mortality data and applied to existing mortality tables to compare expected and …

Normalized Ground State Of A Mixed Dispersion Nonlinear Schrodinger Equation With Combined Power-Type Nonlinearities, Zhouji Ma, Xiaojun Chang, Zhaosheng Feng

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We study the existence of normalized ground state solutions to a mixed dispersion fourth-order nonlinear Schrodinger equation with combined power-type nonlinearities. By analyzing the subadditivity of the ground state energy with respect to the prescribed mass, we employ a constrained minimization method to establish the existence of ground state that corresponds to a local minimum of the associated functional. Under certain conditions, by studying the monotonicity of ground state energy as the mass varies, we apply the constrained minimization arguments on the Nehari-Pohozaev manifold to prove the existence of normalized ground state solutions.

Applications Of Survival Estimation Under Stochastic Order To Cancer: The Three Sample Problem, Sage Vantine

Honors Program Theses and Research Projects

Stochastic ordering of probability distributions holds various practical applications. However, in real-world scenarios, the empirical survival functions extracted from actual data often fail to meet the requirements of stochastic ordering. Consequently, we must devise methods to estimate these distribution curves in order to satisfy the constraint. In practical applications, such as the investigation of the time of death or the progression of diseases like cancer, we frequently observe that patients with one condition are expected to exhibit a higher likelihood of survival at all time points compared to those with a different condition. Nevertheless, when we attempt to fit a …

Doubly Almost Bipartite Leonard Pairs, Shuichi Masuda

Dissertations and Theses

In Linear algebra, the concept of Leonard pair (LP) was motivated by the theory of Q-polynomial distance-regular graphs. In this dissertation, we will first give a brief introduction to LPs and to two closely-related classes of objects: (i) bipartite Leonard pairs (BLPs) and (ii) almost bipartite Leonard pairs (ABLPs). Taking these as departure points, we will introduce a new class of object - doubly almost bipartite Leonard pairs (DABLPs). The primary aim of our work is to fully classify (up to isomorphism) this new family. In addition, since there is known to be a natural correspondence between Leonard pairs …

Analyzing The Influence Of Design And Operating Conditions On Combustion And Emissions In Premixed Turbulent Flames: A Comprehensive Review, Medhat Elkelawy Prof. Dr. Eng., E. A. El Shenawy Prof. Dr., Hagar Alm-Eldin Bastawissi, Ibrahim Ali Mousa Eng., Mohamed M. Abdel-Raouf Ibrahim Dr. Eng.

Journal of Engineering Research

Recently, premixed combustion has dominated the field of combustion research worldwide. The current work is a review that addresses the effects of design and operating regimes on the combustion and emission characteristics of premixed turbulent flames. The study accounts for recent developments aimed at overcoming combustor operability issues that influence emissions and flame stability. Various experimental setups have been utilized in investigations, with results pertaining to performance and emissions concerning premixed turbulent flames. Thus, the objective of this paper is to provide a comprehensive review of the effects of swirl vane angles and equivalence fuel-air ratios for tests conducted both …

On The Subelliptic And Subparabolic Infinity Laplacian In Grushin-Type Spaces, Zachary Forrest

USF Tampa Graduate Theses and Dissertations

This thesis poses the ∞-Laplace equation in Grushin-type spaces. Grushin-type spaces G are defined by the vector fields which serve as a basis for their tangent spaces; by weighting the canonical (Euclidean) directional vectors {∂/∂x_i}ⁿ_i=1 by functions ρ_i that obey certain technical assumptions, we produce a class of metric spaces in which certain directions may not be accessible at all points in the space. We prove the existence and uniqueness of viscosity solutions to both Dirichlet problems and Cauchy-Dirichlet problems involving the∞-Laplacian over bounded Grushin-type domains. The main tool in proving uniqueness of …

The Invariantring Package For Macaulay2, Luigi Ferraro, Federico Galetto, Francesca Gandini, Hang Huang, Matthew Mastroeni, Xianglong Ni

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We describe a significant update to the existing InvariantRing package for Macaulay2. In addition to expanding and improving the methods of the existing package for actions of finite groups, the updated package adds functionality for computing invariants of diagonal actions of tori and finite abelian groups, as well as invariants of arbitrary linearly reductive group actions. The implementation of the package has been completely overhauled with the aim of serving as a unified resource for invariant theory computations in Macaulay2.

A Cohomological Perspective To Nonlocal Operators, Nicholas White

Honors Theses

Nonlocal models have experienced a large period of growth in recent years. In particular, nonlocal models centered around a finite horizon have been the subject of many novel results. In this work we consider three nonlocal operators defined via a finite horizon: a weighted averaging operator in one dimension, an averaging differential operator, and the truncated Riesz fractional gradient. We primarily explore the kernel of each of these operators when we restrict to open sets. We discuss how the topological structure of the domain can give insight into the behavior of these operators, and more specifically the structure of their …

Birkhoff Summation Of Irrational Rotations: A Surprising Result For The Golden Mean, Heather Moore

University Honors Theses

This thesis presents a surprising result that the difference in a certain sums of constant rotations by the golden mean approaches exactly 1/5. Specifically, we focus on the Birkhoff sums of these rotations, with the number of terms equal to squared Fibonacci numbers. The proof relies on the properties of continued fraction approximants, Vajda's identity and the explicit formula for the Fibonacci numbers.

Quasistationary Distribution For The Invasion Model On A Complete Bipartite Graph, Clayton Allard, Iddo Ben-Ari, Shrikant Chand, Van Hovenga, Edith Lee, Julia Shapiro

Journal of Stochastic Analysis

No abstract provided.

The Basel Problem And Summing Rational Functions Over Integers, Pranjal Jain

Rose-Hulman Undergraduate Mathematics Journal

We provide a general method to evaluate convergent sums of the form ∑_{k∈Z} R(k) where R is a rational function with complex coefficients. The method is entirely elementary and does not require any calculus beyond some standard limits and convergence criteria. It is inspired by a geometric solution to the famous Basel Problem given by Wästlund (2010), so we begin by demonstrating the method on the Basel Problem to serve as a pilot application. We conclude by applying our ideas to prove Euler’s factorisation for sin x which he originally used to solve the Basel Problem.

Understanding Waveguides In Resonance, Pieter Johannes Daniel Vandenberge

Dissertations and Theses

Several important classes of modern optical waveguides, including anti-resonant reflecting and photonic bandgap fibers, make use of geometries that guide energy in low refractive index material, a property that makes them of significant interest in numerous applications, notably including high-power delivery and guidance. These waveguides frequently exhibit resonance phenomena, in which their ability to propagate an input signal is sharply curtailed at particular operating frequencies. In this work we detail new advances in understanding these resonance effects and their implications for numerical modeling of these structures.

Part 1 focuses on the fields of slab waveguides, relatively simple structures for which …

Asymptotics Of The Rate Function In The Large Deviation Principle For Sums Of Independent Identically Distributed Random Variables, Iosif Pinelis

Michigan Tech Publications, Part 2

Let Λ∗be the rate function in the large deviation principle for the sums X1 + · · · + Xn of independent identically distributed random variables X1, X2, …. It is shown that Λ∗(x) ∼ − ln P(X1 ≥ x) (as x → ∞) if and only if ln P(X1 ≥ x) ∼ L0(x) for some concave function L0. The main ingredient of the proof is the general, explicit expression of a suitable quasi-minimizer in t ≥ 0 of the Bernstein–Chernoff upper bound e−txEetX1 on P(X1 ≥ x), which is amenable to analysis and, at the same time, is close …