Mathematics Commons^™

Why Self-Esteem Helps To Solve Problems: An Algorithmic Explanation, Oscar Ortiz, Henry Salgado, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that self-esteem helps solve problems. From the algorithmic viewpoint, this seems like a mystery: a boost in self-esteem does not provide us with new algorithms, does not provide us with ability to compute faster -- but somehow, with the same algorithmic tools and the same ability to perform the corresponding computations, students become better problem solvers. In this paper, we provide an algorithmic explanation for this surprising empirical phenomenon.

How To Describe Hypothetic Truly Rare Events (With Probability 0), Luc Longpre, Vladik Kreinovich

Departmental Technical Reports (CS)

In probability theory, rare events are usually described as events with low probability p, i.e., events for which in N observations, the event happens n(N) ~ p*N times. Physicists and philosophers suggested that there may be events which are even rarer, in which n(N) grows slower than N. However, this idea has not been developed, since it was not clear how to describe it in precise terms. In this paper, we propose a possible precise description of this idea, and we use this description to answer a natural question: when two different functions n(N) lead to the same class of …

One More Physics-Based Explanation For Rectified Linear Neurons, Jonatan Contreras, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

The main idea behind artificial neural networks is to simulate how data is processed in the data processing devoice that has been optimized by million-years natural selection -- our brain. Such networks are indeed very successful, but interestingly, the most recent successes came when researchers replaces the original biology-motivated sigmoid activation function with a completely different one -- known as rectified linear function. In this paper, we explain that this somewhat unexpected function actually naturally appears in physics-based data processing.

How To Make Quantum Ideas Less Counter-Intuitive: A Simple Analysis Of Measurement Uncertainty Can Help, Olga Kosheleva, Vladik Kreinovich, Louis Ray Lopez

Departmental Technical Reports (CS)

Our intuition about physics is based on macro-scale phenomena, phenomena which are well described by non-quantum physics. As a result, many quantum ideas sound counter-intuitive -- and this slows down students' learning of quantum physics. In this paper, we show that a simple analysis of measurement uncertainty can make many of the quantum ideas much less counter-intuitive and thus, much easier to accept and understand.

Physical Meaning Often Leads To Natural Derivations In Elementary Mathematics: On The Examples Of Solving Quadratic And Cubic Equations, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Usual derivation of many formulas of elementary mathematics -- such as the formulas for solving quadratic equation -- often leave un unfortunate impression that mathematics is a collection of unrelated unnatural trick. In this paper, on the example of formulas for solving quadratic and cubic equations, we show that these derivations can be made much more natural if we take physical meaning into account.

Why Immunodepressive Drugs Often Make People Happier, Joshua Ramos, Dario Vasquez, Ruth Trejo, Vladik Kreinovich

Departmental Technical Reports (CS)

Many immunodepressive drugs have an unusual side effect on the patient's mood: they often make the patient happier. This side hae been observed for many different immunodepressive drugs, with different chemical composition. Thus, it is natural to conclude that there must be some general reason for this empirical phenomenon, a reason not related to the chemical composition of any specific drug -- but rather with their general functionality. In this paper, we provide such an explanation.

Explaining An Empirical Formula For Bioreaction To Similar Stimuli (Covid-19 And Beyond), Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

A recent comparative analysis of biological reaction to unchanging vs. rapidly changing stimuli -- such as Covid-19 or flu viruses -- uses an empirical formula describing how the reaction to a similar stimulus depends on the distance between the new and original stimuli. In this paper, we provide a from-first-principles explanation for this empirical formula.

A Phenomenological Study Of Perceived Factors Influencing Math Teachers' Technology Self-Efficacy, Jenna Finnegan

Doctoral Dissertations and Projects

The purpose of this study was to explore how a teacher's technology self-efficacy influences decisions they make about the use of technology in their classrooms. The factors influencing teachers' technology self-efficacy were examined through a qualitative transcendental phenomenological research design. This study's theoretical framework was based on the combination of social cognitive theory and self-efficacy theory; both provide insight into how external and internal factors can influence a person's perception of their abilities. Using these theories helped identify and describe the self-efficacy experiences of high school math teachers. This research was conducted during the 2020-2021 school year using a qualitative …

Spectra Of Variants Of Distance Matrices Of Graphs And Digraphs: A Survey, L. Hogben, Carolyn Reinhart

Mathematics & Statistics Faculty Works

Distance matrices of graphs were introduced by Graham and Pollack in 1971 to study a problem in communications. Since then, there has been extensive research on the distance matrices of graphs—a 2014 survey by Aouchiche and Hansen on spectra of distance matrices of graphs lists more than 150 references. In the last 10 years, variants such as the distance Laplacian, the distance signless Laplacian, and the normalized distance Laplacian matrix of a graph have been studied. After a brief description of the early history of the distance matrix and its motivating problem, this survey focuses on comparing and contrasting techniques …

An Introduction To Number Theory, J. J. P. Veerman

PDXOpen: Open Educational Resources

These notes are a preliminary text intended for a graduate course in Number Theory. A thoroughly improved, revised, and expanded version of this book will be published in 2026 through Springer under the title Numbers from All Angles.

Chapters 1-14 of this preliminary text represent almost 3 trimesters of the course. No prior familiarity with number theory is assumed. The current content represents courses the author taught in the academic years 2020-2021 and 2021-2022.

Minimality Of Integer Bar Visibility Graphs, Emily Dehoff

University Honors Theses

A visibility representation is an association between the set of vertices in a graph and a set of objects in the plane such that two objects have an unobstructed, positive-width line of sight between them if and only if their two associated vertices are adjacent. In this paper, we focus on integer bar visibility graphs (IBVGs), which use horizontal line segments with integer endpoints to represent the vertices of a given graph. We present results on the exact widths of IBVGs of paths, cycles, and stars, and lower bounds on trees and general graphs. In our main results, we find …

Existence And Uniqueness Of Minimizers For A Nonlocal Variational Problem, Michael Pieper

Honors Program: Senior Projects (Public)

Nonlocal modeling is a rapidly growing field, with a vast array of applications and connections to questions in pure math. One goal of this work is to present an approachable introduction to the field and an invitation to the reader to explore it more deeply. In particular, we explore connections between nonlocal operators and classical problems in the calculus of variations. Using a well-known approach, known simply as The Direct Method, we establish well-posedness for a class of variational problems involving a nonlocal first-order differential operator. Some simple numerical experiments demonstrate the behavior of these problems for specific choices of …

Number Of Bounded Distance Equivalence Classes In Hulls Of Repetitive Delone Sets, Dirk Frettlöh, Alexey Garber, Lorenzo Sadun

School of Mathematical & Statistical Sciences Faculty Publications

Two Delone sets are bounded distance equivalent to each other if there is a bijection between them such that the distance of corresponding points is uniformly bounded. Bounded distance equivalence is an equivalence relation. We show that the hull of a repetitive Delone set with finite local complexity has either one equivalence class or uncountably many.

Integrable Semi-Discretization Of A Modified Short Wave Equation, Yujuan Zhang, Ruyun Ma, Na Xiong, Bao-Feng Feng

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we are concerned with one of the integrable equations proposed by Hone et al. (2018). We firstly show that it can be derived from the Tzitzeica equation through hodograph or reciprocal transformations and it is linked to the short wave limits of the DP and Novikov equations, respectively, through Miura transformations. Based on the semi-discrete analogue of CKP-Toda hierarchy, we construct the integrable semi-discretization of what we call the modified short wave equation and provide its multi-soliton solution.

The Transition Matrix Between The Specht And ����3 Web Bases Is Unitriangular With Respect To Shadow Containment, Heather M. Russell, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Webs are planar graphs with boundary that describe morphisms in a diagrammatic representation category for ����k ⁠. They are studied extensively by knot theorists because braiding maps provide a categorical way to express link diagrams in terms of webs, producing quantum invariants like the well-known Jones polynomial. One important question in representation theory is to identify the relationships between different bases; coefficients in the change-of-basis matrix often describe combinatorial, algebraic, or geometric quantities (e.g., Kazhdan–Lusztig polynomials). By ”flattening” the braiding maps, webs can also be viewed as the basis elements of a symmetric group representation. In this paper, we define …

Finitely Generated Modules Over A Principal Ideal Domain, Mariam Mutawa Meshaab Shemal Al-Dhaheri

Theses

This thesis covers the main theory of modules: modules, submodules, cosets, quotient modules, homomorphisms, ideals, direct sums, and some related topics. Using these notions, a theorem on the structure of finitely generated modules over domains of principal ideals is proved. As an application of this theorem, the theory of the structure of normal forms of matrices over various fields is presented.

Optimal Quantization For Mixed Distributions Generated By Two Uniform Distributions, Ashley Gomez, Olga Lopez, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, for mixed distributions generated by two uniform distributions we investigate the optimal sets of n-means and the nth quantization errors for all positive integers n. Some conjectures and open problems are also given.

Zero-Knowledge Proof, Deniability And Their Applications In Blockchain, E-Voting And Deniable Secret Handshake Protocols., Somnath Panja Dr.

Doctoral Theses

In this thesis, we propose a cryptographic technique for an authenticated, end-to-end verifiable and secret ballot election. Currently, almost all verifiable e-voting systems require trusted authorities to perform the tallying process except for the DRE-i and DRE-ip systems. We have shown a weaknesses in the DRE-ip system and proposed a solution. We have modified the DRE-ip system so that no adversary can create and post a valid ballot on the public bulletin board without detection. We provide security proofs to prove the security properties of the proposed scheme. We propose two methods to store these ballots using blockchain and cloud …

Essays On Social Networks, Workplace Ties And Labor Productivity., Swati Sharma Dr.

Doctoral Theses

The role of social contacts in finding jobs, career mobility and other labor market outcomes is well acknowledged both theoretically and empirically in labor economics. The existence of individuals’ social networks at their workplace is a pervasive phenomenon, leading to the widespread use of social ties for information, influence and referrals on both sides of the labor market. The central theme of this thesis is to examine the interplay of ‘off-workplace socio-economic interdependence’ and outcomes at the workplace. Understanding this interdependence has the potential to devise policies which can impact labor productivity, especially in developing countries. This thesis divided into …

Revisiting Prejudiced Polygons: Adapting A Familiar Activity During A Time Of Unknowns, Anne M. Ho, Jaime J. Mccauley, Tara T. Craig

Journal of Math Circles

This article describes the design process behind various iterations of Prejudiced Polygons, a Math Circles activity about segregation. In particular, we frame our discussion around two guiding principles from User Experience (UX) Design in thinking about the interconnected components of a Math Circles session, which includes all the people, the physical or virtual setting, the technology, and the world context. Additionally, we describe how we think about developing a “low floor" and “high ceiling" for math content, social issues content, as well as technology and access.

Simian Immunodeficiency Virus Infection Mediated Changes In Jejunum And Peripheral Sars-Cov-2 Receptor Ace2 And Associated Proteins Or Genes In Rhesus Macaques, Nongthombam Boby, Xuewei Cao, Kelsey Williams, Shiva Kumar Goud Gadila, Monica N. Shroyer, Peter J. Didier, Sudesh K. Srivastav, Arpita Das, Kate Baker, Qiuying Sha, Bapi Pahar

Michigan Tech Publications, Part 1

Angiotensin converting enzyme-2 (ACE2) and associated proteins play a pivotal role in various physiological and pathological events, such as immune activation, inflammation, gut barrier maintenance, intestinal stem cell proliferation, and apoptosis. Although many of these clinical events are quite significant in SIV/HIV infection, expression profiling of these proteins has not been well reported. Considering the different pathological consequences in the gut after HIV infection, we hypothesized that the expression of ACE2 and associated proteins of the Renin-angiotensin system (RAS) could be compromised after SIV/HIV infection. We quantified the gene expression of ACE2 as well as AGTR1/2, ADAM17, and TMPRSS2, and …

Recursive Linear Bounds For The Vertex Chromatic Number Of The Pancake Graph, Aldrich Ellis C. Asuncion, Renzo Roel P. Tan, Christian Chan Shio, Kazushi Ikeda

Mathematics Faculty Publications

The pancake graph has been the subject of research. While studies on the various aspects of the graph are abundant, results on the chromatic properties may be further enhanced. Revolving around such context, the paper advances an alternative method to produce novel linear bounds for the vertex chromatic number of the pancake graph. The accompanying demonstration takes advantage of symmetries inherent to the graph, capturing the prefix reversal of subsequences through a homomorphism. Contained within the argument is the incorporation of known vertex chromatic numbers for certain orders of pancake graphs, rendering tighter bounds possible upon the release of new …

The Relative Isolation Probability Of A Vertex In A Multiple-Source Edge-Weighted Graph, Renzo Roel P. Tan, Kyle Stephen S. See, Jun Kawahara, Kazushi Ikeda, Richard De Jesus, Lessandro Estelito Garciano, Agnes Garciano

Mathematics Faculty Publications

Various measures that characterize graphs exist in literature. Insights into the properties of a graph as a whole and its components are revealed largely through graph measures, also called graph metrics. In seeking to interpret a consequential edge metric from a vertex-centric perspective, the paper advances an original measure – the relative isolation probability of a vertex. Concisely, the probability of relative isolation pertains to the likelihood of a vertex to be disconnected from all designated source vertices in a graph with probability-weighted edges. A two-step algorithm for efficient calculation is presented and evaluated. Contained within the procedure is a …

Diversity Of Pseudomonas Aeruginosa Temperate Phages, Genevieve Johnson, Swarnali Banerjee, Catherine Putonti

Mathematics and Statistics: Faculty Publications and Other Works

Modern sequencing technologies have provided insight into the genetic diversity of numerous species, including the human pathogen Pseudomonas aeruginosa. Bacterial genomes often harbor bacteriophage genomes (prophages), which can account for upwards of 20% of the genome. Prior studies have found P. aeruginosa prophages that contribute to their host’s pathogenicity and fitness. These advantages come in many different forms, including the production of toxins, promotion of biofilm formation, and displacement of other P. aeruginosa strains. While several different genera and species of P. aeruginosa prophages have been studied, there has not been a comprehensive study of the overall diversity of P. …

The Cohomology Of The Mod 2 Steenrod Algebra, Robert R. Bruner, John Rognes

Open Data at Wayne State

The dataset contains a minimal resolution of the mod 2 Steenrod algebra in the range 0 < = s < = 128, 0 < = t < = 200, together with chain maps for each cocycle in that range and for the squaring operation Sq^0 in the cohomology of the Steenrod algebra. The included document CohomA2.pdf explains the contents and usage of the dataset in detail (also available as supplemental material in this record).

Dataset is also available at the NIRD Research Data Archive, https://doi.org/10.11582/2021.00077; Data Description also available at arXiv.org, https://doi.org/10.48550/arXiv.2109.13117.

Multi-Breather Solutions To The Sasa-Satsuma Equation, Chengfa Wu, Bo Wei, Changyan Shi, Bao-Feng Feng

School of Mathematical & Statistical Sciences Faculty Publications

General breather solution to the Sasa-Satsuma (SS) equation is systematically investigated in this paper. We firstly transform the SS equation into a set of three Hirota bilinear equations under proper plane wave background. Starting from a specially arranged tau-function of the Kadomtsev-Petviashvili hierarchy and a set of eleven bilinear equations satisfied, we implement a series steps of reduction procedure, i.e., C-type reduction, dimension reduction and complex conjugate reduction, and reduce these eleven equations to three bilinear equations for the SS equation. Meanwhile, general breather solution to the SS equation is found in determinant of even order. The one- and two-breather …

New Limit Theorems For Increments Of Birth-And-Death Processes With Linear Rates, Alexander Ya. Kreinin, Vladimir V. Vinogradov

Journal of Stochastic Analysis

No abstract provided.

Backward Stochastic Differential Equations With No Driving Martingale And Pseudo-Pdes, Adrien Barrasso, Francesco Russo

Journal of Stochastic Analysis

No abstract provided.

Connectedness Of Unit Distance Subgraphs Induced By Closed Convex Sets, Remie Janssen, Leonie Van Steijn

Theory & Applications of Graphs

The unit distance graph G¹_R^d is the infinite graph whose nodes are points in R^d, with an edge between two points if the Euclidean distance between these points is 1. The 2-dimensional version G¹_R2 of this graph is typically studied for its chromatic number, as in the Hadwiger-Nelson problem. However, other properties of unit distance graphs are rarely studied. Here, we consider the restriction of G¹_Rd to closed convex subsets X of R^d. We show that the graph G¹_Rd[X] is connected precisely when the radius of …

Application Of The Combinatorial Nullstellensatz To Integer-Magic Graph Labelings, Richard M. Low, Dan Roberts

Theory & Applications of Graphs

Let A be a nontrivial abelian group and A* = A \ {0}. A graph is A-magic if there exists an edge labeling f using elements of A* which induces a constant vertex labeling of the graph. Such a labeling f is called an A-magic labeling and the constant value of the induced vertex labeling is called an A-magic value. In this paper, we use the Combinatorial Nullstellensatz to show the existence of Ζ_p-magic labelings (prime p ≥ 3 ) for various graphs, without having to construct the Ζ_p-magic labelings. Through many …