Mathematics Commons^™

Consensus Building By Committed Agents, William W. Hackborn, Tetiana Reznychenko, Yihang Zhang

CODEE Journal

One of the most striking features of our time is the polarization, nationally and globally, in politics and religion. How can a society achieve anything, let alone justice, when there are fundamental disagreements about what problems a society needs to address, about priorities among those problems, and no consensus on what constitutes justice itself? This paper explores a model for building social consensus in an ideologically divided community. Our model has three states: two of these represent ideological extremes while the third state designates a moderate position that blends aspects of the two extremes. Each individual in the community is …

Modeling The Spread And Prevention Of Malaria In Central America, Michael Huber

CODEE Journal

In 2016, the World Health Organization (WHO) estimated that there were 216 million cases of Malaria reported in 91 countries around the world. The Central American country of Honduras has a high risk of malaria exposure, especially to United States soldiers deployed in the region. This article will discuss various aspects of the disease, its spread and its treatment and the development of models of some of these aspects with differential equations. Exercises are developed which involve, respectively, exponential growth, logistics growth, systems of first-order equations and Laplace transforms. Notes for instructors are included.

Sir Models: Differential Equations That Support The Common Good, Lorelei Koss

CODEE Journal

This article surveys how SIR models have been extended beyond investigations of biologically infectious diseases to other topics that contribute to social inequality and environmental concerns. We present models that have been used to study sustainable agriculture, drug and alcohol use, the spread of violent ideologies on the internet, criminal activity, and health issues such as bulimia and obesity.

The Mathematics Of Gossip, Jessica Deters, Izabel P. Aguiar, Jacquie Feuerborn

CODEE Journal

How does a lie spread through a community? The purpose of this paper is two-fold: to provide an educational tool for teaching Ordinary Differential Equations (ODEs) and sensitivity analysis through a culturally relevant topic (fake news), and to examine the social justice implications of misinformation. Under the assumption that people are susceptible to, can be infected with, and recover from a lie, we model the spread of false information with the classic Susceptible-Infected-Recovered (SIR) model. We develop a system of ODEs with lie-dependent parameter values to examine the pervasiveness of a lie through a community.

The model presents the opportunity …

Linking Differential Equations To Social Justice And Environmental Concerns

CODEE Journal

Special issue of the CODEE Journal in honor of its founder, Professor Robert Borrelli.

Mathematical Modeling Of Lung Cancer Screening Studies, Deborah L. Goldwasser

Mathematics Colloquium Series

Lung cancer has the second highest cancer incidence, second only to prostate cancer in men and breast cancer in women. Furthermore, more cancer deaths are attributable to lung cancer than any other cancer for both genders. There is a high public health need for effective secondary prevention in the form of early detection and early treatment, complementary to smoking cessation efforts. The U.S. National Lung Screening Trial (NLST) demonstrated that non-small cell lung cancer (NSCLC) mortality can be reduced by 20% through a program of annual CT screening in high-risk individuals. However, CT screening regimens and adherence vary, potentially impacting …

On The Convergence Of Spectral Deferred Correction Methods, Matthew F. Causley, David C. Seal

Mathematics Publications

In this work we analyze the convergence properties of the Spectral Deferred Correction (SDC) method originally proposed by Dutt et al. [BIT, 40 (2000), pp. 241--266]. The framework for this high-order ordinary differential equation (ODE) solver is typically described wherein a low-order approximation (such as forward or backward Euler) is lifted to higher order accuracy by applying the same low-order method to an error equation and then adding in the resulting defect to correct the solution. Our focus is not on solving the error equation to increase the order of accuracy, but on rewriting the solver as an iterative Picard …

Role Of Combinatorial Complexity In Genetic Networks, Sharon Yang

SMU Journal of Undergraduate Research

A common motif found in genetic networks is the formation of large complexes. One difficulty in modeling this motif is the large number of possible intermediate complexes that can form. For instance, if a complex could contain up to 10 different proteins, 210 possible intermediate complexes can form. Keeping track of all complexes is difficult and often ignored in mathematical models. Here we present an algorithm to code ordinary differential equations (ODEs) to model genetic networks with combinatorial complexity. In these routines, the general binding rules, which counts for the majority of the reactions, are implemented automatically, thus the users …

Logics For Rough Concept Analysis, Giuseppe Greco, Peter Jipsen, Krishna Manoorkar, Alessandra Palmigiano, Apostolos Tzimoulis

Mathematics, Physics, and Computer Science Faculty Books and Book Chapters

Taking an algebraic perspective on the basic structures of Rough Concept Analysis as the starting point, in this paper we introduce some varieties of lattices expanded with normal modal operators which can be regarded as the natural rough algebra counterparts of certain subclasses of rough formal contexts, and introduce proper display calculi for the logics associated with these varieties which are sound, complete, conservative and with uniform cut elimination and subformula property. These calculi modularly extend the multi-type calculi for rough algebras to a ‘nondistributive’ (i.e. general lattice-based) setting.

Physics Beyond Catching A Mouse In The Dark: From Big Science To Deep Science, Victor Christianto, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

The Higgs particle has been detected a few years ago, that is what newspapers tell us. For many physicists, the Standard Model of particle physics has accomplished all the jobs. Or to put it simply: The game is over. Is it true? Then some physicists began to ask: can go beyond the Standard Model? Because the supersymmetric extension of the Standard Model has failed. If you feel that theoretical physics is becoming boring, you are not alone. Fortunately, there is good news: a new generation of physicists are doing table-top experiments in their basements. Can we expect new results later?2 …

Singular Ramsey And Turán Numbers, Yair Caro, Zsolt Tuza

Theory & Applications of Graphs

We say that a subgraph F of a graph G is singular if the degrees d_G(v) are all equal or all distinct for the vertices v ∈ V (F). The singular Ramsey number Rs(F) is the smallest positive integer n such that, for every m at least n, in every edge 2-coloring of K_m, at least one of the color classes contains F as a singular subgraph. In a similar flavor, the singular Turán number Ts(n,F) is defined as the maximum number of edges in a graph of order n, …

Decision Theory Explains "Telescoping Effect" -- That Our Time Perception Is Biased, Laxman Bokati, Vladik Kreinovich

Departmental Technical Reports (CS)

People usually underestimate time passed since distant events, and overestimate time passed since recent events. There are several explanations for this "telescoping effect", but most current explanations utilize specific features of human memory and/or human perception. We show that the telescoping effect can be explained on a much basic level of decision theory, without the need to invoke any specific ways we perceive and process time.

Hypersurfaces With Nonnegative Ricci Curvature In Hyperbolic Space, Vincent Bonini, Shiguang Ma, Jie Qing

Mathematics

Based on properties of n-subharmonic functions we show that a complete, noncompact, properly embedded hypersurface with nonnegative Ricci curvature in hyperbolic space has an asymptotic boundary at infinity of at most two points. Moreover, the presence of two points in the asymptotic boundary is a rigidity condition that forces the hypersurface to be an equidistant hypersurface about a geodesic line in hyperbolic space. This gives an affirmative answer to the question raised by Alexander and Currier (Proc Symp Pure Math 54(3):37–44, 1993).

Diagonal Sums Of Doubly Substochastic Matrices, Lei Cao, Zhi Chen, Xuefeng Duan, Selcuk Koyuncu, Huilan Li

Mathematics Faculty Articles

Let Ω_n denote the convex polytope of all n x n doubly stochastic matrices, and ω_n denote the convex polytope of all n x n doubly substochastic matrices. For a matrix A ϵ ω_n, define the sub-defect of A to be the smallest integer k such that there exists an (n + k) x (n + k) doubly stochastic matrix containing A as a submatrix. Let ω_n,k denote the subset of ω_n which contains all doubly substochastic matrices with sub-defect k. For π a permutation of symmetric group of degree …

How To Generate "Nice" Cubic Polynomials -- With Rational Coefficients, Rational Zeros And Rational Extrema: A Fast Algorithm, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Students feel more comfortable with rational numbers than with irrational ones. Thus, when teaching the beginning of calculus, it is desirable to have examples of simple problems for which both zeros and extrema point are rational. Recently, an algorithm was proposed for generating cubic polynomials with this property. However, from the computational viewpoint, the existing algorithm is not the most efficient one: in addition to applying explicit formulas, it also uses trial-and-error exhaustive search. In this paper, we propose a computationally efficient algorithm for generating all such polynomials: namely, an algorithm that uses only explicit formulas.

When Revolutions Happen: Algebraic Explanation, Julio Urenda, Vladik Kreinovich

Departmental Technical Reports (CS)

At first glance, it may seem that revolutions happen when life becomes really intolerable. However, historical analysis shows a different story: that revolutions happen not when life becomes intolerable, but when a reasonably prosperous level of living suddenly worsens. This empirical observation seems to contradict traditional decision theory ideas, according to which, in general, people's happiness monotonically depends on their level of living. A more detailed model of human behavior, however, takes into account not only the current level of living, but also future expectations. In this paper, we show that if we properly take these future expectations into account, …

Estimating The Density Of The Abundant Numbers, Dominic Klyve, Melissa Pidde, Kathryn E. Temple

Mathematics Faculty Scholarship

Mathematicians have been interested in properties of abundant numbers – those which are smaller than the sum of their proper factors – for over 2,000 years. During the last century, one line of research has focused in particular on determining the density of abundant numbers in the integers. Current estimates have brought the upper and lower bounds on this density to within about 10−4, with a value of K ≈ 0.2476, but more precise values seem difficult to obtain. In this paper, we employ computational data and tools from inferential statistics to get more insight into this value. We also …

A Re-Emergent Analysis Of Early Algebraic Learning, Steven Boyce, Diana Moss

Extension Research

In this paper, we discuss a novel approach for collaborative retrospective analysis. One researcher was directly involved in a classroom teaching experiment, adopting an emergent perspective as an interpreter-witness of classroom interactions during a four-week algebra instructional unit with sixth-grade students. The other researcher experienced and analyzed the data in reverse chronological order. We describe how this re-emergent perspective revealed aspects of students’ early algebraic reasoning.

Springer Fibers And Schubert Points, Martha Precup, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Springer fibers are subvarieties of the flag variety parametrized by partitions; they are central objects of study in geometric representation theory. Schubert varieties are subvarieties of the flag variety that induce a well-known basis for the cohomology of the flag variety. This paper relates these two varieties combinatorially. We prove that the Betti numbers of the Springer fiber associated to a partition with at most three rows or two columns are equal to the Betti numbers of a specific union of Schubert varieties.

Proper Orthogonal Decomposition (Pod) Of The Flow Dynamics For A Viscoelastic Fluid In A Four-Roll Mill Geometry At The Stokes Limit, Paloma Gutierrez-Castillo, Becca Thomases

Mathematics Sciences: Faculty Publications

Numerical simulations of viscoelastic fluids in the Stokes limit with a four-roll mill background force were performed at a range of Weissenberg number (non-dimensional relaxation time). For small Weissenberg number the flow is steady and symmetric but upon increasing the Weissenberg number (corresponding to increased elasticity or flow memory time), the flow becomes unstable leading to a variety of temporal evolutions to different periodic and aperiodic solutions. These dynamics were analyzed using a Proper Orthogonal Decomposition (POD) that extracted elastic modes in terms of their contribution to the energy of the system. The temporal behavior of the system, captured by …