Mathematics Commons^™

Limit Distributions Of Products Of Independent And Identically Distributed Random 2 × 2 Stochastic Matrices: A Treatment With The Reciprocal Of The Golden Ratio, Santanu Chakraborty

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Consider a sequence (Xn)n≥1 of i.i.d. 2×2 stochastic matrices with each Xn distributed as μ. This μ is described as follows. Let (Cn,Dn)T denote the first column of Xn and for a given real r with 012 is very challenging. Considering the extreme nontriviality of this case, we stick to a very special such r, namely, r=√5−12 (the reciprocal of the golden ratio), briefly mention the challenges in this nontrivial case, and completely identify λ for a very special situation.

Turing Patterns In A P-Adic Fitzhugh-Nagumo System On The Unit Ball, L. F. Chacón-Cortés, C. A. Garcia-Bibiano, Wilson A. Zuniga-Galindo

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We introduce discrete and p-adic continuous versions of the FitzHugh-Nagumo system on the one-dimensional p-adic unit ball. We provide criteria for the existence of Turing patterns. We present extensive simulations of some of these systems. The simulations show that the Turing patterns are traveling waves in the p-adic unit ball.

The Tor Algebra Of Trimmings Of Gorenstein Ideals, Luigi Ferraro, Alexis Hardesty

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Let (R,\mathfrak m,\Bbbk ) be a regular local ring of dimension 3. Let I be a Gorenstein ideal of R of grade 3. Buchsbaum and Eisenbud proved that there is a skew-symmetric matrix of odd size such that I is generated by the sub-maximal pfaffians of this matrix. Let J be the ideal obtained by multiplying some of the pfaffian generators of I by \mathfrak m; we say that J is a trimming of I. Building on a recent paper of Vandebogert, we construct an explicit free resolution of R/J and compute a partial DG algebra structure on this …

Time Series Based Road Traffic Accidents Forecasting Via Sarima And Facebook Prophet Model With Potential Changepoints, Edmund F. Agyemang, Joseph A. Mensah, Eric Ocran, Enock Opoku, Ezekiel N.N. Nortey

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Road traffic accident (RTA) is a critical global public health concern, particularly in developing countries. Analyzing past fatalities and predicting future trends is vital for the development of road safety policies and regulations. The main objective of this study is to assess the effectiveness of univariate Seasonal Autoregressive Integrated Moving Average (SARIMA) and Facebook (FB) Prophet models, with potential change points, in handling time-series road accident data involving seasonal patterns in contrast to other statistical methods employed by key governmental agencies such as Ghana's Motor Transport and Traffic Unit (MTTU). The aforementioned models underwent training with monthly RTA data spanning …

Rigidity Of Ext And Tor Via Flat–Cotorsion Theory, Lars Winther Christensen, Luigi Ferraro, Peder Thompson

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Let p be a prime ideal in a commutative noetherian ring R and denote by k(p) the residue field of the local ring R_p. We prove that if an R-module M satisfies Ext_R^n(k(p),M) = 0 for some n >= dim R, then Ext_R^i(k(p),M) = 0 holds for all i >= n. This improves a result of Christensen, Iyengar, and Marley by lowering the bound on n. We also improve existing results on Tor-rigidity. This progress is driven by the existence of minimal semi-flat-cotorsion replacements in the derived category as recently proved by Nakamura and Thompson.

A Review Of Cyber Attacks On Sensors And Perception Systems In Autonomous Vehicle, Taminul Islam, Md. Alif Sheakh, Anjuman Naher Jui, Omar Sharif, Md Zobaer Hasan

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Vehicle automation has been in the works for a long time now. Automatic brakes, cruise control, GPS satellite navigation, etc. are all common features seen in today's automobiles. Automation and artificial intelligence breakthroughs are likely to lead to an increase in the usage of automation technologies in cars. Because of this, mankind will be more reliant on computer-controlled equipment and car systems in our daily lives. All major corporations have begun investing in the development of self-driving cars because of the rapid advancement of advanced driver support technologies. However, the level of safety and trustworthiness is still questionable. Imagine what …

The P-Adic Schrödinger Equation And The Two-Slit Experiment In Quantum Mechanics, Wilson A. Zuniga-Galindo

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

p-Adic quantum mechanics is constructed from the Dirac-von Neumann axioms identifying quantum states with square-integrable functions on the N-dimensional p-adic space, Q_{p}^{N}. This choice is equivalent to the hypothesis of the discreteness of the space. The time is assumed to be a real variable. p-Adic quantum mechanics is the response to the question: what happens with the standard quantum mechanics if the space has a discrete nature? The time evolution of a quantum state is controlled by a nonlocal Schrödinger equation obtained from a p-adic heat equation by a temporal Wick rotation. This p-adic heat equation describes a particle performing …

Geometric Aspects Of Quantization And Relationship To Integrability, Paul Bracken

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

It is the case that quantum mechanics has a deep geometric structure and can be presented accordingly. Quantum mechanics is to a certain degree foreshadowed by the geometry inherent in the geometric structure of classical mechanics. The purpose here is to present some new results and proofs which impact the mathematical structure of quantum mechanics. The relationship between integrability and quantum physics is investigated in terms of geometric ideas and structures. Two physical examples are drawn from these mathematical ideas which directly relate to physics. An introduction as to how these ideas can be extended to infinite degrees of freedom …

Conceptualizing Ethics, Authenticity, And Efficacy Of Simulations In Teacher Education, Carrie Wilkerson Lee, Liza Bondurant, Bima Sapkota, Heather Howell, Yvonne Lai

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

This working group was a continuation of working groups in 2019 and 2021 that initially aimed to focus on equity in simulations of practice in mathematics teacher education. We began by discussing our conceptualizations of simulations and equity. Next, we reflected on the lack of work that currently exists at the intersection of simulations and equity as well as our limited collective expertise in this space. We proposed the following areas of potential research: Access,Design, Affective Domains, Teaching Practices, Assessment, Critical Conversations. Attendees self-selected into focus groups and met to discuss their current work and how future work could focus …

On The Vanishing Of The Coefficients Of Cm Eta Quotients, Timothy Huber, Chang Liu, James Mclaughlin, Dongxi Ye, Miaodan Yuan, Sumeng Zhang

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

This work characterizes the vanishing of the Fourier coefficients of all CM (Complex Multiplication) eta quotients. As consequences, we recover Serre’s characterization about that of η(12z)2 and recent results of Chang on the pth coefficients of η(4z)6 and η(6z)4 . Moreover, we generalize the results on the cases of weight 1 to the setting of binary quadratic forms.

Envariance As A Symmetry In Quantum Mechanics And Applications To Statistical Mechanics, Paul Bracken

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

A quantum symmetry called entanglement-assisted invariance, also called envariance, is introduced. It is studied with respect to the process of performing quantum measurements. An apparatus which interacts with other physical systems, which are called environments, exchanges a single state with physical states equal in number to that of the possible outcomes of the experiment. Correlations between the apparatus and environment give rise to a type of selection rule which prohibits the apparatus from appearing in a superposition corresponding to different eigenvalues of the pointer basis of the apparatus. The eigenspaces of this observable form a natural basis for the apparatus …

Explainable Machine Learning Reveals The Relationship Between Hearing Thresholds And Speech-In-Noise Recognition In Listeners With Normal Audiograms, Jithin Raj Balan, Hansapani Rodrigo, Udit Saxena, Srikanta K. Mishra

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Some individuals complain of listening-in-noise difficulty despite having a normal audiogram. In this study, machine learning is applied to examine the extent to which hearing thresholds can predict speech-in-noise recognition among normal-hearing individuals. The specific goals were to (1) compare the performance of one standard (GAM, generalized additive model) and four machine learning models (ANN, artificial neural network; DNN, deep neural network; RF, random forest; XGBoost; eXtreme gradient boosting), and (2) examine the relative contribution of individual audiometric frequencies and demographic variables in predicting speech-in-noise recognition. Archival data included thresholds (0.25–16 kHz) and speech recognition thresholds (SRTs) from listeners with …

Adjusting For Berkson Error In Exposure In Ordinary And Conditional Logistic Regression And In Poisson Regression, Tamer Oraby, Santanu Chakraborty, Siva Sivaganesan, Laurel Kincl, Lesley Richardson, Mary Mcbride, Jack Siemiatycki, Elisabeth Cardis, Daniel Krewski

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Background

INTEROCC is a seven-country cohort study of occupational exposures and brain cancer risk, including occupational exposure to electromagnetic fields (EMF). In the absence of data on individual exposures, a Job Exposure Matrix (JEM) may be used to construct likely exposure scenarios in occupational settings. This tool was constructed using statistical summaries of exposure to EMF for various occupational categories for a comparable group of workers.

Methods

In this study, we use the Canadian data from INTEROCC to determine the best EMF exposure surrogate/estimate from three appropriately chosen surrogates from the JEM, along with a fourth surrogate based on Berkson …

Rogue Waves And Their Patterns In The Vector Nonlinear Schrödinger Equation, Guangxiong Zhang, Peng Huang, Bao-Feng Feng, Chengfa Wu

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this paper, we study the general rogue wave solutions and their patterns in the vector (or M-component) nonlinear Schrödinger (NLS) equation. By applying the Kadomtsev–Petviashvili reduction method, we derive an explicit solution for the rogue wave expressed by τ functions that are determinants of K × K block matrices ( 1 ≤ K ≤ M ) with an index jump of M + 1 . Patterns of the rogue waves for M = 3 , 4 and K = 1 are thoroughly investigated. It is found that when one of the internal parameters is large enough, the wave pattern …

On Explicit Soliton Solutions And Blow-Up For Coupled Variable Coefficient Nonlinear Schrödinger Equations, Jose Escorcia, Erwin Suazo

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

This work is concerned with the study of explicit solutions for a generalized coupled nonlinear Schrödinger equations (NLS) system with variable coefficients. Indeed, we show, employing similarity transformations, the existence of Rogue wave and dark-bright soliton like-solutions for such a generalized NLS system, provided the coefficients satisfy a Riccati system. As a result of the multiparameter solution of the Riccati system, the nonlinear dynamics of the solution can be controlled. Finite-time singular solutions in the L∞ norm for the generalized coupled NLS system are presented explicitly. Finally, an n-dimensional transformation between a variable coefficient NLS coupled system and a constant …

Convergence Of The Two-Point Modulus-Based Matrix Splitting Iteration Method, Ximing Fang, Ze Gu, Zhijun Qiao

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this paper, we discuss the convergence of the two-point modulus-based matrix splitting iteration method for solving the linear complementarity problem. Some convergence conditions are presented from the spectral radius and the matrix norm when the system matrix is a -matrix. Besides, the quasi-optimal cases of the method are studied. Numerical experiments are provided to show the presented results.

A Second Homotopy Group For Digital Images, Gregory Lupton, Oleg R. Musin, Nicholas A. Scoville, P. Christopher Staecker, Jonathan Treviño-Marroquín

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We define a second (higher) homotopy group for digital images. Namely, we construct a functor from digital images to abelian groups, which closely resembles the ordinary second homotopy group from algebraic topology. We illustrate that our approach can be effective by computing this (digital) second homotopy group for a digital 2-sphere.

Rogue Waves In The Massive Thirring Model, Junchao Chen, Bo Yang, Bao-Feng Feng

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this paper, general rogue wave solutions in the massive Thirring (MT) model are derived by using the Kadomtsev–Petviashvili (KP) hierarchy reduction method and these rational solutions are presented explicitly in terms of determinants whose matrix elements are elementary Schur polynomials. In the reduction process, three reduction conditions including one index- and two dimension-ones are proved to be consistent by only one constraint relation on parameters of tau-functions of the KP-Toda hierarchy. It is found that the rogue wave solutions in the MT model depend on two background parameters, which influence their orientation and duration. Differing from many other coupled …

Elementary Mathematics Curriculum: State Policy, Covid-19, And Teachers’ Control, Mona Baniahmadi, Bima Sapkota, Amy M. Olson

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In the U.S., state guidance to schools in response to the COVID-19 pandemic was politicized. We used state-level political affiliation to explore whether access to curricular resources differed pre-pandemic or during pandemic remote teaching and teachers' reported control over curricular resources during pandemic teaching. We found that pre-pandemic the percentage of teachers in Republican states reported higher levels of resources overall, and use of core and teacher-created curricular resources in particular. They also reported having greater control over their curricular decision-making during the pandemic. There were no state-level differences in teachers’ level of preparation for pandemic teaching, but teachers in …

Figured Worlds Of Women Mathematics Education Scholars, Lili Zhou, Ricki L. Geller-Mckee, Brooke Max, Hyunyi Jung, Bima Sapkota, Jill Newton, Lindsay M. Keazer

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Drawing on the concept of figured worlds (Holland et al., 1998), this project focuses on addressing, responding to, and understanding the self within the figured world of the mathematics education community. Specifically, we examine a group of women with diverse backgrounds in terms of race, class, and cultural contexts, who are engaged in various roles as mathematics education scholars, including teachers, teacher educators, and researchers. Using a dialogical self approach, we facilitate both internal and external discourses, exploring personal histories, narratives, and the development of evolving identities. Our findings reveal that culture and social positions, such as gender, class, and …

Semidefinite Programming Bounds For Distance Distribution Of Spherical Codes, Oleg R. Musin

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We present an extension of known semidefinite and linear programming upper bounds for spherical codes. We apply the main result for the distance distribution of a spherical code and show that this method can work effectively In particular, we get a shorter solution to the kissing number problem in dimension 4.

The Taylor Resolution Over A Skew Polynomial Ring, Luigi Ferraro, Desiree Martin, W. Frank Moore

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Let k be a field and let I be a monomial ideal in the polynomial ring Q = k[x1, . . . , xn]. In her thesis, Taylor introduced a complex which provides a finite free resolution for Q/I as a Q-module. Later, Gemeda constructed a differential graded structure on the Taylor resolution. More recently, Avramov showed that this differential graded algebra admits divided powers. We generalize each of these results to monomial ideals in a skew polynomial ring R. Under the hypothesis that the skew commuting parameters defining R are roots of unity, we prove as an application that …

On Angles In Higher Order Brillouin Tessellations And Related Tilings In The Plane, Herbert Edelsbrunner, Alexey Garber, Mohadese Ghafari, Teresa Heiss, Morteza Saghafian

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

For a locally finite set in R 2 , the order-k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k. As an example, a stationary Poisson point process in R 2 is locally finite, coarsely dense, and generic with probability one. For such a set, the distributions of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in …

Multi-Symplectic Simulation On Soliton-Collision For Nonlinear Perturbed Schrödinger Equation, Peijun Zhang, Weipeng Hu, Zhen Wang, Zhijun Qiao

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Seeking solitary wave solutions and revealing their interactional characteristics for nonlinear evolution equations help us lot to comprehend the motion laws of the microparticles. As a local nonlinear dynamic behavior, the soliton-collision is difficult to be reproduced numerically. In this paper, the soliton-collision process in the nonlinear perturbed Schrödinger equation is simulated employing the multi-symplectic method. The multi-symplectic formulations are derived including the multi-symplectic form and three local conservation laws of the nonlinear perturbed Schrödinger equation. Employing the implicit midpoint rule, we construct a multi-symplectic scheme, which is equivalent to the Preissmann box scheme, for the nonlinear perturbed Schrödinger equation. …

Normalized Solutions For Sobolev Critical Schrödinger-Bopp-Podolsky Systems, Yuxin Li, Xiaojun Chang, Zhaosheng Feng

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We study the Sobolev critical Schr¨odinger-Bopp-Podolsky system −∆u + φu = λu + µ|u|p−2u + |u|4u in R3, −∆φ + ∆2φ = 4πu2 in R3, under the mass constraint u2 dx = c R3 for some prescribed c > 0, where 2 < p < 8/3, µ > 0 is a parameter, and λ ∈ R is a Lagrange multiplier. By developing a constraint minimizing approach, we show that the above system admits a local minimizer. Furthermore, we establish the existence of normalized ground state solutions.

Constrained Quantization For A Uniform Distribution With Respect To A Family Of Constraints, Megha Pandey, Mrinal Kanti Roychowdhury

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this paper, with respect to a family of constraints for a uniform probability distribution we determine the optimal sets of n-points and the nth constrained quantization errors for all positive integers n. We also calculate the constrained quantization dimension and the constrained quantization coefficient. The work in this paper shows that the constrained quantization dimension of an absolutely continuous probability measure depends on the family of constraints and is not always equal to the Euclidean dimension of the underlying space where the support of the probability measure is defined.

Stability Analyses On The Effect Of Vaccination And Contact Tracing In Monkeypox Virus Transmission, Solomon Eshun, Richmond Essieku, James Ladzekpo

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Monkeypox is a significant health concern due to its potential for morbidity and occasional mortality. Vaccination and effective contact tracing play pivotal roles in controlling infectious diseases, including monkeypox. This study aims to contribute to our understanding of monkeypox dynamics by developing a comprehensive mathematical model that incorporates key factors such as vaccination, quarantining, and contact tracing. Through rigorous sensitivity analysis, we explore the impact of varying vaccination coverage and contact tracing on the disease’s dynamics. In particular, we investigate the dynamics of the disease in relation to variable vaccination coverage and contact tracing. Our findings highlight the critical role …

Empowering 5g Mmwave: Leveraging Kml Placemarks For Enhanced Rf Design And Deployment Efficiency, Gustavo A. Fernandez

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

This publication explores the significance of Keyhole Markup Language (KML) in telecommunications, particularly in the context of 5G mmWave RF design and planning. With the advent of 5G mmWave technology, the demand for seamless and efficient network deployments has never been greater. The deployment of small cells and repeaters for 5G mmWave necessitates utmost precision in location accuracy and rapid information exchange during site surveys and evaluations. The challenges of mmWave frequencies, including their limited range and susceptibility to attenuation, intensify the complexity and criticality of this process. Network operators must ensure that the chosen location is devoid of obstacles …

Asymptotics And Sign Patterns For Coefficients In Expansions Of Habiro Elements, Ankush Goswami, Abhash Kumar Jha, Byungchan Kim, Robert Osburn

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We prove asymptotics and study sign patterns for coefficients in expansions of elements in the Habiro ring which satisfy a strange identity. As an application, we prove asymptotics and discuss positivity for the generalized Fishburn numbers which arise from the Kontsevich–Zagier series associated to the colored Jones polynomial for a family of torus knots. This extends Zagier’s result on asymptotics for the Fishburn numbers.

Effect Of Total Population, Population Density And Weighted Population Density On The Spread Of Covid-19 In Malaysia, Hui Shan Wong, Md Zobaer Hasan, Omar Sharif, Azizur Rahman

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Since November 2019, most countries across the globe have suffered from the disastrous consequences of the Covid-19 pandemic which redefined every aspect of human life. Given the inevitable spread and transmission of the virus, it is critical to acknowledge the factors that catalyse transmission of the disease. This research investigates the relation of the external demographic parameters such as total population, population density and weighted population density on the spread of Covid-19 in Malaysia. Pearson correlation and simple linear regression were utilized to identify the relation between the population-related variables and the spread of Covid-19 in Malaysia using data from …