# Department of Mathematics and Statistics

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Item Research recast(ed): Dreaming big through research in mathematics and statistics and shaping the minds of young mathematicians with Cristina Anton(2023) Miskiman, Megan; Schabert, Reinette; Anton, CristinaIn today's episode, Cristina Anton, professor of Mathematics and Statistics at MacEwan University, discusses her research on clustering functional data with outliers and toxicity assessments. Cristina discusses involvement in youth math competitions and math labs, encouraging students to dream big. With special thanks to Cristina's research partners and students. Machine learning project for clustering functional data with outliers Iain Smith Malcolm Nielsen. Time Series Project Sandy Julian and Joyce Wu.Item Research recast(ed): MacEwan celebrates month of scholarship - the diffraction of quasicrystals with Dr. Nicolae Strungaru(2023) Miskiman, Megan; Schabert, Reinette; Strungaru, NicolaeIn today’s episode, Associate Professor of Mathematics and Statistics, Dr. Nicolae Strungaru, shares his research and study on a type of mathematical physics - the diffraction of quasicrystals.Item Introduction to applied statistics: open textbook series in statistics(2024) Su, Wanhua; Miller, Dylan; Mewhort, Clarissa; Chipman, Hugh; Fedoruk, JohnThis book aims to provide students taking the first course in introductory statistics with open learning materials to master basic statistical concepts and techniques and to give demonstrations on conducting fundamental statistical analysis using the free statistical software R Commander. Each chapter generally includes a statement of learning outcomes, course notes, review exercises, self-assessment quiz, and homework assignment questions. The book is based on instructor course notes for STAT 151 (Introduction to Applied Statistics) at MacEwan University. In December 2015, the online version of STAT 151, including module notes, quizzes, homework assignment questions and marking rubrics, and a lab manual in R Commander was developed, leading to the creation of this textbook. Each homework assignment has two parts; students must complete Part A by hand and Part B with R Commander. Most data sets for the assignment, assignment questions, and quiz questions are adapted from popular introductory statistics textbooks such as Introductory Statistics by Neil Weiss and Intro STATS by Richard D. De Veaux, Paul F. Velleman, David E. Bock, and Paul D. Velleman. The online STAT 151 was completed and offered for the first time in Spring 2018. This open textbook is the revised and enriched version of that online course. The only prerequisite of this book is high school mathematics; most students take STAT 151 in the first year of their post-secondary education. R Commander is taught instead of R/RStudio as the software for the lab component to avoid focusing on the programming component needed for R/R Studio. In a future edition, there are plans to include a lab manual with command lines in R/RStudio. This book introduces one-sided confidence intervals to help students understand the computer output of hypothesis testing in R Commander.Item Invariant subspace problem for rank-one perturbations: the quantitative version(2022) Tcaciuc, AdiWe show that for any bounded operator T acting on an infinite dimensional complex Banach space, and for any ε > 0, there exists an operator F of rank at most one and norm smaller than ε such that T + F has an invariant subspace of infinite dimension and codimension. A version of this result was proved in [T19] under additional spectral conditions for T or T∗. This solves in full generality the quantitative version of the invariant subspace problem for rank-one perturbations.Item Undergraduate nursing student satisfaction with open educational resources in a professional communication course(2023) Vihos, Jill; Chute, Andrea; Johnston, Sharon; Pawliuk, Brandi; Buro, Karen; Velupillai, Nirudika; Sampaga, CatherineOpen educational resources (OER) are emerging as reference materials in nursing. The purpose of this study was to explore undergraduate nursing students' experiences with OER and the relationship with demographic variables. Findings from this descriptive survey study reveal that the quality of OER materials and learner experience were highly correlated. Integration scores for the youngest cohort was significantly lower compared to other age groups. The integration of quality OER is correlated with positive student experience.Item A modified susceptible-infected-recovered epidemiological model(2022) Bica, Ion; Zhai, Zhichun; Hu, RuiObjectives This paper proposes an infectious disease model incorporating two new model compartments, hospitalization, and intensive care unit. Methods The model dynamics are analyzed using the local and global stability theory of nonlinear systems of ordinary differential equations. For the numerical simulations, we used the Rosenbrock method for stiff initial value problems. We obtained numerical simulations using MAPLE software. The returned MAPLE procedure was called only for points inside the range on which the method evaluated the numerical solution of the system with specied initial conditions. Results We proposed a new model to describe the dynamics of microparasitic infections. Numerical simulations revealed that the proposed model fitted with the expected behaviour of microparasitic infections with "acute epidemicity." The numerical simulations showed consistency in the behaviour of the system. Conclusions The model proposed has "robust" dynamics, supported by the global stability of its endemic state and the consistency of the numerical simulations regarding the model's time evolution behaviour. The introduction of the hospitalization and intensive care unit compartments in the proposed model revealed that it is essential to consider such policies in the case of "acute epidemicity" of microparasitic infections.Item Robust optimal design when missing data happen at random(2023) Hu, Rui; Bica, Ion; Zhai, ZhichunIn this article, we investigate the robust optimal design problem for the prediction of response when the fitted regression models are only approximately specified, and observations might be missing completely at random. The intuitive idea is as follows: We assume that data are missing at random, and the complete case analysis is applied. To account for the occurrence of missing data, the design criterion we choose is the mean, for the missing indicator, of the averaged (over the design space) mean squared errors of the predictions. To describe the uncertainty in the specification of the real underlying model, we impose a neighborhood structure on the regression response and maximize, analytically, the Mean of the averaged Mean squared Prediction Errors (MMPE), over the entire neighborhood. The maximized MMPE is the “worst” loss in the neighborhood of the fitted regression model. Minimizing the maximum MMPE over the class of designs, we obtain robust “minimax” designs. The robust designs constructed afford protection from increases in prediction errors resulting from model misspecifications.Item Explicit pseudo-symplectic Runge-Kutta methods for stochastic Hamiltonian systems(2023) Anton, CristinaWe give conditions for stochastic Runge-Kutta methods to near preserve quadratic invariants, and we discuss the associated simpli ed order conditions. For stochastic Hamiltonian systems we propose a systematic approach to construct explicit stochastic Runge-Kutta pseudo-symplectic schemes. Our approach is based on colored trees and B-series. We construct some pseudosymplectic stochastic Runge-Kutta methods with strong convergence order, and we illustrate numerically the long term performance of the proposed schemes.Item Model-based clustering of functional data via mixtures of t distributions(2023) Anton, Cristina; Smith, IainWe propose a procedure, called T-funHDDC, for clustering multivariate functional data with outliers which extends the functional high dimensional data clustering (funHDDC) method (Schmutz et al, 2020) by considering a mixture of multivariate t distributions. We de ne a family of latent mixture models following the approach used for the parsimonious models considered in funHDDC and also constraining or not the degrees of freedom of the multivariate t distributions to be equal across the mixture components. The parameters of these models are estimated using an expectation maximization (EM) algorithm. In addition to proposing the T-funHDDC method, we add a family of parsimonious models to C-funHDDC, which is an alternative method for clustering multivariate functional data with outliers based on a mixture of contaminated normal distributions (Amovin-Assagba et al, 2022). We compare T-funHDDC, C-funHDDC, and other existing methods on simulated functional data with outliers and for real-world data. T-funHDDC out-performs funHDDC when applied to functional data with outliers, and its good performance makes it an alternative to C-funHDDC. We also apply the T-funHDDC method to the analysis of traffic flow in Edmonton, Canada.Item Why do (weak) Meyer sets diffract?(2023) Strungaru, NicolaeGiven a weak model set in a locally compact Abelian, group we construct a relatively dense set of common Bragg peaks for all its subsets that have non-trivial Bragg spectrum. Next, we construct a relatively dense set of common norm almost periods for the diffraction, pure point, absolutely continuous and singular continuous spectrum, respectively, of all its subsets. We use the Fibonacci model set to illustrate these phenomena. We extend all these results to arbitrary translation bounded weighted Dirac combs supported within some Meyer set. We complete the paper by discussing extensions of the existence of the generalized Eberlein decomposition for measures supported within some Meyer set.Item On higher dimensional arithmetic progressions in Meyer sets(2023) Klick, Anna; Strungaru, NicolaeIn this paper we study the existence of higher dimensional arithmetic progression in Meyer sets. We show that the case when the ratios are linearly dependent over $\ZZ$ is trivial, and focus on arithmetic progressions for which the ratios are linearly independent. Given a Meyer set Λ and a fully Euclidean model set $\oplam(W)$ with the property that finitely many translates of $\oplam(W)$ cover Λ, we prove that we can find higher dimensional arithmetic progressions of arbitrary length with k linearly independent ratios in Λ if and only if k is at most the rank of the $\ZZ$-module generated by $\oplam(W)$. We use this result to characterize the Meyer sets which are subsets of fully Euclidean model sets.Item A note on tempered measures(2023) Baake, Michael; Strungaru, NicolaeThe relation between tempered distributions and measures is analysed and clarified. While this is straightforward for positive measures, it is surprisingly subtle for signed or complex measures.Item Leptin densities in amenable groups(2022) Pogorzelski, Felix; Richard, Christoph; Strungaru, NicolaeConsider a positive Borel measure on a locally compact group. We define a notion of uniform density for such a measure, which is based on a group invariant introduced by Leptin in 1966. We then restrict to unimodular amenable groups and to translation bounded measures. In that case our density notion coincides with the well-known Beurling density from Fourier analysis, also known as Banach density from dynamical systems theory. We use Leptin densities for a geometric proof of the model set density formula, which expresses the density of a uniform regular model set in terms of the volume of its window, and for a proof of uniform mean almost periodicity of such model sets.Item Spectrum of weak model sets with Borel windows(2023) Keller, Gerhard; Richard, Christoph; Strungaru, NicolaeConsider the extended hull of a weak model set together with its natural shift action. Equip the extended hull with the Mirsky measure, which is a certain natural pattern frequency measure. It is known that the extended hull is a measure-theoretic factor of some group rotation, which is called the underlying torus. Among other results, in the article Periods and factors of weak model sets, we showed that the extended hull is isomorphic to a factor group of the torus, where certain periods of the window of the weak model set have been factored out. This was proved for weak model sets having a compact window. In this note, we argue that the same results hold for arbitrary measurable and relatively compact windows. Our arguments crucially rely on Moody’s work on uniform distribution in model sets. We also discuss implications for the diffraction of such weak model sets and discuss a new class of examples which are generic for the Mirsky measure.Item Estimated discharge of microplastics via urban stormwater during individual rain events(2023) Ross, Matthew S.; Loutan, Alyssa; Groeneveld, Tianna M.; Molenaar, Danielle; Kroetch, Kimberly; Bujaczek, Taylor; Kolter, Sheldon; Moon, Sarah; Franczak, Brian C.Urban stormwater runoff is an important pathway for the introduction of microplastics and other anthropogenic pollutants into aquatic environments. Highly variable concentrations of microplastics have been reported globally in runoff, but knowledge of key factors within urban environments contributing to this variability remains limited. Furthermore, few studies to date have quantitatively assessed the release of microplastics to receiving waters via runoff. The objectives of this study were to assess the influence of different catchment characteristics on the type and amount of microplastics in runoff and to provide an estimate of the quantity of microplastics discharged during rain events. Stormwater samples were collected during both dry periods (baseflow) and rain events from 15 locations throughout the city of Calgary, Canada’s fourth largest city. These catchments ranged in size and contained different types of predominant land use. Microplastics were found in all samples, with total concentrations ranging from 0.7 to 200.4 pcs/L (mean = 31.9 pcs/L). Fibers were the most prevalent morphology identified (47.7 ± 33.0%), and the greatest percentage of microplastics were found in the 125–250 µm size range (26.6 ± 22.9%) followed by the 37–125 µm size range (24.0 ± 22.3%). Particles were predominantly black (33.5 ± 33.8%), transparent (22.6 ± 31.3%), or blue (16.0 ± 21.6%). Total concentrations, dominant morphologies, and size distributions of microplastics differed between rain events and baseflow, with smaller particles and higher concentrations being found during rain events. Concentrations did not differ significantly amongst catchments with different land use types, but concentrations were positively correlated with maximum runoff flow rate, catchment size, and the percentage of impervious surface area within a catchment. Combining microplastic concentrations with hydrograph data collected during rain events, we estimated that individual outfalls discharged between 1.9 million to 9.6 billion microplastics to receiving waters per rain event. These results provide further evidence that urban stormwater runoff is a significant pathway for the introduction of microplastics into aquatic environments and suggests that mitigation strategies for microplastic pollution should focus on larger urbanized catchments.Item The isomorphism problem for tensor algebras of multivariable dynamical systems(2022) Katsoulis, Elias G.; Ramsey, ChristopherWe resolve the isomorphism problem for tensor algebras of unital multivariable dynamical systems. Specifically, we show that unitary equivalence after a conjugation for multivariable dynamical systems is a complete invariant for complete isometric isomorphisms between their tensor algebras. In particular, this settles a conjecture of Davidson and Kakariadis, Inter. Math. Res. Not. 2014 (2014), 1289–1311 relating to work of Arveson, Acta Math. 118 (1967), 95–109 from the 1960s, and extends related work of Kakariadis and Katsoulis, J. Noncommut. Geom. 8 (2014), 771–787.Item Estimated discharge of microplastics via urban stormwater during individual rain events(2023) Ross, Matthew S.; Loutan, Alyssa; Groeneveld, Tianna M.; Molenaar, Danielle; Kroetch, Kimberly; Bujaczek, Taylor; Kolter, Sheldon; Moon, Sarah; Huynh, Alan; Khayam, Rosita; Franczak, Brian C.Urban stormwater runoff is an important pathway for the introduction of microplastics and other anthropogenic pollutants into aquatic environments. Highly variable concentrations of microplastics have been reported globally in runoff, but knowledge of key factors within urban environments contributing to this variability remains limited. Furthermore, few studies to date have quantitatively assessed the release of microplastics to receiving waters via runoff. The objectives of this study were to assess the influence of different catchment characteristics on the type and amount of microplastics in runoff and to provide an estimate of the quantity of microplastics discharged during rain events. Stormwater samples were collected during both dry periods (baseflow) and rain events from 15 locations throughout the city of Calgary, Canada’s fourth largest city.Item Sex differences in the inflammatory response of the mouse DRG and its connection to pain in experimental autoimmune encephalomyelitis(2022) Maguire, Aislinn D.; Friedman, Timothy N.; Villarreal Andrade, Dania N.; Haq, Fajr; Dunn, Jacob; Pfeifle, Keiana; Tenorio, Gustavo; Buro, Karen; Plemel, Jason R.; Kerr, Bradley J.Multiple Sclerosis (MS) is an autoimmune disease with notable sex differences. Women are not only more likely to develop MS but are also more likely than men to experience neuropathic pain in the disease. It has been postulated that neuropathic pain in MS can originate in the peripheral nervous system at the level of the dorsal root ganglia (DRG), which houses primary pain sensing neurons (nociceptors). These nociceptors become hyperexcitable in response to inflammation, leading to peripheral sensitization and eventually central sensitization, which maintains pain long-term. The mouse model experimental autoimmune encephalomyelitis (EAE) is a good model for human MS as it replicates classic MS symptoms including pain. Using EAE mice as well as naïve primary mouse DRG neurons cultured in vitro, we sought to characterize sex differences, specifically in peripheral sensory neurons. We found sex differences in the inflammatory profile of the EAE DRG, and in the TNFα downstream signaling pathways activated intracellularly in cultured nociceptors. We also found increased cell death with TNFα treatment. Given that TNFα signaling has been shown to initiate intrinsic apoptosis through mitochondrial disruption, this led us to investigate sex differences in the mitochondria’s response to TNFα. Our results demonstrate that male sensory neurons are more sensitive to mitochondrial stress, making them prone to neuronal injury. In contrast, female sensory neurons appear to be more resistant to mitochondrial stress and exhibit an inflammatory and regenerative phenotype that may underlie greater nociceptor hyperexcitability and pain. Understanding these sex differences at the level of the primary sensory neuron is an important first step in our eventual goal of developing sex-specific treatments to halt pain development in the periphery before central sensitization is established.Item Functional non-parametric mixed effects models for cytotoxicity assessment and clustering(2023) Ma, Tiantian; Richard, Dan; Yang, Yongqing Betty; Kashlak, Adam B.; Anton, CristinaA multitude of natural and synthetic chemicals are present in our environment. Through the study of a compound’s cytotoxicity, researchers can carefully set regulations regarding how much of a certain chemical in the ambient environment is tolerable. In the past, research has focused on point measurements such as the LD50. Instead, we consider entire time-dependent cellular response curves through the application of functional mixed effects models. We identify differences in such curves corresponding to the chemical’s mode of action—i.e. how the compound attacks human cells. Through such analysis, we identify curve features to be used for cluster analysis via application of both k-means and self organizing maps. The data is analyzed by making use of functional principal components as a data driven basis and separately by considering B-splines for identifying local-time features. Our analysis can be used to drastically speed up future cytotoxicity research.Item Pure point spectrum for dynamical systems and mean, Besicovitch and Weyl almost periodicity(2023) Lenz, Daniel; Spindeler, Timo; Strungaru, NicolaeWe consider metrizable ergodic topological dynamical systems over locally compact, σ - compact abelian groups. We study pure point spectrum via suitable notions of almost periodicity for the points of the dynamical system. More specifically, we characterize pure point spectrum via mean almost periodicity of generic points. We then go on and show how Besicovitch almost periodic points determine both eigenfunctions and the measure in this case. After this, we characterize those systems arising from Weyl almost periodic points and use this to characterize weak and Bohr almost periodic systems. Finally, we consider applications to aperiodic order.