Department of Mathematics and Statistics
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Item On eigenmeasures under Fourier transform(2023) Baake, Michael; Spindeler, Timo; Strungaru, NicolaeSeveral classes of tempered measures are characterised that are eigenmeasures of the Fourier transform, the latter viewed as a linear operator on (generally unbounded) Radon measures on Rd. In particular, we classify all periodic eigenmeasures on R, which gives an interesting connection with the discrete Fourier transform and its eigenvectors, as well as all eigenmeasures on R with uniformly discrete support. An interesting subclass of the latter emerges from the classic cut and project method for aperiodic Meyer sets. Finally, we construct a large class of eigenmeasures with locally finite support that is not uniformly discrete and has large gaps around 0.Item Tempered distributions with translation bounded measure as Fourier transform and the generalized Eberlein decomposition(2023) Spindeler, Timo; Strungaru, NicolaeIn this paper, we study the class of tempered distributions whose Fourier transform is a translation bounded measure and show that each such distribution in has order at most 2d. We show the existence of the generalized Eberlein decomposition within this class of distributions, and its compatibility with all previous Eberlein decompositions. The generalized Eberlein decomposition for Fourier transformable measures and properties of its components are discussed. Lastly, we take a closer look at the absolutely continuous spectrum of measures supported on Meyer sets.Item Abstract almost periodicity for group actions on uniform topological spaces(2024) Lenz, Daniel; Spindeler, Timo; Strungaru, NicolaeWe present a unified theory for the almost periodicity of functions with values in an arbitrary Banach space, measures and distributions via almost periodic elements for the action of a locally compact abelian group on a uniform topological space. We discuss the relation between Bohr- and Bochner-type almost periodicity, and similar conditions, and how the equivalence among such conditions relates to properties of the group action and the uniformity. We complete the paper by demonstrating how various examples considered earlier all fit in our framework.Item Inter-model sets in Rd are model sets(2024) Richard, Christoph; Strungaru, NicolaeWe show that any union of finitely many shifted model sets from a given cut-and-project scheme is a model set in some modified cut-and-project scheme. Restricting to direct space Rd, we show that any inter-model set is a model set in some modified cut-and-project scheme with second countable internal space. In both cases, the window in the modified cut-and-project scheme inherits the topological and measure-theoretic properties of the original windows.Item The (reflected) Eberlein convolution of measures(2024) Lenz, Daniel; Spindeler, Timo; Strungaru, NicolaeIn this paper, we study the properties of the Eberlein convolution of measures and introduce a reflected version of it. For functions we show that the reflected Eberlein convolution can be seen as a translation invariant function-valued inner product. We study its regularity properties and show its existence on suitable sets of functions. For translation bounded measures we show that the (reflected) Eberlein convolution always exists along subsequences of the given sequence, and is a weakly almost periodic and Fourier transformable measure. We prove that if one of the two measures is mean almost periodic, then the (reflected) Eberlein convolution is strongly almost periodic. Moreover, if one of the measures is norm almost periodic, so is the (reflected) Eberlein convolution.Item Aspects of aperiodic order(2023) Baake, Michael; Cortez, Maria Isabel; Damanik, David; Strungaru, NicolaeThe theory of aperiodic order expanded and developed significantly since the discovery of quasicrystals, and continues to bring many mathematical disciplines together. The focus of this workshop was on harmonic analysis and spectral theory, dynamical systems and group actions, Schrödinger operators, and their roles in aperiodic order – with links into a full range of problems from number theory to operator theory.Item The Cuntz semigroup as an invariant for C-algebras(2008) Coward, Kristofer T.; Elliott, George A.; Ivanescu, CristianA category is described to which the Cuntz semigroup belongs and as a functor into which it preserves inductive limits.Item Solution of an outstanding conjecture: the non-existence of universal cycles with k= n−2(2002) Stevens, Brett; Buskell, Paul; Ecimovic, Paule; Ivanescu, Cristian; Malik, Abid Muslim; Savu, Anamaria; Vassilev, Tzvetalin S.; Verrall, Helen; Yang, Boting; Zhao, ZhiduoA universal cycle for k-subsets of an n-set, {1,2,…,n}, is a cyclic sequence of integers with the property that each subset of {1,2,…,n} of size k appears exactly once consecutively in the sequence. This problem was first posed by Chung et al. (Discrete Math. 110 (1992) 43) and solved for k=2,3,4,6 by Jackson and Hurlbert (Ph.D. Thesis, Rutgers University, New Brunswick, NJ, 1990; SIAM J. Discrete Math. 7(4) (1994) 598; Discrete Math. 137 (1995) 241; Personal communication, 1999). Both Jackson and Hurlbert noted the difficulty of finding universal cycles with k⩾⌈n/2⌉. Jackson has found some of these but conjectured that universal cycles never exist when k=n−2. We prove this result and give some bounds on the longest word not repeating any (n−2)-subset and also the shortest word that contains all at least once.Item The classification of separable simple C∗-algebras which are inductive limits of continuous-trace C∗-algebras with spectrum homeomorphic to the closed interval [0 , 1](2008) Elliott, George A.; Ivanescu, CristianA classification is given of certain separable nuclear C∗-algebras not necessarily of real rank zero, namely, the class of separable simple C∗-algebras which are inductive limits of continuous-trace C∗-algebras whose building blocks have spectrum homeomorphic to the closed interval [0, 1], or to a disjoint union of copies of this space. Also, the range of the invariant is calculated.Item Noncommutative aspects of Villadsen algebras(2023) Ivanescu, Cristian; Kučerovský, DanIn this short article, we survey the importance of Villadsen algebras in the context of the classification theory, with an emphasis on its non-commutative aspects.Item A Laplace-based model with flexible tail behavior(2024) Tortora, Cristina; Franczak, Brian C.; Bagnato, Luca; Punzo, AntonioThe proposed multiple scaled contaminated asymmetric Laplace (MSCAL) distribution is an extension of the multivariate asymmetric Laplace distribution to allow for a different excess kurtosis on each dimension and for more flexible shapes of the hyper-contours. These peculiarities are obtained by working on the principal component (PC) space. The structure of the MSCAL distribution has the further advantage of allowing for automatic PC-wise outlier detection – i.e., detection of outliers separately on each PC – when convenient constraints on the parameters are imposed. The MSCAL is fitted using a Monte Carlo expectation-maximization (MCEM) algorithm that uses a Monte Carlo method to estimate the orthogonal matrix of eigenvectors. A simulation study is used to assess the proposed MCEM in terms of computational efficiency and parameter recovery. In a real data application, the MSCAL is fitted to a real data set containing the anthropometric measurements of monozygotic/dizygotic twins. Both a skewed bivariate subset of the full data, perturbed by some outlying points, and the full data are considered.Item Mining COVID-19 data to predict the effect of policies on severity of outbreaks(2023) El-Hajj, Mohamad; Anton, Calin; Anton, Cristina; Dobosz, Dominic; Smith, Iain; Deiab, Fattima; Saleh, NagamDuring the years 2020, 2021, and partially 2022, the COVID-19 virus ran rampant across the globe, causing devastating effects on the masses. Using data mining techniques, we explored factors linked to severe cases of COVID-19 and tried to identify the effect of different government policies on the evolution of the severity of infections. Four countries were selected with a date range of the year 2021 to investigate each region’s efforts regarding vaccine distribution and specific policies enacted for COVID-19 suppression. Pearson’s Correlation Coefficients were used to help establish initially relationships between the policies, vaccines, and severe cases. We used the identified factors to predict the number of new COVID-19 cases and hospital ICU admissions. We included all the country data from Our World in Data (OWID) for this phase. Our investigation indicates that, given enough data, long-range trend predictions can be obtained using Random Forest Regressors. A trained Random Forest model can readily explain factors that effectively slow the spread of COVID-19. With proposed policies given as input, the model can return the expected number of cases, thus informing policies without spending multiple weeks tracking results.Item Model based clustering of functional data with mild outliers(2023) Anton, Cristina; Smith, IainWe propose a procedure, called CFunHDDC, for clustering functional data with mild outliers which combines two existing clustering methods: the functional high dimensional data clustering (FunHDDC) [1] and the contaminated normal mixture (CNmixt) [3] method for multivariate data. We adapt the FunHDDC approach to data with mild outliers by considering a mixture of multivariate contaminated normal distributions. To fit the functional data in group-specific functional subspaces we extend the parsimonious models considered in FunHDDC, and we estimate the model parameters using an expectation-conditional maximization algorithm (ECM). The performance of the proposed method is illustrated for simulated and real-world functional data, and CFunHDDC outperforms FunHDDC when applied to functional data with outliers.Item The effect of animal-assisted intervention on undergraduate students’ perception of momentary stress(2023) Chute, Andrea; Vihos, Jill; Johnston, Sharon; Buro, Karen; Velupillai, NirudikaBackground: Student mental wellness is a priority in higher education. Animal Assisted Interventions (AAIs’) are gaining momentum in universities across North America (Dell et al., 2015). Aims: This study explored the relationships between AAIs’, demographic variables, and perceived momentary stress among university students. Methods: Using a descriptive correlational design, students completed a Perceived Momentary Stress questionnaire that included the Stress Numerical Rating Scale-11 (Stress NRS-11) and the Visual Analog Scale (VAS) to measure perceived stress before and after AAIs’. Data were analyzed using R (4.1.2) (R Core Team, Vienna, Austria) to identify relationships between students’ perceptions of momentary stress, AAIs’ and sociodemographic and demographic variables. Results: First-year students, female students, and students identifying as sexual minorities were found to benefit the most from AAIs’. Conclusion: Results from this study reflect relationships between exposure to animal-assisted interventions and student demographic variables.Item Scaffolding information literacy learning for undergraduate nursing students: a mixed-method exploration of student IL self-efficacy(2024) Croxen, Hanneke; Nelson, Jody; McKendrick-Calder, Lisa; Su, WanhuaInformation literacy (IL) competency is an essential component of evidence-informed nursing practice. It is integral to introduce and develop core information literacy competencies for evidence-informed practice within undergraduate education programs. Research has shown undergraduate students may experience challenges with information literacy skills. More research to inform teaching methodologies that effectively enhance students’ skills and abilities, as well as their self-efficacy with these skills, is needed. This article describes an innovative teaching strategy, called journal club, which uses scaffolded learning activities in small groups over one semester.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.