Department of Mathematics and Statistics
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Item A metapopulation SIHURD model(2025) Bica, Ion; Zhai, Zhichun; Hu, Rui; Su, WanhuaWe enhance our SIHURD model published in Bica et-al (2022) and propose a metapopulation infectious disease mathematical monitoring model where individuals move between discrete spatial patches. We divide the environment into a finite number of spatial patches (e.g., adjacent cities), which preserve homogeneity characteristics. We apply the enhanced SIHURD model presented in this article to each spatial patch. The novelty of this model lies in introducing parameters that represent individuals’ travel rates between spatial patches, which depend on their disease status. In addition, it assumes that individuals do not change their disease status while travelling between patches. Our study uses the reproduction number, R0k, for each spatial patch, k = 1, 2, . . . , n, (n > 1) integer, which represents the average number of secondary cases produced by an infected individual in a susceptible population. The system has only a disease-free equilibrium point if R0k ≤ 1. In contrast, if R0k > 1, the system has an endemic equilibrium point. Reproduction numbers R0k are crucial for understanding the spread of infectious diseases and can inform measures to control outbreaks effectively. Migration between patches fundamentally alters the behaviour of the endemic equilibrium within a patch, rendering it unstable in the proposed model.Item Exponential bounds for the density of the law of the solution of a SDE with locally Lipschitz coefficients(2024) Anton, CristinaUnder the uniform Hörmander’s hypothesis we study smoothness and exponential bounds of the density of the law of the solution of a stochastic differential equation (SDE) with locally Lipschitz drift that satisfy a monotonicity condition. To obtain estimates for the Malliavin covariance matrix and its inverse, we extend the approach in to SDEs with non-globally Lipschitz coefficients. As in, to avoid non-integrability problems we use results about Malliavin differentiability based on the concepts of Ray Absolute Continuity and Stochastic Gateâux differentiability.Item Cluster weighted models for functional data(2025) Anton, Cristina; Smith, IainWe propose a method, funWeightClust, based on a family of parsimonious models for clustering heterogeneous functional linear regression data. These models extend cluster weighted models to functional data, and they allow for multi-variate functional responses and predictors. The proposed methodology follows the approach used by the functional high dimensional data clustering (funHDDC) method. We construct an expectation maximization (EM) algorithm for parameter estimation. Using simulated and benchmark data we show that funWeightClust outperforms funHDDC and several two-steps clustering methods. We also use funWeightClust to analyze traffic patterns in Edmonton, Canada.Item A new class of symplectic methods for stochastic Hamiltonian systems(2025) Anton, CristinaWe propose a systematic approach to construct a new family of stochastic symplectic schemes for the strong approximation of the solution of stochastic Hamiltonian systems. Our approach is based both on B-series and generating functions. The proposed schemes are a generalization of the implicit midpoint rule, they require derivatives of the Hamiltonian functions of at most order two, and are constructed by defining a generating function. We construct some schemes with strong convergence order one and a half, and we illustrate numerically their long term performance.Item A multivariate functional data clustering method using parsimonious cluster weighted models(2025) Anton, Cristina; Smith, IainWe propose a method for clustering multivariate functional linear regression data. Our approach extends multivariate cluster weighted models to functional data with multivariate functional response and predictors, based on the ideas used by the funHDDC method. To add model flexibility, we consider several two-component parsimonious models by combining the parsimonious models used for funHDDC with the Gaussian parsimonious clustering models family in. Parameter estimation is carried out within the expectation maximization (EM) algorithm framework. The proposed method outperforms funHDDC on simulated and real-world data.Item Exponential bounds for the density of the law of the solution of an SDE with locally Lipschitz coefficients(2025) Anton, CristinaUnder the uniform Hörmander hypothesis, we study the smoothness and exponential bounds of the density of the law of the solution of a stochastic differential equation (SDE) with locally Lipschitz drift that satisfies a monotonicity condition. We extend the approach used for SDEs with globally Lipschitz coefficients and obtain estimates for the Malliavin covariance matrix and its inverse. Based on these estimates and using the Malliavin differentiability of any order of the solution of the SDE, we prove exponential bounds of the solution’s density law. These results can be used to study the convergence of implicit numerical schemes for SDEs.Item Preliminary results on using clustering of functional data to identify patients with alzheimer’s disease by analyzing brain MRI scans(2025) Anton, Calin; Anton, Cristina; El-Hajj, Mohamad; Craner, Matthew; Lui, RichardThis study delves into the effectiveness of funWeightClust, a sophisticated model-based clustering technique that leverages functional linear regression models to pinpoint patients diagnosed with Alzheimer’s Disease. Our research entailed a thorough analysis of voxelwise fractional anisotropy data derived from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) dataset, with a particular emphasis on the Cingulum and Corpus Callosum, which are critical regions of interest in understanding the disease’s impact on brain structure. Through a series of experiments, we established that funWeightClust is efficient at distinguishing between patients with Alzheimer’s Disease and healthy control subjects. Notably, the clustering model yielded even more pronounced and accurate results when we focused our analysis on specific brain regions, such as the Left Hippocampus and the Splenium. We postulate that integrating additional biomarkers could significantly enhance the accuracy and reliability of funWeightClust in identifying patients who exhibit signs of Alzheimer’s Disease.Item A note on positive bilinear maps(2023) Davey, Aaron C. H.; Ivanescu, Cristian; Tcaciuc, AdiThis paper concerns positive maps between C∗-algebras, particularly when those positive maps are multilinear. We construct examples of positive bilinear maps that are not 2-positive, and therefore are not completely positive bilinear maps. Paulsen and Smith showed that completely bounded bilinear maps are in one-to-one correspondence with completely bounded linear maps. We show that a similar correspondence does not hold for positive bilinear and linear maps. In particular, we observe that a similar correspondence does not hold if we replace completely bounded with completely positive.Item Villadsen idempotents(2024) Ivanescu, Cristian; Kucerovsky, Dan; Markin, Marat V.; Nikolaev, Igor V.; Trunk, CarstenC*-algebras are rings, sometimes nonunital, obeying certain axioms that ensure a very well-behaved representation theory upon Hilbert space. Moreover, there are some wellknown features of the representation theory leading to subtle questions about norms on tensor products of C*-algebras, and thus to the subclass of nuclear C*-algebras. The question whether all separable nuclear C*-algebras satisfy the Universal Coefficient Theorem (UCT) remains one of the most important open problems in the structure and classification theory of such algebras. One of the most promising ways to test the UCT conjecture depends on finding C*- algebras that behave as idempotents under the tensor product, and satisfy certain additional properties. Briefly put, if there exists a simple, separable, and nuclear C*-algebra that is an idempotent under the tensor product, satisfies a certain technical property, and is not one of the already known such elements {𝑂∞,𝑂2,UHF∞, 𝐽,𝑍,ℂ,} then the UCT fails. Although we do not disprove the UCT in this publication, we do find new idempotents in the class of Villadsen algebras.Item Unsupervised classification with a family of parsimonious contaminated shifted asymmetric Laplace mixtures(2024) McLaughlin, Paul; Franczak, Brian C.; Kashlak, Adam B.A family of parsimonious contaminated shifted asymmetric Laplace mixtures is developed for unsupervised classification of asymmetric clusters in the presence of outliers and noise. A series of constraints are applied to a modified factor analyzer structure of the component scale matrices, yielding a family of twelve models. Application of the modified factor analyzer structure and these parsimonious constraints makes these models effective for the analysis of high-dimensional data by reducing the number of free parameters that need to be estimated. A variant of the expectation-maximization algorithm is developed for parameter estimation with convergence issues being discussed and addressed. Popular model selection criteria like the Bayesian information criterion and the integrated complete likelihood (ICL) are utilized, and a novel modification to the ICL is also considered. Through a series of simulation studies and real data analyses, that includes comparisons to well-established methods, we demonstrate the improvements in classification performance found using the proposed family of models.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.