Department of Mathematics and Statistics

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https://hdl.handle.net/20.500.14078/10

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Capacities and embeddings of Besov spaces via general convolution kernels
(2024) Hu, Rui; Zhai, Zhichun; Li, Pentago
This note is denoted to establish equivalent characterizations of Carleson embeddings of fractional Besov spaces Λp,q β (Rn) into the Lorentz spaces Lq0,pμ (R1+n+) induced by general convolution kernels Φt(·). When (p, q) ∈ (1, n/β) x (1,∞), the embeddings will be characterized in terms of capacitary type inequalities for open subsets of Rn. When p = q ∈ (0, 1], the embeddings will be characterized in terms of fractional Besov capacities or the associated variational functional of a nonnegative Radon measure μ. Especially, when p = q = 1 and β ∈ (0, 1), the characterization can be also established in terms of fractional perimeters.
Crossed products and C* -covers of semi-Dirichlet operator algebras
(2025) Humeniuk, Adam; Katsoulis, Elias G.; Ramsey, Christopher
In this paper, we show that the semi-Dirichlet C∗-covers of a semi-Dirichlet operator algebra form a complete lattice, establishing that there is a maximal semi-Dirichlet C∗-cover. Given an operator algebra dynamical system we prove a dilation theory that shows that the full crossed product is isomorphic to the relative full crossed product with respect to this maximal semi-Dirichlet cover. In this way, we can show that every semi-Dirichlet dynamical system has a semi-Dirichlet full crossed product.
On numerical diameters and linear maps
(2024) de Beaudrap, Niel; Ramsey, Christopher
This paper studies the diameter of the numerical range of bounded operators on Hilbert space and the induced seminorm, called the numerical diameter, on bounded linear maps between operator systems which is sensible in the case of unital maps and their scalar multiples. It is shown that the completely bounded numerical diameter is a norm that is comparable but not equal to the completely bounded norm. This norm is particularly interesting in the case of unital completely positive maps and their sections.
The lattice of C*-covers of an operator algebra
(2025) Humeniuk, Adam; Ramsey, Christopher
In this article, it is shown that the lattice of C ∗ -covers of an operator algebra does not contain enough information to distinguish operator algebras up to completely isometric isomorphism. In addition, four natural equivalences of the lattice of C ∗ -covers are developed and proven to be distinct. The lattice of C ∗ -covers of direct sums and tensor products are studied. Along the way key examples are found of operator algebras, each of which generates exactly n C ∗ -algebras up to ∗-isomorphism, and a simple operator algebra that is not similar to a C ∗ -algebra.
Sobolev trace-type inequalities via time-space fractional heat equations
(2025) Tang, Yongrui; Li, Pengtao; Hu, Rui; Zhai, Zhichun
This article aims to establish fractional Sobolev trace inequalities, logarithmic Sobolev trace inequalities, and Hardy trace inequalities associated with time-space fractional heat equations. The key steps involve establishing dedicated estimates for the fractional heat kernel, regularity estimates for the solution of the time-space fractional equations, and characterizing the norm of W˙ ν/2p (Rn) in terms of the solution u(x, t). Additionally, fractional logarithmic Gagliardo–Nirenberg inequalities are proven, leading to Lp−logarithmic Sobolev inequalities for W˙ ν/2p (Rn). As a byproduct, Sobolev affine trace-type inequalities for H˙ −ν/2(Rn) and local Sobolev-type trace inequalities for Qν/2(Rn) are established.
Embryological incubation temperature modulates behaviour in larval white sturgeon (Acispencer transmontanus)
(2025) Hamilton, Trevor; Cheung, Katherine; Hudson, James; Szaszkiewicz, Joshua; Ingraham, Erica; Krook, Jeffrey; Johnson, Andréa; Franczak, Brian; McAdam, Steve; Brauner, Colin J.
White sturgeon inhabit large rivers and estuaries along the Pacific coast of North America and play important ecological and cultural roles. As with other poikilotherms, the expected rise in global temperatures will create physiological challenges for sturgeon, including direct effects on physiological rates, and may also cause changes in behaviour that impact survival. In this study, we investigated ‘carryover effects’ in white sturgeon (Acipenser transmontanus) by incubating fish at one of three embryological temperatures (12 °C, 15 °C, or 18 °C) from fertilization to hatch, and then holding them at 15 °C for the following 30 days post-hatch (dph). We investigated the effect of these incubation temperatures on subsequent behaviours, including locomotion and anxiety-like behaviour, at 18, 24 and 30 dph. The open field test was used to quantify distance moved by the fish and time spent near the outer wall of the arena (to quantify thigmotaxis, an indicator of anxiety-like behaviour) and recorded with motion-tracking software. This test was also used to compare behaviour during the day versus night (at 21 dph) and hourly at night (at 28 dph). Additionally, a light/dark test commonly used in rodents and other fish species was performed for the first time on larval sturgeon at 24 dph. We found a significant family effect; the impact of embryo rearing temperature on locomotion and thigmotaxis varied depending on the family lineage of the fish. Furthermore, we provide evidence that larval sturgeon exhibit greater locomotion at 24 and 30 dph compared to 18 dph, and have a preference for the light zone of the light/dark test. Our findings identify a more nuanced effect of incubation temperature on later life stages that varies with family lineage and underscores the need to consider such variation in research and conservation programs.
Handling skewness and directional tails in model-based clustering
(2025) Tortora, Cristina; Punzo, Antonio; Franczak, Brian C.
Model-based clustering is a powerful approach used in data analysis to unveil underlying patterns or groups within a data set. However, when applied to clusters that exhibit skewness, heavy tails, or both, the classification of data points becomes more challenging. In this study, we introduce two models considering two component-wise transformations of the observed data within a mixture of multiple scaled contaminated normal (MSCN) distributions. MSCN distributions are designed to enable a different tail behavior in each dimension and directional outlier detection in the direction of the principal components. Using the transformed MSCN distributions as components of a mixture, we obtain model-based clustering techniques that allow for 1) flexible cluster shapes in terms of skewness and kurtosis and 2) component-wise and directional outlier detection. We assess the efficacy of the proposed techniques by comparing them with model-based clustering methods that perform global or component-wise outlier detection using simulated and real data sets. This comparative analysis aims to demonstrate which practical clustering scenarios using the proposed MSCN-based approaches are advantageous.
Relative position in binary substitutions
(2025) Coons, Michael; Ramsey, Christopher; Strungaru, Nicolae
Given an infinite word on a finite alphabet, an immediate question arises: can we understand the frequency of letters in that word? For words that are the fixed points of substitutions, the answer to this question is often 'yes'—the details and methods of these answers have been well-documented. In this paper, toward a better understanding of the fixed points of binary substitutions, we delve deeper by investigating, in fine detail, the position of letters by defining various position functions and proving results about their behavior. Our analysis reveals new information about the Fibonacci substitution and the extended Pisa family of substitutions, as well as a new characterization of the Thue-Morse sequence.
Diffraction of the primes and other sets of zero density
(2025) Humeniuk, Adam; Ramsey, Christopher; Strungaru, Nicolae
In this paper, we show that the diffraction of the primes is absolutely continuous, showing no bright spots (Bragg peaks). We introduce the notion of counting diffraction, extending the classical notion of (density) diffraction to sets of density zero. We develop the counting diffraction theory and give many examples of sets of zero density of all possible spectral types.
Pure point diffraction and almost periodicity
(2024) Lenz, Daniel; Spindeler, Timo; Strungaru, Nicolae
This article deals with pure point diffraction and its connection to various notions of almost periodicity. We explain why the Fibonacci chain does not fit into the classical concept of Bohr almost periodicity and how it fits into the classes of mean, Besicovitch and Weyl almost periodic point sets. We report on recent results which characterize pure point diffraction as mean almost periodicity of the underlying structure, and discuss how the complex amplitudes fit into this picture.
A metapopulation SIHURD model
(2025) Bica, Ion; Zhai, Zhichun; Hu, Rui; Su, Wanhua
We 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.
Exponential bounds for the density of the law of the solution of a SDE with locally Lipschitz coefficients
(2024) Anton, Cristina
Under 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.
Cluster weighted models for functional data
(2025) Anton, Cristina; Smith, Iain
We 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.
A new class of symplectic methods for stochastic Hamiltonian systems
(2025) Anton, Cristina
We 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.
A multivariate functional data clustering method using parsimonious cluster weighted models
(2025) Anton, Cristina; Smith, Iain
We 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.
Exponential bounds for the density of the law of the solution of an SDE with locally Lipschitz coefficients
(2025) Anton, Cristina
Under 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.
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, Richard
This 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.
A note on positive bilinear maps
(2023) Davey, Aaron C. H.; Ivanescu, Cristian; Tcaciuc, Adi
This 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.
Villadsen idempotents
(2024) Ivanescu, Cristian; Kucerovsky, Dan; Markin, Marat V.; Nikolaev, Igor V.; Trunk, Carsten
C*-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.
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.

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