Browsing by Author "Farenick, Douglas"

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Open Access
Algebraic and Measurable Sub-Product Systems
(Faculty of Graduate Studies and Research, University of Regina, 2018-12) Krumer, Daniel; Floricel, Remus; Argerami, Martin; Farenick, Douglas; Mobed, Nader
Our main goal, in this thesis, is to conduct a thorough investigation of the math- ematical concept of sub-product system in relation to both quantum dynamical sys- tems and product systems. This notion originates in W. Arveson's pioneering work in non-commutative dynamics theory [5, 7]. The concept has been further extended and analyzed from various perspectives by D. Markiewicz [22], O. Shalit and B. Solel [28], and B.V.R. Bhat and M. Mukherjee [10], among others. The fundamental theme of this thesis is the analysis of relationship between sub- product systems and quantum dynamical semigroups, with emphasis on the role played by certain measurable structures, a role which has often been neglected in the literature. i
Open Access
Complete Order Equivalence of Spin Unitaries
(Faculty of Graduate Studies and Research, University of Regina, 2020-07) Masanika, Adili Julius; Farenick, Douglas; Plosker, Sarah; Floricel, Remus; Grinyer, Gwen
The main objective of this thesis is to explore linear spaces of matrices and linear maps on matrix spaces that arise from spin systems, or spin unitaries, which are finite sets S of selfadjoint unitary matrices such that any two distinct unitaries in S anticommute. We are interested in examining finite-dimensional spin unitaries obtained from spin systems and complete order isomorphisms between the operator systems associated with spin systems.
Open Access
Decoherence of Quantum Markov Systems
(Faculty of Graduate Studies and Research, University of Regina, 2013-08) Zhang, Tian Li; Floricel, Remus; Farenick, Douglas; Argerami, Martin; Mobed, Nader
The main purpose of this thesis is to discuss quantum decoherence in the frame- work of operator algebras. We will define several types of decoherence of quantum Markov systems based on the original notion introduced by Ph. Blanchard and R. Olkiewicz; then we will give a number of suffiecient conditions for a quantum Markov system to display decoherence, and explain the relation between them.
Open Access
The Hahn-Banach Separation Theorem in Free Convexity
(Faculty of Graduate Studies and Research, University of Regina, 2015-12) Cui, Bo; Farenick, Douglas; Floricel, Remus
This thesis presents and develops theorems on separation in the context of free convexity and matrix convex sets. After presenting a proof of the Hahn-Banach Separation Theorem, the main work in this thesis is a treatment of the Effros-Winkler Hahn-Banach Separation Theorem in the setting of matrix convex sets. This result is subsequently interpreted as a theorem concerning noncommutative sets in free analysis. Lastly, a new and complete characterization (in terms of liner pencils) of the matrix convex hull of a g-tuple of complex matrices is established.
Open Access
Image Quality Assessment Using Level-of-Detail
(Faculty of Graduate Studies and Research, University of Regina, 2012-07) Dosselmann, Richard Wayne; Yang, Xue-Dong; Gerhard, David; Yao, Yiyu; Farenick, Douglas; Barron, John
Image quality assessment is a very challenging problem in image processing. Most image quality metrics are currently based on super cial variations in pixel values, perceptual models and structural changes. As well, most are full-reference metrics in which a corrupted image is compared with an \original" or \perfect" version of that image. In many practical settings, however, this \perfect" image is not available. At this time, no metric is able to genuinely replicate human perception of quality. This research therefore introduces a new image quality model, one that centers on level-of-detail. Furthermore, the proposed techniques operate without any \original" image, making them ideal for real-world applications such as television monitoring. In this research, level-of-detail is employed in both the initial detection and subsequent psychological evaluation of noise, blurring and compression. A given image is tested for noise and blurring using a new measure of detail in the frequency domain. For perhaps the rst time, it enables a machine to separate noisy, blurred and otherwise uncorrupted images. This decision is based on the slope of the cumulative histogram of the spectral energy values. Moreover, noise and blur are appropriately organized as \opposite" phenomena in this model, with noise corresponding to high levels of detail and blur being paired with lesser amounts. Once detected, the precise magnitude and speci c type of noise, whether it be random, Gaussian or salt-and-pepper, or blur, either isotropic or motion, is determined using individual noise and blur metrics. Using an alternate notion of level-of-detail based on the prominent mean shift segmentation algorithm, this time in the spatial domain, blocking and ringing metrics to detect and gauge the e ects of JPEG and JPEG2000 compression errors are de ned. All errors uncovered in an image are psychologically weighed in relation to surrounding image content. The goal is to determine which errors are perceptible and which are masked by neighboring pixels. Further motivating this research are the ndings of an investigation of a popular full-reference metric, namely the structural similarity index. Experimental test cases, a numerical analysis and a theoretical study link this method to the conventional mean squared error.
Open Access
Learning Pattern Languages from a Small Number of Helpfully Chosen Examples
(Faculty of Graduate Studies and Research, University of Regina, 2013-08) Mazadi, Zeinab; Zilles, Sandra; Yang, Boting; Hilderman, Robert; Farenick, Douglas
A pattern is a string containing variable symbols and constants. The language of a pattern is the set of all strings obtained by replacing all variables in the pattern with non-empty strings. Patterns and their languages were introduced by Angluin in 1980. Since that time, learning of pattern languages has been a topic of great interest in the research area of computational learning theory, mainly because of its relevance for many applications. Areas in which patterns are suitable for modelling data are for example bioinformatics (e.g., when representing sets of amino acid sequences) or text mining (e.g., for automated information extraction). This thesis studies learning of pattern languages in the context of computational learning theory. Computational learning theory studies various models of learning and investigates the learnability of classes of languages using each learning model. More- over, determining the number of data points (sample complexity) and the amount of computational time (time complexity) required for learning a particular class of languages in a particular model is a main goal in computational learning theory. In this thesis we focus on learning classes of pattern languages in a model called “learning from helpful examples” or “learning from teachers”. In particular, we are interested in determining the worst case number of examples that need to be com- municated between a teacher and a learner to identify the target language from all other languages in the class. The sample complexity measures we use are the teaching dimension in the classic teaching model and the recursive teaching dimension in the cooperative teaching model that was introduced later. We are interested in computing these parameters for certain classes of pattern languages over various sizes of alphabets from which the constant symbols can be taken. In particular, we aim to determine the effect of the alphabet size on these parameters. We study arbitrary patterns, regular patterns and one-variable patterns over three types of alphabets, namely finite alphabets of size at least two, singleton alphabets and infinite alphabets. Our results show that sometimes the alphabet size does in- fluence the sample complexity parameters and sometimes it does not influence them. Moreover, we demonstrate that more advanced models of teaching, like the recursive teaching model, can be more sample-efficient than the classic teaching model when learning certain kinds of pattern languages.
Open Access
Majorization and the Schur-Horn Theorem.
(Faculty of Graduate Studies and Research, University of Regina, 2013-01) Albayyadhi, Maram; Argerami, Martin; Farenick, Douglas
We study majorization in Rn and some of its properties. The concept of majorization plays an important role in matrix analysis by producing several useful relationships. We find out that there is a strong relationship between majorization and doubly stochastic matrices; this relation has been perfectly described in Birkhoff's Theorem. On the other hand, majorization characterizes the connection between the eigenvalues and the diagonal elements of self adjoint matrices. This relation is summarized in the Schur-Horn Theorem. Using this theorem, we prove versions of Kadison's Carpenter's Theorem. We discuss A. Neumann's extension of the concept of majorization to in_nite dimension to that provides a Schur-Horn Theorem in this context. Finally, we detail the work of W. Arveson and R.V. Kadison in proving a strict Schur-Horn Theorem for positive trace-class operators.
Open Access
Optimizing Inference in Bayesian Networks: From Join Tree Propagation to Deep Learning
(Faculty of Graduate Studies and Research, University of Regina, 2020-03) dos Santos, Andre Evaristo; Butz, Cortney J.; Mouhoub, Malek; Yao, Yiyu; Farenick, Douglas; Santos Jr., Eugene
Many di erent platforms, techniques, and concepts can be employed while modelling and reasoning with Bayesian networks (BNs). A problem domain is modelled initially as a directed acyclic graph (DAG) and the strengths of relationships are quanti ed by conditional probability tables (CPTs). We consider four perspectives to BN inference. First, a central task in discrete BN inference is to determine those variables relevant to answer a given query. Two linear algorithms for this task explore the possibly relevant and active parts of a BN, respectively. We empirically compare these two methods along with a variation of each. Second, we start with BN inference using message passing in a join tree. We start with BN inference using message passing in a join tree. Here, we propose Simple Propagation (SP) as a new join tree propagation algorithm for exact inference in discrete Bayesian networks. We establish the correctness of SP. The striking feature of SP is that its message construction exploits the factorization of potentials at a sending node but without the overhead of building and examining graphs as done in Lazy Propagation (LP). Experimental results on optimal (or close to optimal) join trees built from numerous benchmark Bayesian networks show that SP is often faster than LP. Sum-Product Networks (SPNs) are a probabilistic graphical model with deep learning applications. A key feature in an SPN is that inference is tractable under certain structural constraints. This is a strong feature when compared to BNs, where inference is NP-hard. SPNs internal nodes can be understood as marginal inference recording of a BN. This is the third perspective. SPNs have shown to excel at many tasks, achieving even state-of-the-art in important tasks such as image completion. Even though SPNs pose a clear advantage against BNs, the later still have a clearer illustration of in uence among the variables they represent. To take advantage of both, we use an SPN to perform inference, but utilize BNs to observe i the bacterial relationships in soil datasets. We rst learn a BN to read independencies in linear time between bacterial community characteristics. These independencies are useful in understanding the semantic relationships between bacteria within communities. Next, we learn an SPN to perform e cient inference. Here, inference can be conducted to answer traditional queries, involving marginals, or most probable expla- nation (MPE) queries, requesting the most likely values of the non-evidence variables given evidence. Our results extend the literature by showing that known relationships between soil bacteria holding in one or a few datasets, in fact, hold across at least 3500 diverse datasets. This study paves the way for future large-scale studies of agricultural, health, and environmental applications, for which data are publicly available. In an SPN, leaf nodes are indicator variables for each value that a random variable can assume and the remaining nodes are either sum or product. As contribution to SPN inference, we derive partial propagation (PP), which performs SPN exact inference without requiring a full propagation over all nodes in the SPN as currently done. Experimental results on SPN datasets demonstrate that PP has several advantages over full propagation in SPNs, including relative time savings, absolute time savings in large SPNs, and scalability. Finally, as the fourth perspective to BN inference, we give conditions under which convolutional neural networks (CNNs) de ne valid SPNs. One subclass, called con- volutional SPNs (CSPNs), can be implemented using tensors but also can su er from being too shallow. Fortunately, tensors can be augmented while maintaining valid SPNs. This yielded a larger subclass of CNNs, which is called deep convolutional SPNs (DCSPNs), where the convolutional and sum-pooling layers form rich directed acyclic graph structures. One salient feature of DCSPNs is that they keep the rigorousness probabilistic model. As such, they can exploit multiple kinds of probabilistic reasoning, including marginal inference and MPE inference. This allowed an alternative method for learning DCSPNs using vectorized di erentiable MPE, which plays a similar role to the generator in generative adversarial networks (GANs). Image sampling is yet another application demonstrating the robustness of DCSPNs. The results on image sampling were encouraging and sampled images exhibited variability, a salient attribute.
Open Access
Properties of Pure Completely Positive Linear Maps of Operator Systems
(Faculty of Graduate Studies and Research, University of Regina, 2020-02) Tessier, Ryan Brett; Farenick, Douglas; Floricel, Remus; Plosker, Sarah; Zilles, Sandra; Clouatre, Raphael
If Sn denotes the (2n + 1)-dimensional operator system spanned in the group C*-algebra C*(Fn) by the n generators of the free group Fn and their inverses, then the identity map in : Sn -> Sn is shown to be a pure completely positive map. Similarly, the identity map jn : NC(n) -> NC(n) on the noncommutative n-cube NC(n) in the group C_-algebra of the free product of n copies of Z2 is also shown to be pure. Further results on the purity of the reductions of the tensor product of pure completely positive maps are given. Some previously unrecorded generic features of pure completely positive linear maps are also presented, including a result on the pure extendibility of pure completely positive linear maps on operator systems with values in an injective von Neumann algebra.
Open Access
Quantum Fidelity and the Bures Metric in Operator Algebras
(Faculty of Graduate Studies and Research, University of Regina, 2017-06) Rahaman, Mizanur; Farenick, Douglas; Floricel, Remus; Mobed, Nader; Mare, Augustin-Liviu; Pereira, Rajesh
This dissertation undertakes the study of quantum delity, a distinguishability measure in the context of quantum mechanics, from the operator algebraic viewpoint. The notion of delity provides a quantitative measure of how close one state of a quantum system is to another state. High delity occurs when the two states are very close to each other. Evidently, this concept is closely related to a metric on the quantum states which is known as the Bures metric. In this thesis, delity and the Bures metric have been studied in the context of (i) unital C -algebras that possess a faithful positive trace functional and (ii) semi nite von Neumann algebras. In addition, these notions have been analysed in the matrix algebras in an e ort to relate to the quantum information theory literature.
Open Access
Quasi-Pure E0-Semigroups
(Faculty of Graduate Studies and Research, University of Regina, 2021-03) Wood, Clifford Tyler; Floricel, Remus; Farenick, Douglas; Argerami, Martin; Mobed, Nader; Brenken, Berndt
We introduce the class of quasi-pure E0-semigroups acting on a von Neumann algebra as that with equal tail and fixed-point algebras and we describe these notable algebras for E0-semigroups of B(H) where H is a separable Hilbert space. We consider the classes of pure, quasi-pure and ergodic E0-semigroups induced by essential states of the spectral C+-algebra of a product system and we characterize spatial product systems in terms of the existence of certain essential states of the corresponding spectral C+-algebra.
Open Access
A Riparian Buffer at 10 Years: The Effect of Black Plastic Mulch on Soil Variables, Nutirent Stocks and Tree Biomass
(Faculty of Graduate Studies and Research, University of Regina, 2021-03) Jones, Amy Lee; Gagnon, Daniel; Finlay, Kerri; Vanderwel, Mark; Farenick, Douglas; Laroque, Colin P.
Riparian zones are the interface between terrestrial and aquatic ecosystems. Riparian buffers are an important component of this interface environment because the vegetation of the buffer can provide many ecosystem services. Establishing riparian buffers on agricultural land can help reduce agricultural pollutants such as excessive nitrogen and phosphorus, but also provide opportunities to sequester carbon and produce biomass. Sampling was done after 10 years of growth in a riparian buffer of five tree species, hybrid poplar (Populus × canadensis), red ash (Fraxinus pennsylvanica), bur oak (Quercus macrocarpa), red oak (Quercus rubra) and white pine (Pinus strobus) to measure the long-term effects of black plastic mulch on soil variables, tree growth, tree nutrient sequestration and biomass production, as well as the species effect on these variables. Specifically, this study measured: (1) the physical, chemical and biological soil properties, and the fine root biomass of soils in the buffer to determine the long-term effects of black plastic mulch, and; (2) tree biomass production and the carbon and nutrient sequestration potential of the five tree species, as well as to determine if black plastic mulch can cause an increase or decrease in these variables. After 10 years, the use of black plastic mulch decreased soil organic matter, total soil carbon, total soil nitrogen, earthworm biomass and abundance. Soil NO3 concentration was three times greater under the mulch, despite higher tree growth in this treatment. This study shows the importance of the total pool of fine roots on the maintenance and enhancement of soil carbon and earthworm biomass. White pine growth was only slightly enhanced by the black plastic mulch, while red oak was the tree species that ii benefited the most from the mulch treatment, especially for survival. The mulch treatment produced a greater average tree height, survival and basal diameter than the control treatment. Foliage, branch and stem biomass were significantly higher in the mulch treatment, as were the sequestration of carbon, nitrogen, potassium, phosphorus, sulphur, magnesium and calcium. Hybrid poplar accumulated the greatest biomass of all species in all its compartments and had the highest quantity of carbon and all measured nutrients. Hybrid poplar was the best species to produce rapid tree biomass. Alternative mulching options that do not degrade important soil properties and other native tree species should be investigated in other studies to determine their value for future use in riparian buffers. The results from this study will help in designing agroforestry riparian buffers that will maximize nutrient capture and sequestration, and tree biomass production. Such riparian buffers will deliver more ecosystem services (reducing agricultural nitrogen and phosphorus input to watersheds; creating new habitats for flora and fauna) and produce wood biomass for timber or bioenergy.
Open Access
Shift and Quasi-Shift Endomorphisms Associated with Representations of Cuntz Algebras
(Faculty of Graduate Studies and Research, University of Regina, 2012-12-14) Wood, Clifford Tyler; Floricel, Remus; Argerami, Martin; Farenick, Douglas; Mobed, Nader
We present an overview of the main literature regarding *-endomorphisms of the von Neumann algebra B(H), the bounded linear operators on a separable, in nite- dimensional Hilbert space H. In doing so, we follow the established technique of ex- ploiting the surjective correspondence with non-degenerate representations of Toeplitz- Cuntz algebras. Moreover, we introduce an unstudied class of endomorphism, which we call quasi-shift endomorphisms, and show that they are determined by the well- known class of shift endomorphisms. This is original research adapted from our recent pre-print Quasi-shift Endomorphisms Associated with Representations of Cuntz Alge- bras [13]. i
Open Access
The Structure of Operator Systems on Finite-Dimensional Hilbert Spaces
(Faculty of Graduate Studies and Research, University of Regina, 2012-04) Mwangangi, Sadia Hassan; Argerami, Martin; Farenick, Douglas; Floricel, Remus; Mobed, Nader
The purpose of this thesis is to describe in detail the structure of arbitrary operator systems S B(H), where H is assumed to be of finite dimension, using Arveson's non-commutative Choquet theory, and to determine the C* -envelope of S in certain special cases of interest. Arveson classifies these operator systems as either reduced or non-reduced, and we look at these classifications in detail. S is said to be reduced when its boundary ideal is {0} and non-reduced otherwise. We will give examples of 2-dimensional and 3-dimensional operator systems; show what Arveson's parametrization would be in such cases; and determine the C*-envelopes and boundary ideals.
Open Access
Subproduct and Product Systems of C* - Algebras
(Faculty of Graduate Studies and Research, University of Regina, 2019-08) Ketelboeter, Brian; Floricel, Remus; Farenick, Douglas; Argerami, Martin; Mobed, Nader; Wang, Jiun-Chau
This thesis initiates the theory of Tsirelson C -subproduct and product systems on a solid basis of results, which extends Arveson's theory of product systems of Hilbert spaces, with emphasis on Tsirelson's two parameter version of product systems of Hilbert spaces. Within we prove that every Tsirelson C -subproduct system admits a Bhat-type dilation to a Tsirelson C -product system, analogous to the Bhat dilation of a conservative quantum dynamical semigorup to an E0-semigroup. We also construct the universal C -algebra of an arbitrary unital Tsirelson C -subproduct system that, together with its associated quantum L evy process, reflect the behavior of the system. Finally, we provide a Arveson-Powers classi cation type scheme of unital Tsirelson C -subproduct and product systems, based on the concept of unit. We also give a description of the units of a spatial Tsirelson C -subproduct system at the level of the state-space of its universal C -algebra, and show that every unit gives rise to an algebraic Tsirelson subproduct system of Hilbert spaces via the GNS construction.
Open Access
Subproduct Systems of Quasi-Free Quantum Dynamical Semigroups
(Faculty of Graduate Studies and Research, University of Regina, 2016-12) Alzulaibani, Alaa Awad; Floricel, Remus; Farenick, Douglas; Mobed, Nader; Argerami, Martin; Samei, Ebrahim
In this thesis, we construct a class of quantum dynamical semigroups, called quasifree quantum dynamical semigroups, of the von Neumann algebra generated by an arbitrary Weyl system, with respect to some compatible families of quasi-measures and C0-semigroups. We then describe in concrete terms their associated subproduct systems, in the case of irreducible Weyl systems.
Open Access
Universality of Weyl Unitaries
(Faculty of Graduate Studies and Research, University of Regina, 2021-07) Ojo, Oluwatobi Ruth; Farenick, Douglas; Plosker, Sarah; Fallat, Shaun; Grinyer, Gwen
Weyl's unitaries are p×p unitary matrices given by a diagonal matrix having primitive p-th roots of unity as its entries and a cyclic shift matrix. The Weyl unitaries, which we denote by u and v, satisfy u^p= v^p=1_p(the p×p identity matrix) and the commutation relation uv=ζvu, where ζ is a primitive p-th root of unity. In this work, we prove that the Weyl unitaries are universal in the sense that if u and v are any d×d unitary matrices such that u^p=v^p=1_d and uv=ζvu, for some ζ, then there exists a unital completely positive linear map Φ:〖 M〗_p (C ) →〖 M〗_d (C) such that Φ(u)=u and Φ(v)=v. Also, we show that any two pairs of p-th order unitary matrices (not just the Weyl unitaries) satisfying the commutation relation are completely order equivalent. However, we show in this work that the analogous result does not hold for triples of p-th order unitary matrices satisfying the Weyl commutation relation. In conclusion, we show that the Weyl matrices are extremal in their matrix range, using recent ideas from noncommutative convexity theory.
Open Access
University of Regina Community Authors 2015-2016
(2016) Bates-Hardy, Courtney; Benning, Sheri; Bundock, Chris; Chattopadhyay, Sutapa; Cronlund Anderson, Mark; Dahms, Tanya E. S.; Donovan, Darcy; Eaton, Emily; Elshakankiri, Maher; Farenick, Douglas; Fletcher, Amber J.; Kubik, Wendee; Garneau, David; Gidluck, Lynn; Gregory, David; Raymond-Seniuk, Christy; Patrick, Linda; Stephen, Tracy; Hillabold, Jean R. (pen name: Jean Roberta); Hurlbert, Margot; Diaz, Harry; Warren, James; James, Anne; Mather, Philippe; Rheault, Sylvain; McMullin, Brooks; McNeil, Barbara; Meagher, Karen; Montgomery, H. Monty; Ramsay, Christine; Robertson, Carmen; Steen, Sandra; Stonechild, Blair; Wolvengrey, Arok
Open Access
Weak Expectation Properties of C*-Algebras and Operator Systems
(Faculty of Graduate Studies and Research, University of Regina, 2013-11) Bhattacharya, Angshuman; Farenick, Douglas; Argerami, Martin; Floricel, Remus; Mobed, Nader; Samei, Ebrahim
The purpose of this dissertation is two fold. Firstly, we prove a permanence result involving C*-algebras with the weak expectation property. More speci cally, we show that if is an amenable action of a discrete group G on a unital C*-algebra A, then the crossed-product C*-algebra Ao G has the weak expectation property if and only if A has this property. Secondly, the concept of a relatively weakly injective pair of operator systems is introduced and studied, motivated by relative weak injectivity in the C*-algebra category. E. Kirchberg [14] proved that the C*-algebra C (F1) of the free group F1 on countably many generators characterizes relative weak injectivity for pairs of C*-algebras by means of the maximal tensor product. One of the main results in the latter part of this thesis is to show that C (F1) also characterizes relative weak injectivity in the operator system category. A key tool is the theory of operator system tensor products [12, 13].