This thesis consists of three papers on the following topics in functional analysis and probability theory: Riesz bases and frames, weakly stationary stochastic processes and analysis of set-valued stochastic processes. In the first paper we investigate Uniformly Bounded Linearly Stationary stochastic processes from the point of view of the theory of Riesz bases. By regarding these stochastic processes as generalized Riesz bases we are able to gain some new insight into there structure. Special attention is paid to regular UBLS processes as well as perturbations of weakly stationary processes. An infinite sequence of subspaces of a Hilbert space is called regular if it is decreasing and zero is the only element in its intersection. In the second paper we ask for conditions under which the regularity of a sequence of subspaces is preserved when the sequence undergoes a deformation by a linear and bounded operator. Linear, bounded and surjective operators are closely linked with frames and we also investigate when a frame is a regular sequence of vectors. A multiprocess is a stochastic process whose values are compact sets. As generalizations of the class of subharmonic processes and the class of subholomorphic processesas introduced by Thomas Ransford, in the third paper of this thesis we introduce the general notions of a gauge of processes and a multigauge of multiprocesses. Compositions of multiprocesses with multifunctions are discussed and the boundary crossing property, related to the intermediate-value property, is investigated for general multiprocesses. Time changes of multiprocesses are investigated in the environment of multigauges and we give a multiprocess version of the Dambis-Dubins-Schwarz Theorem.

We study two classes of stochastic systems, the limited processor sharing system and the multi-server system. They share the common feature that multiple jobs/customers are being processed simultaneously, which makes the study of them intrinsically difficult. In the limited processor sharing system, a limited number of jobs can equally share a single server, and the excess ones wait in a first-in-first-out buffer. The model is mainly motivated by computer related applications, such as database servers and packet transmission over the Internet. This model is studied in the first part of the thesis. The multi-server queue is mainly motivated by call centers, where each customer is handled by an agent. The number of customers being served at any time is limited by number of agents employed. Customers who can not be served upon arrival wait in a first-in-first-out buffer. This model is studied in the second part of the thesis.

Stochastic differential inclusions can be considered as a generalisation of stochastic differential equations. In particular a multivalued mapping describes the set of equations, in which a solution has to be found. This paper presents an existence result for a special parabolic stochastic inclusion. The proof is based on the method of upper and lower solutions. In the deterministic case this method was effectively introduced by S. Carl.

This thesis proposes a novel directed hypergraph based model for any database. We introduce the notion of association rules for multi-valued attributes, which is an adaptation of the definition of quantitative association rules known in the literature. The association rules for multi-valued attributes are integrated in building the directed hypergraph model. This model allows to capture attribute-level associations and their strength. Basing on this model, we provide association-based similarity notions between any two attributes and present a method for finding clusters of similar attributes. We then propose algorithms to identify a subset of attributes known as a leading indicator that influences the values of almost all other attributes. Finally, we present an association-based classifier that can be used to predict values of attributes. We demonstrate the effectiveness of our proposed model, notions, algorithms, and classifier through experiments on a financial time-series data set (S&P 500).

We explore the connection between quantum error correction and quantum cryptography through the notion of conjugate (or complementary) channels. This connection is at the level of subspaces and operator subsystems; if we use a more general form of subsystem, the link between the two topics breaks down. We explore both the subspace and subsystem settings. Error correction arises as a means of addressing the issue of the introduction of noise to a message being sent from one party to another. Noise also plays a role in quantum measurement theory: If one wishes to measure a system that is in a particular state via a measurement apparatus, one can first act upon the system by a quantum channel, which can be thought of as a noise source, and then measure the resulting system using a different measurement apparatus. Such a setup amounts to the introduction of noise to the measurement process, yet has the advantage of preserving the measurement statistics. Preprocessing by a quantum channel leads to the partial order "cleaner than" on quantum probability measures. Other meaningful partial orders on quantum probability measures exist, and we shall investigate that of cleanness as well as that of absolute continuity. Lastly, we investigate partial orders on vectors corresponding to quantum states; such partial orders, namely majorization and trumping, have been linked to entanglement theory. We characterize trumping first by means of yet another partial order, power majorization, which gives rise to a family of examples. We then characterize trumping through the complete monotonicity of certain Dirichlet polynomials corresponding to the states in question. This not only generalizes a recent characterization of trumping, but the use of such mathematical objects simpli es the derivation of the result. / The Natural Sciences and Engineering Research Council of Canada (NSERC)

In recent years ternary reversible logic has caught the attention of researchers because of its enormous potential in different fields, in particular quantum computing. It is desirable that any future reversible technology should be fault tolerant and have low power consumption; hence developing testing techniques in this area is of great importance. In this work we propose a design for an online testable ternary reversible circuit. The proposed design can implement almost all of the ternary logic operations and is also capable of testing the reversible ternary network in real time (online). The error detection unit is also constructed in a reversible manner, which results in an overall circuit which meets the requirements of reversible computing. We have also proposed an upgrade of the initial design to make the design more optimized. Several ternary benchmark circuits have been implemented using the proposed approaches. The number of gates required to implement the benchmarks for each approach have also been compared. To our knowledge this is the first such circuit in ternary with integrated online testability feature. / xii, 92 leaves : ill. ; 29 cm

The main contribution of this thesis, is to define similarity measures between two images with the main focus on content-based image retrieval (CBIR). Each image is considered as a set of visual elements that can be described with a set of visual descriptions (features). The similarity between images is then defined as the nearness between sets of elements based on a tolerance and a fuzzy tolerance relation. A tolerance relation is used to describe the approximate nature of the visual perception. A fuzzy tolerance relation is adopted to eliminate the need for a sharp threshold and hence model the gradual changes in perception of similarities. Three real valued similarity measures as well as a fuzzy valued similarity measure are proposed. All of the methods are then used in two CBIR experiments and the results are compared with classical measures of distance (namely, Kantorovich, Hausdorff and Mahalanobis). The results are compared with other published research papers. An important advantage of the proposed methods is shown to be their effectiveness in an unsupervised setting with no prior information. Eighteen different features (based on color, texture and edge) are used in all the experiments. A feature selection algorithm is also used to train the system in choosing a suboptimal set of visual features.

A composition algebra is a non-zero algebra endowed with a strictly non-degenerate, multiplicative quadratic form. Finite-dimensional composition algebras exist only in dimension 1, 2, 4 and 8 and are in general not associative or unital. Over the real numbers, such algebras are division algebras if and only if they are absolute valued, i.e. equipped with a multiplicative norm. The problem of classifying all absolute valued algebras and, more generally, all composition algebras of finite dimension remains unsolved. In dimension eight, this is related to the triality phenomenon. We approach this problem using a categorical language and tools from representation theory and the theory of algebraic groups. We begin by considering the category of absolute valued algebras of dimension at most four. In Paper I we determine the morphisms of this category completely, and describe their irreducibility and behaviour under the actions of the automorphism groups of the algebras. We then consider the category of eight-dimensional absolute valued algebras, for which we provide a description in Paper II in terms of a group action involving triality. Then we establish general criteria for subcategories of group action groupoids to be full, and applying this to the present setting, we obtain hitherto unstudied subcategories determined by reflections. The reflection approach is further systematized in Paper III, where we obtain a coproduct decomposition of the category of finite-dimensional absolute valued algebras into blocks, for several of which the classification problem does not involve triality. We study these in detail, reducing the problem to that of certain group actions, which we express geometrically. In Paper IV, we use representation theory of Lie algebras to completely classify all finite-dimensional absolute valued algebras having a non-abelian derivation algebra. Introducing the notion of quasi-descriptions, we reduce the problem to the study of actions of rotation groups on products of spheres. We conclude by considering composition algebras over arbitrary fields of characteristic not two in Paper V. We establish an equivalence of categories between the category of eight-dimensional composition algebras with a given quadratic form and a groupoid arising from a group action on certain pairs of outer automorphisms of affine group schemes

We study multiple valued functions with values in a Hilbert space. We introduce a possibledefinition of Sobolev spaces and the rightful notion of p energy. We prove the existence of pminimizers. Then we consider two-valued real functions of two variables which are stationarywith respect to both domain and range transformations. We prove their local Lipschitzcontinuity and use it to establish strong convergence in W1,2 to their unique blow-up at anypoint. We claim that the branch set of any such function consists of finitely many real analyticcurves meeting at nod points with equal angles. We also provide an example showing thatstationarity with respect to domain transformations only does not imply continuity.In a second part, we prove that there does not exist a uniformly continuous retractionfrom the space of continuous vector fields onto the subspace of vector fields whose divergencevanishes in the distributional sense. We then generalise this result using the concept of mcharges on any subset X _ Rn satisfying a mild geometric condition, there is no uniformlycontinuous representation operator for mcharges in X.

The main contribution of this thesis, is to define similarity measures between two images with the main focus on content-based image retrieval (CBIR). Each image is considered as a set of visual elements that can be described with a set of visual descriptions (features). The similarity between images is then defined as the nearness between sets of elements based on a tolerance and a fuzzy tolerance relation. A tolerance relation is used to describe the approximate nature of the visual perception. A fuzzy tolerance relation is adopted to eliminate the need for a sharp threshold and hence model the gradual changes in perception of similarities. Three real valued similarity measures as well as a fuzzy valued similarity measure are proposed. All of the methods are then used in two CBIR experiments and the results are compared with classical measures of distance (namely, Kantorovich, Hausdorff and Mahalanobis). The results are compared with other published research papers. An important advantage of the proposed methods is shown to be their effectiveness in an unsupervised setting with no prior information. Eighteen different features (based on color, texture and edge) are used in all the experiments. A feature selection algorithm is also used to train the system in choosing a suboptimal set of visual features.