C*-algebras from tilings are of particular interest. In 1998 J. Anderson and I. Putnam introduced a C*-algebra obtained from a substitution tiling that is viewed today as a standard invariant for this tilings. In this thesis we introduce another C*-algebra associated to a substitution tiling. We expect this C*-algebra to be in some sense a dual C*-algebra to the one introduced by Anderson and Putnam. but we were not able to make a precise statement. In our effort to characterize this new C*-algebras we prove that they are simple and can be constructed as an inductive limit of recursive subhomogenous algebras. We finish with K-theory computations for a number of examples.

This thesis studies the linear model, estimators of the treatment means, and opti¬mality criteria for designs and analysis of spatially arranged experiments. Four types of commonly used spatial correlation structures are discussed, and a neighbourhood of covariance matrices is investigated. Various properties about the neighbourhood are explored. When the covariance matrix of the error process is unknown, but be-longs to a neighbourhood of a covariance matrix, a modified generalized least squares estimator (GLSE) is proposed. This estimator seems more efficient than the ordinary least squares estimator in many practical applications. We also propose a criterion to find minimax designs that are efficient for a neighbourhood of correlations. When the number of plots is small, minimax designs can be computed exactly. When the number of plots is large, a simulated annealing algorithm is applied to find minimax or near minimax designs. Minimax designs for the least squares and generalized least squares estimators are compared in details. In general, we recommend using GLSE and the minimax design based on GLSE.

The invariant subspace problem is the long-standing question whether every operator on a Hilbert space of dimension greater than one has a non-trivial invariant subspace. Although the problem is unsolved in the Hilbert space case, there are counter-examples for operators acting on certain well-known non-reflexive Banach spaces. These counter-examples are constructed by considering a single orbit and then extending continuously to a hounded linear map on the entire space. Based on this process, we introduce an operator which has properties closely linked with an orbit. We call this operator the orbit operator. In the first part of the thesis, examples and basic properties of the orbit operator are discussed. Next, properties linking invariant subspaces to properties of the orbit operator are presented. Topics include the kernel and range of the orbit operator, compact operators, dilation theory, and Rotas theorem. Finally, we extend results obtained for strict contractions to contractions.

We present a class of multi-lane traffic models of Vlasov-Fokker-Planck type incorporating non-local and time-delayed braking/acceleration, diffu¬sion and lane changing terms whose dependencies are based on empirical guidelines. By investigating the spatially homogeneous case with non-zero passing probability incorporated in the braking term. we are left with the drift diffusion equation. which leads to a multi-valued fundamental diagram. As a novelty of this thesis. we find out that the monotonicity of the quotient between the braking/acceleration and the diffusion term in average speed guarantees the single-valued fundamental diagram. We study the large time behavior of the time-dependent kinetic density by convex entropy methods based on [3]. With a positive "residual" diffusion, convergence results remain with fewer assumptions. Two simplified examples are studied to illustrate the application of entropy methods.

Singly-periodic (SP) and doubly-periodic (DP) graphs arc infinite graphs which have translational symmetries in one and two dimensions, respectively. The problem of counting the number of connected components in such graphs is investigated. A method for determining whether or not an SP graph is k-colourable for a given positive integer k is given, and the question of deciding k-colourability of DP graphs is discussed. Colourings of SP and DP graphs can themselves be either periodic or aperiodic, and properties which determine the symmetries of their colourings arc also explored.

The principal-agent problem is an important model in the field of Economics of Information. In this thesis we study only a particular type of principal-agent problem which is called moral hazard model and by the principal-agent problem we mean it is moral hazard model. The moral hazard model actually belongs to the class of bilevel programming problems in Mathematics. In Economics. the first order approach is used to reduce the principal-agent problem to a single level optimization problem. However, this approach is only valid under some strong conditions. Moreover the approach can only be used under the assumption that the optimal action of the principal-agent problem and its relaxed problem appears only at an interior point. In this thesis, we consider a new relaxed problem. Under more general assumptions. we can solve the principal-agent problem without restricting the optimal action of the agent to be in the interior.

There are various kinds of noise that affects DNA array data and it is a challenge to obtain high quality data from an experiment. Background correction is a first step for DNA array data processing; however, current methods for background correction have disadvantages. In this thesis, I propose a mixture model to facilitate the background correction for DNA array data generated by ImaGene. The corresponding software is also provided.

In this thesis, we study cluster analysis of time series and develop a new mixture model which is effective for such analysis. Our study is motivated by a real life problem of clustering time series of electricity load (demand) for BC Hydro customers. BC Hydro collects electricity load data for selected customers for the purpose of grouping customers into homogeneous classes in terms of the load. Such homogeneous classes or clusters are useful for rate setting and long term generation capacity planning. The BC Hydro data set that we use in this thesis contains 923 load series representing 923 BC Hydro customers. Each load series consists of repeated hourly load measurements over a one year period and thus is a long time series. There are a number of clustering methods in the literature for clustering general multivariate data but these are not effective for clustering such long time series. This is because time series such as the BC Hydro customer’s load series typically have high dimensions and special covariance structures. Existing clustering methods are not designed to accommodate these special characteristics. The contributions of this thesis are the following: We first develop a mixture model based clustering method for time series which cannot only handle their high dimensions but also makes effective use of their special covariance structures. Our method is based on the random effects mixture model, a mixture model which we develop specifically for time series. We devise a special EM algorithm based on the AECM algorithm of Meng and van Dyk (1997) to handle the computation of the random effects mixture model. Once the model is computed, we assign individual time series to clusters by their posterior probabilities of belonging to the components of the mixture model. Then to demonstrate the application of our method, we apply it to analyse BC Hydro data. We obtain a new clustering of the BC Hydro sample which is superior to the existing clustering in terms of relevance and interpretability.

This thesis introduces a new strategy of defending the vertices of a graph - secure paired domination, where guards are required to be paired and, when a vertex is attacked, one or two guards move to defend the attacked vertex, while keeping the graph dominated and the guards paired after the move. We propose nine possible definitions of secure paired domination, compare and contrast each with the others, and obtain properties and inequalities of the secure paired domination (SPD) numbers associated with the definitions. Based on each of the nine definitions, the SPD numbers of five types of special graphs, namely paths, cycles, spiders, ladders and grid graphs, are studied. We then compare the SPD number of an arbitrary isolate-free graph to various other parameters such as clique partition number, independence number, vertex- covering number, secure domination number and paired domination number. We establish that, for any graph without isolated vertices, its SPD number does not exceed twice the value of any of its other parameters mentioned above. Also, we give classes of trees for which some of the bounds are achieved. As conclusion, some open problems and directions for further studies regarding secure paired domination are listed.

We present results evolving from established connections between zeta functions and different systems of polynomials, particularly the Riemann and Hurwitz zeta functions and the Bernoulli and Euler polynomials. In particular we develop certain results related to Apostol's deformation of the Bernoulli polynomials and obtain identities of Carlitz by a novel approach using generating functions instead of difference equations. In the last two chapters we work out new rapidly convergent series expansions of the Riemann zeta function, find coefficient symmetries of a polynomial sequence obtained from the cyclotomic polynomials by a linear fractional transformation of argument and obtain an expression for the constant term in an identity involving the gamma function.