We study positive solutions to nonlinear elliptic systems of the form: \begin{eqnarray*} -\Delta u =\lambda f(v) \mbox{ in }\Omega\\-\Delta v =\lambda g(u) \mbox{ in }\Omega\\\quad~~ u=0=v \mbox{ on }\partial\Omega \end{eqnarray*} where $\Delta u$ is the Laplacian of $u$, $\lambda$ is a positive parameter and $\Omega$ is a bounded domain in $R^n$ with smooth boundary $\partial\Omega$. In particular, we will analyze the combined effects of the nonlinearities on the existence and multiplicity of positive solutions. We also study systems with multiparameters and stronger coupling. We extend our results to $p$-$q$-Laplacian systems and to $n\times n$ systems. We mainly use sub- and super-solutions to prove our results.

We study positive solutions to classes of nonlinear elliptic singular problems of the form: -Δpu = λ g(u) uα in Ω u = 0 on δΩ where Ω is a bounded domain in ℝN, N ≥ 1 with smooth boundary δΩ, &lambda¸ is a positive parameter, α ∈(0; 1), Δpu := div(⌊∇u⌋p-2 ∇u); p > 1 is the p-Laplacian operator, and g is a smooth function. Such elliptic problems naturally arise in the study of steady state reaction diffusion processes. In particular, we will be interested in the challenging new class of problems when g(0) < 0 (hence lims→0+g(s) sα = - ∞ which we refer to as infinite semipositone problems. Our focus is on existence results. We obtain results for the single equation case as well as to the case of systems. We use the method of sub-super solutions to prove our results. The results in this dissertation provide a solid foundation for the analysis of such infinite semipositone problems.

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In this thesis, we reexamine the classical problem of video super-resolution, with an aim to reproduce fine edge/texture details of acquired digital videos. In general, the video super-resolution reconstruction is an ill-posed inverse problem, because of an insufficient number of observations from registered low-resolution video frames. To stabilize the problem and make its solution more accurate, we develop two video super-resolution techniques: 1) a 2D autoregressive modeling and interpolation technique for video super-resolution reconstruction, with model parameters estimated from multiple registered low-resolution frames; 2) the use of image model as a regularization term to improve the performance of the traditional video super-resolution algorithm. We further investigate the interactions of various unknown variables involved in video super-resolution reconstruction, including motion parameters, high-resolution pixel intensities and the parameters of the image model used for regularization. We succeed in developing a joint estimation technique that infers these unknowns simultaneously to achieve statistical consistency among them. / Thesis / Master of Applied Science (MASc)

Super Duplex Stainless Steels (SDSS) are widely used in offshore oil and gas industrial components. They are dual phase materials consisting of ferrite and austenite in similar ratios with high contents of chromium and presence of molybdenum. This combination of microstructure and chemical composition results in enhanced mechanical strength and corrosion resistance. However, this material has poor machinability, exhibiting the following characteristics: (i) tendency to strain-harden; (ii) extreme adhesive behaviour; and (iii) high cutting temperatures. These circumstances not only result in high tool wear rates, but also lead to poor surface integrity due to the work hardening effect, high roughness and tensile residual stress. To minimize these detrimental effects, PVD coating technologies have been widely applied to cutting tools due to their tribological properties exhibited during cutting, which reduce friction and diminish heat. In this work, three different PVD coatings were tested during the turning of super duplex stainless steel of grade UNS S32750. In addition to the tool performance, surface integrity was assessed by surface texture analysis, residual stresses and hardness profile. The electrochemical behaviour of the machined surface was evaluated by potentiodynamic anodic polarization measurements. Stress cracking corrosion (SCC) tests were also performed. Results indicate a relationship between the tool performance and surface electrochemical behaviour, where the tool with best cutting performance, AlTiN, also presented the best electrochemical behaviour. Stress cracking corrosion was found to be associated with residual stresses on the workpiece, among the three tested PVD coated tools the AlCrN/TiSiN showed lowest tensile residual stresses and lowest SCC susceptibility. The surface generated by AlTiN coated tool presented the highest levels of tensile residual stresses, resulting in a higher SCC susceptibility. / Thesis / Master of Applied Science (MASc)

The strain energy absorption of shape memory alloy (SMA) bars and beams under tension and bending loading was studied. A theoretical model is presented that can give quantitative relations between the martensite fraction, the applied load, and the strain energy absorbed in the shape memory alloy (SMA). It was found analytically that the super-elastic SMA demonstrates a high strain energy absorption capability. The closed- form solution of the strain energy absorption capability of SMA bars is a simple and useful tool in the design of energy absorption applications of super-elastic SMA. The nonlinear equations for SMA hybrid composite plates, which can be used for low velocity impact or quasi-static contact loading, are derived. The governing equations include the transverse shear deformation to the first-order, large deformation of the plates, and SMA/epoxy lamina. The equations are derived in the general form with general boundary conditions and general stack of angle ply. The equations can be simplified to special forms in the specific applications. A theoretical study of the impact force and the strain energy absorption of an SMA/graphite/epoxy composite beam under a low-velocity impact has been performed. The contact deformation, the global bending deformation, the transverse shear deformation, and the martensitic phase transformation of the super-elastic SMA fibers are studied. The energy absorbed by the SMA hybrid composite is calculated for each task of the absorption mechanisms: contact deformation, global bending deformation, and The analysis methods and models developed in this dissertation are the first reported research in modeling SMA composite under low velocity impact and quasi-static loading. The models and methods developed here can be used for further study and design of SMA composites for low velocity impact or quasi-static loading in failure process. / Ph. D.