We study blowup of solutions of one-dimensional nonlinear heat equations (NLH). We consider two cases: a power nonlinearity and initial conditions having two equal absolute maxima and a polynomial nonlinearity and initial conditions having a single global maximum. We show in both cases that for a certain open set of initial conditions solutions of the NLH blow up in finite time and we find asymptotical behavior of blowup frofiles. In the first case the blowup occurs at two points while in the second case, at one point.

In their exploratory study Muniz & O’Guinn (2001) found three markers of a brand community: a sense of belonging, rituals and tradition and a sense of duty toward the community. Two of the three markers of community have been included in conceptual models on brand communities. However, the third marker (sense of duty) has not been implemented up to now. Hence, the objective of this thesis is to extend Bagozzi & Dholakia’s (2006) brand community model by incorporating the construct “sense of duty”. In this research, a conceptual model of brand communities is developed. Overall, the findings support the conceptual model. The results show that sense of duty is a decisive mediator of brand community behaviours and that sense of duty is divided into three distinct components: new member integration, product usage and member retention. Further, this research indicates that community-related behavioural intentions are not significantly related to purchase intentions.

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Under some conditions in real world, precise parameters and/or initial values of dynamic systems are hard to be determined. Fuzzy Differential Equation (FDE) is a powerful tool to model dynamical systems with the uncertainty of impreciseness. This thesis presents the first numerical solution for Fuzzy Differential Equations with multiple fuzzy parameters and initial Values (FDEPIV) problems. Previous approaches for solving the FDEs only focused on FDEs with single fuzzy condition. In this thesis, we applied the proper fuzzy arithmetic on Runge-Kutta method for solving the FDEPIV problems with multiple fuzzy parameters and initial conditions. Furthermore, comparing with directly applying the extension principle in solving FDEPIV, the complexity of the proposed method is much lower, and parallelization of the proposed algorithm is feasible. Numerical examples of the FDEPIV problems are presented to demonstrate the effectiveness of the proposed method.

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Under some conditions in real world, precise parameters and/or initial values of dynamic systems are hard to be determined. Fuzzy Differential Equation (FDE) is a powerful tool to model dynamical systems with the uncertainty of impreciseness. This thesis presents the first numerical solution for Fuzzy Differential Equations with multiple fuzzy parameters and initial Values (FDEPIV) problems. Previous approaches for solving the FDEs only focused on FDEs with single fuzzy condition. In this thesis, we applied the proper fuzzy arithmetic on Runge-Kutta method for solving the FDEPIV problems with multiple fuzzy parameters and initial conditions. Furthermore, comparing with directly applying the extension principle in solving FDEPIV, the complexity of the proposed method is much lower, and parallelization of the proposed algorithm is feasible. Numerical examples of the FDEPIV problems are presented to demonstrate the effectiveness of the proposed method.

In principle Quantum Mechanics allows the creation of macroscopic mass superposition states - so called "Schrödinger Cat States". This has not been confirmed experimentally largely due to the difficulty of isolating such states from environmental decoherence. It is of interest to create massive superpositions both in order to test Quantum Mechanics and to shed light on the elusive 'measurement problem'. This thesis presents the theoretical analysis of, and the initial experimental steps towards, an ambitious proposal to test the superposition principle of Quantum Mechanics at the 10^-12 kg-scale, approximately nine orders of magnitude more massive than any superposition observed to date. The experimental principle is that a small mirror mounted on a micro-mechanical oscillator (cantilever) forms one end of a high-finesse cavity in one arm of a Michelson interferometer and is coupled to a single photon by radiation pressure. The photon, in a superposition of each arm, and the cantilever evolve into a superposition involving two distinct locations of the cantilever. By observing the interference of the photon only, one can study the creation and decoherence of the combined state. Firstly, a detailed analysis of the experimental requirements is given based on (1) the need for sufficient momentum transfer from the photon to displace the micro-mirror/cantilever to a distinguishable degree, (2) the need to isolate the cantilever to avoid significant environmental decoherence, and (3) the need to have sufficient interferometric stability to perform the measurement. An iterative analysis was performed to optimise these to a set that is feasible with current technology. This demands: (1) cavity mirrors with a reflectivity of R ≥ 0.9999998 at visible wavelength, (2) a system temperature of ≤ 3mK, (3) a cantilever mechanical quality Q ≥ 10⁵ , (4) a vacuum with gas particle density of 1012/m3, (5) a relative position stability of the cavity mirrors of ≤ 10^-13 m/min, and (6) optical mirror switching to 50% for ≤ lμs. Whilst extremely demanding, all of these goals appear to be within reach of current technology. Secondly, initial experimental results are described: (1) the fabrication of a 10μm radius dielectric mirror designed for peak reflectivity R > 0.99997 and the attaching of this to an AFM-type cantilever of mechanical quality Q > 4 x 10⁴ ; (2) the alignment of a cavity of length 2.5cm involving this micro-mirror/cantilever at one end and the demonstration of a finesse of F > 1000 using two independent measurement techniques. The diffraction losses for the cavity are calculated numerically to be < 10^-6 . Other mechanisms limiting the finesse are investigated and the dominant one is determined to be accoustic noise which can be alleviated by placing the cavity into a vacuum. In addition, results demonstrating ultra-fast optical switching of high reflectivity mirrors are shown.

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In this thesis, an integral representation theorem is obtained for non-negative solutions of the heat equation on X = (//R)('n-1) x (0,(INFIN)) x (0,T) and their boundary behaviour is investigated by using the abstract Fatou-Naim-Doob theorem. The boundary behaviour of positive solutions of the equation Lu = 0 on Y = (//R)('n) x (0,T), where L is a uniformly parabolic second-order differential operator in divergence form is also studied. / In particular, the notion of semi-thinness is introduced for the corresponding potential theories on X and Y and relationships between fine, semi-fine and parabolic limits are obtained. / Results of Kemper specialised to X are obtained by means of fine convergence and a Carleson-type local Fatou theorem is obtained for solutions of Lu = 0 on a union of parabolic regions.