Application of real and functional analysis to solve boundary value problems

This thesis is about using appropriate tools in functional analysis arid classical analysis to tackle the problem of existence and uniqueness of nonlinear partial differential equations. There being no unified strategy to deal with these equations, one approaches each equation with an appropriate method, depending on the characteristics of the equation. The correct setting of the problem in appropriate function spaces is the first important part on the road to the solution. Here, we choose the setting of Sobolev spaces. The second essential part is to choose the correct tool for each equation. In the first part of this thesis (Chapters 3 and 4) we consider a variety of nonlinear hyperbolic partial differential equations with mixed boundary and initial conditions. The methods of compactness and monotonicity are used to prove existence and uniqueness of the solution (Chapter 3). Finding a priori estimates is the main task in this analysis. For some types of nonlinearity, these estimates cannot be easily obtained, arid so these two methods cannot be applied directly. In this case, we first linearise the equation, using linear recurrence (Chapter 4). In the second part of the thesis (Chapter 5), by using an appropriate tool in functional analysis (the Sobolev Imbedding Theorem), we are able to improve previous results on a posteriori error estimates for the finite element method of lines applied to nonlinear parabolic equations. These estimates are crucial in the design of adaptive algorithms for the method, and previous analysis relies on, what we show to be, unnecessary assumptions which limit the application of the algorithms. Our analysis does not require these assumptions. In the last part of the thesis (Chapter 6), staying with the theme of choosing the most suitable tools, we show that using classical analysis in a proper way is in some cases sufficient to obtain considerable results. We study in this chapter nonexistence of positive solutions to Laplace's equation with nonlinear Neumann boundary condition. This problem arises when one wants to study the blow-up at finite time of the solution of the corresponding parabolic problem, which models the heating of a substance by radiation. We generalise known results which were obtained by using more abstract methods.

Focusing on solutions : the evaluation and development of individual university teachers

This paper outlines the rationale for a current research project that focuses on teaching development in individual university lecturers. The ways in which student evaluation of university teaching (SET) results are best used to bring about positive changes in teaching are discussed. It is argued that providing individualised evaluation feedback coupled with consultation to teachers is potentially valuable in terms of achieving positive and measurable impacts on teaching and learning at several levels within higher education systems.

Content based image retrieval using unclean positive examples

Conventional content-based image retrieval (CBIR) schemes employing relevance feedback may suffer from some problems in the practical applications. First, most ordinary users would like to complete their search in a single interaction especially on the web. Second, it is time consuming and difficult to label a lot of negative examples with sufficient variety. Third, ordinary users may introduce some noisy examples into the query. This correspondence explores solutions to a new issue that image retrieval using unclean positive examples. In the proposed scheme, multiple feature distances are combined to obtain image similarity using classification technology. To handle the noisy positive examples, a new two-step strategy is proposed by incorporating the methods of data cleaning and noise tolerant classifier. The extensive experiments carried out on two different real image collections validate the effectiveness of the proposed scheme.

Bubble-solid interactions in water and electrolyte solutions

Surface forces between an air bubble and a flat mica surface immersed in aqueous electrolyte solutions have been investigated using a modified surface force apparatus. An analysis of the deformation of the air bubble with respect to the mutual position of the bubble and the mica surface, the capillary pressure, and the disjoining pressure allows the air-liquid surface electrical potential to be determined. The experiments show that a long-range, double-layer repulsion acts between the mica (which is negatively charged) and an air bubble in water and in various electrolyte solutions at low concentration, thereby indicating that the air bubble surface is negatively charged. However, there is clear evidence that charge regulation occurs at the air-water interface to maintain a constant surface potential, and as a result of this, the charge at this interface changes from negative to positive as the bubble approaches the mica surface. Because of the attraction that arises as a result of the charge reversal, a finite force is required to separate the bubble from the mica, though the mica remains wetted by the aqueous phase. At the low concentrations investigated, the potential on the gas-liquid interface is independent of the electrolyte type within experimental uncertainty.

The electrochemical surface forces apparatus : the effect of surface roughness, electrostatic surface potentials, and anodic oxide growth on interaction forces, and friction between dissimilar surfaces in aqueous solutions

We present a newly designed electrochemical surface forces apparatus (EC-SFA) that allows control and measurement of surface potentials and interfacial electrochemical reactions with simultaneous measurement of normal interaction forces (with nN resolution), friction forces (with μN resolution), and distances (with Å resolution) between apposing surfaces. We describe three applications of the developed EC-SFA and discuss the wide-range of potential other applications. In particular, we describe measurements of (1) force–distance profiles between smooth and rough gold surfaces and apposing self-assembled monolayer-covered smooth mica surfaces; (2) the effective changing thickness of anodically growing oxide layers with Å-accuracy on rough and smooth surfaces; and (3) friction forces evolving at a metal–ceramic contact, all as a function of the applied electrochemical potential. Interaction forces between atomically smooth surfaces are well-described using DLVO theory and the Hogg–Healy–Fuerstenau approximation for electric double layer interactions between dissimilar surfaces, which unintuitively predicts the possibility of attractive double layer forces between dissimilar surfaces whose surface potentials have similar sign, and repulsive forces between surfaces whose surface potentials have opposite sign. Surface roughness of the gold electrodes leads to an additional exponentially repulsive force in the force–distance profiles that is qualitatively well described by an extended DLVO model that includes repulsive hydration and steric forces. Comparing the measured thickness of the anodic gold oxide layer and the charge consumed for generating this layer allowed the identification of its chemical structure as a hydrated Au(OH)3 phase formed at the gold surface at high positive potentials. The EC-SFA allows, for the first time, one to look at complex long-term transient effects of dynamic processes (e.g., relaxation times), which are also reflected in friction forces while tuning electrochemical surface potentials.

Existence and global asymptotic stability of positive periodic solution of delayed Cohen-Grossberg neural networks

In this paper, a class of periodic Cohen-Grossberg neural networks with discrete and distributed time-varying delays is considered. By an extension of the Lyapunov-Krasovskii functional method, a novel criterion for the existence and uniqueness and global asymptotic stability of positive periodic solution is derived in terms of M-matrix without any restriction on uniform positiveness of the amplification functions. Comparison and illustrative examples are given to illustrate the effectiveness of the obtained results. © 2014 Elsevier Inc. All rights reserved.