The theory of viscosity solutions was developed for certain types of nonlinear first-order and second-order partial differential equations. It has been particularly useful in describing the solutions of partial differential equations associated with deterministic and stochastic optimal control problems [16], [53]. In its classical formulation, see [16], the theory deals with solutions which are continuous functions. The concept of continuous viscosity solutions was further generalized in various ways to include discontinuous solutions with the definition of Ishii given in [71] playing a pivotal role. In this thesis we propose a new approach for the treatment of discontinuous solutions of first-order Hamilton-Jacobi equations, namely, by involving Hausdorff continuous interval valued functions. The advantages of the proposed approach are justified by demonstrating that the main ideas within the classical theory of continuous viscosity solutions can be extended almost unchanged to the wider space of Hausdorff continuous functions and the existing theory of discontinuous viscosity solutions is a particular case of that developed in this thesis in terms of Hausdorff continuous interval valued functions. Two approaches to numerical solutions for Hamilton-Jacobi equations are presented. The first one is a monotone scheme for Hamilton-Jacobi equations while the second is based on preserving total variation diminishing property for conservation laws. In the first approach, we couple the finite element method with the nonstandard finite difference method which is based on the Mickens’ rule of nonlocal approximation [9]. The scheme obtained in this way is unconditionally monotone. In the second approach, computationally simple implicit schemes are derived by using nonlocal approximation of nonlinear terms. Renormalization of the denominator of the discrete derivative is used for deriving explicit schemes of first or higher order. Unlike the standard explicit methods, the solutions of these schemes have diminishing total variation for any time step size. / Thesis (PhD (Mathematical Science))--University of Pretoria, 2007. / Mathematics and Applied Mathematics / unrestricted

Thirty years since the removal of lead from gasoline, lead still poses a health risk. Children are most at-risk for adverse health outcomes caused by lead toxicity due to both behavioural (e.g., hand-to-mouth behaviour) and physiological differences (i.e., increased intake of lead by body weight, higher uptake rate and a higher vulnerability to the effects of lead) compared to adults (Yeoh et al., 2009). As a result, governments must identify children that may be at-risk of lead poisoning and develop practical methods to mitigate lead exposure. Before a government can develop a policy to help mitigate exposure of lead for children, we need to understand the spatial distribution of lead within the city. A popular spatial model used within air pollution research may allow more accurate, and more localized predictions than the most common interpolation method, kriging. Land use regression (LUR) is a technique leveraging multiple predictor variables to help estimate the spatial distribution of the dependent variable. By using historical sources of lead, LUR can be used to model soil lead levels (SLL) with localized variation. Unfortunately, spurious relationships can be the basis of a LUR model, which may lead to an overfitted spatial model resulting in a model with little generalizability and questionable ability to estimate the dependent variable at unobserved locations. Ultimately, Empirical Bayesian Kriging may be the best option for soil contamination research due to its ability to provide a smoothed prediction surface and its dependence on the spatial structure of the data to provide estimations. The benefit to society and the return on investment (ROI) is often the justification for lead remediation. Gould (2009) estimates a $17 to $221 ROI for every dollar spent on lead hazard control. One of the main components of this estimate of ROI comes from the decrease in intelligence quotient (IQ) that a child may experience as a consequence of lead toxicity. There are three main ways that a decrease in IQ can negatively impact the economy, (i) lower potential lifetime earnings, (ii) reduced tax revenues, and (iii) higher spending on special education (Gould, 2009). Since IQ has such a significant role in the ROI estimates, chapter 3 seeks to achieve a greater understanding of the relationship between blood lead levels (BLLs) and IQ. The loss of IQ points for an increase in blood lead concentration proposed by Lanphear et al. (2005) and referenced by Gould (2009) is significantly higher than what we found in our meta-analysis. Thus, the projected ROI proposed by Gould (2009) may be much lower than previously calculated. In the final chapter, the cost associated with permanent lead abatement is investigated based on ROI projections as a case study in Hamilton, Ontario. We show that, in most cases, permanent lead remediation is far too expensive for a municipal government. Furthermore, the capital initially invested may not be distributed back into the local economy, as the ROI suggests. We suggest that municipal governments make decisions based on need, rather than basing remediation decisions on ROI projections. Furthermore, we recommend the use of hazard quotient maps to help justify lead remediation as a more accurate representation of potential lead toxicity, instead of only looking at SLL exceedances across the city. / Dissertation / Doctor of Philosophy (PhD)

Throughout the last several decades many techniques have been developed in establishing solutions to nonlinear partial differential equations (NPDE). These techniques are characterized by their limited reach in solving large classes of NPDE. This body of work will study the analysis of NPDE using two of the most ubiquitous techniques developed in the last century. In this body of work, the analysis and techniques herein are applied to unsolved physical problems in both the fields of variational approximations and inverse scattering transform. Additionally, a new technique for estimating the error of a variational approximation is established. Note that the material in chapter 2, "Quantitative Measurements of Variational Approximations" has recently been published. Variational problems have long been used to mathematically model physical systems. Their advantage has been the simplicity of the model as well as the ability to deduce information concerning the functional dependence of the system on various parameters embedded in the variational trial functions. However, the only method in use for estimating the error in a variational approximation has been to compare the variational result to the exact solution. In this work, it is demonstrated that one can computationally obtain estimates of the errors in a one-dimensional variational approximation, without any a priori knowledge of the exact solution. Additionally, this analysis can be done by using only linear techniques. The extension of this method to multidimensional problems is clearly possible, although one could expect that additional difficulties would arise. One condition for the existence of a localized soliton is that the propagation constant does not fall into the continuous spectrum of radiation modes. For a higher order dispersive systems, the linear dispersion relation exhibits a multiple branch structure. It could be the case that in a certain parameter region for which one of the components of the solution has oscillations (i.e., is in the continuous spectrum), there exists a discrete value of the propagation constant, k(ES), for which the oscillations have zero amplitude. The associated solution is referred to as an embedded soliton (ES). This work examines the ES solutions in a CHI(2):CHI(3), type II system. The method employed in searching for the ES solutions is a variational method recently developed by Kaup and Malomed [Phys. D 184, 153-61 (2003)] to locate ES solutions in a SHG system. The variational results are validated by numerical integration of the governing system. A model used for the 1-D longitudinal wave propagation in microstructured solids is a KdV-type equation with third and fifth order dispersions as well as first and third order nonlinearities. Recent work by Ilison and Salupere (2004) has identified certain types of soliton solutions in the aforementioned model. The present work expands the known family of soliton solutions in the model to include embedded solitons. The existence of embedded solitons with respect to the dispersion parameters is determined by a variational approximation. The variational results are validated with selected numerical solutions.

<p> Multivariate statistical analysis of the ratios of single family dwelling assessed value to current market value or sale price, has been used to evaluate the administration of the property tax assessment function in Hamilton, Ontario. The Hamilton assessment administration has been found to exhibit an unacceptable level of non-uniformity in the assessment of single family dwellings. It is shown that tax burdens will vary substantially within and between price classes of single family dwellings; thus, both horizontal and vertical inequities are attributed to the administration of the assessment function.</p> <p> The spatial distribution of property tax assessment inequities in Hamilton is presented. It has been shown that certain districts of the city are overassessed relative to others. The spatial pattern is distinct; the area proximate to the industrial waterfront and the central business district have assessment-sales ratios above the mean ratio determined for the city as a whole, while the peripheral or fringe areas of the city are generally characterized by lower assessment-sales ratios.</p> / Thesis / Master of Arts (MA)

This paper suggests a ghettoization of Lodging Homes has developed in Hamiton, Ontario. Through an examination of Residential Care Facilities (R.C.F.s), much can be learned about this phenomenon. Specifically, the processes which cause the ghetto to develop, and its impact on the environment. A model representing the processes which lead to the development of the ghetto clarifies this picture. As a result of deinstitutionalization, there are new client demands which are met by the public and private sectors. Any R.C.F. operator is limited by the physical and social infrastructures. The model suggests these factors combine to create the ghetto. The examination of the pre and post By-law periods in Hamilton provides a two stage test of the model. The findings of this study indicate the Lodging Home Ghetto is seen as a social problem by both government and local citizen groups. This opposition initiated the creation of Lodging Home legislation in Hamilton. However, the ghetto is still present. The client has a minor role in this process, while the social service and land use professionals impact is also not as significant as expected. Hence, problems associated with the use of social programs within a free market structure are apparent. / Thesis / Candidate in Philosophy

This thesis was submitted to the Faculty of Arts and Sciences in partial fulfillment of the requirements for the degree Bachelor of Science. / Field observation at seven outcrops and subsequent polished section studies reveal that lead and zinc mineralization in the Hamilton vicinity occurred after the lithification of the host carbonates. The availability of open structures and the presence of potential reducing agents are the main controlling factors of sulphide deposition, The absence of an igneous source demands an origin of the metals from the sedimentary country rocks. Analysis of 14 shales and 12 dolomites by atomic absorption for Cu, Pb, and Zn indicates that the argillaceous rocks are potential source of the heavy metals. Furthermore, a simple leaching experiment has demonstrated that concentrated brine (sodium chloride solution) can preferentially leach lead and zinc from the shales. A mode of origin involving preferential_leaching can then adequately explain the scarcity of copper minerals in the Mississippi Valley-Type ore. It is suggested that better controlled leaching experiments at various temperatures should be done in conjunction with more extensive field work to check the plausibility of such a theory. / Thesis / Bachelor of Science (BSc)

The purpose of the research is to observe and describe the ethnic composition of Hamilton, using 1986 census data. To identify salient spatial trends, two main types of analyses are employed: The Segregation Index and the Location Quotient. Elementary statistics, such as mean and percentages were also calculated. Eight ethnic groups and 88 census tracts were used. These techniques involved the use of 'ethnic origin' data, from Canada's 1986 census. This was the first enumeration that included multiple ethnic origins, previously only the single, paternal lineage was recorded. The reseach demonstrates that Hamilton's ethnic groups exhibit varing degrees of clustering. The city can therefore, be described as a cultural mosaic, with high concentrations of ethnic groups occurring throughout. Conclusions shall be reached as to the possible causes of the observed patterns and shall be related to the choices and contstraints of the particular ethnic groups involved. / Thesis / Candidate in Philosophy

<p> This research identifies the socio-economic, demographic and ethnic composition changes that have occurred within the City of Hamilton from 1961 to 1981. Within this twenty-year time period, Hamilton has undergone growth, downtown renewal and gentrification. A statistical analysis of census data shows the extent to which these changes are reflected in the recent evolution of the socio-demographic structure of Hamilton.</p> <p> A factor analysis was performed on the data for each of the census years: 1961, 1971 and 1981. The factor scores were mapped to show the spatial distribution of socio-economic status, family status and ethnic composition. In general, the distributions conformed to the classical sectoral, zonal, and multiple nuclei models. The results revealed two areas where changes on these dimensions were most significant: the inner city and the Hamilton mountain.</p> </p> Statistical tests confirmed these changes. A significant increase in socio-economic status for the inner city area was observed in the period (1971-1981) following the renewal and gentrification. The mountain area retained its high family status throughout the twenty year time period, and ethnic composition became more dispersed throughout the city.</p> / Thesis / Candidate in Philosophy

<p> Field observation at seven outcrops and subsequent polished section studies reveal that lead and zinc mineralization in the Hamilton vicinity occurred after the lithification of the host carbonates. The availability of open structures and the presence of potential reducing agents are the main controlling factors of sulphide deposition, The absence of an igneous source demands an origin of the metals from the sedimentary country rocks. Analysis of 14 shales and 12 dolomites by atomic absorption for Cu, Pb, and Zn indicates that the argillaceous rocks are potential source of the heavy metals. Furthermore, a simple leaching experiment has demonstrated that concentrated brine (sodium chloride solution) can preferentially leach lead and zinc from the shales. A mode of origin involving preferential.leaching can then adequately explain the scarcity of copper minerals in the Mississippi Valley-Type ore, It is suggested that better controlled leaching experiments at various temperatures should be done in conjunction with more extensive field work to check the plausibility of such a theory. </p> / Thesis / Bachelor of Science (BSc)

<p> Hamilton has been affected by waves of foreign immigrants moving into the city since the beginning of this century. The resulting admixture of ethnic groups has produced a varied spatial pattern which forms the focus of analysis in this paper. </p> <p> A review of the literature on various ethnic enclaves in different cities of the world serves to focus this ecological study of ethnic groups in Hamilton. </p> <p> The historical and cultural background of Hamilton and the distributional pattern of the major ethnic communities are described. The factors influencing the distribution of ethnic groups in the city are then examined by means of factor analysis and regression analysis of 1971 census data. </p> <p> The analysis shows well distributed homogenous ethnic enclaves varying in their spatial concentration. As a whole it may be described as a mosaic with a high concentration of ethnic groups in the northern part of the city, north~east and west end. </p> <p> The results confirm the importance of ethnic dimensions within the social and spatial structure of the city and the importance of socioeconomic status and housing characteristics in influencing the distribution of ethnic groups. </p> / Thesis / Master of Arts (MA)