Cubature formulas, geometrical designs, reproducing kernels, and Markov operators

Cubature formulas and geometrical designs are described in terms of reproducing kernels for Hilbert spaces of functions on the one hand, and Markov operators associated to orthogonal group representations on the other hand. In this way, several known results for spheres in Euclidean spaces, involving cubature formulas for polynomial functions and spherical designs, are shown to generalize to large classes of finite measure spaces $(\Omega,\sigma)$ and appropriate spaces of functions inside $L^2(\Omega,\sigma)$. The last section points out how spherical designs are related to a class of reflection groups which are (in general dense) subgroups of orthogonal groups

Determination of an Ultimate Pit Limit Utilising Fractal Modelling to Optimise NPV

The speed and complexity of globalisation and reduction of natural resources on the one hand, and interests of large multinational corporations on the other, necessitates proper management of mineral resources and consumption. The need for scientific research and application of new methodologies and approaches to maximise Net Present Value (NPV) within mining operations is essential. In some cases, drill core logging in the field may result in an inadequate level of information and subsequent poor diagnosis of geological phenomenon which may undermine the delineation or separation of mineralised zones. This is because the interpretation of individual loggers is subjective. However, modelling based on logging data is absolutely essential to determine the architecture of an orebody including ore distribution and geomechanical features. For instance, ore grades, density and RQD values are not included in conventional geological models whilst variations in a mineral deposit are an obvious and salient feature. Given the problems mentioned above, a series of new mathematical methods have been developed, based on fractal modelling, which provide a more objective approach. These have been established and tested in a case study of the Kahang Cu-Mo porphyry deposit, central Iran. Recognition of different types of mineralised zone in an ore deposit is important for mine planning. As a result, it is felt that the most important outcome of this thesis is the development of an innovative approach to the delineation of major mineralised (supergene and hypogene) zones from ‘barren’ host rock. This is based on subsurface data and the utilisation of the Concentration-Volume (C-V) fractal model, proposed by Afzal et al. (2011), to optimise a Cu-Mo block model for better determination of an ultimate pit limit. Drawing on this, new approaches, referred to Density–Volume (D–V) and RQD-Volume (RQD-V) fractal modelling, have been developed and used to delineate rock characteristics in terms of density and RQD within the Kahang deposit (Yasrebi et al., 2013b; Yasrebi et al., 2014). From the results of this modelling, the density and RQD populations of rock types from the studied deposit showed a relationship between density and rock quality based on RQD values, which can be used to predict final pit slope. Finally, the study introduces a Present Value-Volume (PV-V) fractal model in order to identify an accurate excavation orientation with respect to economic principals and ore grades of all determined voxels within the obtained ultimate pit limit in order to achieve an earlier pay-back period.Institute of Materials, Minerals and Mining, the global network IOM3Cornish Institute of EngineersWhittle Consulting (Business Optimisation for the Mining Industry

Resultados espectrais relacionados com a estrutura dos grafos

Doutoramento em MatemáticaNesta tese são estabelecidas novas propriedades espectrais de grafos com estruturas específicas, como sejam os grafos separados em cliques e independentes e grafos duplamente separados em independentes, ou ainda grafos com conjuntos (κ,τ)-regulares. Alguns invariantes dos grafos separados em cliques e independentes são estudados, tendo como objectivo limitar o maior valor próprio do espectro Laplaciano sem sinal. A técnica do valor próprio é aplicada para obter alguns majorantes e minorantes do índice do espectro Laplaciano sem sinal dos grafos separados em cliques e independentes bem como sobre o índice dos grafos duplamente separados em independentes. São fornecidos alguns resultados computacionais de modo a obter uma melhor percepção da qualidade desses mesmos extremos. Estudamos igualmente os grafos com um conjunto (κ,τ)-regular que induz uma estrela complementar para um valor próprio não-principal $. Além disso, é mostrado que$=κ-τ. Usando uma abordagem baseada nos grafos estrela complementares construímos, em alguns casos, os respectivos grafos maximais. Uma caracterização dos grafos separados em cliques e independentes que envolve o índice e as entradas do vector principal é apresentada tal como um majorante do número da estabilidade dum grafo conexo.In this thesis new spectral properties of graphs with a specific structure (as split graphs, nested split and double split graphs as well as graphs with (κ,τ)-regular sets) are deduced. Some invariants of nested split graphs are studied in order to bound the largest eigenvalue of signless Laplacian spectra. The eigenvalue technique is applied to obtain some lower and upper bounds on the index of signless Laplacian spectra of nested split graphs as well as on the index of double nested graphs. Computational results are provided in order to gain a better insight of quality of these bounds. The graphs having a (κ,τ)-regular set which induces a star complement for a non-main eigenvalue $are studied. Furthermore, it is shown that$= κ-τ. By the star complement technique, in some cases, maximal graphs with desired properties are constructed. A spectral characterization of families of split graphs involving its index and the entries of the principal eigenvector is given as well as an upper bound on the stability number of a connected graph

Feature-based validation reasoning for intent-driven engineering design

Feature based modelling represents the future of CAD systems. However, operations such as modelling and editing can corrupt the validity of a feature-based model representation. Feature interactions are a consequence of feature operations and the existence of a number of features in the same model. Feature interaction affects not only the solid representation of the part, but also the functional intentions embedded within features. A technique is thus required to assess the integrity of a feature-based model from various perspectives, including the functional intentional one, and this technique must take into account the problems brought about by feature interactions and operations. The understanding, reasoning and resolution of invalid feature-based models requires an understanding of the feature interaction phenomena, as well as the characterisation of these functional intentions. A system capable of such assessment is called a feature-based representation validation system. This research studies feature interaction phenomena and feature-based designer's intents as a medium to achieve a feature-based representation validation system. [Continues.

Feature-Based Uncertainty Visualization

While uncertainty in scientific data attracts an increasing research interest in the visualization community, two critical issues remain insufficiently studied: (1) visualizing the impact of the uncertainty of a data set on its features and (2) interactively exploring 3D or large 2D data sets with uncertainties. In this study, a suite of feature-based techniques is developed to address these issues. First, a framework of feature-level uncertainty visualization is presented to study the uncertainty of the features in scalar and vector data. The uncertainty in the number and locations of features such as sinks or sources of vector fields are referred to as feature-level uncertainty while the uncertainty in the numerical values of the data is referred to as data-level uncertainty. The features of different ensemble members are indentified and correlated. The feature-level uncertainties are expressed as the transitions between corresponding features through new elliptical glyphs. Second, an interactive visualization tool for exploring scalar data with data-level and two types of feature-level uncertainties — contour-level and topology-level uncertainties — is developed. To avoid visual cluttering and occlusion, the uncertainty information is attached to a contour tree instead of being integrated with the visualization of the data. An efficient contour tree-based interface is designed to reduce users’ workload in viewing and analyzing complicated data with uncertainties and to facilitate a quick and accurate selection of prominent contours. This thesis advances the current uncertainty studies with an in-depth investigation of the feature-level uncertainties and an exploration of topology tools for effective and interactive uncertainty visualizations. With quantified representation and interactive capability, feature-based visualization helps people gain new insights into the uncertainties of their data, especially the uncertainties of extracted features which otherwise would remain unknown with the visualization of only data-level uncertainties