Thesis (Ph.D.)--Mississippi State University. Department of Physics and Astronomy. / Title from title screen. Includes bibliographical references.

Understanding joint and causal relationships among multiple endogenous variables has been of much interest to researchers in the field of activity and travel behavior modeling. Structural equation models have been widely developed for modeling and analyzing the causal relationships among travel time, activity duration, car ownership, trip frequency and activity frequency. In the model, travel time and activity duration are treated as continuous variables, while car ownership, trip frequency and activity frequency as ordered discrete variables. However, many endogenous variables of interest in travel behavior are not continuous or ordered discrete but unordered discrete in nature, such as mode choice, destination choice, trip chaining pattern and time-of-day choice (it can be classified into a few categories such as AM peak, midday, PM peak and off-peak). A modeling methodology with involvement of unordered discrete variables is highly desired for better understanding the causal relationships among these variables. Under this background, the proposed dissertation study will be dedicated into seeking an appropriate modeling methodology which aids in identifying the causal relationships among activity and travel variables including unordered discrete variables. In this dissertation, the proposed modeling methodologies are applied for modeling the causal relationship between three pairs of endogenous variables: trip chaining pattern vs. mode choice, activity timing vs. duration and trip departure time vs.mode choice. The data used for modeling analysis is extracted from Swiss Travel Microcensus 2000. Such models provide us with rigorous criteria in selecting a reasonable application sequence of sub-models in the activity-based travel demand model system.

This thesis studies the approximation of solutions to partial differential equations (PDEs) over domains discretized by the dual of a simplicial mesh. While `primal' methods associate degrees of freedom (DoFs) of the solution with specific geometrical entities of a simplicial mesh (simplex vertices, edges, faces, etc.), a `dual discretization method' associates DoFs with the geometric duals of these objects. In a tetrahedral mesh, for instance, a primal method might assign DoFs to edges of tetrahedra while a dual method for the same problem would assign DoFs to edges connecting circumcenters of adjacent tetrahedra. Dual discretization methods have been proposed for various specific PDE problems, especially in the context of electromagnetics, but have not been analyzed using the full toolkit of modern numerical analysis as is considered here. The recent and still-developing theories of finite element exterior calculus (FEEC) and discrete exterior calculus (DEC) are shown to be essential in understanding the feasibility of dual methods. These theories treat the solutions of continuous PDEs as differential forms which are then discretized as cochains (vectors of DoFs) over a mesh. While the language of DEC is ideal for describing dual methods in a straightforward fashion, the results of FEEC are required for proving convergence results. Our results about dual methods are focused on two types of stability associated with PDE solvers: discretization and numerical. Discretization stability analyzes the convergence of the approximate solution from the discrete method to the continuous solution of the PDE as the maximum size of a mesh element goes to zero. Numerical stability analyzes the potential roundoff errors accrued when computing an approximate solution. We show that dual methods can attain the same approximation power with regard to discretization stability as primal methods and may, in some circumstances, offer improved numerical stability properties. A lengthier exposition of the approach and a detailed description of our results is given in the first chapter of the thesis. / text

Title: Generalized stable distributions and their applications Author: Mgr. Lenka Slámová, MSc. Department: Department of probability and mathematical statistics Supervisor: Prof. Lev Klebanov, DrSc. Abstract: This thesis deals with different generalizations of the strict stability property with a particular focus on discrete distributions possessing some form of stability property. Three possible definitions of discrete stability are introduced, followed by a study of some particular cases of discrete stable distributions and their properties. The random normalization used in the definition of discrete stability is applicable for continuous random variables as well. A new concept of casual stability is introduced by replacing classical normalization in the definition of stability by random normalization. Examples of casual stable distributions, both discrete and continuous, are given. Discrete stable distributions can be applied in discrete models that exhibit heavy tails. Applications of discrete stable distributions on rating of scientific work and financial time series modelling are presented. A method of parameter estimation for discrete stable family is also introduced. Keywords: discrete stable distribution, casual stability, discrete approximation of stable distribution

The South African National Defence Force (SANDF) currently requires a system of systems simulation capability for supporting the different phases of a Ground Based Air Defence System (GBADS) acquisition program. A non-distributed, fast-as-possible simulator and its architectural predecessors developed by the Council for Scientific and Industrial Research (CSIR) was able to provide the required capability during the concept and definition phases of the acquisition life cycle. The non-distributed simulator implements a 100Hz logical time Discrete Time System Specification (DTSS) in support of the existing models. However, real-time simulation execution has become a prioritised requirement to support the development phase of the acquisition life cycle. This dissertation is about the ongoing migration of the non-distributed simulator to a practical simulation architecture that supports the real-time requirement. The simulator simulates a synthetic environment inhabited by interacting GBAD systems and hostile airborne targets. The non-distributed simulator was parallelised across multiple Commod- ity Off the Shelf (COTS) PC nodes connected by a commercial Gigabit Eth- ernet infrastructure. Since model reuse was important for cost effectiveness, it was decided to reuse all the existing models, by retaining their 100Hz logical time DTSSs. The large scale and event-based High Level Architecture (HLA), an IEEE standard for large-scale distributed simulation interoperability, had been identified as the most suitable distribution and parallelisation technology. However, two categories of risks in directly migrating to the HLA were iden- tified. The choice was made, with motivations, to mitigate the identified risks by developing a specialised custom distributed architecture. In this dissertation, the custom discrete time, distributed, peer-to-peer, message-passing architecture that has been built by the author in support of the parallelised simulator requirements, is described and analysed. It reports on empirical studies in regard to performance and flexibility. The architecture is shown to be a suitable and cost effective distributed simulator architecture for supporting a speed-up of three to four times through parallelisation of the 100 Hz logical time DTSS. This distributed architecture is currently in use and working as expected, but results in a parallelisation speed-up ceiling irrespective of the number of distributed processors. In addition, a hybrid discrete-time/discrete-event modelling approach and simulator is proposed that lowers the distributed communication and time synchronisation overhead—to improve on the scalability of the discrete time simulator—while still economically reusing the existing models. The pro- posed hybrid architecture was implemented and its real-time performance analysed. The hybrid architecture is found to support a parallelisation speed- up that is not bounded, but linearly related to the number of distributed pro- cessors up to at least the 11 processing nodes available for experimentation. / Dissertation (MSc)--University of Pretoria, 2009. / Computer Science / unrestricted

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The master's thesis is concerned with digital image denoising methods. The theoretical part explains some elementary terms related to image processing, image noise, categorization of noise and quality determining criteria of denoising process. There are also particular denoising methods described, mentioning their advantages and disadvantages in this paper. The practical part deals with an implementation of the selected denoising methods in a Java, in the environment of application RapidMiner. In conclusion, the results obtained by different methods are compared.

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This thesis concerns the mathematics and application of various methods for approximating matrices, with a particular eye towards the role that such methods play in image compression. An image is stored as a matrix of values with each entry containing a value recording the intensity of a corresponding pixel, so image compression is essentially equivalent to matrix approximation. First, we look at the singular value decomposition, one of the central tools for analyzing a matrix. We show that, in a sense, the singular value decomposition is the best low-rank approximation of any matrix. However, the singular value decomposition has some serious shortcomings as an approximation method in the context of digital images. The second method we consider is the discrete Fourier transform, which does not require the storage of basis vectors (unlike the SVD). We describe the fast Fourier transform, which is a remarkably efficient method for computing the discrete cosine transform, and how we can use this method to reduce the information in a matrix. Finally, we look at the discrete cosine transform, which reduces the complexity of the calculation further by restricting to a real basis. We also look at how we can apply a filter to adjust the relative importance of the data encoded by the discrete cosine transform prior to compression. In addition, we developed code implementing the ideas explored in the thesis and demonstrating examples.

[pt] A Geometria Diferencial Discreta tem por objetivo desenvolver uma teoria discreta que respeite os aspectos fundamentais da teoria suave. Com isto em mente, são apresentados incialmente resultados da teoria suave da Geometria Afim que terão suas versões discretas tratadas a posteriori. O primeiro objetivo deste trabalho é construir uma estrutura afim discreta para as redes assintóticas definidas no espaço tridimensional, com métrica de Blaschke indefinida e parâmetros assintóticos. Com este intuito, são definidos um campo conormal, que satisfaz as equações de Lelieuvre e está associado a um parâmetro real, e um normal afim que define a forma cúbica da rede e torna a estrutura bem definida. Esta estrutura permite, por exemplo, o estudo das superfícies regradas, com ênfase nas esferas afins impróprias. Além disso, propõe-se uma definição para as singularidades no caso das esferas afins impróprias discretas a partir da construção centrocorda. Outro objetivo deste trabalho é propor uma definição para as superfícies afins discretas com curvatura afim média constante (CAMC), de forma que englobe as superfícies afins mínimas e as esferas afins. As superfícies afins mínimas discretas recebem uma caracterização geométrica bastante interessane e ligada diretamente às quádricas de Lie discretas. O trabalho se completa com o principal resultado, referente à versão discreta das superfícies de Cayley, esferas afins impróprias regradas caracterizadas a partir da conexão afim induzida: uma rede assintótica com CAMC é congruente equiafim à uma superfície de Cayley se, e somente se, a forma cúbica é não nula e a conexão afim induzida é paralela. / [en] Discrete Differential Geometry aims to develop a discrete theory which respects fundamental aspects of smooth theory. With this in mind, some results of smooth theory of Affine Geometry are firstly introduced since their discrete counterparts shall be treated a posteriori. The first goal of this work is construct a discrete affine structure for nets in a three-dimensional space with indefinite Blaschke metric and asymptotic parameters. For this purpose, one defines a conormal vector field, which satisfies Lelieuvre s equations and it is associated to a real parameter; and an affine normal vector field, which defines the cubic form of the net and makes the structure well defined. This structure allows to study, e.g., ruled surfaces with emphasis on improper affine spheres. Moreover, a definition for singularities is proposed in the case of discrete improper affine spheres from the center-chord construction. Another goal here is to propose a definition for an asymptotic net with constant affine mean curvature (CAMC), in a way that encompasses discrete affine minimal surfaces and discrete affine spheres. Discrete affine minimal surfaces receive a beautiful geometrical characterization directly linked to discrete Lie quadrics. This work is completed with the main result about a discrete version of Cayley surfaces, which are ruled improper affine spheres that can be characterized by the induced connection as: an asymptotic net with CAMC is equiaffinely congruent to a Cayley surface if and only if the cubic form does not vanish and the affine induced connection is parallel.