Mannheim, Univ., Diss., 2005.

This report analyzes and compares Barycentric Lagrange interpolation to Cardinal Trigonometric interpolation, with regards to computational cost and accuracy. It also covers some edge case scenarios which may interfere with the accuracy and stability. Later on, these two interpolation methods are applied on parameterized curves and surfaces, to compare and contrast differences with the standard one dimensional scenarios. The report also contains analysis of and comparison with regular Lagrange interpolation. The report concludes that Barycentric Lagrange interpolation is generally speaking more computationally efficient, and that the inherent need for periodicity makes Cardinal Trigonometric interpolation less reliable in comparison. On the other hand, Barycentric Lagrange interpolation is difficult to implement for higher dimensional problems, and also relies heavily on Chebyshev spaced nodes, something which can cause issues in a practical application of interpolation. Given ideal scenarios, Cardinal Trigonometric interpolation is more accurate, and for periodic functions generally speaking better than Barycentric Lagrange interpolation. Regular Lagrange interpolation is found to be unviable due to the comparatively big computational cost.

With growing trend in "cloud computing" and increase in the data moving into the Internet, the need to store large amounts of data by service providers such as Google, Yahoo and Microsoft has increased over time. Now, more than ever, there is a need to efficiently and securely store large amounts of data. This thesis presents an implementation of a Ramp Scheme that confidentially splits a data file into a configurable number of parts or shares of equal size such that a subset of those shares can recover the data entirely. Furthermore, the implementation supports a threshold for data compromise and data verification to verify that the data parts have not been tampered with. This thesis addresses two key problems faced in large-scale data storage, namely, data availability and confidentiality.

In 2010, Prof. Chang and Prof. Lee applied Lagrange interpolation formula to decode a class of binary cyclic codes, but they did not provide an effective way to calculate the Lagrange interpolation formula. In this thesis, we use the least common multiple of polynomials to compute it effectively. Let E be an extension field of degree m over F = F_p and £] be a primitive nth root of unity in E. For a nonzero element r in E, the minimal polynomial of r over F is denoted by m_r(x). Then, let Min (r, F) denote the least common multiple of m_r£]^i(x) for i = 0, 1,..., n-1 and be called the generalized minimal polynomial of over F. For any binary quadratic residue code mentioned in this thesis, the set of all its correctable error patterns can be partitioned into root sets of some generalized minimal polynomials over F. Based on this idea, we can develop an effective method to calculate the Lagrange interpolation formula.

Mannheim, Univ., Diss., 2006.

A mixture experiment is an experiment in which the k ingredients are nonnegative and subject to the simplex restriction £Ux_i=1 on the (k-1)-dimensional probability simplex S^{k-1}. This dissertation discusses optimal designs for linear and quadratic log contrast models for experiments with mixtures suggested by Aitchison and Bacon-Shone (1984), where the experimental domain is restricted further as in Chan (1992). In this study, firstly, an essentially complete class of designs under the Kiefer ordering for linear log contrast models with mixture experiments is presented. Based on the completeness result, £X_p-optimal designs for all p, -¡Û<p≤1 including D- and A-optimal are obtained, where the eigenvalues of the design moment matrix are used. By using the approach presented here, we gain insight on how these £X_p-optimal designs behave. Following that, the exact N-point D-optimal designs for linear log contrast models with three and four ingredients are further investigated. The results show that for k=3 and N=3p+q ,1 ≤q≤2, there is an exact N-point D-optimal design supported at the points of S_1 (S_2) with equal weight n/N, 0≤n≤p , and puts the remaining weight (N-3n)/N uniformly on the points of S_2 (S_1) as evenly as possible, where S_1 and S_2 are sets of the supports of the approximate D-optimal designs. When k=4 and N=6p+q , 1 ≤q≤5, an exact N-point design which distributes the weights as evenly as possible among the supports of the approximate D-optimal design is proved to be exact D-optimal. Thirdly, the approximate D_s-optimal designs for discriminating between linear and quadratic log contrast models for experiments with mixtures are derived. It is shown that for a symmetric subspace of the finite dimensional simplex, there is a D_s-optimal design with the nice structure that puts a weight 1/(2^{k-1}) on the centroid of this subspace and the remaining weight is uniformly distributed on the vertices of the experimental domain. Moreover, the D_s-efficiency of the D-optimal design for quadratic model and the design given by Aitchison and Bacon-Shone (1984) are also discussed Finally, we show that an essentially complete class of designs under the Kiefer ordering for the quadratic log contrast model is the set of all designs in the boundary of T or origin of T . Based on the completeness result, numerical £X_p -optimal designs for some p, -¡Û<p≤1 are obtained.

Submitted by Geandra Rodrigues (geandrar@gmail.com) on 2018-07-12T11:34:57Z No. of bitstreams: 1 henriqueluismoreiramonteiro.pdf: 6497394 bytes, checksum: e4ba6289e2d5a09f203b42d5461ff1b1 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-07-17T11:53:38Z (GMT) No. of bitstreams: 1 henriqueluismoreiramonteiro.pdf: 6497394 bytes, checksum: e4ba6289e2d5a09f203b42d5461ff1b1 (MD5) / Made available in DSpace on 2018-07-17T11:53:38Z (GMT). No. of bitstreams: 1 henriqueluismoreiramonteiro.pdf: 6497394 bytes, checksum: e4ba6289e2d5a09f203b42d5461ff1b1 (MD5) Previous issue date: 2018-04-04 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho é proposto um método para estimar a impedância de maneira invasiva, utilizando uma estrutura com controle de distúrbio, filtro para eliminar os componentes harmônicos contidos na rede e um interpolador em caso de variação de frequência. A impedância é um parâmetro importante para a análise da estabilidade de uma rede elétrica, pois através de sua estimação pode-se verificar, por exemplo, se há a presença de ressonância. Para provocar os distúrbios nos sinais de tensão e corrente utiliza-se o Sinal da Gaussiana Modulada (SGM), que pode ter seu decaimento controlado pelos parâmetros da função. Em relação à presença dos componentes harmônicos de fundo, que influenciam na estimação da impedância, é proposta uma versão do filtro Sliding Window Recursive Discrete Fourier Transform (SWRDFT), para estimar e eli-minar esses componentes, com o objetivo de permanecer somente o sinal de distúrbio. Sobre o desvio de frequência, outra questão que compromete a estimação da impedância, é proposta uma estrutura de interpolação, utilizando o método de Lagrange. A função da interpolação é estabelecer a sincronização dos sinais de corrente e tensão antes de serem processados para determinar o valor da impedância. Para validar os resultados, são realizados testes em redes no Simulink, RTDS, dSPACE e utilizando em um circuito experimental. Para os resultados são considerados Sistemas Elétricos com ressonância, com presença de componentes harmônicos de fundo e com variação de frequência. Para todos os casos a estimação obteve resultados satisfatórios. / This work proposes a method to estimate the power grid impedance in an invasive form, using a disturbance control structure, a filter to eliminate the harmonic components contained in the power grid and an interpolator in case of frequency variation. The impedance is an important parameter for the power grid stability analysis because through its estimation it can be verified, for example, if there is the presence of resonance. To insert the disturbances in the voltage and current signals, the Gaussian Modulated Signal (GMS) is used, which may have its decay controlled by the function parameters. Regarding the presence of the harmonic components, which influence the estimation of impedance, a version of the Sliding Window Recursive Discrete Fourier Transform (SWRDFT) is proposed to estimate and eliminate these components in order to remain only the disturbance signal. Regarding the frequency deviation, another issue that affects the impedance estimation, an interpolation structure is proposed, using the Lagrange method. The function of the interpolation is to establish the synchronization of the current and voltage signals before being processed to determine the impedance value. To validate the results, tests with grids on Simulink, RTDS, dSpace and using an experimental circuit are performed. Results consider Electrical Systems with resonance, with the presence of background harmonic components and frequency variation. For all cases, the estimation results were satisfactory.