Thesis (Ph. D.)--Ohio State University, 2005. / Title from first page of PDF file. Includes bibliographical references (p. 93-97).

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Estimators for the individual error variance were derived in a nonreplicated two-way classification by the use of the model x_ij = μ_i + β_ij + ε_ij, i=1,2,...n; j=1,2,...,r, where x_ij = observation on the i^th treatment of the j^th block, μ_i = true mean of the i^th treatment, β_j = bias of the j^th block, ε_ij = random error, distributed normally with means zero and variance σ²_j, and E(x_ij) = μ_i + β_j. The estimator σ̂²_t, for σ²_t, t=1,2,3,...,r, was derived for n ≥ 2 and r = 3, by applying the principle of maximum likelihood to a set of (n-1)(r-1) transformed variables usually ascribed to error. Equations were derived for the maximum likelihood estimators for n ≥ 2 and r ≥ 3. A general quadratic form was used and when four reasonable assumptions were applied, estimators of the variances were obtained in for form of Q_t = [r(r-1)∑_i(x_ij-x_i.-x_.t+x_..)²-∑_i∑_j(x_ij-x_i.-x_.j+x_..)²] ÷ [(n-1)(r-1)(r-2)] where x_i., x_.j and x_.. are the means of i^th treatment, j^th block and grand mean respectively. σ̂²_t and Q_t were shown to be identical when σ²_t was being estimated for the case n ≥ 2, r = 3. It was noted that the derived estimator Q_j is equal to the estimators proposed by Grubbs [J.A.S.A., Vol. 43 (1948)] and Ehrenberd [Biometrika, Vol 37. (1950).] It was shown that Q_t/σ² = [(r-1)²x_(n-1)²-x_(n-1)(r-2)²]/[(n-1)(r-1)(r-2)], a linear difference of two independent central chi-square variates. The statistic Q/E was derived such that Q_t/E = [(((r-1)²)/(1+(r-2)F))-1]/[(n-1)(r-1)(r-2)] with F, a central F-statistic with (n-1)(r-2) and (n-1) degrees of freedom in the numerator and denominator respectively and E =∑_i∑_j(x_ij-x_i.-x_.j+x_..)². It was noted that this statistic may be used to test H_o: σ²_t = σ²against one of H_a₁: σ²_t > σ²; H_a₂: σ²_t < σ² and H_a₃: σ²_t ≠ σ² assuming σ²_j = σ², j≠t, j=1,2,...,r. A final test was of homogeneity of variances when r = 3 and was based on - 2 ln λ = (n-1)[2 ln (n-1) + ln(Q₁Q₂+Q₁Q₃+Q₂Q₃) - 2 ln E + ln 4/3], where λ is a likelihood ratio and -2 ln λ is approximately distributed as x² with 2 degrees of freedom for large n. A more general statistic for testing homogeneity of variance for r ≥ 3 was proposed and its distribution discussed in a special case. / Ph. D.

Thesis (Ph.D. (Mathematics)) -- University of Limpopo, 2021 / A finite group G is called (l, m, n)-generated, if it is a quotient group of the triangle group T(l, m, n) = x, y, z|xl = ym = zn = xyz = 1-. In [43], Moori posed the question of finding all the (p, q, r) triples, where p, q and r are prime numbers, such that a non-abelian finite simple group G is a (p, q, r)-generated. In this thesis, we will establish all the (p, q, r)-generations of the following groups, the Mathieu sporadic simple group M23, the alternating group A11 and the symplectic group Sp(6, 2). Let X be a conjugacy class of a finite group G. The rank of X in G, denoted by rank(G : X), is defined to be the minimum number of elements of X generating G. We investigate the ranks of the non-identity conjugacy classes of the above three mentioned finite simple groups. The Groups, Algorithms and Programming (GAP) [26] and the Atlas of finite group representatives [55] are used in our computation / University of Limpopo

The PID controller is regarded as a dependable and reliable controller for process industry systems. Many researchers have devoted time and attention to PID controller tuning and they all agree that PID controllers are very important for control systems. A PID equation is very sensitive; its parameters must always be varied following the specific application to increase performance, such as by increasing the system’s responsiveness. PID controllers still have many problems despite their importance for control systems in industries. The problem of big overshoot on the conventional gain tuning is one of the serious problems. Researchers use the PSO algorithm to try and overcome those problems. The tuning of the MIMO PID controller based on the PSO algorithm shows many disadvantages such as high-quality control with a short settle time, steady-state error, and periodical step response. The traditional PSO algorithm is very sensitive and it sometimes affects the quality of good PID controller tuning. This research has proposed a new equation for improving the PSO algorithm. The proposed algorithm is the combination of linearly decreasing inertia weight and chaotic inertia weight, after which a control factor was introduced as an exponential factor. This was very useful for simulations as it is adjustable. The Matlab simulation results of the experiments show that the simulations as it is adjustable. The Matlab simulation results of the experiments show that the new proposed equation converges faster and it gives the best fitness compared to linear inertia weight and oscillating inertia weight and other old equations. The MIMO PID controller system that consists of four plants was tuned based on the new proposed equation for the PSO algorithm (LCPSO). The optimized results show the best rise time, settling time, time delays, and steady-state compared to the systems that are tuned using the old equations. The exploration was directed at considering the impact of using the PSO calculation as an instrument for MIMO PID tuning. The results obtained in the examination reveal that the PSO tuning output improved reactions and can be applied to various system models in the measure control industry. The results for the MIMO PID controller tuned using PSO were assessed using integral square error (ISE), integral absolute error (IAE), and the integral of time expanded by absolute error (ITAE). The five well-known benchmark functions were also used to endorse the feasibility of the improved PSO and excellent results in terms of convergence and best fitness were attained. / Electrical and Mining Engineering / M. Tech. (Electrical Engineering)

Diploma thesis focus on creation of neural network’s internal structure with goal of creation Artificial Neural Network capable of machine state monitoring and predicting its remaining usefull life. Main goal is creation of algorithm’s and library for design and learning of Artificial Neural Network, and deeper understanding of the problematics in the process, then by utilising existing libraries. Selected method was forward-propagation network with multi-layered perceptron architecture, and backpropagation learning. Achieved results was, that the network was able to determine parts state from vibration measurement and on its basis predict remaining usefull life.