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.

We give some background information about the Cauchy-Schwarz inequality including its history. We then continue by providing a number of proofs for the inequality in its classical form using various proof techniques, including proofs without words. Next we build up the theory of inner product spaces from metric and normed spaces and show applications of the Cauchy-Schwarz inequality in each content, including the triangle inequality, Minkowski's inequality and Hölder's inequality. In the final part we present a few problems with solutions, some proved by the author and some by others.

Submitted by Jean Medeiros (jeanletras@uepb.edu.br) on 2017-11-06T12:17:01Z No. of bitstreams: 1 PDF - Josenildo da Cunha Lima.pdf: 21330682 bytes, checksum: b04b7f38635dc46f7b4cfbfc949c318b (MD5) / Approved for entry into archive by Secta BC (secta.csu.bc@uepb.edu.br) on 2017-11-08T16:45:20Z (GMT) No. of bitstreams: 1 PDF - Josenildo da Cunha Lima.pdf: 21330682 bytes, checksum: b04b7f38635dc46f7b4cfbfc949c318b (MD5) / Made available in DSpace on 2017-11-08T16:45:20Z (GMT). No. of bitstreams: 1 PDF - Josenildo da Cunha Lima.pdf: 21330682 bytes, checksum: b04b7f38635dc46f7b4cfbfc949c318b (MD5) Previous issue date: 2017-05-05 / This work presents activities that can be carried out with the High School classes and that do have basic notions of the functions, area, volume and means inequality. We present a didactic sequence, composed of several activities, with the use of GeoGebra software, so that in each of them the student can conjecture an optimization result in a classroom application. In some activities aim the elaboration of a file of type .ggb to discover an optimal value for a certain geometric element. In each activities we seek to optimize geometric elements such as segments, angles, areas and volumes. By performing these activities, students will learn geometry contents dynamically and this will provide them with a view next of what actually occurs in search to optimize of elements such geometric elements. This study aims to show that optimization problems can be worked on in High School and the results found in resolutions of these problems are demonstrated with theorems involving mathematical contents of Basic Education. / Neste trabalho apresentamos atividades que podem ser realizadas com turmas do Ensino Médio e que tenham noções básicas de funções, área, volume e desigualdade das médias. Apresentamos uma sequência didática, composta por diversas atividades, com a utilização do software GeoGebra, de modo que em cada uma delas, o aluno possa conjecturar um resultado de otimização numa aplicação em sala de aula. Algumas dessas atividades têm como objetivo a elaboração de um arquivo do tipo .ggb para se descobrir um valor ótimo para determinado elemento geométrico. Em todas as atividades buscamos otimizar elementos geométricos como segmentos, ângulos, áreas e volumes. Realizando essas atividades, os estudantes aprenderão conteúdos de geometria de forma dinâmica e isso os proporcionará uma visão próxima do que concretamente ocorre na busca por otimizar tais elementos geo- métricos. Este estudo tem como finalidade mostrar que problemas de otimização podem ser trabalhados no Ensino Médio e que os resultados usados nas resoluções desses problemas são demonstrados com teoremas envolvendo conteúdos matemáticos do Ensino Básico.