Humboldt-Univ., Habil.-Schr.--Berlin. / Literaturverz. S. [100] - 109.

Orientador: Sandra Augusta Santos / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T19:53:49Z (GMT). No. of bitstreams: 1 Souza_MariliaFranceschinellide_M.pdf: 3298789 bytes, checksum: cd2737cf2f9747a3856c42aee3ac5d83 (MD5) Previous issue date: 2014 / Resumo: O currículo de Matemática do Ensino Médio está atualmente muito denso e quase não permite ao professor explorar outros trabalhos que fujam das aulas expositivas, nas quais o papel do aluno é o de apenas escutar, anotar e reproduzir. Esse esquema antiquado desperta pouco interesse dos alunos pelas disciplinas, especialmente pela Matemática, tida por muitos como a vilã, bastante difícil de ser compreendida. Neste trabalho apresentamos uma proposta de projeto para ser trabalhado no Ensino Médio. O problema de otimização da área de quadriláteros de perímetro constante é abordado, utilizando essencialmente o conteúdo do Ensino Básico. O ponto de partida é um problema simples, presente na maioria dos livros-texto. A análise é ampliada gradativamente por situações mais próximas da realidade, oferecendo ao aluno a oportunidade de utilizar diversos conceitos estudados em uma aplicação da Matemática. Os problemas são abordados tanto de forma algébrica quanto geométrica, oferecendo elementos para que o aluno processe informações, anteveja possibilidades, analise o caso geral, exemplifique situações específicas e de fato possa compreender os problemas e interpretar as soluções obtidas / Abstract: The mathematics curriculum of the Brazilian High School is currently very dense. As a result, it is hard to explore alternative ways of teaching that allow the students to effectively participate in lessons, instead of just listening to the teacher and taking notes. The old fashioned expositive method, in general, does not encourage the interest of the students, especially in Mathematics, considered as a villain by many, because of it is intrinsic difficult. In this work it is presented the proposal of a project for the High School level. The problem of maximizing the area of quadrilaterals with constant perimeter is approached, using essentially the content of Basic Education. The starting point is a simple problem, present in most textbooks. The analysis is extended for situations closer to reality, offering students the opportunity to use many concepts already studied, in an application of mathematics. The problems are treated algebraically and geometrically, providing elements for the student to process information, anticipates possibilities, consider the general case, exemplify specific situations, so that they might indeed understand the problems and interpret the obtained solutions / Mestrado / Matemática em Rede Nacional / Mestra em Matemática em Rede Nacional

This dissertation deals with the use of Shewhart control charts, modified to have variable sampling intervals, to simultaneously monitor a set of parameters. Fixed sampling interval control charts are modified to utilize sampling intervals that vary depending on what is being observed from the data. Two problems are emphasized, namely, the simultaneous monitoring of the mean and the variance and the simultaneous monitoring of several means. For each problem, two basic strategies are investigated. One strategy uses separate control charts for each parameter. A second strategy uses a single statistic which combines the information in the entire sample and is sensitive to shifts in any of the parameters. Several variations on these two basic strategies are studied. Numerical studies investigate the optimal number of sampling intervals and the length of the sampling intervals to be used. Each procedure is compared to corresponding fixed interval procedures in terms of time and the number of samples taken to signal. The effect of correlation on multiple means charts is studied through simulation. For both problems, it is seen that the variable sampling interval approach is substantially more efficient than fixed interval procedures, no matter which strategy is used. / Ph. D.

Thesis (M.S.)--University of North Carolina Wilmington, 2009. / Title from PDF title page (February 16, 2010) Includes bibliographical references (p. 34-35)

We derive a lower bound for the Wehrl entropy in the setting of SU(1,1). For asymptotically high values of the quantum number k, this bound coincides with the analogue of the Lieb-Wehrl conjecture for SU(1,1) coherent states. The bound on the entropy is proved via a sharp norm bound. The norm bound is deduced by using an interesting identity for Fisher information of SU(1,1) coherent state transforms on the hyperbolic plane and a new family of sharp Sobolev inequalities on the hyperbolic plane. To prove the sharpness of our Sobolev inequality, we need to first prove a uniqueness theorem for solutions of a semi-linear Poisson equation (which is actually the Euler-Lagrange equation for the variational problem associated with our sharp Sobolev inequality) on the hyperbolic plane. Uniqueness theorems proved for similar semi-linear equations in the past do not apply here and the new features of our proof are of independent interest, as are some of the consequences we derive from the new family of Sobolev inequalities. We also prove Fisher information identities for the groups SU(n,1) and SU(n,n).

<p>The goal of this thesis is to establish generalized isoperimetric inequalities on first exit time distributions as well as expectations of L\'evy processes.</p> <p>Firstly, we prove inequalities on first exit time distributions in the case that the L\'evy process is an $\alpha$-stable symmetric process $A_t$ on $\R^d$, $\alpha\in(0,2]$. Given $A_t$ and a bounded domain $D\subset\R^d$, we present a proof, based on the classical Brascamp-Lieb-Luttinger inequalities for multiple integrals, that the distribution of the first exit time of $A_t$ from $D$ increases under Steiner symmetrization. Further, it is shown that when a sequence of domains $\{D_m\}$ each contained in a ball $B\subset\R^d$ and satisfying the $\varepsilon$-cone property converges to a domain $D'$ with respect to the Hausdorff metric, the sequence of distributions of first exit times for Brownian motion from  $D_m$  converges to the distribution of the exit time of Brownian motion from $D'$. The second set of results in this thesis extends the theorems from \cite{BanMen} by proving generalized isoperimetric inequalities on expectations of L\'evy processes in the case of Steiner symmetrization.% using the Brascamp-Lieb-Luttinger inequalities used above. </p> <p>These results will then be used to establish inequalities involving distributions of first exit times of $\alpha$-stable symmetric processes $A_t$ from triangles and quadrilaterals. The primary application of these inequalities is verifying a conjecture from Ba\~nuelos for these planar domains. This extends a classical result of P\'olya and Szeg\"o to the fractional Laplacian with Dirichlet boundary conditions.</p>

The presented work deals with two distinct problems in the field of Mathematical Physics. The first part is dedicated to an 'almost' solution of Villani's conjecture, a known conjecture related to a Statistical Mechanics model invented by Kac in 1956, giving a rigorous explanation of some simple cases of the Boltzmann equation. In 2003 Villani conjectured that the time it will take the system of particles in Kac's model to equilibrate is proportional to the number of particles in the system. Our main result in this part is a proof, up to an epsilon, of that conjecture, showing that for all practical purposes we can consider it to be true. The second part of the presentation is based on a joint work with Prof. Michael Loss and is dedicated to a newly developed trace inequality for the fractional Laplacian, connecting between the fractional Laplacian of a function and its restriction to intersection of hyperplanes. The newly found inequality is sharp and the functions that attain equality in it are completely classified.