Telegram from Maria Joao Sande Lemos, Social Democratic Party of Portugal, to Geraldine Ferraro

Telegram from Maria Joao Sande Lemos, Social Democratic Party of Portugal, to Geraldine Ferraro. Telegram has handwritten notes.https://ir.lawnet.fordham.edu/vice_presidential_campaign_correspondence_1984_international/1363/thumbnail.jp

Approximation in FEM, DG and IGA: A Theoretical Comparison

In this paper we compare approximation properties of degree $p$ spline spaces with different numbers of continuous derivatives. We prove that, for a given space dimension, \smooth {p-1} splines provide better a priori error bounds for the approximation of functions in $H^{p+1}(0,1)$. Our result holds for all practically interesting cases when comparing \smooth {p-1} splines with \smooth {-1} (discontinuous) splines. When comparing \smooth {p-1} splines with \smooth 0 splines our proof covers almost all cases for $p\ge 3$, but we can not conclude anything for $p=2$. The results are generalized to the approximation of functions in $H^{q+1}(0,1)$ for $q<p$, to broken Sobolev spaces and to tensor product spaces.Comment: 21 pages, 4 figures. Fixed typos and improved the presentatio

Optimal spline spaces for L2L^2 nn-width problems with boundary conditions

In this paper we show that, with respect to the $L^2$ norm, three classes of functions in $H^r(0,1)$, defined by certain boundary conditions, admit optimal spline spaces of all degrees $\geq r-1$, and all these spline spaces have uniform knots.Comment: 17 pages, 4 figures. Fixed a typo. Article published in Constructive Approximatio

Biliteral Crossword

A crossword puzzle

Análise das provas de exame de Geografia a 10º e 11º anos - 2008 a 2014

Neste artigo abordar-se-á a análise estatística dos exames nacionais da disciplina de Geografia A, dos 10º e 11º anos, desde 2008 a 2014. Esta análise terá em conta não só os principais temas de cada um dos anos referidos como também o tipo de questão, isto é, se se trata de uma pergunta de escolha múltipla ou de resposta aberta. A discriminação por temas mas também por tipo de questão é fundamental para se equacionar não só a eventual diferença no grau de exigência das noções e conceitos desenvolvidos em cada tema, como também se o fator mais condicionante é o tipo de pergunta colocada ao aluno e os desempenhos que dele se espera na compreensão e respetiva resposta a essas questões

Sharp error estimates for spline approximation: explicit constants, nn-widths, and eigenfunction convergence

In this paper we provide a priori error estimates in standard Sobolev (semi-)norms for approximation in spline spaces of maximal smoothness on arbitrary grids. The error estimates are expressed in terms of a power of the maximal grid spacing, an appropriate derivative of the function to be approximated, and an explicit constant which is, in many cases, sharp. Some of these error estimates also hold in proper spline subspaces, which additionally enjoy inverse inequalities. Furthermore, we address spline approximation of eigenfunctions of a large class of differential operators, with a particular focus on the special case of periodic splines. The results of this paper can be used to theoretically explain the benefits of spline approximation under $k$-refinement by isogeometric discretization methods. They also form a theoretical foundation for the outperformance of smooth spline discretizations of eigenvalue problems that has been numerically observed in the literature, and for optimality of geometric multigrid solvers in the isogeometric analysis context.Comment: 31 pages, 2 figures. Fixed a typo. Article published in M3A