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Let \(\alpha,\beta\in(0,1)\) and
\[
K^{\alpha,\beta}:=\left\{a\in L^\infty(\mathbb{T}):\
\sum_{k=1}^\infty |\widehat{a}(-k)|^2 k^{2\alpha}<\infty,\
\sum_{k=1}^\infty |\widehat{a}(k)|^2 k^{2\beta}<\infty
\right\}.
\]
Mark Krein proved in 1966 that \(K^{1/2,1/2}\) forms a Banach algebra. He also observed that this algebra is important in the asymptotic theory of finite Toeplitz matrices. Ten years later, Harold Widom extended
earlier results of Gabor Szegö for scalar symbols and established the asymptotic trace formula
\[
\operatorname{trace}f(T_n(a))=(n+1)G_f(a)+E_f(a)+o(1)
\quad\text{as}\ n\to\infty
\]
for finite Toeplitz matrices \(T_n(a)\) with matrix symbols \(a\in K^{1/2,1/2}_{N\times N}\). We show that if \(\alpha+\beta\ge 1\) and \(a\in K^{\alpha,\beta}_{N\times N}\), then the Szegö-Widom asymptotic trace formula holds with \(o(1)\) replaced by \(o(n^{1-\alpha-\beta})\).

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Four different localities from the Late Jurassic of Lourinhã formation with eggshells and embryos were studied: Paimogo (lower Amoreira-Porto Novo member), Peralta (Praia Azul member), Porto das barcas (Bombarral member) and Casal da Rôla (Amoreira-Porto Novo member). All but Casal da Rôla have embryonic material. Preliminary results show that eggshells from Paimogo correspond to obliquiprismatic morphotype (0.92mm thick), similar to those from Morrison Formation. Within Paimogo site a different type of eggshell was discovered, having a radial section of 153 μm with a mammilary layer measuring 65 μm. Porto das Barcas eggshells represent a discretispherulitic morphotype (1,23 mm thick).
This locality presents a nest 60-cm diameter containing many eggshells but an indeterminate number of eggs. Some embryonic bones were discovered between the eggshells including teeth and skull bones showing that the eggs belong to a saurischian, tentatively a sauropod dinosaur. Peralta nest eggshells are preliminary ascribed to obliquiprismatic morphotype (column: 0,56mm and mammilla: 0,21mm) probably related to Paimogo’s nest taxon (Lourinhanosaurus). Peralta site bears embryonic bones namely small theropod teeth associated with bone fragments, and unidentifiable dinosaur vertebra. Only eggshells have been collected at Casal da Rôla (ML1194). The eggshells (0,78mm thick) are prismatic morphotype and it was impossible to determine the pore system, the outer surface is smooth with no ornamentation.
Lourinhã formation has the oldest sauropod and theropod nest with embryos known so far.

The rapid growth of electric and electronic equipment waste (WEEE) transformed this waste stream into a worldwide problem. The Directive 2002/96/EC on electrical and electronic waste aims at the reduction of the environmental impact of WEEE, encouraging end-of-life management, eco design, life cycle analyses and extended producer responsibility. However, this legislation may not produce the results the legislator aimed for. In this work we analyse the environmental impact of a WEEE recovery network in the Portuguese context. With this aim, a model, previously developed by the authors for the optimal design of this network using economic indicators (Salema, 2007), was now adapted to design the network subjected to the minimization of environmental performance indices. The original mathematical formulation was found to be flexible and easily adapted to the two types of indices and the major differences between the optimal network configurations obtained, were identified and discussed.

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