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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 \(o(n^{1-\alpha-\beta})\).

Fiber reinforced polymers (FRP) are often used to strengthen RC structures.
Despite intense research, durability of reinforced concrete (RC) retrofitted with FRP remains insufficiently known. Long time behavior of the bond laminate-concrete, in flexural strengthening, under environmental action is not well known, conditioning Codes and engineers. An experimental program that subjected RC beams, externally reinforced with Glass FRP (GFRP) strips, to temperature and salt water cycles, for up to 10000h is reported.
At selected intermediate times, the RC beams were loaded to failure in bending. Rupture took place, normally, by tensile failure of concrete at a short distance from the interface with GFRP. The results showed that freeze-thaw cycles were the most severe of the environmental
conditions. The study also generated also non-linear bond-slip relationships from the experimental data. Numerical modeling has been undertaken, based on a commercial code.
The model is based on smeared cracking. Parameters needed for the characterization, namely cohesion and friction angle, were obtained from shear tests conceived for the effect.

Analisa-se e caracteriza-se por via experimental a ligação entre elementos de betão armado e materiais compósitos, nomeadamente com base nas fibras de vidro. Fabricaram-se vigas de betão armado que foram exteriormente reforçadas com GFRP. Os resultados obtidos experimentalmente foram comparados com os resultados conseguidos por intermédio de modelação computacional, recorrendo-se ao programa de cálculo ATENA 2D. Para melhor modelação de elementos de interface, foram realizados ensaios de corte tendo-se obtido valores que permitiram caracterizar a lei de rotura de Mohr-Coulomb. Os parâmetros estudados foram a evolução das forças máximas absorvidas pelo reforço; as tensões de aderência máximas; a distribuição das tensões de aderência.

A mixture of 70% (w/w) pine biomass and 30% (w/w) plastics (mixture of polypropylene, polyethylene, and polystyrene) was subjected to pyrolysis at 400 °C, for 15 min, with an initial pressure of 40 MPa. Part of the solid residue produced was subjected to extraction with dichloromethane (DCM). The extracted residue (residue A) and raw residue (residue B) were analyzed by weight loss combustion and submitted to the leaching test ISO/TS 21268-2 using two different leachants: DCM (0.2%, v/v) and calcium chloride (0.001 mol/L). The concentrations of the heavy metals Cd, Cr, Ni, Zn, Pb and Cu were determined in the eluates and in the two residues. The eluates were further characterized by determining their pH and the concentrations of benzene, toluene, ethylbenzene and xylenes (BTEX). The presence of other organic contaminants in the eluates was qualitatively evaluated by gas chromatography, coupled with mass spectrometry. An ecotoxicological characterization was also performed by using the bio-indicator Vibrio fischeri. The chemical and ecotoxicological results were analyzed according to the French proposal of Criteria on the Evaluation Methods of Waste Ecotoxicity (CEMWE). Residue A was not considered to be ecotoxic by the ecotoxicological criterion (EC50 (30 min) ≥10%), but it was considered to be ecotoxic by the chemical criterion (Ni ≥ 0.5 mg/L). Residue B was considered to be ecotoxic by the ecotoxicological criterion: EC50 (30 min) ≤ 10%. Besides that, residue B was considered to be hazardous according the European legislation (BTEX concentrations higher than 100 ppb). The results indicate that volatile organic contaminants can be present in sufficient amounts in these residues and their eluates to induce ecotoxicity levels. The extraction of the pyrolysis residue with DCM was an efficient method for removing lighter organic contaminants.

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The co-combustion of sewage sludge (SS) and coal is widely used for the treatment and thermal valorization of SS produced in wastewater treatment plants. The chemical and ecotoxicological properties of the ashes produced in this thermal treatment have not been fully studied. Two combustion tests were performed in a fluidized bed combustor. Colombian coal was used as fuel in test A. A blend (1 + 1) of this coal and a stabilized SS (Biogran®) was used in a second test B. Samples of the bottom and fly ashes trapped in two sequential cyclones were collected. The characterization of the ashes was focused on two main aspects: (1) the bulk content of a set of metals and (2) the characterization of eluates produced according to the European Standard leaching test EN 12457-2. The eluates were submitted to an ecotoxicological characterization for two bio-indicators. In what concerns the bulk content of ashes, both combustion tests have produced ashes with different compositions. The ashes formed during the co-combustion test have shown higher concentrations of metals, namely Cr, Cu, Ni, Pb, Zn and Fe for all ashes. The leaching test has shown low mobility of these elements from the by-products produced during the combustion and co-combustion tests. Cr and Cr(VI) were mainly detected in the eluates of the 1st cyclone ashes produced in both combustion tests and in the 2nd cyclone ashes produced in the co-combustion test.

Considering the ecotoxicity assays, the eluates of bottom and fly ashes for both combustion and co-combustion tests have shown low ecotoxic levels. The micro-crustacean Daphnia magna was generally more sensitive than the bacterium Vibrio fischeri. CEMWE criterion has allowed to classify the bottom ashes for both combustion and co-combustion tests as non-toxic residues and the fly ashes collected in both cyclones as toxic.

The problem of finding the independent number of an undirected graph is formulated as two equivalent Mathematical Programs with Linear Complementarity Constraints (MPLCC). A multistarting Lemke’s method
is introduced for dealing with the first formulation and is able to find a good approximate of the independent
number in a finite number of iterations. A sequential complementary algorithm is also discussed for the second formulation and can find the independent number at least in theory. Some computational experience is included to highlight the efficacy of the complementary approaches for computing the independent number of graphs from the Dimacs collection.

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A method for the determination of 15 aromatic hydrocarbons in eluates from solid residues produced during the co-pyrolysis of plastics and pine biomass was developed. In a first step, several sampling techniques (headspace solid phase microextraction (HS-SPME), static headspace sampling (HS), and dispersive liquid–liquid microextraction (DLLME) were compared in order to evaluate their sensitivity towards these analytes. HS-SPME and HS sampling had the better performance, but DLLME was itself as a technique able to extract volatiles with a significant enrichment factor.

HS sampling coupled with GC–MS was chosen for method validation for the analytes tested. Calibration curves were constructed for each analyte with correlation coefficients higher than 0.999. The limits of detection were in the range of 0.66–37.85 ng/L. The precision of the HS method was evaluated and good repeatability was achieved with relative standard deviations of 4.8–13.2%. The recoveries of the analytes were evaluated by analysing fortified real eluate samples and were in the range of 60.6–113.9%.

The validated method was applied in real eluate samples. Benzene, toluene, ethylbenzene and xylenes (BTEX) were the compounds in higher concentrations.

The DLLME technique coupled with GC–MS was used to investigate the presence of less volatile contaminants in eluate samples. This analysis revealed the presence of significant amounts of alkyl phenols and other aromatic compounds with appreciable water solubility.

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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