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Let G be a simple graph. A clique is a subgraph of G isomorphic to a complete graph, and a t-friendship graph is a graph consisting of t edge-disjoint cliques sharing one vertex. A t-friendship decomposition of G for a fixed value of t is a set F of edge-disjoint t-friendship subgraphs of G such that every edge of G belongs to exactly one element of F. It is proved that any graph of order n admits a t-friendship decomposition with at most n2/4t+n/4t+n elements for all fixed t≥2. For t=2,3 exact bounds are given. For t=2 the bound is ⌈n2/8⌉ for n even and (n2−1)/8 for n odd; for t=3 the bound is ⌈n2/12⌉ for n even and ⌈(n2−1)/12⌉ for n odd.

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We prove a higher order asymptotic formula for traces of finite block Toeplitz matrices with symbols belonging to Hölder-Zygmund spaces. The remainder in this formula goes to zero very rapidly for very smooth symbols. This formula refines previous asymptotic trace formulas by Szegő and Widom and complement higher order asymptotic formulas for determinants of finite block Toeplitz matrices due to Böttcher and Silbermann.

We prove higher-order asymptotic formulas for determinants and traces of finite block Toeplitz matrices generated by matrix functions belonging to generalized Hölder spaces with characteristic functions from the Bari-Stechkin class. We follow the approach of Böttcher and Silbermann and generalize their results for symbols in standard Hölder spaces.

The Upper Jurassic Lourinhã Formation (Lusitanian Basin, Portugal) contains a diverse dinosaur fauna comprising theropods, sauropods, stegosaurs, ankylosaurs and several genera of ornithopods. The sedimentology in the area favours preservation of tracksways, and tracks from most of the dinosaurs are also represented by skeletal remains. During fieldwork in the summer of 2003 a new, large, tridactyl track was found at the beach of Vale Frades, approximately 6 km north of Lourinhã (central west Portugal). The track was found together with a stegosaur track on a clay bed exposed within the intertidal zone. Due to the immediate danger of erosion, the track was collected and is now on display at Museu da Lourinhã. The track is 70 cm long and 69 cm wide, the toes are short and broad, with indications of short blunt claws, and there is a high angle of divarication between the outer digits. The shape and dimensions of the track identifies it as deriving from an ornithopod dinosaur with an estimated hip height around three metres. Although very large ornithopods are known from the Cretaceous, the largest known Jurassic ornithopod is Camptosaurus from North America, and the largest known from Portugal is the camptosaurid Draconyx loureiroi. Neither of these reached the body size suggested by the new track. So far the track described herein is the only evidence for a Jurassic ornithopod of that size.

A previous paper studied the stability of soldier-pile walls in clay under vertical loading using upper bound analyses. A classical Tresca yield criterion was assumed in that analysis. This paper extends that study by considering a tension truncated Tresca yield criterion in an upper bound numerical analysis of the problem. It shows that assuming zero tension soil strength has a significant influence on the values of the minimum soldier-pile resistance required to ensure stability.

A ligação pilar-fundação tem um papel fundamental no comportamento das estruturas pré-fabricadas, em particular, na presença de acções horizontais, como a acção dos sismos e do vento, que introduzem esforços elevados na base dos pilares. Um dos sistemas de ligação pilar-fundação usados em estruturas pré-fabricadas consiste em deixar na extremidade inferior dos pilares armaduras salientes que são introduzidas em negativos executados nas fundações. Estes negativos são posteriormente injectados com grout para garantir a ligação. No presente trabalho foram estudadas diferentes soluções de negativos para ligação das armaduras do pilar à fundação, tendo sido realizados ensaios monotónicos para analisar o seu comportamento. Nesta comunicação apresentam-se e discutem-se os resultados obtidos da análise experimental efectuada.

A ligação pilar-fundação tem um papel fundamental no comportamento das estruturas pré-fabricadas,
em particular, na presença de acções horizontais, como a acção dos sismos e do vento. Um dos sistemas de ligação pilar-fundação usados em estruturas pré-fabricadas consiste em deixar na extremidade inferior dos pilares armaduras salientes que são introduzidas em negativos executados nas fundações, sejam elas sapatas ou maciços de encabeçamento de estacas. Estes negativos são posteriormente injectados com “grout” para garantir a ligação. Foram estudados diferentes soluções de negativos para ligação das armaduras do pilar na fundação. Foram realizados ensaios monotónicos e cíclicos, para analisar não só o comportamento às acções estáticas mas também para a acção sísmica. No presente trabalho apresentam-se os resultados obtidos com a análise experimental efectuada. O estudo realizado permite apresentar recomendações para projecto e quais os tipos de ligações com melhor desempenho.

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{Summary: Heavy tailed-models are quite useful in many fields, like insurance, finance, telecommunications, internet traffic, among others, and it is often necessary to estimate a high quantile, i.e., a value that is exceeded with a probability $p$, small. The semiparametric estimation of this parameter relies essentially on the estimation of the tail index, the primary parameter in statistics of extremes. Classical semi-parametric estimators of extreme parameters show usually a severe bias and are known to be very sensitive to the number $k$ of top order statistics used in the estimation. For $k$ small they have a high variance, and for large $k$ a high bias. Recently, new second-order ``shape'' and ``scale'' estimators allowed the development of second-order reduced-bias estimators, which are much less sensitive to the choice of $k$. Here we study, under a third order framework, minimum-variance reduced-bias (MVRB) tail index estimators, recently introduced in the literature, and dependent on an adequate estimation of second order parameters. The improvement comes from the asymptotic variance, which is kept equal to the asymptotic variance of the classical Hill estimator [ıt B. Hill}, Ann. Stat. 3, 1163–1174 (1975; Zbl 0323.62033)] provided that we estimate the second order parameters at a level of a larger order than the level used for the estimation of the first order parameter. The use of those MVRB tail index estimators enables us to introduce new classes of reduced-bias high quantile estimators. These new classes are compared among themselves and with previous ones through the use of a small-scale Monte Carlo simulation.}

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