Let D n be the pseudovariety of all ÿnite semigroups in which products of length n are right zeros and let D = n≥1 Dn. It is shown in this paper that, if V is a pseudovariety of semigroups whose global gV is ÿnitely based, then V * Dn (n ≥ 1) and V * D are also ÿnitely based. Moreover, if V is itself ÿnitely based and contains the aperiodic ÿve-element Brandt semigroup, then gV is also ÿnitely based. As a further application, it is proved that the ÿnite basis properties for gV, V * D and V * D n (n ≥ 1) are all equivalent for an arbitrary non-group monoidal pseudovariety V:

a b s t r a c tWe study the existence of solutions of stationary variational and quasivariational inequalities with curl constraint, Neumann type boundary condition and a p-curl type operator. These problems are studied in bounded, not necessarily simply connected domains, with a special geometry, and the functional framework is the space of divergencefree functions with curl in L p and null tangential or normal traces.The analogous variational or quasivariational inequalities with gradient constraint are also studied, considering Neumann or Dirichlet non-homogeneous boundary conditions. The existence of a generalized solution for a Lagrange multiplier problem with homogeneous Dirichlet boundary condition and the equivalence with the variational inequality is proved in the linear case, for an arbitrary gradient constraint.

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The Birnbaum-Saunders (BS) distribution has been largely studied and applied. A random variable with BS distribution is a transformation of another random variable with standard normal distribution. Generalized BS distributions are obtained when the normally distributed random variable is replaced by another symmetrically distributed random variable. This allows us to obtain a wide class of positively skewed models with lighter and heavier tails than the BS model. Its failure rate admits several shapes, including the unimodal case, with its change-point being able to be used for different purposes. For example, to establish the reduction in a dose, and then in the cost of the medical treatment. We analyze the failure rates of generalized BS distributions obtained by the logistic, normal and Student-t distributions, considering their shape and change-point, estimating them, evaluating their robustness, assessing their performance by simulations, and applying the results to real data from different areas.Special cases of the t distribution are the Cauchy and normal distributions, when ν = 1 and ν → +∞, respectively.

In this paper we consider a stationary variational inequality with nonconstant gradient constraint and we prove the existence of solution of a Lagrange multiplier, assuming that the bounded open not necessarily convex set Ω has a smooth boundary. If the gradient constraint g is sufficiently smooth and satisfies ∆g 2 ≤ 0 and the source term belongs to L ∞ (Ω), we are able to prove that the Lagrange multiplier belongs to L q (Ω), for 1 < q < ∞, even in a very degenerate case. Fixing q ≥ 2, the result is still true if ∆g 2 is bounded from above by a positive sufficiently small constant that depends on Ω, q, min Ω g and max Ω g. Without the restriction on the sign of ∆g 2 we are still able to find a Lagrange multiplier, now belonging to L ∞ (Ω). We also prove that if we consider the variational inequality with coercivity constant δ and constraint g, then the family of solutions (λ δ , u δ) δ>0 of our problem has a subsequence that converges weakly to (λ 0 , u 0), which solves the transport equation. Résumé Dans cet article, nous considérons une inégalité variationnelle stationnaire avec une restriction non-constante sur le gradient et nous prouvons l'existence d'un multiplicateur de Lagrange, en supposant que l'ensemble ouvert et borné Ω, pas nécessairement convexe, a une frontière régulière. Si la restriction du gradient g est suffisamment régulière et satisfait ∆g 2 ≤ 0 et le terme source appartient a L ∞ (Ω), nous pouvons prouver que le multiplicateur de Lagrange appartientà L q (Ω), pour 1 < q < ∞, même dans un cas très dégénéré. Si nous fixons q ≥ 2, le résultat est aussi vrai si ∆g 2 est borné par une constante positive et suffisamment petite qui dépend de Ω, q, min Ω g et max Ω g. Sans la restriction sur le signe de ∆g 2 nous sommes capables de trouver un multiplicateur de Lagrange, maintenant appartenantà L ∞ (Ω). Nous montrons aussi que si l'on considère l'inégalité variationnelle avec la coercitivité constante δ et la restriction g, alors la famille des solutions (λ δ , u δ) δ>0 de notre problème a une sous-suite qui converge faiblement vers (λ 0 , u 0), ce qui résout l'équation de transport.