## António Caetano

## Growth envelopes for Besov spaces on h-sets

The growth envelopes for Besov spaces on h-sets (a kind of fractal sets) are presented. The proof relies on atomic representations and explores some similarities with a previously derived proof where the growth envelopes for Besov spaces of generalized smoothness on Euclidean spaces were obtained. In some sense this sums up the developments in this direction that have been obtained in the last years.

Eugénio Rocha

## Can mathematics boost petrochemical industry?

We briefly describe an example where the interaction between mathematics and chemistry in a research work connected with the petrochemical industry was fruitful, namely, in the study of the origin of the lower critical solution temperature (LCST) on the liquid-liquid equilibrium of thiophene with ionic liquids. Ionic liquids (ILs) are a novel class of salts possessing unique physical and chemical properties, which have been in recent years the focus of a great interest for both academic and industrial researchers, and are an excellent choice to replace common molecular organic solvents in several applications, namely, extractive solvents for the desulfurization of refined products, as fuel.

Other examples, related with the development of geophysical navigation methods with applications to the navigation of autonomous underwater vehicles (AUVs), may also be mentioned.

Other examples, related with the development of geophysical navigation methods with applications to the navigation of autonomous underwater vehicles (AUVs), may also be mentioned.

Saburou Saitoh

## Aveiro discretization method in mathematics: a new discretization principle

We will present a very general discretization method for solving wide classes of mathematical problems by applying the theory of reproducing kernels. An illustration of the generality of the method will be performed by considering several distinct classes of problems to which the method is applied. In fact, one of the advantages of the present method is its global nature and no need of special or very particular data conditions. Numerical experiments have been made, and consequent results will be exhibited. Due to the powerful results which arise from the application of the present method, we consider that this method has everything to become one of the next generation methods of solving general analytical problems by using computers. In particular, we would like to point out that we will be able to solve very global partial differential equations satisfying very general boundary conditions or initial values (and in a somehow independent way of the boundary and domain). Furthermore, we will be able to give an ultimate sampling theory and an ultimate realization of the consequent general reproducing kernel Hilbert spaces.

The talk is based on joint work with L.P. Castro, H. Fujiwara, M.M. Rodrigues and V.K. Tuan.

The talk is based on joint work with L.P. Castro, H. Fujiwara, M.M. Rodrigues and V.K. Tuan.

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Vasile Staicu

## Dirichlet problems with singular and superlinear terms

We consider a parametric nonlinear Dirichlet problem driven by the p-Laplacian, with a singular term and a p- superlinear perturbation, which needs not satisfy the usual in such cases Ambrosetti-Rabinowitz condition. Using variational methods together with truncation techniques, we prove a bifurcation-type theorem describing the behavior of the set of positive solutions as the parameter varies.