The eigenvalue problem for the resonances of finite-dimensional Friedrichs models on the positive half line is solved using an appropriate Gelʼfand triplet. The associated Gamow vectors are uniquely determined by the corresponding eigenantilinear forms. They turn out to be the restriction of the eigenantilinear forms to the Hardy space part of the Gelʼfand space. Conditions are presented such that there are only finitely many resonances and all resonances are simple poles of the scattering matrix.

Spacelike constant mean curvature (CMC) surfaces in Minkowski 3-space $\mathbb{L}^3$ have an infinite dimensional generalized Weierstrass representation. This is analogous to that given by Dorfmeister, Pedit and Wu for CMC surfaces in Euclidean space, replacing the group $\mathrm{SU}(2)$ with $\mathrm{SU}(1,1)$. The non-compactness of the latter group, however, means that the Iwasawa decomposition of the loop group, used to construct the surfaces, is not global. The construction is described here, with an emphasis on the difference from the Euclidean case.

In this survey we describe a general method to deal with the variational problem associated to the generalized elastic curves, paying special attention to closed critical curves in real space forms due to its special geometric significance. We illustrate the method by studying particular choices of this energy in some more detail. Finally, we will review also some interesting applications of generalized elasticae to other higher dimensional variational problems in Physics, Biophysics and the Theory of Submanifolds.

In this paper we study some properties of a Lie algebroid and its prolongation over the vector bundle projection of the dual bundle. We generalize some results on Poisson manifolds to the level of a Lie algebroid. The notions of canonical Poisson bivector and horizontal lift are studied and their compatibility conditions are pointed out.

A thin, flexible, one side-built-in rod under a concentrated terminal force is studied in its elastic equilibrium configuration. In order to make the problem more tractable, a proper set of state variables is chosen, facing with a second order, nonlinear, but autonomous boundary value problem, in the rotation $\varphi (s)$ pertaining to each $s$-section. The search of the free end rotation $\varphi _{0}$, following the isoperimetric assumption, leads to a numerical sub-problem inside the main BVP. Furthermore, if $x(s)$ and $y(s)$ mean the elastica coordinates parametrized on the arclength $s$, one obtains $x^{\prime }(s)$ and $y^{\prime }(s)$ as elliptic functions of $s$. Finally, some minor changes have been shown in order to pass from a loading force to a more general free-end load combination, consisting of a force and a couple.