The energy-critical defocusing NLS on T^3

We prove global well-posedness in H^1(T^3) for the energy-critical defocusing NLS.Comment: 24 page

On the particle paths and the stagnation points in small-amplitude deep-water waves

In order to obtain quite precise information about the shape of the particle paths below small-amplitude gravity waves travelling on irrotational deep water, analytic solutions of the nonlinear differential equation system describing the particle motion are provided. All these solutions are not closed curves. Some particle trajectories are peakon-like, others can be expressed with the aid of the Jacobi elliptic functions or with the aid of the hyperelliptic functions. Remarks on the stagnation points of the small-amplitude irrotational deep-water waves are also made.Comment: to appear in J. Math. Fluid Mech. arXiv admin note: text overlap with arXiv:1106.382

Nonlinear cross Gramians and gradient systems

We study the notion of cross Gramians for non-linear gradient systems, using the characterization in terms of prolongation and gradient extension associated to the system. The cross Gramian is given for the variational system associated to the original nonlinear gradient system. We obtain linearization results that precisely correspond to the notion of a cross Gramian for symmetric linear systems. Furthermore, first steps towards relations with the singular value functions of the nonlinear Hankel operator are studied and yield promising results.

Rhythms of division of some algae in situ and in laboratory culture. [Translation from: Studii Cerc.Biol.(Bot.) 21, 61-65, 1969. ]

The rhythm of division of 9 species belonging to different groups of algae were analysed in situ and in the laboratory. The research which developed in different environmental conditions attempted to establish the capacity for multiplication and assimilation of chlorophyll on the part of the algae under study with a view to placing them in a culture. The results obtained showed that the green multicellular algae (eg. Ulothrix) and the blue algae (eg. Lyngbya, Oscillatoria) are able to produce an appreciable quantity of dry matter, just as the unicellular algae. At the same time it arises that amongst the numerous factors of the environment, temperature plays one of the most important roles in the process of multiplication