Lower semi-continuity of integrals with G-quasiconvex potential

This paper introduces the proper notion of variational quasiconvexity associated to a group of diffeomorphisms. We prove a lower semicontinuity theorem connected to this notion. In the second part of the paper we apply this result to a class of functions introduced in [5]. Such functions are $GL(n,R)^{+}$ quasiconvex, hence they induce lower semicontinuous integrals

What is a space? Computations in emergent algebras and the front end visual system

With the help of link diagrams with decorated crossings, I explain computations in emergent algebras, introduced in arXiv:0907.1520, as the kind of computations done in the front end visual system.Comment: comments welcome

Curvature of sub-Riemannian spaces

To any metric spaces there is an associated metric profile. The rectifiability of the metric profile gives a good notion of curvature of a sub-Riemannian space. We shall say that a curvature class is the rectifiability class of the metric profile. We classify then the curvatures by looking to homogeneous metric spaces. The classification problem is solved for contact 3 manifolds, where we rediscover a 3 dimensional family of homogeneous contact manifolds, with a distinguished 2 dimensional family of contact manifolds which don't have a natural group structure

Braided spaces with dilations and sub-riemannian symmetric spaces

Braided sets which are also spaces with dilations are presented and explored in this paper, in the general frame of emergent algebras arxiv:0907.1520. Examples of such spaces are the sub-riemannian symmetric spaces. Keywords: braided sets, quandles; emergent algebras; dilatation structures (spaces with dilations); sub-riemannian symmetric spaces.Comment: Final version, appeared in: Geometry. Exploratory Workshop on Differential Geometry and its Applications, eds. D. Andrica, S. Moroianu, Cluj-Napoca 2011, 21-3