Algebra versus analysis in the theory of flexible polyhedra

Two basic theorems of the theory of flexible polyhedra were proven by completely different methods: R. Alexander used analysis, namely, the Stokes theorem, to prove that the total mean curvature remains constant during the flex, while I.Kh. Sabitov used algebra, namely, the theory of resultants, to prove that the oriented volume remains constant during the flex. We show that none of these methods can be used to prove the both theorems. As a by-product, we prove that the total mean curvature of any polyhedron in the Euclidean 3-space is not an algebraic function of its edge lengths.Comment: 5 pages, 5 figures; condition (iii) in Theorem 5 is correcte

On the total mean curvature of non-rigid surfaces

Using Green's theorem we reduce the variation of the total mean curvature of a smooth surface in the Euclidean 3-space to a line integral of a special vector field and obtain the following well-known theorem as an immediate consequence: the total mean curvature of a closed smooth surface in the Euclidean 3-space is stationary under an infinitesimal flex.Comment: 4 page

Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds

We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold M. Under the assumption that the sectional curvature of M is strictly positive, we prove the existence of a smoothly immersed sphere minimizing the L^{2} integral of the second fundamental form. Assuming instead that the sectional curvature is less than or equal to 2, and that there exists a point in M with scalar curvature bigger than 6, we obtain a smooth 2-sphere minimizing the integral of 1/4|H|^{2} +1, where H is the mean curvature vector

Market Power and Collusion on Interconnection Phone Market in Tunisia : What Lessons from International Experiences

We try in this paper to characterize the state of mobile phone market in Tunisia. Our study is based on a survey of foreign experience (Europe) in detecting collusive behavior and a comparison of the critical threshold of collusion between operators in developing countries like Tunisia. The market power is estimated based on the work of Parker Roller (1997) and the assumption of "Balanced Calling Pattern". We use then the model of Friedman (1971) to compare the critical threshold of collusion. We show that the "conduct parameter" measuring the intensity of competition is not null during the period 1993-2011. Results show also that collusion is easier on the Tunisian market that on the Algerian, Jordanian, or Moroccan one