On the reconstruction of conductivity of bordered two-dimensional surface in R^3 from electrical currents measurements on its boundary

An electrical potential U on bordered surface X (in Euclidien three-dimensional space) with isotropic conductivity function sigma>0 satisfies equation d(sigma d^cU)=0, where d^c is real operator associated with complex (conforme) structure on X induced by Euclidien metric of three-dimensional space. This paper gives exact reconstruction of conductivity function sigma on X from Dirichlet-to-Neumann mapping (for aforementioned conductivity equation) on the boundary of X. This paper extends to the case of the Riemann surfaces the reconstruction schemes of R.Novikov (1988) and of A.Bukhgeim (2008) given for the case of domains in two-dimensional Euclidien space. The paper extends and corrects the statements of Henkin-Michel (2008), where the inverse boundary value problem on the Riemann surfaces was firstly considered

Estimates for solutions of Burgers type equations and some applications

We obtain precise large time asymptotics for the Cauchy problem for Burgers type equations satisfying shock profile condition. The proofs are based on the exact a priori estimates for (local) solutions of these equations and a recent result of the first and second authors.Comment: to appear in J. Math. Pure et App

An Evolutionary Model with Interaction between Development and Adoption of New Technologies

We propose a difference-differential equation that reflects interactions between innovation and imitation processes to describe the evolution of the distribution curve of firms by efficiency levels. An explicit solution of this equation is obtained for arbitrary finite initial conditions. It is shown that this equation admits one-parametric family of logistic waves, and that arbitrary solution exponentially converges to one of the waves. This result explains two stylized empirical facts: the "logistic" shape of diffusion curves and the stable form of production capacities distribution by efficiency levels. Possible generalizations, modifications and applications are discussed.innovation; imitation; diffusion; logistic distribution; efficiency; wave solution; stability; Burgers equation