The reconstruction in the Calderon inverse conductivity problem is considered. A suitable linearization of the inverse problem allows us to define the so-called Born approximation. This provides a good reconstruction only for the high frequencies of the conductivity. In a recent work, we established a new formula relating this Born approximation with the Dirichlet to Neumann map and the momenta of the conductivity coefficient. Based on this formula, we give a new convergent iterative algorithm that improves the numerical reconstruction given by the Born approximation. We illustrate the process with numerical experiments for radial and non-radial potentials.

 This is part of a collaboration work with J.A. Barceló, F. Macià, D. Mendoza,  and C. Meroño.

 The Hansen and Spies laminated beam model is a model that de-

scribes the interaction between two identical Timoshenko beams that are bonded together by means of an adhesive layer of negligible thickness and mass, allowing interfacial sliding. It is a system of three coupled hy-perbolic equations. In this lecture, we will present several recent results on the stability and lack of stability of the semigroup associated with this model when subjected to the action of various dissipative mechanisms.

In the 1970s, H.O Fattorini and D.L Russell, [4], [5], linked the method of moments to the study of the controllability at time T > 0 of the heat equation. The basic argument is the construction and estimation of bi-orthogonal families in L2(0T) a {exp- landakt, k>= 1} where landak, k>= 1 is the sequence of eigenvalues of Sturm Liouville operators. Muntzs theorem and Weyls formula require us to restrict ourselves to the dimension 1 of space. In recent years works such as, for example, [1], [3], [2] have shown that even in the case of parabolic equations, controllability to 0 at time T could require T >=T0 with T0 > 0. These results, based on high frequency phenomena, are obtained by deepening the work of [5] on the construction and estimation of families bi-orthogonal in L2(0T) a { exp- landakt,landa in F} where F contained in C. Consequently, they are restricted to the dimension 1 of space. The objective of this talk is to take a rst step in the construction of bi-orthogonal families in L2((0T)Xw) (in space-time) a  { exp- landakt,landa in F} . F in R, Phi landa,landa in F contained in L2(Omega ),w contained in Omega,Omega contained in  Rd,d >=1.

 I’ll present the so called Localized Induction Approximation that de- scribes the dynamics of a vortex filament according to the Binormal Curvature Flow (BF). I’ll give a result about the desingularization of the Biot-Savart in- tegral proved with Marco A. Fontelos within the framework of Navier-Stokes equations. Some particular examples regarding BF obtained with Valeria Ban- ica will be also considered. These examples allow to connect BF with the so- called Riemann non-differentiable function and the Frisch-Parisi approach to turbulence. 

 In this talk, we will introduce and analyze the concept of tracking controllability, where the objective is for the state of a control system, or some projection thereof, to follow a given trajectory within a specified time horizon. By leveraging duality, we characterize tracking controllability through a non-standard observability inequality for the adjoint system. This characterization facilitates the development of a systematic approach for constructing controls with minimal norm. However, achieving such an observability property is challenging. To address this, we utilize the classical Brunovsky canonical form for control systems, specifically focusing on the case of scalar controls. Our findings reveal that tracking control inherently involves a loss of regularity in the controls. This phenomenon is highly sensitive to the system’s structure and the interaction between the projection to be tracked and the system.

We consider the Cauchy problem for the wave  equation with fast-oscillating coefficient and localized initial perturbation in the whole space. This problem contains two small parameters. The first parameter is the localization parameter and it describes the wavelength of the propagating wave. The second parameter 

is the parameter of the fast oscillations of the coefficient.

The global task is to describe the asymptotic solution of the given problem. To do so, we reduce the initial problem to the homogenized one with the equation with variable and smooth coefficients. The localization of the initial conditions allows us to use the semiclassical approximation for the asymptotic description of the solution 

of the homogenized equation.

It turns out that depending on the ratio between two given small parameters the dispersion effects may appear during the asymptotic description of the solution of the homogenized problem.

Our aim in the present talk is to discuss the mechanism of appearance of these dispersion effects and to derive the effective homogenized equations which describe the asymptotic solution with and without dispersion effects.

In this talk we present some recent result about the long time behavior
of the solutions for some diffusion processes on a metric graph. We study evolution problems on a metric connected finite graph in which some of the edges have infinity length. We show that the asymptotic behaviour of the solutions of the heat equation (or even some nonlocal diffusion problems) is given by the solution of the heat equation, but on a star shaped graph in which there is only one node and as many infinite edges as in the original graph. In this way we obtain that the compact component that consists in all the vertices and all the edges of finite length can be reduced to a single point when looking at the asymptotic behaviour of the solutions. We prove that when time is large the solution behaves like a gaussian profile on the infinite edges. When the nonlinear convective part is present we obtain similar results but only on a star shaped tree.
Acknowledgment: this is a joint work with Cristian Cazacu (University
of Bucharest), Ademir Pazoto (Federal University of Rio de Janeiro), Julio D. Rossi (University of Buenos Aires) and Angel San Antolin (University of Alicante).