Nonlinear equations involving the square root of the Laplacian

In this paper we discuss the existence and non-existence of weak solutions to parametric fractional equations involving the square root of the Laplacian $A_{1/2}$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$ ($n\geq 2$) and with zero Dirichlet boundary conditions. Namely, our simple model is the following equation \begin{equation*} \left\{ \begin{array}{ll} A_{1/2}u=\lambda f(u) & \mbox{ in } \Omega\\ u=0 & \mbox{ on } \partial\Omega. \end{array}\right. \end{equation*} The existence of at least two non-trivial $L^{\infty}$-bounded weak solutions is established for large value of the parameter $\lambda$ requiring that the nonlinear term $f$ is continuous, superlinear at zero and sublinear at infinity. Our approach is based on variational arguments and a suitable variant of the Caffarelli-Silvestre extension method

The Monge problem in Wiener Space

We address the Monge problem in the abstract Wiener space and we give an existence result provided both marginal measures are absolutely continuous with respect to the infinite dimensional Gaussian measure {\gamma}

Monte Carlo simulation of an experiment looking for radiative solar neutrino decays

We analyse the possibility of detecting visible photons from a hypothetical radiative decay of solar neutrinos. Our study is focused on the simulation of such measurements during total solar eclipses and it is based on the BP2000 Standard Solar Model and on the most recent experimental information concerning the neutrino properties.Comment: 13 pages, 10 figures, accepted by Astropart. Phy

A model for the quasi-static growth of brittle fractures based on local minimization

We study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo. The main feature of our model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the $L^2$-distance between the approximate solutions at two consecutive times. We study the continuous-time version of this model, obtained by passing to the limit as the time step tends to zero, and show that it satisfies (for almost every time) some minimality conditions which are slightly different from those considered in Francfort and Marigo and in our previous paper, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffith's criterion holds at the crack tips. We prove also that, if no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this model, provided the penalization term is large enough.Comment: 20 page

A new multiparametric topological method for determining the primary cosmic ray mass composition in the knee energy region

The determination of the primary cosmic ray mass composition from the characteristics of extensive air showers (EAS), obtained at an observation level in the lower half of the atmosphere, is still an open problem. In this work we propose a new method of the Multiparametric Topological Analysis and show its applicability for the determination of the mass composition of the primary cosmic rays at the PeV energy region.Comment: 8 pages, 4 figures, talk given at Vulcano 2004 Workshop 'Frontier Objects in Physics and Astrophysics', Vulcano, Italy, 24-29.05.04, to be published in the Proceedings of the Worksho

Mathematics Education as a response to societal needs

In this paper I will discuss alternative proposals for Mathematics Education, taking into account its impact in the social processes. I start with discussing some of the recognized values for Mathematics Education. Essentially, the following set of six basic values are accepted: 1. Utilitarian; 2. Academic; 3. Formative (of reasoning); 4. Cultural; 5. Sociological; 6. Aesthetic. Affecting all these values, the question of assessment is of fundamental importance

El programa Etnomatemáticas: bases cognitivas, antropológicas, históricas y socioculturales

This document was originally presented at the Segundo Coloquio Internacional de la Teoría de la Objetivación, Laurentian University, Canadá; Universidad Distrital Francisco José de Caldas, Colombia; Universidad Pedagógica Nacional, Mexico: Instituto Politécnico Nacional, México, 17 al 20 de enero de 2017.We discuss the concept of knowledge and the cognitive process backing the evolution of knowledge in the human species. This leads to a reflection about the meaning peace. In simple terms, peace is the capability of dealing with conflicts [which are unavoidable as a result of individual differences] without the resource to arrogance and to bigotry, which culminates in aggression and violent confrontation. A road to peace is the reformulation of education. A proposal of a new concept of curriculum is presented and there is a brief discussion of the Program Ethnomathematics as the realization of the proposal.Discutimos el concepto de conocimiento y el proceso cognitivo de soporte a la evolución del conocimiento en la especie humana. Esto conduce a una reflexión sobre el significado de paz. En términos simples, la paz es la capacidad de lidiar con conflictos [que son inevitables como consecuencia de las diferencias individuales] sin el recurso a la arrogancia y fanatismo, que culmina en agresión y la confrontación violenta. Un camino hacia la paz es la reformulación de la educación. Se presenta una propuesta de un nuevo concepto de currículo y hay una breve discusión del Programa Etnomatemáticas como la realización de la propuesta.Universidad de Granada. Grupo de Investigación Didáctica de la Matemática: Pensamiento Numérico (FQM-193

Geodesics in the space of measure-preserving maps and plans

We study Brenier's variational models for incompressible Euler equations. These models give rise to a relaxation of the Arnold distance in the space of measure-preserving maps and, more generally, measure-preserving plans. We analyze the properties of the relaxed distance, we show a close link between the Lagrangian and the Eulerian model, and we derive necessary and sufficient optimality conditions for minimizers. These conditions take into account a modified Lagrangian induced by the pressure field. Moreover, adapting some ideas of Shnirelman, we show that, even for non-deterministic final conditions, generalized flows can be approximated in energy by flows associated to measure-preserving maps

Precise determination of muon and electromagnetic shower contents from shower universality property

We consider two new aspects of Extensive Air Shower development universality allowing to make accurate estimation of muon and electromagnetic (EM) shower contents in two independent ways. In the first case, to get muon (or EM) signal in water Cherenkov tanks or in scintillator detectors it is enough to know the vertical depth of shower maximum and the total signal in the ground detector. In the second case, the EM signal can be calculated from the primary particle energy and the zenith angle. In both cases the parametrizations of muon and EM signals are almost independent on primary particle nature, energy and zenith angle. Implications of the considered properties for mass composition and hadronic interaction studies are briefly discussed. The present study is performed on 28000 of proton, oxygen and iron showers, generated with CORSIKA 6.735 for $E^{-1}$ spectrum in the energy range log(E/eV)=18.5-20.0 and uniformly distributed in cos^2(theta) in zenith angle interval theta=0-65 degrees for QGSJET II/Fluka interaction models.Comment: Submitted to Phys. Rev.