María del Mar González

Voltammetric determination of linuron at a carbon-paste electrode modified with sepiolite

The determination of linuron by differential pulse voltammetry with a carbon-paste electrode modified with 20% w/w sepiolite has been studied. The linuron is preconcentrated under open-circuit conditions at pH 2.0. With 0.01M potassium nitrate at pH 1.7 in the measurement cell, a sweep rate of 30 mV/sec and a pulse amplitude of 100 mV, an oxidation wave with a peak potential of 1.2 V is obtained. Under these conditions, determination limits of 75 ng/ml have been obtained, with a relative error of +2.8% and a relative standard deviation of 8.0%. The method has been applied to the direct determination of linuron in river water with no previous separation of the pesticide. Determination in sea-water is not possible, as chloride interferes at high concentration.

Singular sets of a class of locally conformally flat manifolds

We look at complete, locally conformally flat metrics defined on a domain Ω ⊂ S. There is a lot of information about the singular set ∂Ω contained in the positivity of σk and, in particular, we obtain a bound for the Hausdorff dimension of ∂Ω, in relation to Schoen-Yau’s work for the scalar curvature. On the other hand, since some locally conformally flat manifolds can be embedded into S, this dimension bound implies several topological corollaries.

Some constructions for the fractional Laplacian on noncompact manifolds

We give a denition of the fractional Laplacian on some noncompact manifolds, through an extension problem introduced by Caarelli-Silvest re. While this denition in the compact case is straightforward, in the noncompact setting one needs to have a precise control of the behavior of the metric at innity and geometry plays a crucial role. First we give explicit calculations in the hyperbolic space, including a formula for the kernel and a trace Sobolev inequality. Then we consider more general noncompact manifolds, where the problem reduces to obtain suitable upper bounds for the heat kernel.

Global Existence and Uniqueness of Solutions to a Model of Price Formation

We study a model, due to J. M. Lasry and P. L. Lions, describing the evolution of a scalar price which is realized as a free boundary in a one-dimensional diffusion equation with dynamically evolving, nonstandard sources. We establish global existence and uniqueness.

Classical Solutions for a Nonlinear Fokker-Planck Equation Arising in Computational Neuroscience

In this paper we analyze the global existence of classical solutions to the initial boundary-value problem for a nonlinear parabolic equation describing the collective behavior of an ensemble of neurons. These equations were obtained as a diffusive approximation of the mean-field limit of a stochastic differential equation system. The resulting nonlocal Fokker-Planck equation presents a nonlinearity in the coefficients depending on the probability flux through the boundary. We show by an appropriate change of variables that this parabolic equation with nonlinear boundary conditions can be transformed into a non standard Stefan-like free boundary problem with a Dirac-delta source term. We prove that there are global classical solutions for inhibitory neural networks, while for excitatory networks we give local well-posedness of classical solutions together with a blow up criterium. Surprisingly, we will show that the spectrum for the operator in the linear case, that corresponding to a system of uncoupled networks, does not give any information about the large time asymptotic behavior.

A nonlocal diffusion problem on manifolds

ABSTRACT In this paper we study a nonlocal diffusion problem on a manifold. These kinds of equations can model diffusions when there are long range effects and have been widely studied in Euclidean space. We first prove existence and uniqueness of solutions and a comparison principle. Then, for a convenient rescaling we prove that the operator under consideration converges to a multiple of the usual Heat-Beltrami operator on the manifold. Next, we look at the long time behavior on compact manifolds by studying the spectral properties of the operator. Finally, for the model case of hyperbolic space we study the long time asymptotics and find a different and interesting behavior.

Existence and Uniqueness to a Fully Nonlinear Version of the Loewner–Nirenberg Problem

We consider the problem of finding on a given Euclidean domain

Fractional Laplacian in conformal geometry

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