Institute Ciencias Matematicas ICMAT

Madrid, Spain

Institute Ciencias Matematicas ICMAT

Madrid, Spain
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Docampo R.,Institute Ciencias Matematicas ICMAT | Nigro A.,Federal University of Fluminense
Advances in Mathematics | Year: 2017

We study the arc space of the Grassmannian from the point of view of the singularities of Schubert varieties. Our main tool is a decomposition of the arc space of the Grassmannian that resembles the Schubert cell decomposition of the Grassmannian itself. Just as the combinatorics of Schubert cells is controlled by partitions, the combinatorics in the arc space is controlled by plane partitions (sometimes also called 3d partitions). A combination of a geometric analysis of the pieces in the decomposition and a combinatorial analysis of plane partitions leads to invariants of the singularities. As an application we reduce the computation of log canonical thresholds of pairs involving Schubert varieties to an easy linear programming problem. We also study the Nash problem for Schubert varieties, showing that the Nash map is always bijective in this case. © 2016 Elsevier Inc.

Breschi G.,Institute Ciencias Matematicas ICMAT | Fontelos M.A.,Institute Ciencias Matematicas ICMAT
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences | Year: 2017

We provide a method to compute self-similar solutions for various fragmentation equations and use it to compute their asymptotic behaviours. Our procedure is applied to specific cases: (i) the case of mitosis, where fragmentation results into two identical fragments, (ii) fragmentation limited to the formation of sufficiently large fragments, and (iii) processes with fragmentation kernel presenting a power-like behaviour. © 2017 The Author(s) Published by the Royal Society. All rights reserved.

Kim H.J.,Pohang University of Science and Technology | Kim H.J.,The Interdisciplinary Center | Fontelos M.A.,Institute Ciencias Matematicas ICMAT | Hwang H.J.,Pohang University of Science and Technology
Journal of Mathematical Fluid Mechanics | Year: 2017

We study the capillary oscillations of the surface of a 2D drop attached to a fan-shaped pillar. The fluid flow is modeled by means of a velocity potential and we assume a no-flux condition at the liquid–solid interface. The natural oscillation frequencies and oscillation modes are computed for two different physical situations depending on the contact line behavior: (1) free-end, when the contact line moves along the solid with a constant contact angle and (2) pinned-end when the contact line is pinned to the solid and does not move. We also study the linearized initial value problem and prove well-posedness results in both free-end and pinned-end cases. Hence, for capillary oscillations when the fluid is in partial contact with a solid, not only initial conditions must be prescribed but also the behavior of the contact line. © 2016, Springer International Publishing.

Fontelos M.A.,Institute Ciencias Matematicas ICMAT | De La Hoz F.,University of the Basque Country
Journal of Fluid Mechanics | Year: 2010

We describe, by means of asymptotic methods and direct numerical simulation, the structure of singularities developing at the interface between two perfect, inviscid and irrotational fluids of different densities 1 and 2 and under the action of gravity. When the lighter fluid is on top of the heavier fluid, one encounters the water-wave problem for fluids of different densities. In the limit when the density of the lighter fluid is zero, one encounters the classical water-wave problem. Analogously, when the heavier fluid is on top of the lighter fluid, one encounters the Rayleigh-Taylor problem for fluids of different densities, with this being the case when one of the densities is zero for the classical Rayleigh-Taylor problem. We will show that both water-wave and Rayleigh-Taylor problems develop singularities of the Moore-type (singularities in the curvature) when both fluid densities are non-zero. For the classical water-wave problem, we propose and provide evidence of the development of a singularity in the form of a logarithmic spiral, and for the classical Rayleigh-Taylor problem no singularities were found. The regularizing effects of surface tension are also discussed, and estimates of the size and wavelength of the capillary waves, bubbles or blobs that are produced are provided. © 2010 Cambridge University Press.

Fontelos M.A.,Institute Ciencias Matematicas ICMAT | Hong S.H.,Pohang University of Science and Technology | Hwang H.J.,Pohang University of Science and Technology
Archive for Rational Mechanics and Analysis | Year: 2015

We study the problem of evaporating drops contracting to a point. Going back to Maxwell and Langmuir, the existence of a spherical solution for which evaporating drops collapse to a point in a self-similar manner is well established in the physical literature. The diameter of the drop follows the so-called D2 law: the second power of the drop-diameter decays linearly in time. In this study we provide a complete mathematical proof of this classical law. We prove that evaporating drops which are initially small perturbations of a sphere collapse to a point and the shape of the drop converges to a self-similar ellipsoid whose center, orientation, and semi-axes are determined by the initial shape. © 2014, Springer-Verlag Berlin Heidelberg.

PubMed | Institute Ciencias Matematicas ICMAT, Technical University of Madrid and Comision Nacional de la Energia Atomica
Type: Journal Article | Journal: The Journal of chemical physics | Year: 2017

In this paper, we extend a method recently reported [F. Revuelta et al., Phys. Rev. E 87, 042921 (2013)] for the calculation of the eigenstates of classically highly chaotic systems to cases of mixed dynamics, i.e., those presenting regular and irregular motions at the same energy. The efficiency of the method, which is based on the use of a semiclassical basis set of localized wave functions, is demonstrated by applying it to the determination of the vibrational states of a realistic molecular system, namely, the LiCN molecule.

Palazuelos C.,Institute Ciencias Matematicas ICMAT
Physical Review Letters | Year: 2012

In this Letter we show that quantum nonlocality can be superactivated. That is, one can obtain violations of Bell inequalities by tensorizing a local state with itself. In the second part of this work we study how large these violations can be. In particular, we show the existence of quantum states with very low Bell violation but such that five copies of them give very large violations. In fact, this gap can be made arbitrarily large by increasing the dimension of the states. © 2012 American Physical Society.

Eggers J.,University of Bristol | Fontelos M.A.,Institute Ciencias Matematicas ICMAT | Josserand C.,CNRS Jean Le Rond d'Alembert Institute | Zaleski S.,CNRS Jean Le Rond d'Alembert Institute
Physics of Fluids | Year: 2010

We study the impact of a fluid drop onto a planar solid surface at high speed so that at impact, kinetic energy dominates over surface energy and inertia dominates over viscous effects. As the drop spreads, it deforms into a thin film, whose thickness is limited by the growth of a viscous boundary layer near the solid wall. Owing to surface tension, the edge of the film retracts relative to the flow in the film and fluid collects into a toroidal rim bounding the film. Using mass and momentum conservation, we construct a model for the radius of the deposit as a function of time. At each stage, we perform detailed comparisons between theory and numerical simulations of the Navier-Stokes equation. © 2010 American Institute of Physics.

Breschi G.,Institute Ciencias Matematicas ICMAT | Fontelos M.A.,Institute Ciencias Matematicas ICMAT
Nonlinearity | Year: 2014

We study the self-similar solutions of Smoluchowski's equation with kernel K(x, y) = x1-εy1-ε for ε > 0 and ε ≪ 1. We prove that by choosing the similarity exponents as suitable functions of ε, the self-similar solutions present correct behaviours at the origin and at infinity, which amounts to solving a nonlinear eigenvalue problem. This characterizes the self-similar solution found as being of the second kind in the notation introduced by Barenblatt. © 2014 IOP Publishing Ltd & London Mathematical Society.

Fontelos M.A.,Institute Ciencias Matematicas ICMAT | Kindelan U.,Technical University of Madrid
IOP Conference Series: Materials Science and Engineering | Year: 2014

In this work we study the static shape of charged drops of a conducting fluid placed over a solid substrate, surrounded by a gas, and in absence of gravitational forces. The problem can be posed, since Gauss, in a variational setting consisting of obtaining the configurations of a given mass of fluid that minimize (or in general make extremal) a certain energy involving the areas of the solid-liquid interface and of the liquid-gas interface, as well as the electric capacity of the drop. In [6] we have found, as a function of two parameters, Young's angle θYand the potential at the drop's surface V0, the axisymmetric minimizers of the energy. In the same article we have also described their shape and showed the existence of symmetry-breaking bifurcations such that, for given values of θYand V0, configurations for which the axial symmetry is lost are energetically more favorable than axially symmetric configurations. We have proved the existence of such bifurcations in the limits of very flat and almost spherical equilibrium shapes. In this work we study all other cases numerically. When dealing with radially perturbed equilibrium shapes we lose the axially symmetric properties and need to do a full three-dimensional approximation in order to compute area and capacity and hence the energy. We use a boundary element method that we have already implemented in [3] to compute the surface charge density. From the surface charge density we can obtain the capacity of the body. One conclusion of this study is that axisymmetric drops cannot spread indefinitely by introducing sufficient amount of electric charges, but can reach only a limiting (saturation) size, after which the axial symmetry would be lost and finger-like shapes energetically preferred. © 2010 IOP Publishing Ltd.

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