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Berselli L.C.,University of Pisa | Cordoba D.,Institute Ciencias Matematicas CSIC UAM UC3M UCM | Granero-Belinchon R.,University of California at Davis
Interfaces and Free Boundaries | Year: 2014

In this work we study the evolution of the free boundary between two different fluids in a porous medium where the permeability is a two dimensional step function. The medium can fill the whole plane R2 or a bounded strip S D Rx (-π/2, π/2). The system is in the stable regime if the denser fluid is below the lighter one. First, we show local existence in Sobolev spaces by means of energy method when the system is in the stable regime. Then we prove the existence of curves such that they start in the stable regime and in finite time they reach the unstable one. This change of regime (turning) was first proven in [5] for the homogenous Muskat problem with infinite depth. © European Mathematical Society 2014.

Ascasibar Y.,Autonomous University of Madrid | Granero-Belinchon R.,Institute Ciencias Matematicas CSIC UAM UC3M UCM | Granero-Belinchon R.,University of California at Davis | Moreno J.M.,Institute Ciencias Matematicas CSIC UAM UC3M UCM
Physica D: Nonlinear Phenomena | Year: 2013

This work studies a simplified model of the gravitational instability of an initially homogeneous infinite medium, represented by Td, based on the approximation that the mean fluid velocity is always proportional to the local acceleration. It is shown that, mathematically, this assumption leads to the restricted Patlak-Keller-Segel model considered by Jäger and Luckhaus or, equivalently, the Smoluchowski equation describing the motion of self-gravitating Brownian particles, coupled to the modified Newtonian potential that is appropriate for an infinite mass distribution. We discuss some of the fundamental properties of a non-local generalization of this model where the effective pressure force is given by a fractional Laplacian with 0<α<2 and illustrate them by means of numerical simulations. Local well-posedness in Sobolev spaces is proven, and we show the smoothing effect of our equation, as well as a Beale-Kato-Majda-type criterion in terms of â€-ρâ-L. © 2013 Elsevier B.V. All rights reserved.

Bernardi O.,University Paris - Sud | Rue J.,Institute Ciencias Matematicas CSIC UAM UC3M UCM
European Journal of Combinatorics | Year: 2012

It is well-known that the triangulations of the disc with n+2 vertices on its boundary are counted by the nth Catalan number C(n)=1/n+1 ( 2n n). This paper deals with the generalisation of this problem to any compact surface S with boundaries. We obtain the asymptotic number of simplicial decompositions of the surface S with n vertices on its boundary. More generally, we determine the asymptotic number of dissections of S{double-struck} when the faces are δ-gons with δ belonging to a set of admissible degrees δ⊆ δ {3, 4, 5,...}. We also give the limit laws for certain parameters of such dissections. © 2011 Elsevier Ltd.

Iglesias D.,University of La Laguna | Marrero J.C.,University of La Laguna | Vaquero M.,Institute Ciencias Matematicas CSIC UAM UC3M UCM
Letters in Mathematical Physics | Year: 2013

In this paper we introduce poly-Poisson structures as a higher-order extension of Poisson structures. It is shown that any poly-Poisson structure is endowed with a polysymplectic foliation. It is also proved that if a Lie group acts polysymplectically on a polysymplectic manifold then, under certain regularity conditions, the reduced space is a poly-Poisson manifold. In addition, some interesting examples of poly-Poisson manifolds are discussed. © 2013 Springer Science+Business Media Dordrecht.

Gomez-Ullate D.,Complutense University of Madrid | Gomez-Ullate D.,Institute Ciencias Matematicas CSIC UAM UC3M UCM | Grandati Y.,University of Lorraine | Milson R.,Dalhousie University
Journal of Physics A: Mathematical and Theoretical | Year: 2014

We prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free, and therefore can be obtained by applying a finite number of state-deleting Darboux transformations on the harmonic oscillator. Equivalently, every exceptional orthogonal polynomial system of Hermite type can be obtained by applying a Darboux-Crum transformation to the classical Hermite polynomials. Exceptional Hermite polynomial systems only exist for even codimension 2 m, and they are indexed by the partitions λ of m. We provide explicit expressions for their corresponding orthogonality weights and differential operators and a separate proof of their completeness. Exceptional Hermite polynomials satisfy a 2ℓ + 3 recurrence relation where ℓ is the length of the partition λ. Explicit expressions for such recurrence relations are given. © 2014 IOP Publishing Ltd.

De Leon M.,Institute Ciencias Matematicas CSIC UAM UC3M UCM | Vilarino S.,Centro Universitario Of La Defensa
International Journal of Geometric Methods in Modern Physics | Year: 2014

In this paper, we extend the geometric formalism of the Hamilton-Jacobi theory for time-dependent Mechanics to the case of classical field theories in the k-cosymplectic framework. © 2014 World Scientific Publishing Company.

Colombo L.,University of Michigan | De Diego D.M.,Institute Ciencias Matematicas CSIC UAM UC3M UCM
Journal of Geometric Mechanics | Year: 2014

In this paper, we describe a geometric setting for higher-order Lagrangian problems on Lie groups. Using left-trivialization of the higher-order tangent bundle of a Lie group and an adaptation of the classical Skinner- Rusk formalism, we deduce an intrinsic framework for this type of dynamical systems. Interesting applications as, for instance, a geometric derivation of the higher-order Euler-Poincaré equations, optimal control of underactuated control systems whose configuration space is a Lie group are shown, among others, along the paper © American Institute of Mathematical Sciences.

Lopez-Fernandez M.,Institute Ciencias Matematicas CSIC UAM UC3M UCM
BIT Numerical Mathematics | Year: 2010

We present a quadrature-based method to evaluate exponential-like operators required by different kinds of exponential integrators. The method approximates these operators by means of a quadrature formula that converges like O(e-cK), with K the number of quadrature nodes, and it is useful when solving parabolic equations. The approach allows also the evaluation of the associated scalar mappings. The method is based on numerical inversion of sectorial Laplace transforms. Several numerical illustrations are provided to test the algorithm, including examples with a mass matrix and the application of the method inside the MATLAB package EXP4, an adaptive solver based on an exponential Runge-Kutta method. © 2010 Springer Science + Business Media B.V.

Luca R.,Institute Ciencias Matematicas CSIC UAM UC3M UCM | Rogers K.M.,Institute Ciencias Matematicas CSIC UAM UC3M UCM
Communications in Mathematical Physics | Year: 2016

We consider Carleson’s problem regarding convergence for the Schrödinger equation in dimensions (Formula presented.). We show that if the solution converges almost everywhere with respect to (Formula presented.)-Hausdorff measure to its initial datum as time tends to zero, for all data (Formula presented.), then (Formula presented.). This strengthens and generalises results of Bourgain and Dahlberg–Kenig. © 2016 Springer-Verlag Berlin Heidelberg

Garcia-Fernandez M.,Institute Ciencias Matematicas CSIC UAM UC3M UCM
Communications in Mathematical Physics | Year: 2014

This work revisits the notions of connection and curvature in generalized geometry, with emphasis on torsion-free generalized connections on a transitive Courant algebroid. As an application, we provide a mathematical derivation of the equations of motion of heterotic supergravity in terms of the Ricci tensor of a generalized metric, inspired by the work of Coimbra, Strickland-Constable and Waldram. © 2014 Springer-Verlag Berlin Heidelberg.

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