arXiv subjects

R. Orive

Publications and source records attributed to R. Orive.

6 recordsLinked to original sources

arXiv2026

Coulomb equilibrium in the external field of an attractive-repellent pair of charges

The aim of this paper is to provide a complete analysis of the Coulomb equilibrium problem in the euclidean space $\mathbb{R}^d$, $d\geq2$, associated to the kernel $1/|x|^{d-2}$, with a non-convex external field created by an attractive-repellent pair of charges placed in $\mathbb{R}^{d+1} \setminus \mathbb{R}^d$. We consider the admissible setting, where the equilibrium measure is compactly supported, as well as the limiting weakly admissible setting, with a weaker external field at infinity, where the existence of the equilibrium measure still holds but possibly with an unbounded support. The main tools for our analysis are the notions of signed equilibrium and balayage of measures. We note that for certain configurations of charges and distances to the conductor, the support of the equilibrium measure is a shell (multidimensional annulus).

math.CA↗

arXiv2016

Probability, Minimax Approximation and Nash-Equilibrium. Estimating the Parameter of a Biased Coin

This paper deals with the application of Approximation Theory type techniques to study a classical problem in Probability: estimating the parameter of a biased coin. For this purpose, a Minimax Estimation problem is considered and the characterization of the optimal estimator is shown, together with the weak asymptotics of such optimal choices as the number of coin tosses approaches infinity; in addition, a number of numerical examples and graphs are displayed. At the same time, the problem is also discussed from the Game Theory viewpoint, as a non-cooperative, two-player game, and a Nash-equilibrium is established. The particular case of n=2 tosses is completely solved.

math.PR↗

arXiv2013

Phase transitions and equilibrium measures in random matrix models

The paper is devoted to a study of phase transitions in the Hermitian random matrix models with a polynomial potential. In an alternative equivalent language, we study families of equilibrium measures on the real line in a polynomial external field. The total mass of the measure is considered as the main parameter, which may be interpreted also either as temperature or time. Our main tools are differentiation formulas with respect to the parameters of the problem, and a representation of the equilibrium potential in terms of a hyperelliptic integral. Using this combination we introduce and investigate a dynamical system (system of ODE's) describing the evolution of families of equilibrium measures. On this basis we are able to systematically derive a number of new results on phase transitions, such as the local behavior of the system at all kinds of phase transitions, as well as to review a number of known ones.

math.CA↗

arXiv2008

Incompressible flow in porous media with fractional diffusion

In this paper we study the heat transfer with a general fractional diffusion term of an incompressible fluid in a porous medium governed by Darcy's law. We show formation of singularities with infinite energy and for finite energy we obtain existence and uniqueness results of strong solutions for the sub-critical and critical cases. We prove global existence of weak solutions for different cases. Moreover, we obtain the decay of the solution in $L^p$, for any $p\geq2$, and the asymptotic behavior is shown. Finally, we prove the existence of an attractor in a weak sense and, for the sub-critical dissipative case with $\alpha\in (1,2]$, we obtain the existence of the global attractor for the solutions in the space $H^s$ for any $s > (N/2)+1-\alpha$.

math.AP↗

arXiv2004

Riemann-Hilbert analysis for Jacobi polynomials orthogonal on a single contour

Classical Jacobi polynomials $P_{n}^{(α,β)}$, with $α, β>-1$, have a number of well-known properties, in particular the location of their zeros in the open interval $(-1,1)$. This property is no longer valid for other values of the parameters; in general, zeros are complex. In this paper we study the strong asymptotics of Jacobi polynomials where the real parameters $α_n,β_n$ depend on $n$ in such a way that $$ \lim_{n\to\infty}\frac{α_{n}}{n}=A, \quad \lim_{n\to\infty}\frac{β_{n}}{n}=B, $$ with $A,B \in \mathbb{R}$. We restrict our attention to the case where the limits $A,B$ are not both positive and take values outside of the triangle bounded by the straight lines A=0, B=0 and $A+B+2=0$. As a corollary, we show that in the limit the zeros distribute along certain curves that constitute trajectories of a quadratic differential. The non-hermitian orthogonality relations for Jacobi polynomials with varying parameters lie in the core of our approach; in the cases we consider, these relations hold on a single contour of the complex plane. The asymptotic analysis is performed using the Deift-Zhou steepest descent method based on the Riemann-Hilbert reformulation of Jacobi polynomials.

math.CA↗

arXiv2003

Orthogonality of Jacobi polynomials with general parameters

In this paper we study the orthogonality conditions satisfied by Jacobi polynomials $P_n^{(\alpha,\beta)}$ when the parameters $\alpha$ and $\beta$ are not necessarily $>-1$. We establish orthogonality on a generic closed contour on a Riemann surface. Depending on the parameters, this leads to either full orthogonality conditions on a single contour in the plane, or to multiple orthogonality conditions on a number of contours in the plane. In all cases we show that the orthogonality conditions characterize the Jacobi polynomial $P_n^{(\alpha, \beta)}$ of degree $n$ up to a constant factor.

math.CA↗