Percolation on trees as a Brownian excursion: from Gaussian to Kolmogorov-Smirnov to Exponential statistics

We calculate the distribution of the size of the percolating cluster on a tree in the subcritical, critical and supercritical phase. We do this by exploiting a mapping between continuum trees and Brownian excursions, and arrive at a diffusion equation with suitable boundary conditions. The exact solution to this equation can be conveniently represented as a characteristic function, from which the following distributions are clearly visible: Gaussian (subcritical), Kolmogorov-Smirnov (critical) and exponential (supercritical). In this way we provide an intuitive explanation for the result reported in R. Botet and M. Ploszajczak, Phys. Rev. Lett 95, 185702 (2005) for critical percolation.Comment: 5 pages, 4 fiure

Improving QED-Tutrix by Automating the Generation of Proofs

The idea of assisting teachers with technological tools is not new. Mathematics in general, and geometry in particular, provide interesting challenges when developing educative softwares, both in the education and computer science aspects. QED-Tutrix is an intelligent tutor for geometry offering an interface to help high school students in the resolution of demonstration problems. It focuses on specific goals: 1) to allow the student to freely explore the problem and its figure, 2) to accept proofs elements in any order, 3) to handle a variety of proofs, which can be customized by the teacher, and 4) to be able to help the student at any step of the resolution of the problem, if the need arises. The software is also independent from the intervention of the teacher. QED-Tutrix offers an interesting approach to geometry education, but is currently crippled by the lengthiness of the process of implementing new problems, a task that must still be done manually. Therefore, one of the main focuses of the QED-Tutrix' research team is to ease the implementation of new problems, by automating the tedious step of finding all possible proofs for a given problem. This automation must follow fundamental constraints in order to create problems compatible with QED-Tutrix: 1) readability of the proofs, 2) accessibility at a high school level, and 3) possibility for the teacher to modify the parameters defining the "acceptability" of a proof. We present in this paper the result of our preliminary exploration of possible avenues for this task. Automated theorem proving in geometry is a widely studied subject, and various provers exist. However, our constraints are quite specific and some adaptation would be required to use an existing prover. We have therefore implemented a prototype of automated prover to suit our needs. The future goal is to compare performances and usability in our specific use-case between the existing provers and our implementation.Comment: In Proceedings ThEdu'17, arXiv:1803.0072

The Post-Quasistatic Approximation as a test bed for Numerical Relativity

It is shown that observers in the standard ADM 3+1 treatment of matter are the same as the observers used in the matter treatment of Bondi: they are comoving and local Minkowskian. Bondi's observers are the basis of the post--quasitatic approximation (PQSA) to study a contracting distribution of matter. This correspondence suggests the possibility of using the PQSA as a test bed for Numerical Relativity. The treatment of matter by the PQSA and its connection with the ADM 3+1 treatment are presented, for its practical use as a calibration tool and as a test bed for numerical relativistic hydrodynamic codes.Comment: 4 pages; to appear as a Brief Report in Physical Review

Numerical 3+1 general relativistic magnetohydrodynamics: a local characteristic approach

We present a general procedure to solve numerically the general relativistic magnetohydrodynamics (GRMHD) equations within the framework of the 3+1 formalism. The work reported here extends our previous investigation in general relativistic hydrodynamics (Banyuls et al. 1997) where magnetic fields were not considered. The GRMHD equations are written in conservative form to exploit their hyperbolic character in the solution procedure. All theoretical ingredients necessary to build up high-resolution shock-capturing schemes based on the solution of local Riemann problems (i.e. Godunov-type schemes) are described. In particular, we use a renormalized set of regular eigenvectors of the flux Jacobians of the relativistic magnetohydrodynamics equations. In addition, the paper describes a procedure based on the equivalence principle of general relativity that allows the use of Riemann solvers designed for special relativistic magnetohydrodynamics in GRMHD. Our formulation and numerical methodology are assessed by performing various test simulations recently considered by different authors. These include magnetized shock tubes, spherical accretion onto a Schwarzschild black hole, equatorial accretion onto a Kerr black hole, and magnetized thick accretion disks around a black hole prone to the magnetorotational instability.Comment: 18 pages, 8 figures, submitted to Ap

Nonlinear r-modes in Rapidly Rotating Relativistic Stars

The r-mode instability in rotating relativistic stars has been shown recently to have important astrophysical implications (including the emission of detectable gravitational radiation, the explanation of the initial spins of young neutron stars and the spin-distribution of millisecond pulsars and the explanation of one type of gamma-ray bursts), provided that r-modes are not saturated at low amplitudes by nonlinear effects or by dissipative mechanisms. Here, we present the first study of nonlinear r-modes in isentropic, rapidly rotating relativistic stars, via 3-D general-relativistic hydrodynamical evolutions. Our numerical simulations show that (1) on dynamical timescales, there is no strong nonlinear coupling of r-modes to other modes at amplitudes of order one -- unless nonlinear saturation occurs on longer timescales, the maximum r-mode amplitude is of order unity (i.e., the velocity perturbation is of the same order as the rotational velocity at the equator). An absolute upper limit on the amplitude (relevant, perhaps, for the most rapidly rotating stars) is set by causality. (2) r-modes and inertial modes in isentropic stars are predominantly discrete modes and possible associated continuous parts were not identified in our simulations. (3) In addition, the kinematical drift associated with r-modes, recently found by Rezzolla, Lamb and Shapiro (2000), appears to be present in our simulations, but an unambiguous confirmation requires more precise initial data. We discuss the implications of our findings for the detectability of gravitational waves from the r-mode instability.Comment: 4 pages, 4 eps figures, accepted in Physical Review Letter

Influence of self-gravity on the runaway instability of black hole-torus systems

Results from the first fully general relativistic numerical simulations in axisymmetry of a system formed by a black hole surrounded by a self-gravitating torus in equilibrium are presented, aiming to assess the influence of the torus self-gravity on the onset of the runaway instability. We consider several models with varying torus-to-black hole mass ratio and angular momentum distribution orbiting in equilibrium around a non-rotating black hole. The tori are perturbed to induce the mass transfer towards the black hole. Our numerical simulations show that all models exhibit a persistent phase of axisymmetric oscillations around their equilibria for several dynamical timescales without the appearance of the runaway instability, indicating that the self-gravity of the torus does not play a critical role favoring the onset of the instability, at least during the first few dynamical timescales.Comment: To appear on Phys.Rev.Let

Mirror Symmetry for Two Parameter Models -- II

We describe in detail the space of the two K\"ahler parameters of the Calabi--Yau manifold $\P_4^{(1,1,1,6,9)}[18]$ by exploiting mirror symmetry. The large complex structure limit of the mirror, which corresponds to the classical large radius limit, is found by studying the monodromy of the periods about the discriminant locus, the boundary of the moduli space corresponding to singular Calabi--Yau manifolds. A symplectic basis of periods is found and the action of the $Sp(6,\Z)$ generators of the modular group is determined. From the mirror map we compute the instanton expansion of the Yukawa couplings and the generalized $N=2$ index, arriving at the numbers of instantons of genus zero and genus one of each degree. We also investigate an $SL(2,\Z)$ symmetry that acts on a boundary of the moduli space.Comment: 57 pages + 9 figures using eps