345 research outputs found
Setting the scene for Betti characters
Finite group actions on free resolutions and modules arise naturally in many
interesting examples. Understanding these actions amounts to describing the
terms of a free resolution or the graded components of a module as group
representations which, in the non modular case, are completely determined by
their characters. With this goal in mind, we introduce a Macaulay2 package for
computing characters of finite groups on free resolutions and graded components
of finitely generated graded modules over polynomial rings.Comment: 7 page
Torsional solutions of convection in rotating fluid spheres
A numerical study of the nonlinear torsional solutions of convection in rotating, internally heated, self-gravitating fluid spheres is presented. Their dependence on the Rayleigh number has been found for two pairs of Ekman (E) and small Prandtl (Pr) numbers in the region of parameters where, according to Zhang et al. [J. Fluid Mech. 813, R2 (2017)], the linear stability of the conduction state predicts that they can be preferred at the onset of convection. The bifurcation to periodic torsional solutions is supercritical for sufficiently small Pr. They are not rotating waves, unlike the nonaxisymmetric case. Therefore they have been computed by using continuation methods for periodic orbits. Their stability with respect to axisymmetric perturbations and physical characteristics have been analyzed. It was found that the time- and space-averaged equatorially antisymmetric part of the kinetic energy of the stable orbits splits into equal poloidal and toroidal parts, while the symmetric part is much smaller. Direct numerical simulations for E=10-4 at higher Rayleigh numbers (Ra) show that this trend is also valid for the nonperiodic flows and that the mean values of the energies remain almost constant with Ra. However, the modulated oscillations bifurcated from the quasiperiodic torsional solutions reach a high amplitude, compared with that of the periodic, increasing slowly and decaying very fast. This repeated behavior is interpreted as trajectories near heteroclinic chains connecting unstable periodic solutions. The torsional flows give rise to a meridional propagation of the kinetic energy near the outer surface and an axial oscillation of the hot nucleus of the metallic fluid sphere.Postprint (published version
Simulations of recoiling black holes: adaptive mesh refinement and radiative transfer
(Abridged) We here continue our effort to model the behaviour of matter when
orbiting or accreting onto a generic black hole by developing a new numerical
code employing advanced techniques geared solve the equations of in
general-relativistic hydrodynamics. The new code employs a number of
high-resolution shock-capturing Riemann-solvers and reconstruction algorithms,
exploiting the enhanced accuracy and the reduced computational cost of AMR
techniques. In addition, the code makes use of sophisticated ray-tracing
libraries that, coupled with general-relativistic radiation-transfer
calculations, allow us to compute accurately the electromagnetic emissions from
such accretion flows. We validate the new code by presenting an extensive
series of stationary accretion flows either in spherical or axial symmetry and
performed either in 2D or 3D. In addition, we consider the highly nonlinear
scenario of a recoiling black hole produced in the merger of a supermassive
black hole binary interacting with the surrounding circumbinary disc. In this
way we can present, for the first time, ray-traced images of the shocked fluid
and the light-curve resulting from consistent general-relativistic
radiation-transport calculations from this process. The work presented here
lays the ground for the development of a generic computational infrastructure
employing AMR techniques to deal accurately and self-consistently with
accretion flows onto compact objects. In addition to the accurate handling of
the matter, we provide a self-consistent electromagnetic emission from these
scenarios by solving the associated radiative-transfer problem. While magnetic
fields are presently excluded from our analysis, the tools presented here can
have a number of applications to study accretion flows onto black holes or
neutron stars.Comment: 20 pages, 20 figures, accepted for publication in A&
On the Geometric Interpretation of N = 2 Superconformal Theories
We clarify certain important issues relevant for the geometric interpretation
of a large class of N = 2 superconformal theories. By fully exploiting the
phase structure of these theories (discovered in earlier works) we are able to
clearly identify their geometric content. One application is to present a
simple and natural resolution to the question of what constitutes the mirror of
a rigid Calabi-Yau manifold. We also discuss some other models with unusual
phase diagrams that highlight some subtle features regarding the geometric
content of conformal theories.Comment: 25 pages, note adde
ECHO: an Eulerian Conservative High Order scheme for general relativistic magnetohydrodynamics and magnetodynamics
We present a new numerical code, ECHO, based on an Eulerian Conservative High
Order scheme for time dependent three-dimensional general relativistic
magnetohydrodynamics (GRMHD) and magnetodynamics (GRMD). ECHO is aimed at
providing a shock-capturing conservative method able to work at an arbitrary
level of formal accuracy (for smooth flows), where the other existing GRMHD and
GRMD schemes yield an overall second order at most. Moreover, our goal is to
present a general framework, based on the 3+1 Eulerian formalism, allowing for
different sets of equations, different algorithms, and working in a generic
space-time metric, so that ECHO may be easily coupled to any solver for
Einstein's equations. Various high order reconstruction methods are implemented
and a two-wave approximate Riemann solver is used. The induction equation is
treated by adopting the Upwind Constrained Transport (UCT) procedures,
appropriate to preserve the divergence-free condition of the magnetic field in
shock-capturing methods. The limiting case of magnetodynamics (also known as
force-free degenerate electrodynamics) is implemented by simply replacing the
fluid velocity with the electromagnetic drift velocity and by neglecting the
matter contribution to the stress tensor. ECHO is particularly accurate,
efficient, versatile, and robust. It has been tested against several
astrophysical applications, including a novel test on the propagation of large
amplitude circularly polarized Alfven waves. In particular, we show that
reconstruction based on a Monotonicity Preserving filter applied to a fixed
5-point stencil gives highly accurate results for smooth solutions, both in
flat and curved metric (up to the nominal fifth order), while at the same time
providing sharp profiles in tests involving discontinuities.Comment: 20 pages, revised version submitted to A&
GRHydro: a new open-source general-relativistic magnetohydrodynamics code for the Einstein toolkit
We present the new general-relativistic magnetohydrodynamics (GRMHD) capabilities of the Einstein toolkit, an open-source community-driven numerical relativity and computational relativistic astrophysics code. The GRMHD extension of the toolkit builds upon previous releases and implements the evolution of relativistic magnetized fluids in the ideal MHD limit in fully dynamical spacetimes using the same shock-capturing techniques previously applied to hydrodynamical evolution. In order to maintain the divergence-free character of the magnetic field, the code implements both constrained transport and hyperbolic divergence cleaning schemes. We present test results for a number of MHD tests in Minkowski and curved spacetimes. Minkowski tests include aligned and oblique planar shocks, cylindrical explosions, magnetic rotors, Alfvén waves and advected loops, as well as a set of tests designed to study the response of the divergence cleaning scheme to numerically generated monopoles. We study the code's performance in curved spacetimes with spherical accretion onto a black hole on a fixed background spacetime and in fully dynamical spacetimes by evolutions of a magnetized polytropic neutron star and of the collapse of a magnetized stellar core. Our results agree well with exact solutions where these are available and we demonstrate convergence. All code and input files used to generate the results are available on http://einsteintoolkit.org. This makes our work fully reproducible and provides new users with an introduction to applications of the code
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