Constructing Involutive Tableaux with Guillemin Normal Form

Involutivity is the algebraic property that guarantees solutions to an analytic and torsion-free exterior differential system or partial differential equation via the Cartan-K\"ahler theorem. Guillemin normal form establishes that the prolonged symbol of an involutive system admits a commutativity property on certain subspaces of the prolonged tableau. This article examines Guillemin normal form in detail, aiming at a more systematic approach to classifying involutive systems. The main result is an explicit quadratic condition for involutivity of the type suggested but not completed in Chapter IV, \S 5 of the book Exterior Differential Systems by Bryant, Chern, Gardner, Goldschmidt, and Griffiths. This condition enhances Guillemin normal form and characterizes involutive tableaux.Comment: This article co-evolved with "Degeneracy of the Characteristic Variety," arXiv:1410.6947 and most notation is shared. However, be aware that the meaning of the indices i,j,k,l and the space Y is not the same between these article

The first supermassive black holes

We briefly review the historical development of the ideas regarding the first supermassive black hole seeds, the physics of their formation and radiative feedback, recent theoretical and observational progress, and our outlook for the future.Comment: Invited review, Astronomy & Geophysics, 6 pages, 6 figures, updated to reflect the printed versio

Extending the Coinvariant Theorems of Chevalley, Shephard--Todd, Mitchell and Springer

We extend in several directions invariant theory results of Chevalley, Shephard and Todd, Mitchell and Springer. Their results compare the group algebra for a finite reflection group with its coinvariant algebra, and compare a group representation with its module of relative coinvariants. Our extensions apply to arbitrary finite groups in any characteristic.Comment: The applications and Examples in section 4 have been extende

Precession and Nutation in the eta Carinae binary system: Evidences from the X-ray light curve

It is believed that eta Carinae is actually a massive binary system, with the wind-wind interaction responsible for the strong X-ray emission. Although the overall shape of the X-ray light curve can be explained by the high eccentricity of the binary orbit, other features like the asymmetry near periastron passage and the short quasi-periodic oscillations seen at those epochs, have not yet been accounted for. In this paper we explain these features assuming that the rotation axis of eta Carinae is not perpendicular to the orbital plane of the binary system. As a consequence, the companion star will face eta Carinae on the orbital plane at different latitudes for different orbital phases and, since both the mass loss rate and the wind velocity are latitude dependent, they would produce the observed asymmetries in the X-ray flux. We were able to reproduce the main features of the X-ray light curve assuming that the rotation axis of eta Carinae forms an angle of 29 degrees with the axis of the binary orbit. We also explained the short quasi-periodic oscillations by assuming nutation of the rotation axis, with amplitude of about 5 degrees and period of about 22 days. The nutation parameters, as well as the precession of the apsis, with a period of about 274 years, are consistent with what is expected from the torques induced by the companion star.Comment: 9 pages, 8 figures, MNRAS accepte

A Dynamic Atomistic-Continuum Method for the Simulation of Crystalline Materials

We present a coupled atomistic-continuum method for the modeling of defects and interface dynamics of crystalline materials. The method uses atomistic models such as molecular dynamics near defects and interfaces, and continuum models away from defects and interfaces. We propose a new class of matching conditions between the atomistic and continuum regions. These conditions ensure the accurate passage of large scale information between the atomistic and continuum regions and at the same time minimize the reflection of phonons at the atomistic-continuum interface. They can be made adaptive if we choose appropriate weight functions. We present applications to dislocation dynamics, friction between two-dimensional crystal surfaces and fracture dynamics. We compare results of the coupled method and the detailed atomistic model.Comment: 48 pages, 20 figure