Structurable superalgebras of Cartan type

We describe the simple Lie superalgebras arising from the unital structurable superalgebras of characteristic 0 and construct four series of the unital simple structurable superalgebras of Cartan type. We give a classification of simple structurable superalgebras of Cartan type over an algebraically closed field F of characteristic 0. Together with the Faulkner theorem on the classification of classical such superalgebras, it gives a classification of the simple structurable superalgebras over F. Crown Copyright (C) 2010 Published by Elsevier Inc. All rights reserved.

A carbon isotope challenge to the snowball Earth

The snowball Earth hypothesis postulates that the planet was entirely covered by ice for millions of years in the Neoproterozoic era, in a self-enhanced glaciation caused by the high albedo of the ice-covered planet. In a hard-snowball picture, the subsequent rapid unfreezing resulted from an ultra-greenhouse event attributed to the buildup of volcanic carbon dioxide (CO(2)) during glaciation(1). High partial pressures of atmospheric CO(2) (p(CO2); from 20,000 to 90,000 p. p. m. v.) in the aftermath of the Marinoan glaciation (similar to 635 Myr ago) have been inferred from both boron and triple oxygen isotopes(2,3). These p(CO2) values are 50 to 225 times higher than present-day levels. Here, we re-evaluate these estimates using paired carbon isotopic data for carbonate layers that cap Neoproterozoic glacial deposits and are considered to record post-glacial sea level rise(1). The new data reported here for Brazilian cap carbonates, together with previous ones for time-equivalent units(4-8), provide p(CO2) estimates lower than 3,200 p. p. m. v.-and possibly as low as the current value of similar to 400 p. p. m. v. Our new constraint, and our reinterpretation of the boron and triple oxygen isotope data, provide a completely different picture of the late Neoproterozoic environment, with low atmospheric concentrations of carbon dioxide and oxygen that are inconsistent with a hard-snowball Earth.

SINGULARITY THEORY AND FORCED SYMMETRY BREAKING IN EQUATIONS

A theory of bifurcation equivalence for forced symmetry breaking bifurcation problems is developed. We classify (O(2), 1) problems of corank 2 of low codimension and discuss examples of bifurcation problems leading to such symmetry breaking.