About a conjecture for uniformly isochronous polynomial centers
We study a specific family of uniformly isochronous polynomial systems. Our results permit to solve a problem about centers of such systems.
Poincaré's Reversibility Condition
We consider a real planar analytic vector field, X, such that the origin, O, is a centre for the linearization of X. Poincare's condition of reversibility with respect to a line passing through O is then a sufficient condition for O to be a centre for the vector field X. We provide necessary and sufficient conditions, involving the vanishing of certain polynomials in the coefficients in the expansion of X, for reversibility. We also show that if the linearization, L(x), of the divergence of X is non-trivial, then the only possible reversibility line is given by L(x) = 0; in such cases, this provides the basis for a simple test of reversibility. We examine the consequences of our various tests for quadratic and cubic vector fields; all non-Hamiltonian cases are discussed. When L(x) ≡ 0 in cubic systems, it is possible for the reversibility line (if it exists) to be unique, but it is also possible for there to be two such lines. These possibilities are characterized algebraically, and a prescription is provided for determining the reversibility line(s) in each case.
O2 constraints from Paleoproterozoic detrital pyrite and uraninite
Redox-sensitive detrital grains such as pyrite and uraninite in sedimentary successions provide one of the most conspicuous geological clues to a different composition of the Archean and early Paleoproterozoic atmosphere. Today, these minerals are rapidly chemically weathered within short transport distances. Prior to the rise of oxygen, low O2 concentrations allowed their survival in siliciclastic deposits with grain erosion tied only to physical transport processes. After the rise of oxygen, redox-sensitive detrital grains effectively vanish from the sedimentary record. To get a better understanding of the timing of this transition, we examined sandstones recorded in a scientifi c drill core from the South African 2.415 Ga Koegas Subgroup, a mixed siliciclastic and iron formation–bearing unit deposited on the western deltaic margin of the Kaapvaal craton in early Paleoproterozoic time. We observed detrital pyrite and uraninite grains throughout all investigated sandstone beds in the section, indicating the rise of oxygen is younger than 2.415 Ga. To better understand how observations of detrital pyrite and uraninite in sedimentary rocks can quantitatively constrain Earth surface redox conditions, we constructed a model of grain erosion from chemical weathering and physical abrasion to place an upper limit on ancient environmental O2 concentrations. Even conservative model calculations for deltaic depositional systems with suffi cient transport distances (approximately hundreds of kilometers) show that redox-sensitive detrital grains are remarkably sensitive to environmental O2 concentrations, and they constrain the Archean and early Paleoproterozoic atmosphere to have <3.2 × 10 –5 atm of molecular O2. These levels are lower than previously hypothesized for redox-sensitive detrital grains, but they are consistent with estimates made from other redox proxy data, including the anomalous fractionation of sulfur isotopes. The binary loss of detrital pyrite and uraninite from the sedimentary record coincident with the rise of oxygen indicates that atmospheric O2 concentrations rose substantially at this time and were never again suffi ciently low (<0.01 atm) to enable survival and preservation of these grains in short transport systems.