A free subalgebra of the algebra of matroids

This paper is an initial inquiry into the structure of the Hopf algebra of matroids with restriction-contraction coproduct. Using a family of matroids introduced by Crapo in 1965, we show that the subalgebra generated by a single point and a single loop in the dual of this Hopf algebra is free.Comment: 19 pages, 3 figures. Accepted for publication in the European Journal of Combinatorics. This version incorporates a few minor corrections suggested by the publisher

The Free product of Matroids

We introduce a noncommutative binary operation on matroids, called free product. We show that this operation respects matroid duality, and has the property that, given only the cardinalities, an ordered pair of matroids may be recovered, up to isomorphism, from its free product. We use these results to give a short proof of Welsh's 1969 conjecture, which provides a progressive lower bound for the number of isomorphism classes of matroids on an n-element set.Comment: 5 pages, 1 figure. Accepted for publication in the European Journal of Combinatorics. See also arXiv:math.CO/040902

A unique factorization theorem for matroids

We study the combinatorial, algebraic and geometric properties of the free product operation on matroids. After giving cryptomorphic definitions of free product in terms of independent sets, bases, circuits, closure, flats and rank function, we show that free product, which is a noncommutative operation, is associative and respects matroid duality. The free product of matroids $M$ and $N$ is maximal with respect to the weak order among matroids having $M$ as a submatroid, with complementary contraction equal to $N$. Any minor of the free product of $M$ and $N$ is a free product of a repeated truncation of the corresponding minor of $M$ with a repeated Higgs lift of the corresponding minor of $N$. We characterize, in terms of their cyclic flats, matroids that are irreducible with respect to free product, and prove that the factorization of a matroid into a free product of irreducibles is unique up to isomorphism. We use these results to determine, for K a field of characteristic zero, the structure of the minor coalgebra $\cal C$ of a family of matroids $\cal M$ that is closed under formation of minors and free products: namely, $\cal C$ is cofree, cogenerated by the set of irreducible matroids belonging to $\cal M$.Comment: Dedicated to Denis Higgs. 25 pages, 3 figures. Submitted for publication in the Journal of Combinatorial Theory (A). See arXiv:math.CO/0409028 arXiv:math.CO/0409080 for preparatory work on this subjec

51 Pegasi - a planet-bearing Maunder minimum candidate

We observed 51 Peg, the first detected planet-bearing star, in a 55 ks XMM-Newton pointing and in 5 ks pointings each with Chandra HRC-I and ACIS-S. The star has a very low count rate in the XMM observation, but is clearly visible in the Chandra images due to the detectors' different sensitivity at low X-ray energies. This allows a temperature estimate for 51 Peg's corona of T<1MK; the detected ACIS-S photons can be plausibly explained by emission lines of a very cool plasma near 200eV. The constantly low X-ray surface flux and the flat-activity profile seen in optical CaII data suggest that 51 Peg is a Maunder minimum star; an activity enhancement due to a Hot Jupiter, as proposed by recent studies, seems to be absent. The star's X-ray fluxes in different instruments are consistent with the exception of the HRC Imager, which might have a larger effective area below 200eV than given in the calibration.Comment: accepted by A&

Quiescent and flaring X-ray emission from the nearby M/T dwarf binary SCR 1845-6357

We investigate an XMM-Newton observation of SCR 1845-6357, a nearby, ultracool M8.5/T5.5 dwarf binary. The binary is unresolved in the XMM detectors, however the X-ray emission is very likely from the M8.5 dwarf. We compare its flaring emission to those of similar very low mass stars and additionally present an XMM observation of the M8 dwarf VB 10. We detect quasi-quiescent X-ray emission from SCR 1845-6357 at soft X-ray energies in the 0.2-2.0 keV band, as well as a strong flare with a count rate increase of a factor of 30 and a duration of only 10 minutes. The quasi-quiescent X-ray luminosity of log L_x = 26.2 erg/s and the corresponding activity level of log L_x/L_bol = -3.8 point to a fairly active star. Coronal temperatures of up to 5 MK and frequent minor variability support this picture. During the flare, that is accompanied by a significant brightening in the near-UV, plasma temperatures of 25-30 MK are observed and an X-ray luminosity of L_x= 8 x 10^27 erg/s is reached. SCR 1845-6357 is a nearby, very low mass star that emits X-rays at detectable levels in quasi-quiescence, implying the existence of a corona. The high activity level, coronal temperatures and the observed large flare point to a rather active star, despite its estimated age of a few Gyr.Comment: Accepted by A&A, 6 pages, 5 figure

Coronal activity cycles in nearby G and K stars - XMM-Newton monitoring of 61 Cygni and Alpha Centauri

We use X-ray observations of the nearby binaries 61 Cyg A/B (K5V and K7V) and Alpha Cen A/B (G2V and K1V) to study the long-term evolution of magnetic activity in weakly to moderately active G + K dwarfs over nearly a decade. Specifically we search for X-ray activity cycles and related coronal changes and compare them to the solar behavior. For 61 Cyg A we find a regular coronal activity cycle analog to its 7.3 yr chromospheric cycle. The X-ray brightness variations are with a factor of three significantly lower than on the Sun, yet the changes of coronal properties resemble the solar behavior with larger variations occurring in the respective hotter plasma components. 61 Cyg B does not show a clear cyclic coronal trend so far, but the X-ray data matches the more irregular chromospheric cycle. Both Alpha Cen stars exhibit significant long-term X-ray variability. Alpha Cen A shows indications for cyclic variability of an order of magnitude with a period of about 12-15 years; the Alpha Cen B data suggests an X-ray cycle with an amplitude of about six to eight and a period of 8-9 years. The sample stars exhibit X-ray luminosities ranging between Lx < 1x10^26 - 3x10^27 erg s^-1 in the 0.2-2.0 keV band and have coronae dominated by cool plasma with variable average temperatures of around 1.0-2.5 MK. We find that coronal activity cycles are apparently a common phenomenon in older, slowly rotating G and K stars. The spectral changes of the coronal X-ray emission over the cycles are solar-like in all studied targets.Comment: 11 pages, 9 figures, accepted by Astronomy and Astrophysic