Counting matroids in minor-closed classes

A flat cover is a collection of flats identifying the non-bases of a matroid. We introduce the notion of cover complexity, the minimal size of such a flat cover, as a measure for the complexity of a matroid, and present bounds on the number of matroids on $n$ elements whose cover complexity is bounded. We apply cover complexity to show that the class of matroids without an $N$-minor is asymptotically small in case $N$ is one of the sparse paving matroids $U_{2,k}$, $U_{3,6}$, $P_6$, $Q_6$, or $R_6$, thus confirming a few special cases of a conjecture due to Mayhew, Newman, Welsh, and Whittle. On the other hand, we show a lower bound on the number of matroids without $M(K_4)$-minor which asymptoticaly matches the best known lower bound on the number of all matroids, due to Knuth.Comment: 13 pages, 3 figure

The Westerbork HI Survey of Spiral and Irregular Galaxies I. HI Imaging of Late-type Dwarf Galaxies

Neutral hydrogen observations with the Westerbork Synthesis Radio Telescope are presented for a sample of 73 late-type dwarf galaxies. These observations are part of the WHISP project (Westerbork HI Survey of Spiral and Irregular Galaxies). Here we present HI maps, velocity fields, global profiles and radial surface density profiles of HI, as well as HI masses, HI radii and line widths. For the late-type galaxies in our sample, we find that the ratio of HI extent to optical diameter, defined as 6.4 disk scale lengths, is on average 1.8+-0.8, similar to that seen in spiral galaxies. Most of the dwarf galaxies in this sample are rich in HI, with a typical M_HI/L_B of 1.5. The relative HI content M_HI/L_R increases towards fainter absolute magnitudes and towards fainter surface brightnesses. Dwarf galaxies with lower average HI column densities also have lower average optical surface brightnesses. We find that lopsidedness is as common among dwarf galaxies as it is in spiral galaxies. About half of the dwarf galaxies in our sample have asymmetric global profiles, a third has a lopsided HI distribution, and about half shows signs of kinematic lopsidedness.Comment: Accepted for publication in A&A. 18 pages. 39 MB version with all figures is available http://www.robswork.net/publications/WHISPI.ps.g

On the number of matroids

We consider the problem of determining $m_n$, the number of matroids on $n$ elements. The best known lower bound on $m_n$ is due to Knuth (1974) who showed that $\log \log m_n$ is at least $n-3/2\log n-1$. On the other hand, Piff (1973) showed that $\log\log m_n\leq n-\log n+\log\log n +O(1)$, and it has been conjectured since that the right answer is perhaps closer to Knuth's bound. We show that this is indeed the case, and prove an upper bound on $\log\log m_n$ that is within an additive $1+o(1)$ term of Knuth's lower bound. Our proof is based on using some structural properties of non-bases in a matroid together with some properties of independent sets in the Johnson graph to give a compressed representation of matroids.Comment: Final version, 17 page

An entropy argument for counting matroids

We show how a direct application of Shearers' Lemma gives an almost optimum bound on the number of matroids on $n$ elements.Comment: Short note, 4 page

Encapsulation of DNA by cationic diblock copolymer vesicles

Encapsulation of dsDNA fragments (contour length 54 nm) by the cationic diblock copolymer poly(butadiene-b-N-methyl 4-vinyl pyridinium) [PBd-b-P4VPQ] has been studied with phase contrast, polarized light, and fluorescence microscopy, as well as scanning electron microscopy. Encapsulation was achieved with a single emulsion technique. For this purpose, an aqueous DNA solution is emulsified in an organic solvent (toluene) and stabilized by the amphiphilic diblock copolymer. The PBd block forms an interfacial brush, whereas the cationic P4VPQ block complexes with DNA. A subsequent change of the quality of the organic solvent results in a collapse of the PBd brush and the formation of a capsule. Inside the capsules, the DNA is compacted as shown by the appearance of birefringent textures under crossed polarizers and the increase in fluorescence intensity of labeled DNA. The capsules can also be dispersed in aqueous medium to form vesicles, provided they are stabilized with an osmotic agent (polyethylene glycol) in the external phase. It is shown that the DNA is released from the vesicles once the osmotic pressure drops below 105 N/m2 or if the ionic strength of the supporting medium exceeds 0.1 M. The method has also proven to be efficient to encapsulate pUC18 plasmid in sub-micron sized vesicles and the general applicability of the method has been demonstrated by the preparation of the charge inverse system: cationic poly(ethylene imine) encapsulated by the anionic diblock poly(styrene-b-acrylic acid).Comment: 35 pages, 11 figures, accepted for publication in Langmui

The black hole mass distribution in early-type galaxies: cusps in HST photometry interpreted through adiabatic black hole growth

The surface brightness profiles of early-type galaxies have central cusps. Two characteristic profile types are observed with HST: `core' profiles have a break at a resolved radius and logarithmic cusp slope gamma < 0.3 inside that radius; `power-law' profiles have no clear break and gamma > 0.3. With few exceptions, galaxies with M_V -20.5 have power-law profiles. Both profile types occur in galaxies with -22 < M_V < -20.5. We show that these results are consistent with the hypothesis that: (i) all early-type galaxies have black holes (BHs) that grew adiabatically in homogeneous isothermal cores; and (ii) these `progenitor' cores followed scaling relations similar to those of the fundamental plane. The models studied here are the ones first proposed by Young. Models with BH masses and progenitor cores that obey established scaling relations predict (at Virgo) that galaxies with M_V < -21.2 have core profiles and galaxies with M_V > -21.2 have power-law profiles. This reproduces both the sense and the absolute magnitude of the observed transition. Intrinsic scatter in BH and galaxy properties can explain why both types of galaxies are observed around the transition magnitude. The observed bimodality in cusp slopes may be due to a bimodality in M_bh/L, with rapidly rotating disky galaxies having larger M_bh/L than slowly rotating boxy galaxies. Application to individual galaxies with HST photometry yields a roughly linear correlation between BH mass and V-band galaxy luminosity, log M_bh = -1.83 + log L (solar units). This agrees with the average relation for nearby galaxies with kinematically determined BH masses, and also with predictions from quasar statistics (shortened abstract).Comment: 41 pages, LaTeX, with 11 PostScript figures. Submitted to the Astronomical Journal. Postscript version also available from http://sol.stsci.edu/~marel/abstracts/abs_R23.htm

Observational Constraints on Disk Heating as a Function of Hubble Type

Current understanding of the secular evolution of galactic disks suggests that this process is dominated by two or more heating mechanisms, which increase the random motions of stars in the disk. In particular, the gravitational influence of giant molecular clouds and irregularities in the spiral potential have been proposed to explain the observed velocity dispersions in the solar neighborhood. Each of these mechanisms acts on different components of the stellar velocities, which affects the ratio of the vertical and radial components of the stellar velocity dispersion since the relative strengths of giant molecular clouds and spiral irregularities vary with Hubble type. A study of this ratio as function of Hubble type has the potential to provide strong constraints on disk heating mechanisms. We present major and minor axis stellar kinematics for four spiral galaxies of Hubble type from Sa to Sbc, and use the data to infer the ratios sigma_z/sigma_R in the galaxy disks. The results combined with two galaxies studied previously and with Milky Way data show that the ratio is generally in the range 0.5 - 0.8. There is a marginally significant trend of decreasing ratio with advancing Hubble type, consistent with the predictions of disk heating theories. However, the errors on individual measurements are large, and the absence of any trend is consistent with the data at the 1-sigma level. As a byproduct of our study, we find that three of the four galaxies in our sample have a central drop in their stellar line-of-sight velocity dispersion, a phenomenon that is increasingly observed in spiral galaxies. [ABRIDGED]Comment: 24 pages, LaTeX, 5 Postscript figures, to appear in AJ (Dec 2003