Fork-decompositions of matroids

For the abstract of this paper, please see the PDF file

On matroids of branch-width three

For the abstract of this paper, please see the PDF file

The structure of the 3-separations of 3-connected matroids II

The authors showed in an earlier paper that there is a tree that displays, up to a natural equivalence, all non-trivial 3-separations of a 3-connected matroid. The purpose of this paper is to show that if certain natural conditions are imposed on the tree, then it has a uniqueness property. In particular; suppose that, from every pair of edges that meet at a degree-2 vertex and have their other ends of degree at least three, one edge is contracted. Then the resulting tree is unique

The structure of the 3-separations of 3-connected matroids

Special Issue Dedicated to Professor W.T. TutteTutte defined a k-separation of a matroid M to be a partition (A,B) of the ground set of M such that ∣A∣,∣B∣ ≥ k and r(A) + r(B) − r(M) < k. If, for all m < n, the matroid M has no m-separations, then M is n-connected. Earlier, Whitney showed that (A,B) is a 1-separation of M if and only if A is a union of 2-connected components of M. When M is 2-connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2-separations. When M is 3-connected, this paper describes a tree decomposition of M that displays, up to a certain natural equivalence, all non-trivial 3-separations of M

Mass fluctuation kinetics: analysis and computation of equilibria and local dynamics

The mass fluctuation kinetics (MFK) model is a set of coupled ordinary differential equations approximating the time evolution of means and covariances of species concentrations in chemical reaction networks. It generalises classical mass action kinetics (MAK), in which fluctuations around the mean are ignored. MFK may be used to approximate stochasticity in system trajectories when stochastic simulation methods are prohibitively expensive computationally. This study presents a set of tools to aid in the analysis of systems within the MFK framework. A closed-form expression for the MFK Jacobian matrix is derived. This expression facilitates the computation of MFK equilibria and the characterisation of the dynamics of small deviations from the equilibria (i.e. local dynamics). Software developed in MATLAB to analyse systems within the MFK framework is also presented. The authors outline a homotopy continuation method that employs the Jacobian for bifurcation analysis, that is, to generate a locus of steady-state Jacobian eigenvalues corresponding to changing a chosen MFK parameter such as system volume or a rate constant. This method is applied to study the effect of small-volume stochasticity on local dynamics at equilibria in a pair of example systems, namely the formation and dissociation of an enzyme-substrate complex and a genetic oscillator. For both systems, this study reveals volume regimes where MFK provides a quantitatively and/or qualitatively correct description of system behaviour, and regimes where the MFK approximation is inaccurate. Moreover, our analysis provides evidence that decreasing volume from the MAK regime (infinite volume) has a destabilising effect on system dynamics

A Search for Binary Active Galactic Nuclei: Double-Peaked [OIII] AGN in the Sloan Digital Sky Survey

We present AGN from the Sloan Digital Sky Survey (SDSS) having double-peaked profiles of [OIII] 5007,4959 and other narrow emission-lines, motivated by the prospect of finding candidate binary AGN. These objects were identified by means of a visual examination of 21,592 quasars at z < 0.7 in SDSS Data Release 7 (DR7). Of the spectra with adequate signal-to-noise, 148 spectra exhibit a double-peaked [OIII] profile. Of these, 86 are Type 1 AGN and 62 are Type 2 AGN. Only two give the appearance of possibly being optically resolved double AGN in the SDSS images, but many show close companions or signs of recent interaction. Radio-detected quasars are three times more likely to exhibit a double-peaked [OIII] profile than quasars with no detected radio flux, suggesting a role for jet interactions in producing the double-peaked profiles. Of the 66 broad line (Type 1) AGN that are undetected in the FIRST survey, 0.9% show double peaked [OIII] profiles. We discuss statistical tests of the nature of the double-peaked objects. Further study is needed to determine which of them are binary AGN rather than disturbed narrow line regions, and how many additional binaries may remain undetected because of insufficient line-of-sight velocity splitting. Previous studies indicate that 0.1% of SDSS quasars are spatially resolved binaries, with typical spacings of ~10 to 100 kpc. If a substantial fraction of the double-peaked objects are indeed binaries, then our results imply that binaries occur more frequently at smaller separations (< 10 kpc). This suggests that simultaneous fueling of both black holes is more common as the binary orbit decays through these spacings.Comment: 33 pages, 5 figures, LaTeX. Major revisions. Accepted for publication in ApJ

Statistical Properties of Radio Emission from the Palomar Seyfert Galaxies

We have carried out an analysis of the radio and optical properties of a statistical sample of 45 Seyfert galaxies from the Palomar spectroscopic survey of nearby galaxies. We find that the space density of bright galaxies (-22 mag <= M_{B_T} <= -18 mag) showing Seyfert activity is (1.25 +/- 0.38) X 10^{-3} Mpc^{-3}, considerably higher than found in other Seyfert samples. Host galaxy types, radio spectra, and radio source sizes are uncorrelated with Seyfert type, as predicted by the unified schemes for active galaxies. Approximately half of the detected galaxies have flat or inverted radio spectra, more than expected based on previous samples. Surprisingly, Seyfert 1 galaxies are found to have somewhat stronger radio sources than Seyfert 2 galaxies at 6 and 20 cm, particularly among the galaxies with the weakest nuclear activity. We suggest that this difference can be accommodated in the unified schemes if a minimum level of Seyfert activity is required for a radio source to emerge from the vicinity of the active nucleus. Below this level, Seyfert radio sources might be suppressed by free-free absorption associated with the nuclear torus or a compact narrow-line region, thus accounting for both the weakness of the radio emission and the preponderance of flat spectra. Alternatively, the flat spectra and weak radio sources might indicate that the weak active nuclei are fed by advection-dominated accretion disks.Comment: 18 pages using emulateapj5, 13 embedded figures, accepted by Ap

A Potts/Ising Correspondence on Thin Graphs

We note that it is possible to construct a bond vertex model that displays q-state Potts criticality on an ensemble of phi3 random graphs of arbitrary topology, which we denote as ``thin'' random graphs in contrast to the fat graphs of the planar diagram expansion. Since the four vertex model in question also serves to describe the critical behaviour of the Ising model in field, the formulation reveals an isomorphism between the Potts and Ising models on thin random graphs. On planar graphs a similar correspondence is present only for q=1, the value associated with percolation.Comment: 6 pages, 5 figure

Active Galactic Nuclei with Candidate Intermediate-Mass Black Holes

We present an initial sample of 19 intermediate-mass black hole candidates in active galactic nuclei culled from the First Data Release of the Sloan Digital Sky Survey. Using the linewidth-luminosity-mass scaling relation established for broad-line active nuclei, we estimate black hole masses in the range of 8 x 10^4 - 10^6 solar masses, a regime in which only two objects are currently known. The absolute magnitudes are faint for active galactic nuclei, ranging from M_g of -15 to -18 mag, while the bolometric luminosities are all close to the Eddington limit. The entire sample formally satisfies the linewidth criterion for so-called narrow-line Seyfert 1 galaxies; however, they display a wider range of FeII and [OIII] (5007) line strengths than is typically observed in this class of objects. Although the available imaging data are of insufficient quality to ascertain the detailed morphologies of the host galaxies, it is likely that the majority of the hosts are relatively late-type systems. The host galaxies have estimated g-band luminosities ~ 1 mag fainter than M* for the general galaxy population at z of 0.1. Beyond simply extending the known mass range of central black holes in galactic nuclei, these objects provide unique observational constraints on the progenitors of supermassive black holes. They are also expected to contribute significantly to the integrated signal for future gravitational wave experiments.Comment: ApJ Accepted, 13 pages, 9 figures, uses emulateapj.cl