Average degree conditions forcing a minor

Mader first proved that high average degree forces a given graph as a minor. Often motivated by Hadwiger's Conjecture, much research has focused on the average degree required to force a complete graph as a minor. Subsequently, various authors have consider the average degree required to force an arbitrary graph $H$ as a minor. Here, we strengthen (under certain conditions) a recent result by Reed and Wood, giving better bounds on the average degree required to force an $H$-minor when $H$ is a sparse graph with many high degree vertices. This solves an open problem of Reed and Wood, and also generalises (to within a constant factor) known results when $H$ is an unbalanced complete bipartite graph

Do stiffness and asymmetries predict change of direction performance?

Change of direction speed (CODS) underpins performance in a wide range of sports but little is known about how stiffness and asymmetries affect CODS. Eighteen healthy males performed unilateral drop jumps to determine vertical, ankle, knee and hip stiffness, and a CODS test to evaluate left and right leg cutting performance during which ground reaction force data were sampled. A step-wise regression analysis was performed to ascertain the determinants of CODS time. A two-variable regression model explained 63% (R-2 = 0.63; P = 0.001) of CODS performance. The model included the mean vertical stiffness and jump height asymmetry determined during the drop jump. Faster athletes (n = 9) exhibited greater vertical stiffness (F = 12.40; P = 0.001) and less asymmetry in drop jump height (F = 6.02; P = 0.026) than slower athletes (n = 9); effect sizes were both "large" in magnitude. Results suggest that overall vertical stiffness and drop jump height asymmetry are the strongest predictors of CODS in a healthy, non-athletic population

Disk Properties and Density Structure of the Star-Forming Dense Core B335

We present subarcsecond resolution dust continuum observations of the protostellar collapse candidate B335 made with the IRAM Plateau de Bure Interferometer at wavelengths of 1.2 and 3.0 mm. These observations probe to < 100 AU size scales and reveal a compact source component that we identify with a circumstellar disk. We analyze these data in concert with previous lower resolution interferometer observations and find a best fit density structure for B335 that consists of a power law envelope with index p=1.55 +/- 0.04 (r < 5000 AU) together with a disk (r < 100 AU) of flux F_{1.2 mm}=21 +/-2 mJy. We estimate a systematic uncertainty in the power law index delta(p) < 0.15, where the largest error comes from the assumed form of the dust temperature falloff with radius. This determination of the inner density structure of B335 has a precision unique amongst protostellar cores, and it is consistent with the r^{-1.5} profile of gravitational free-fall, in accord with basic expectations for the formation of a star. The flux (and implied mass) of the compact component in B335 is typical of the disks around T Tauri stars.Comment: 16 pages, 2 figures. Accepted to the Astrophysical Journal, sched v596 (2003 Oct 10

Evidence for J and H-band excess in classical T Tauri stars and the implications for disk structure and estimated ages

We argue that classical T Tauri stars (cTTs) possess significant non- photospheric excess in the J and H bands. We first show that normalizing the spectral energy distributions (SEDs) of cTTs to the J-band leads to a poor fit of the optical fluxes, while normalizing the SEDs to the Ic-band produces a better fit to the optical bands and in many cases reveals the presence of a considerable excess at J and H. NIR spectroscopic veiling measurements from the literature support this result. We find that J and H-band excesses correlate well with the K-band excess, and that the J-K and H-K colors of the excess emission are consistent with that of a black body at the dust sublimation temperature (~ 1500-2000 K). We propose that this near-IR excess originates at a hot inner rim, analogous to those suggested to explain the near-IR bump in the SEDs of Herbig Ae/Be stars. To test our hypothesis, we use the model presented by Dullemond et al. (2001) to fit the photometry data between 0.5 um and 24 um of 10 cTTs associated with the Chamaeleon II molecular cloud. The models that best fit the data are those where the inner radius of the disk is larger than expected for a rim in thermal equilibrium with the photospheric radiation field alone. In particular, we find that large inner rims are necessary to account for the mid infrared fluxes (3.6-8.0 um) obtained by the Spitzer Space Telescope. Finally, we argue that deriving the stellar luminosities of cTTs by making bolometric corrections to the J-band fluxes systematically overestimates these luminosities. The overestimated luminosities translate into underestimated ages when the stars are placed in the H-R diagram. Thus, the results presented herein have important implications for the dissipation timescale of inner accretion disks.Comment: 45 pages, 13 figure