Reflection Groups and Polytopes over Finite Fields, III

When the standard representation of a crystallographic Coxeter group is reduced modulo an odd prime p, one obtains a finite group G^p acting on some orthogonal space over Z_p . If the Coxeter group has a string diagram, then G^p will often be the automorphism group of a finite abstract regular polytope. In parts I and II we established the basics of this construction and enumerated the polytopes associated to groups of rank at most 4, as well as all groups of spherical or Euclidean type. Here we extend the range of our earlier criteria for the polytopality of G^p . Building on this we investigate the class of 3-infinity groups of general rank, and then complete a survey of those locally toroidal polytopes which can be described by our construction.Comment: Advances in Applied Mathematics (to appear); 19 page

Reflection groups and polytopes over finite fields, II

When the standard representation of a crystallographic Coxeter group $\Gamma$ is reduced modulo an odd prime $p$, a finite representation in some orthogonal space over $\mathbb{Z}_p$ is obtained. If $\Gamma$ has a string diagram, the latter group will often be the automorphism group of a finite regular polytope. In Part I we described the basics of this construction and enumerated the polytopes associated with the groups of rank 3 and the groups of spherical or Euclidean type. In this paper, we investigate such families of polytopes for more general choices of $\Gamma$, including all groups of rank 4. In particular, we study in depth the interplay between their geometric properties and the algebraic structure of the corresponding finite orthogonal group.Comment: 30 pages (Advances in Applied Mathematics, to appear

Semisymmetric graphs from polytopes

AbstractEvery finite, self-dual, regular (or chiral) 4-polytope of type {3,q,3} has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by dropping self-duality, we obtain a construction for semisymmetric trivalent graphs (which are edge- but not vertex-transitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4-polytope

The Carnegie Hubble Program: The Distance and Structure of the SMC as Revealed by Mid-infrared Observations of Cepheids

Using Spitzer observations of classical Cepheids we have measured the true average distance modulus of the SMC to be $18.96 \pm 0.01_{stat} \pm 0.03_{sys}$ mag (corresponding to $62 \pm 0.3$ kpc), which is $0.48 \pm 0.01$ mag more distant than the LMC. This is in agreement with previous results from Cepheid observations, as well as with measurements from other indicators such as RR Lyrae stars and the tip of the red giant branch. Utilizing the properties of the mid--infrared Leavitt Law we measured precise distances to individual Cepheids in the SMC, and have confirmed that the galaxy is tilted and elongated such that its eastern side is up to 20 kpc closer than its western side. This is in agreement with the results from red clump stars and dynamical simulations of the Magellanic Clouds and Stream.Comment: Accepted for publication in ApJ. 38 Pages, 11 figures. Figure 9 is interactive. Spitzer photometry for all Cepheids available as online tabl

The Carnegie Hubble Program

We present an overview of and preliminary results from an ongoing comprehensive program that has a goal of determining the Hubble constant to a systematic accuracy of 2%. As part of this program, we are currently obtaining 3.6 micron data using the Infrared Array Camera (IRAC) on Spitzer, and the program is designed to include JWST in the future. We demonstrate that the mid-infrared period-luminosity relation for Cepheids at 3.6 microns is the most accurate means of measuring Cepheid distances to date. At 3.6 microns, it is possible to minimize the known remaining systematic uncertainties in the Cepheid extragalactic distance scale. We discuss the advantages of 3.6 micron observations in minimizing systematic effects in the Cepheid calibration of the Hubble constant including the absolute zero point, extinction corrections, and the effects of metallicity on the colors and magnitudes of Cepheids. We are undertaking three independent tests of the sensitivity of the mid-IR Cepheid Leavitt Law to metallicity, which when combined will allow a robust constraint on the effect. Finally, we are providing a new mid-IR Tully-Fisher relation for spiral galaxies

Calibration of the Mid-Infrared Tully-Fisher Relation

Distance measures on a coherent scale around the sky are required to address the outstanding cosmological problems of the Hubble Constant and of departures from the mean cosmic flow. The correlation between galaxy luminosities and rotation rates can be used to determine distances to many thousands of galaxies in a wide range of environments potentially out to 200 Mpc. Mid-infrared (3.6 microns) photometry with the Spitzer Space Telescope is particularly valuable as the source of the luminosities because it provides products of uniform quality across the sky. From a perch above the atmosphere, essentially the total magnitude of targets can be registered in exposures of a few minutes. Extinction is minimal and the flux is dominated by the light from old stars which is expected to correlate with the mass of the targets. In spite of the superior photometry, the correlation between mid-infrared luminosities and rotation rates extracted from neutral hydrogen profiles is slightly degraded from the correlation found with I band luminosities. A color correction recovers a correlation that provides comparable accuracy to that available at I band (~20% 1sigma in an individual distance) while retaining the advantages identified above. Without the color correction the relation between linewidth and [3.6] magnitudes is M^{b,i,k,a}_{[3.6]} = -20.34 - 9.74 (log W_{mx}^{i} -2.5). This description is found with a sample of 213 galaxies in 13 clusters that define the slope and 26 galaxies with Cepheid or tip of the red giant branch distances that define the zero point. A color corrected parameter M_{C_{[3.6]}} is constructed that has reduced scatter: M_{C_{[3.6]}} = -20.34 - 9.13 (log W_{mx}^{i} -2.5). Consideration of the 7 calibration clusters beyond 50 Mpc, outside the domain of obvious peculiar velocities, provides a preliminary Hubble Constant estimate of H_0=74+/-5 km/s/Mpc.Comment: Accepted for publication in The Astrophysical Journal, 14 pages, 11 figures, 4 table

Carnegie Hubble Program: A Mid-Infrared Calibration of the Hubble Constant

Using a mid-infrared calibration of the Cepheid distance scale based on recent observations at 3.6 um with the Spitzer Space Telescope, we have obtained a new, high-accuracy calibration of the Hubble constant. We have established the mid-IR zero point of the Leavitt Law (the Cepheid Period-Luminosity relation) using time-averaged 3.6 um data for ten high-metallicity, Milky Way Cepheids having independently-measured trigonometric parallaxes. We have adopted the slope of the PL relation using time-averaged 3.6 um data for 80 long-period Large Magellanic Cloud (LMC) Cepheids falling in the period range 0.8 < log(P) < 1.8. We find a new reddening-corrected distance to the LMC of 18.477 +/- 0.033 (systematic) mag. We re-examine the systematic uncertainties in H0, also taking into account new data over the past decade. In combination with the new Spitzer calibration, the systematic uncertainty in H0 over that obtained by the Hubble Space Telescope (HST) Key Project has decreased by over a factor of three. Applying the Spitzer calibration to the Key Project sample, we find a value of H0 = 74.3 with a systematic uncertainty of +/-2.1 (systematic) km/s/Mpc, corresponding to a 2.8% systematic uncertainty in the Hubble constant. This result, in combination with WMAP7 measurements of the cosmic microwave background anisotropies and assuming a flat universe, yields a value of the equation of state for dark energy, w0 = -1.09 +/- 0.10. Alternatively, relaxing the constraints on flatness and the numbers of relativistic species, and combining our results with those of WMAP7, Type Ia supernovae and baryon acoustic oscillations yields w0 = -1.08 +/- 0.10 and a value of N_eff = 4.13 +/- 0.67, mildly consistent with the existence of a fourth neutrino species.Comment: 27 pages, 8 figures, Accepted for publication in Ap