Globular Cluster Luminosity Functions and the Hubble Constant from WFPC2 Imaging: Galaxies in the Coma I Cloud

The membership of some galaxies in the nearby (d ~ 12 Mpc) Coma I cloud is uncertain. Here we present globular cluster luminosity functions (GCLFs) from the HST for two bright ellipticals which may belong to this group. After fitting the GCLF, we find a turnover magnitude of m_V^0 = 23.23 +/- 0.11 for NGC 4278 and m_V^0 = 23.07 +/- 0.13 for NGC 4494. Our limiting magnitude is about two magnitudes fainter than these values, making this data among the most complete GCLFs published to date. The fitted GCLF dispersions (~ 1.1 mag.) are somewhat smaller than typical values for other ellipticals. Assuming an absolute turnover magnitude of M_V^0 = -7.62, and after applying a small metallicity correction, we derive distance modulii of (m -- M) = 30.61 +/- 0.14 for NGC 4278 and 30.50 +/- 0.15 for NGC 4494. These distance estimates are compared to other methods, and lie within the published range of values. We conclude that both galaxies lie at the same distance and are both members of the Coma I cloud.Comment: 13 pages, Latex. Full paper also available at http://www.ucolick.org/~forbes/home.htm

Completing the design spectra for graphs with six vertices and eight edges

Apart from two possible exceptions, the design spectrum has been determined for every graph with six vertices and at most eight edges. The purpose of this note is to establish the existence of the two missing designs, both of order 32

Improved Soundness for QMA with Multiple Provers

We present three contributions to the understanding of QMA with multiple provers: 1) We give a tight soundness analysis of the protocol of [Blier and Tapp, ICQNM '09], yielding a soundness gap Omega(1/N^2). Our improvement is achieved without the use of an instance with a constant soundness gap (i.e., without using a PCP). 2) We give a tight soundness analysis of the protocol of [Chen and Drucker, ArXiV '10], thereby improving their result from a monolithic protocol where Theta(sqrt(N)) provers are needed in order to have any soundness gap, to a protocol with a smooth trade-off between the number of provers k and a soundness gap Omega(k^2/N), as long as k>=Omega(log N). (And, when k=Theta(sqrt(N)), we recover the original parameters of Chen and Drucker.) 3) We make progress towards an open question of [Aaronson et al., ToC '09] about what kinds of NP-complete problems are amenable to sublinear multiple-prover QMA protocols, by observing that a large class of such examples can easily be derived from results already in the PCP literature - namely, at least the languages recognized by a non-deterministic RAMs in quasilinear time.Comment: 24 pages; comments welcom

UBRI Photometry of Globular Clusters in the Leo Group Galaxy NGC 3379

We present wide area UBRI photometry for globular clusters around the Leo group galaxy NGC 3379. Globular cluster candidates are selected from their B-band magnitudes and their (U-B)o vs (B-I)o colours. A colour-colour selection region was defined from photometry of the Milky Way and M31 globular cluster systems. We detect 133 globular cluster candidates which, supports previous claims of a low specific frequency for NGC 3379. The Milky Way and M31 reveal blue and red subpopulations, with (U-B)o and (B-I)o colours indicating mean metallicities similar to those expected based on previous spectroscopic work. The stellar population models of Maraston (2003) and Brocato etal (2000) are consistent with both subpopulations being old, and with metallicities of [Fe/H] \~ -1.5 and -0.6 for the blue and red subpopulations respectively. The models of Worthey (1994) do not reproduce the (U-B)o colours of the red (metal-rich) subpopulation for any modelled age. For NGC 3379 we detect a blue subpopulation with similar colours and presumably age/metallicity, to that of the Milky Way and M31 globular cluster systems. The red subpopulation is less well defined, perhaps due to increased photometric errors, but indicates a mean metallicity of [Fe/H] ~ -0.6.Comment: 12 pages, Latex, 10 figures, 1 table, submitted for publication in MNRAS, Fig. 11 available in source file or from [email protected]

Icosahedron designs

It is known from the work of Adams and Bryant that icosahedron designs of order v exist for v ≡ 1 (mod 60) as well as for v = 16. Here we prove that icosahedron designs exist if and only if v ≡ 1, 16, 21 or 36 (mod 60), wit

Functional lower bounds for arithmetic circuits and connections to boolean circuit complexity

We say that a circuit $C$ over a field $F$ functionally computes an $n$-variate polynomial $P$ if for every $x \in \{0,1\}^n$ we have that $C(x) = P(x)$. This is in contrast to syntactically computing $P$, when $C \equiv P$ as formal polynomials. In this paper, we study the question of proving lower bounds for homogeneous depth-$3$ and depth-$4$ arithmetic circuits for functional computation. We prove the following results : 1. Exponential lower bounds homogeneous depth-$3$ arithmetic circuits for a polynomial in $VNP$. 2. Exponential lower bounds for homogeneous depth-$4$ arithmetic circuits with bounded individual degree for a polynomial in $VNP$. Our main motivation for this line of research comes from our observation that strong enough functional lower bounds for even very special depth-$4$ arithmetic circuits for the Permanent imply a separation between ${\#}P$ and $ACC$. Thus, improving the second result to get rid of the bounded individual degree condition could lead to substantial progress in boolean circuit complexity. Besides, it is known from a recent result of Kumar and Saptharishi [KS15] that over constant sized finite fields, strong enough average case functional lower bounds for homogeneous depth-$4$ circuits imply superpolynomial lower bounds for homogeneous depth-$5$ circuits. Our proofs are based on a family of new complexity measures called shifted evaluation dimension, and might be of independent interest