Discrepancy of Symmetric Products of Hypergraphs

For a hypergraph ${\mathcal H} = (V,{\mathcal E})$, its $d$--fold symmetric product is $\Delta^d {\mathcal H} = (V^d,\{E^d |E \in {\mathcal E}\})$. We give several upper and lower bounds for the $c$-color discrepancy of such products. In particular, we show that the bound ${disc}(\Delta^d {\mathcal H},2) \le {disc}({\mathcal H},2)$ proven for all $d$ in [B. Doerr, A. Srivastav, and P. Wehr, Discrepancy of {C}artesian products of arithmetic progressions, Electron. J. Combin. 11(2004), Research Paper 5, 16 pp.] cannot be extended to more than $c = 2$ colors. In fact, for any $c$ and $d$ such that $c$ does not divide $d!$, there are hypergraphs having arbitrary large discrepancy and ${disc}(\Delta^d {\mathcal H},c) = \Omega_d({disc}({\mathcal H},c)^d)$. Apart from constant factors (depending on $c$ and $d$), in these cases the symmetric product behaves no better than the general direct product ${\mathcal H}^d$, which satisfies ${disc}({\mathcal H}^d,c) = O_{c,d}({disc}({\mathcal H},c)^d)$.Comment: 12 pages, no figure

Approximate Hypergraph Coloring under Low-discrepancy and Related Promises

A hypergraph is said to be $\chi$-colorable if its vertices can be colored with $\chi$ colors so that no hyperedge is monochromatic. $2$-colorability is a fundamental property (called Property B) of hypergraphs and is extensively studied in combinatorics. Algorithmically, however, given a $2$-colorable $k$-uniform hypergraph, it is NP-hard to find a $2$-coloring miscoloring fewer than a fraction $2^{-k+1}$ of hyperedges (which is achieved by a random $2$-coloring), and the best algorithms to color the hypergraph properly require $\approx n^{1-1/k}$ colors, approaching the trivial bound of $n$ as $k$ increases. In this work, we study the complexity of approximate hypergraph coloring, for both the maximization (finding a $2$-coloring with fewest miscolored edges) and minimization (finding a proper coloring using fewest number of colors) versions, when the input hypergraph is promised to have the following stronger properties than $2$-colorability: (A) Low-discrepancy: If the hypergraph has discrepancy $\ell \ll \sqrt{k}$, we give an algorithm to color the it with $\approx n^{O(\ell^2/k)}$ colors. However, for the maximization version, we prove NP-hardness of finding a $2$-coloring miscoloring a smaller than $2^{-O(k)}$ (resp. $k^{-O(k)}$) fraction of the hyperedges when $\ell = O(\log k)$ (resp. $\ell=2$). Assuming the UGC, we improve the latter hardness factor to $2^{-O(k)}$ for almost discrepancy-$1$ hypergraphs. (B) Rainbow colorability: If the hypergraph has a $(k-\ell)$-coloring such that each hyperedge is polychromatic with all these colors, we give a $2$-coloring algorithm that miscolors at most $k^{-\Omega(k)}$ of the hyperedges when $\ell \ll \sqrt{k}$, and complement this with a matching UG hardness result showing that when $\ell =\sqrt{k}$, it is hard to even beat the $2^{-k+1}$ bound achieved by a random coloring.Comment: Approx 201

Fermion loop simulation of the lattice Gross-Neveu model

We present a numerical simulation of the Gross-Neveu model on the lattice using a new representation in terms of fermion loops. In the loop representation all signs due to Pauli statistics are eliminated completely and the partition function is a sum over closed loops with only positive weights. We demonstrate that the new formulation allows to simulate volumes which are two orders of magnitude larger than those accessible with standard methods

The propagation of uncertainties in stellar population synthesis modeling III: model calibration, comparison, and evaluation

Stellar population synthesis (SPS) provides the link between the stellar and dust content of galaxies and their observed spectral energy distributions. In the present work we perform a comprehensive calibration of our own flexible SPS (FSPS) model against a suite of data. Several public SPS models are intercompared, including the models of Bruzual & Charlot (BC03), Maraston (M05) and FSPS. The relative strengths and weaknesses of these models are evaluated, with the following conclusions: 1) The FSPS and BC03 models compare favorably with MC data at all ages, whereas M05 colors are too red and the age-dependence is incorrect; 2) All models yield similar optical and near-IR colors for old metal-poor systems, and yet they all provide poor fits to the integrated J-K and V-K colors of both MW and M31 star clusters; 4) All models predict ugr colors too red, D4000 strengths too strong and Hdelta strengths too weak compared to massive red sequence galaxies, under the assumption that such galaxies are composed solely of old metal-rich stars; 5) FSPS and, to a lesser extent, BC03 can reproduce the optical and near-IR colors of post-starburst galaxies, while M05 cannot. Reasons for these discrepancies are explored. The failure at predicting the ugr colors, D4000, and Hdelta strengths can be explained by some combination of a minority population of metal-poor stars, young stars, blue straggler and/or blue horizontal branch stars, but not by appealing to inadequacies in either theoretical stellar atmospheres or canonical evolutionary phases (e.g., the main sequence turn-off). We emphasize that due to a lack of calibrating star cluster data in regions of the metallicity-age plane relevant for galaxies, all of these models continue to suffer from serious uncertainties that are difficult to quantify. (ABRIDGED)Comment: 26 pages, 16 figures, submitted to ApJ. The FSPS code can be downloaded at http://www.astro.princeton.edu/~cconroy/SPS

Examination of the mass-dependent Li depletion hypothesis by the Li abundances of the very metal-poor double-lined spectroscopic binary G166-45

The Li abundances of the two components of the very metal-poor ([Fe/H]=-2.5) double-lined spectroscopic binary G166-45 (BD+26 2606) are determined separately based on high resolution spectra obtained with the Subaru Telescope High Dispersion Spectrograph and its image slicer. From the photometric colors and the mass ratio the effective temperatures of the primary and secondary components are estimated to be 6350+/-100K and 5830+/-170K, respectively. The Li abundance of the primary (A(Li)=2.23) agrees well with the Spite plateau value, while that of the secondary is slightly lower (A(Li)=2.11). Such a discrepancy of the Li abundances between the two components is previously found in the extremely metal-poor, double-lined spectroscopic binary CS22876-032, however, the discrepancy in G166-45 is much smaller. The results agree with the trends found for Li abundance as a function of effective temperature (and of stellar mass) of main-sequence stars with -3.0<[Fe/H]<-2.0, suggesting that the depletion of Li at Teff ~ 5800K is not particularly large in this metallicity range. The significant Li depletion found in CS22876-032B is a phenomenon only found in the lowest metallicity range ([Fe/H]<-3).Comment: 3 figures, 1 table, to appear in ApJ