Strong coprimality and strong irreducibility of Alexander polynomials

A polynomial f(t) with rational coefficients is strongly irreducible if f(t^k) is irreducible for all positive integers k. Likewise, two polynomials f and g are strongly coprime if f(t^k) and g(t^l) are relatively prime for all positive integers k and l. We provide some sufficient conditions for strong irreducibility and prove that the Alexander polynomials of twist knots are pairwise strongly coprime and that most of them are strongly irreducible. We apply these results to describe the structure of the subgroup of the rational knot concordance group generated by the twist knots and to provide an explicit set of knots which represent linearly independent elements deep in the solvable filtration of the knot concordance group.Comment: 16 pages, 6 figure

CONTRIBUTIONS OF THE AXIAL SPINE TO KICKING BIOMECHANICS IN THE DIPPING KICK AMONG ELITE SOCCER PLAYERS

The aim of this study was to develop normative data for thoracic, lumbar and pelvic range of motion (ROM) during a soccer dipping kick among five NCAA Division I and high-level youth soccer players, comparing successful and unsuccessful kicks. The “dipping” kick is a complex, skill whereby a player strikes the ball so that it initially rises, but due to its top spin subsequently “dips” toward the intended target. From a repeated measures, cross-sectional design, successful kicks had a lower thoracic rotation at ball contact and average maximum thoracic rotation at 31.1±26.5º compared to the average maximum value for unsuccessful kicks at 43.7±28.6º, although not statistically significant. This study suggests that twisting the thoracic spine away from the target in an effort to “whip” and dip the ball may be suboptimal. The thoracic spine is more in line with the pelvis in successful kicks

Global Properties of M31's Stellar Halo from the SPLASH Survey: III. Measuring the Stellar Velocity Dispersion Profile

We present the velocity dispersion of red giant branch (RGB) stars in M31's halo, derived by modeling the line of sight velocity distribution of over 5000 stars in 50 fields spread throughout M31's stellar halo. The dataset was obtained as part of the SPLASH (Spectroscopic and Photometric Landscape of Andromeda's Stellar Halo) Survey, and covers projected radii of 9 to 175 kpc from M31's center. All major structural components along the line of sight in both the Milky Way (MW) and M31 are incorporated in a Gaussian Mixture Model, including all previously identified M31 tidal debris features in the observed fields. The probability an individual star is a constituent of M31 or the MW, based on a set of empirical photometric and spectroscopic diagnostics, is included as a prior probability in the mixture model. The velocity dispersion of stars in M31's halo is found to decrease only mildly with projected radius, from 108 km/s in the innermost radial bin (8.2 to 14.1 kpc) to $\sim 80$ to 90 km/s at projected radii of $\sim 40$ to 130 kpc, and can be parameterized with a power-law of slope $-0.12\pm 0.05$. The quoted uncertainty on the power-law slope reflects only the precision of the method, although other sources of uncertainty we consider contribute negligibly to the overall error budget.Comment: Submitted to the Astrophysical Journa

The Role of Dwarf Galaxies in Building Large Stellar Halos

The hierarchical theory of galaxy formation rests on the idea that smaller galactic structures merge to form the galaxies that we see today. The past decade has provided remarkable observational support for this scenario, driven in part by advances in spectroscopic instrumentation. Multi-object spectroscopy enabled the discovery of kinematically cold substructures around the Milky Way and M31 that are likely the debris of disrupting satellites. Improvements in high-resolution spectroscopy have produced key evidence that the abundance patterns of the Milky Way halo and its dwarf satellites can be explained by Galactic chemical evolution models based on hierarchical assembly. These breakthroughs have depended almost entirely on observations of nearby stars in the Milky Way and luminous red giant stars in M31 and Local Group dwarf satellites. In the next decade, extremely large telescopes will allow observations far down the luminosity function in the known dwarf galaxies, and they will enable observations of individual stars far out in the Galactic halo. The chemical abundance census now available for the Milky Way will become possible for our nearest neighbor, M31. Velocity dispersion measurements now available in M31 will become possible for systems beyond the Local Group such as Sculptor and M81 Group galaxies. Detailed studies of a greater number of individual stars in a greater number of spiral galaxies and their satellites will test hierarchical assembly in new ways because dynamical and chemical evolution models predict different outcomes for halos of different masses in different environments.Comment: Astro2010 Decadal Survey White Paper, 8 page

Power spectrum for the small-scale Universe

The first objects to arise in a cold dark matter universe present a daunting challenge for models of structure formation. In the ultra small-scale limit, CDM structures form nearly simultaneously across a wide range of scales. Hierarchical clustering no longer provides a guiding principle for theoretical analyses and the computation time required to carry out credible simulations becomes prohibitively high. To gain insight into this problem, we perform high-resolution (N=720^3 - 1584^3) simulations of an Einstein-de Sitter cosmology where the initial power spectrum is P(k) propto k^n, with -2.5 < n < -1. Self-similar scaling is established for n=-1 and n=-2 more convincingly than in previous, lower-resolution simulations and for the first time, self-similar scaling is established for an n=-2.25 simulation. However, finite box-size effects induce departures from self-similar scaling in our n=-2.5 simulation. We compare our results with the predictions for the power spectrum from (one-loop) perturbation theory and demonstrate that the renormalization group approach suggested by McDonald improves perturbation theory's ability to predict the power spectrum in the quasilinear regime. In the nonlinear regime, our power spectra differ significantly from the widely used fitting formulae of Peacock & Dodds and Smith et al. and a new fitting formula is presented. Implications of our results for the stable clustering hypothesis vs. halo model debate are discussed. Our power spectra are inconsistent with predictions of the stable clustering hypothesis in the high-k limit and lend credence to the halo model. Nevertheless, the fitting formula advocated in this paper is purely empirical and not derived from a specific formulation of the halo model.Comment: 30 pages including 10 figures; accepted for publication in MNRA

The SPLASH Survey: A Spectroscopic Portrait of Andromeda's Giant Southern Stream

The giant southern stream (GSS) is the most prominent tidal debris feature in M31's stellar halo. The GSS is composed of a relatively metal-rich, high surface-brightness "core" and a lower metallicity, lower surface brightness "envelope." We present Keck/DEIMOS spectroscopy of red giant stars in six fields in the vicinity of M31's GSS and one field on Stream C, an arc-like feature on M31's SE minor axis at R=60 kpc. Several GSS-related findings and measurements are presented here. We present the innermost kinematical detection of the GSS core to date (R=17 kpc). This field also contains the continuation of a second kinematically cold component originally seen in a GSS core field at R=21 kpc. The velocity gradients of the GSS and the second component in the combined data set are parallel over a radial range of 7 kpc, suggesting a possible bifurcation in the line-of-sight velocities of GSS stars. We also present the first kinematical detection of substructure in the GSS envelope. Using kinematically identified samples, we show that the envelope debris has a ~0.7 dex lower mean photometric metallicity and possibly higher intrinsic velocity dispersion than the GSS core. The GSS is also identified in the field of the M31 dSph satellite And I; the GSS in this field has a metallicity distribution identical to that of the GSS core. We confirm the presence of two kinematically cold components in Stream C, and measure intrinsic velocity dispersions of ~10 and ~4 km/s. This compilation of the kinematical (mean velocity, intrinsic velocity dispersion) and chemical properties of stars in the GSS core and envelope, coupled with published surface brightness measurements and wide-area star-count maps, will improve constraints on the orbit and internal structure of the dwarf satellite progenitor.Comment: Accepted for publication in Ap

Linking dwarf galaxies to halo building blocks with the most metal-poor star in Sculptor

Current cosmological models indicate that the Milky Way's stellar halo was assembled from many smaller systems. Based on the apparent absence of the most metal-poor stars in present-day dwarf galaxies, recent studies claimed that the true Galactic building blocks must have been vastly different from the surviving dwarfs. The discovery of an extremely iron-poor star (S1020549) in the Sculptor dwarf galaxy based on a medium-resolution spectrum cast some doubt on this conclusion. However, verification of the iron-deficiency and measurements of additional elements, such as the alpha-element Mg, are mandatory for demonstrating that the same type of stars produced the metals found in dwarf galaxies and the Galactic halo. Only then can dwarf galaxy stars be conclusively linked to early stellar halo assembly. Here we report high-resolution spectroscopic abundances for 11 elements in S1020549, confirming the iron abundance of less than 1/4000th that of the Sun, and showing that the overall abundance pattern mirrors that seen in low-metallicity halo stars, including the alpha-elements. Such chemical similarity indicates that the systems destroyed to form the halo billions of years ago were not fundamentally different from the progenitors of present-day dwarfs, and suggests that the early chemical enrichment of all galaxies may be nearly identical.Comment: 16 pages, including 2 figures. Accepted for publication in Nature. It is embargoed for discussion in the press until formal publication in Natur

Global Properties of M31's Stellar Halo from the SPLASH Survey. II. Metallicity Profile

We present the metallicity distribution of red giant branch (RGB) stars in M31's stellar halo, derived from photometric metallicity estimates for over 1500 spectroscopically confirmed RGB halo stars. The stellar sample comes from 38 halo fields observed with the Keck/DEIMOS spectrograph, ranging from 9 to 175 kpc in projected distance from M31's center, and includes 52 confirmed M31 halo stars beyond 100 kpc. While a wide range of metallicities is seen throughout the halo, the metal-rich peak of the metallicity distribution function becomes significantly less prominent with increasing radius. The metallicity profile of M31's stellar halo shows a continuous gradient from 9 to ~100 kpc, with a magnitude of ~ – 0.01 dex kpc–1. The stellar velocity distributions in each field are used to identify stars that are likely associated with tidal debris features. The removal of tidal debris features does not significantly alter the metallicity gradient in M31's halo: a gradient is maintained in fields spanning 10-90 kpc. We analyze the halo metallicity profile, as well as the relative metallicities of stars associated with tidal debris features and the underlying halo population, in the context of current simulations of stellar halo formation. We argue that the large-scale gradient in M31's halo implies M31 accreted at least one relatively massive progenitor in the past, while the field to field variation seen in the metallicity profile indicates that multiple smaller progenitors are likely to have contributed substantially to M31's outer halo