TAX REFORM AND REVENUE SHARING CHANGES: FISCAL IMPACTS ON SMALL, RURAL MASSACHUSETTS TOWNS

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Accurate fundamental parameters for Lower Main Sequence Stars

We derive an empirical effective temperature and bolometric luminosity calibration for G and K dwarfs, by applying our own implementation of the InfraRed Flux Method to multi-band photometry. Our study is based on 104 stars for which we have excellent BVRIJHK photometry, excellent parallaxes and good metallicities. Colours computed from the most recent synthetic libraries (ATLAS9 and MARCS) are found to be in good agreement with the empirical colours in the optical bands, but some discrepancies still remain in the infrared. Synthetic and empirical bolometric corrections also show fair agreement. A careful comparison to temperatures, luminosities and angular diameters obtained with other methods in literature shows that systematic effects still exist in the calibrations at the level of a few percent. Our InfraRed Flux Method temperature scale is 100K hotter than recent analogous determinations in the literature, but is in agreement with spectroscopically calibrated temperature scales and fits well the colours of the Sun. Our angular diameters are typically 3% smaller when compared to other (indirect) determinations of angular diameter for such stars, but are consistent with the limb-darkening corrected predictions of the latest 3D model atmospheres and also with the results of asteroseismology. Very tight empirical relations are derived for bolometric luminosity, effective temperature and angular diameter from photometric indices. We find that much of the discrepancy with other temperature scales and the uncertainties in the infrared synthetic colours arise from the uncertainties in the use of Vega as the flux calibrator. Angular diameter measurements for a well chosen set of G and K dwarfs would go a long way to addressing this problem.Comment: 34 pages, 20 figures. Accepted by MNRAS. Landscape table available online at http://users.utu.fi/luccas/IRFM

Covering collections and a challenge problem of Serre

We answer a challenge of Serre by showing that every rational point on the projective curve X$^4$ + Y$^4$ = 17 Z$^4$ is of the form ($\pm$1, $\pm$2, 1) or ($\pm$2, $\pm$1, 1). Our approach builds on recent ideas from both Nils Bruin and the authors on the application of covering collections and Chabauty arguments to curves of high rank. This is the only value of c$\le$81 for which the Fermat quartic X$^4$ + Y$^4$ = c Z$^4$ cannot be solved trivially, either by local considerations or maps to elliptic curves of rank 0, and it seems likely that our approach should give a method of attack for other nontrivial values of c

Finding rational points on bielliptic genus 2 curves

We discuss a technique for trying to find all rational points on curves of the form $Y^2 = f_3 X^6 + f_2 X^4 + f_1 X^2 + f_0$, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty's Theorem may be applied. However, we shall concentrate on the situation when the rank is at least 2. In this case, we shall derive an associated family of elliptic curves, defined over a number field Q(a). If each of these elliptic curves has rank less than the degree of Q(a) : Q, then we shall describe a Chabauty-like technique which may be applied to try to find all the points (x,y) defined over Q(a) on the elliptic curves, for which x is in Q. This in turn allows us to find all Q-rational points on the original genus 2 curve. We apply this to give a solution to a problem of Diophantus (where the sextic in X is irreducible over Q), which simplifies the recent solution of Wetherell. We also present two examples where the sextic in X is reducible over Q

Lattice-Constrained Parametrizations of Form Factors for Semileptonic and Rare Radiative B Decays

We describe the form factors for semileptonic B to rho l nu and radiative B to K* gamma decays with just two parameters and the two form factors for semileptonic B to pi l nu decays with three parameters. The parametrizations are constrained by lattice results and are consistent with heavy quark symmetry, kinematic constraints and light cone sum rule scaling relations.Comment: 3 pages, latex, 2 eps files, uses epsf.sty and espcrc2.sty, poster presented at Lattice 97, Edinburgh, 22-26 July 199

The arithmetic of hyperelliptic curves

We summarise recent advances in techniques for solving Diophantine problems on hyperelliptic curves; in particular, those for finding the rank of the Jacobian, and the set of rational points on the curve

Galactic Archaeology and Minimum Spanning Trees

Chemical tagging of stellar debris from disrupted open clusters and associations underpins the science cases for next-generation multi-object spectroscopic surveys. As part of the Galactic Archaeology project TraCD (Tracking Cluster Debris), a preliminary attempt at reconstructing the birth clouds of now phase-mixed thin disk debris is undertaken using a parametric minimum spanning tree (MST) approach. Empirically-motivated chemical abundance pattern uncertainties (for a 10-dimensional chemistry-space) are applied to NBODY6-realised stellar associations dissolved into a background sea of field stars, all evolving in a Milky Way potential. We demonstrate that significant population reconstruction degeneracies appear when the abundance uncertainties approach 0.1 dex and the parameterised MST approach is employed; more sophisticated methodologies will be required to ameliorate these degeneracies.Comment: To appear in "Multi-Object Spectroscopy in the Next Decade: Big Questions, Large Surveys and Wide Fields"; Held: Santa Cruz de La Palma, Canary Islands, Spain, 2-6 Mar 2015; ed. I Skillen & S. Trager; ASP Conference Series (Figures now optimised for B&W printing