25 research outputs found
Relative Equilibria in the Four-Vortex Problem with Two Pairs of Equal Vorticities
We examine in detail the relative equilibria in the four-vortex problem where
two pairs of vortices have equal strength, that is, \Gamma_1 = \Gamma_2 = 1 and
\Gamma_3 = \Gamma_4 = m where m is a nonzero real parameter. One main result is
that for m > 0, the convex configurations all contain a line of symmetry,
forming a rhombus or an isosceles trapezoid. The rhombus solutions exist for
all m but the isosceles trapezoid case exists only when m is positive. In fact,
there exist asymmetric convex configurations when m < 0. In contrast to the
Newtonian four-body problem with two equal pairs of masses, where the symmetry
of all convex central configurations is unproven, the equations in the vortex
case are easier to handle, allowing for a complete classification of all
solutions. Precise counts on the number and type of solutions (equivalence
classes) for different values of m, as well as a description of some of the
bifurcations that occur, are provided. Our techniques involve a combination of
analysis and modern and computational algebraic geometry
Volumes of polytopes in spaces of constant curvature
We overview the volume calculations for polyhedra in Euclidean, spherical and
hyperbolic spaces. We prove the Sforza formula for the volume of an arbitrary
tetrahedron in and . We also present some results, which provide a
solution for Seidel problem on the volume of non-Euclidean tetrahedron.
Finally, we consider a convex hyperbolic quadrilateral inscribed in a circle,
horocycle or one branch of equidistant curve. This is a natural hyperbolic
analog of the cyclic quadrilateral in the Euclidean plane. We find a few
versions of the Brahmagupta formula for the area of such quadrilateral. We also
present a formula for the area of a hyperbolic trapezoid.Comment: 22 pages, 9 figures, 58 reference
The Ks-band Tully-Fisher Relation - A Determination of the Hubble Parameter from 218 ScI Galaxies and 16 Galaxy Clusters
The value of the Hubble Parameter (H0) is determined using the
morphologically type dependent Ks-band Tully-Fisher Relation (K-TFR). The slope
and zero point are determined using 36 calibrator galaxies with ScI morphology.
Calibration distances are adopted from direct Cepheid distances, and group or
companion distances derived with the Surface Brightness Fluctuation Method or
Type Ia Supernova. Distances are determined to 16 galaxy clusters and 218 ScI
galaxies with minimum distances of 40.0 Mpc. From the 16 galaxy clusters a
weighted mean Hubble Parameter of H0=84.2 +/-6 km s-1 Mpc-1 is found. From the
218 ScI galaxies a Hubble Parameter of H0=83.4 +/-8 km s-1 Mpc-1 is found. When
the zero point of the K-TFR is corrected to account for recent results that
find a Large Magellanic Cloud distance modulus of 18.39 +/-0.05 a Hubble
Parameter of 88.0 +/-6 km s-1 Mpc-1 is found. A comparison with the results of
the Hubble Key Project (Freedman et al 2001) is made and discrepancies between
the K-TFR distances and the HKP I-TFR distances are discussed. Implications for
Lamda-CDM cosmology are considered with H0=84 km s-1 Mpc-1. (Abridged)Comment: 37 pages including 12 tables and 7 figures. Final version accepted
for publication in the Journal of Astrophysics & Astronom