A Lower Limit to the Universal Density of Metals at z \sim 3

Column density distribution functions of CIV with 12.05 < log (N) < 14.35 and SiIV with 11.70 < log (N) < 13.93 have been obtained using 81 CIV absorbers and 35 SiIV absorbers redward of the Ly alpha forest in the lines of sight to seven quasars with 2.518 < z(em) < 3.78. These distribution functions have been directly integrated to yield ion densities at z = 3 to 3.5 of Omega(CIV) = (2.0 \pm 0.5) x 10(-8) and Omega(SiIV) = (7.0 \pm 2.6) x 10(-9) with H0 = 65 km/s/Mpc and q0 = 0.02 (1 sigma errors). A larger sample of 11 quasar lines of sight was used to measure CII/CIV, SiIII/SiIV, and NV/CIV ratios, which suggest that CIV and SiIV are the dominant ionization stages and that corrections to Omega(Carbon) and Omega(Silicon) are no more than a factor of two. Normalizing the alpha-process elements to silicon and the Fe-coproduction elements to carbon gives a density of heavy elements in these forest clouds of Omega(metals) = (3.3 \pm 0.8) x 10(-7) (H0 = 65, q0 = 0.02). The implications for the amount of star formation and for the ionization of the IGM prior to z = 3 are discussed.Comment: 12 pages LaTeX (aaspp4.sty) with 3 encapsulated postscript figures. To be published in ApJ Letters (accepted September 3, 1997

Physical conditions in broad and associated narrow absorption-line systems toward APM 08279+5255

Results of a careful analysis of the absorption systems with zabs = zem seen toward the bright, z_em ~ 3.91, gravitationally lensed quasar APM 08279+5255 are presented. Two of the narrow-line systems, at z_abs = 3.8931 and z_abs = 3.9135, show absorptions from singly ionized species with weak or no NV and O V absorptions at the same redshift. Absorption due to fine structure transitions of C II and Si II (excitation energies corresponding to, respectively, 156$\mu$m and 34$\mu$m) are detected at z_abs = 3.8931. Excitation by IR radiation is favored as the column density ratios are consistent with the shape of APM 08279+5255 IR spectrum. The low-ionization state of the system favors a picture where the cloud is closer to the IR source than to the UV source, supporting the idea that the extension of the IR source is larger than ~ 200 pc. The absence of fine structure lines at z_abs = 3.9135 suggests that the gas responsible for this system is farther away from the IR source. Abundances are ~ 0.01 and 1$Z_{\odot}$ at z_abs = 3.913 and 3.8931 and aluminum could be over-abundant with respect to silicon and carbon by at least a factor of two and five. All this suggests that whereas the \zabs = 3.8931 system is probably located within 200 pc from the QSO and ejected at a velocity larger than 1000 kms^{-1}, the \zabs = 3.9135 system is farther away and part of the host-galaxy. (abridged)Comment: 15 pages with 15 figures (psfiles), To appear in A&

Equivariantly uniformly rational varieties

We introduce equivariant versions of uniform rationality: given an algebraic group G, a G-variety is called G-uniformly rational (resp. G-linearly uniformly rational) if every point has a G-invariant open neighborhood equivariantly isomorphic to a G-invariant open subset of the affine space endowed with a G-action (resp. linear G-action). We establish a criterion for Gm-uniform rationality of affine variety equipped with hyperbolic Gm-action with a unique fixed point, formulated in term of their Altmann-Hausen presentation. We prove the Gm-uniform rationality of Koras-Russel threefolds of the first kind and we also give example of non Gm-uniformly rational but smooth rational Gm-threefold associated to pairs of plane rational curves birationally non equivalent to a union of lines

Automorphism groups of Koras-Russell threefolds of the second kind

We determine the automorphism groups of Koras-Russell threefolds of the second kind. In particular we show that these groups are semi-direct products of two subgroups, one given by the multiplicative group and the other isomorphic to a polynomial ring in two variables with the addition law. We also show that these groups are generated by algebraic subgroups isomorphic to Gm and Ga

About the Algebraic Solutions of Smallest Enclosing Cylinders Problems

Given n points in Euclidean space E^d, we propose an algebraic algorithm to compute the best fitting (d-1)-cylinder. This algorithm computes the unknown direction of the axis of the cylinder. The location of the axis and the radius of the cylinder are deduced analytically from this direction. Special attention is paid to the case d=3 when n=4 and n=5. For the former, the minimal radius enclosing cylinder is computed algebrically from constrained minimization of a quartic form of the unknown direction of the axis. For the latter, an analytical condition of existence of the circumscribed cylinder is given, and the algorithm reduces to find the zeroes of an one unknown polynomial of degree at most 6. In both cases, the other parameters of the cylinder are deduced analytically. The minimal radius enclosing cylinder is computed analytically for the regular tetrahedron and for a trigonal bipyramids family with a symmetry axis of order 3.Comment: 13 pages, 0 figure; revised version submitted to publication (previous version is a copy of the original one of 2010