Algebraic Properties of the Real Quintic Equation for a Binary Gravitational Lens

It has been recently shown that the lens equation for a binary gravitational lens, which is apparently a coupled system, can be reduced to a real fifth-order (quintic) algebraic equation. Some algebraic properties of the real quintic equation are revealed. We find that the number of images on each side of the separation axis is independent of the mass ratio and separation unless the source crosses the caustics. Furthermore, the discriminant of the quintic equation enables us to study changes in the number of solutions, namely in the number of images. It is shown that this discriminant can be factorized into two parts: One represents the condition that the lens equation can be reduced to a single quintic equation, while the other corresponds to the caustics.Comment: 7 pages (PTPTeX); accepted for publication in Prog. Theor. Phy

Images for an Isothermal Ellipsoidal Gravitational Lens from a Single Real Algebraic Equation

We present explicit expressions for the lens equation for a cored isothermal ellipsoidal gravitational lens as a single real sixth-order algebraic equation in two approaches; 2-dimensional Cartesian coordinates and 3-dimensional polar ones. We find a condition for physical solutions which correspond to at most five images. For a singular isothermal ellipsoid, the sixth-order equation is reduced to fourth-order one for which analytic solutions are well-known. Furthermore, we derive analytic criteria for determining the number of images for the singular lens, which give us simple expressions for the caustics and critical curves. The present formulation offers a useful way for studying galaxy lenses frequently modeled as isothermal ellipsoids.Comment: 5 pages; accepted for publication in A&

Age of the Universe: Influence of the Inhomogeneities on the global Expansion-Factor

For the first time we calculate quantitatively the influence of inhomogeneities on the global expansion factor by averaging the Friedmann equation. In the framework of the relativistic second-order Zel'dovich-approximation scheme for irrotational dust we use observational results in form of the normalisation constant fixed by the COBE results and we check different power spectra, namely for adiabatic CDM, isocurvature CDM, HDM, WDM, Strings and Textures. We find that the influence of the inhomogeneities on the global expansion factor is very small. So the error in determining the age of the universe using the Hubble constant in the usual way is negligible. This does not imply that the effect is negligible for local astronomical measurements of the Hubble constant. Locally the determination of the redshift-distance relation can be strongly influenced by the peculiar velocity fields due to inhomogeneities. Our calculation does not consider such effects, but is contrained to comparing globally homogeneous and averaged inhomogeneous matter distributions. In addition we relate our work to previous treatments.Comment: 10 pages, version accepted by Phys. Rev.

Hic-5, an adaptor-like nuclear receptor coactivator

In recent years, numerous nuclear receptor-interacting proteins have been identified that influence nuclear transcription through their direct modification of chromatin. Along with coactivators that possess histone acetyltransferase (HAT) or methyltransferase activity, other coactivators that lack recognizable chromatin-modifying activity have been discovered whose mechanism of action is largely unknown. The presence of multiple protein-protein interaction motifs within mechanistically undefined coactivators suggests that they function as adaptor molecules, either recruiting or stabilizing promoter-specific protein complexes. This perspective will focus on a family of nuclear receptor coactivators (i.e., group III LIM domain proteins related to paxillin) that appear to provide a scaffold to stabilize receptor interactions with chromatin-modifying coregulators

One-variable word equations in linear time

In this paper we consider word equations with one variable (and arbitrary many appearances of it). A recent technique of recompression, which is applicable to general word equations, is shown to be suitable also in this case. While in general case it is non-deterministic, it determinises in case of one variable and the obtained running time is O(n + #_X log n), where #_X is the number of appearances of the variable in the equation. This matches the previously-best algorithm due to D\k{a}browski and Plandowski. Then, using a couple of heuristics as well as more detailed time analysis the running time is lowered to O(n) in RAM model. Unfortunately no new properties of solutions are shown.Comment: submitted to a journal, general overhaul over the previous versio