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&

Propagation of a magnetic domain wall in magnetic wires with asymmetric notches

The propagation of a magnetic domain wall (DW) in a submicron magnetic wire consisting of a magnetic/nonmagnetic/magnetic trilayered structure with asymmetric notches was investigated by utilizing the giant magnetoresistance effect. The propagation direction of a DW was controlled by a pulsed local magnetic field, which nucleates the DW at one of the two ends of the wire. It was found that the depinning field of the DW from the notch depends on the propagation direction of the DW.Comment: 12 pages, 3 figure

Current-driven domain wall motion in magnetic wires with asymmetric notches

Current-driven domain wall (DW) motion in magnetic wires with asymmetric notches was investigated by means of magnetic force microscopy. It was found that the critical current density necessary for the current-driven DW motion depended on the propagation direction of the DW. The DW moved more easily in the direction along which the slope of the asymmetric notch was less inclined.Comment: 11 pages, 2 figure

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.

Lagrangian description of the fluid flow with vorticity in the relativistic cosmology

We develop the Lagrangian perturbation theory in the general relativistic cosmology, which enables us to take into account the vortical effect of the dust matter. Under the Lagrangian representation of the fluid flow, the propagation equation for the vorticity as well as the density is exactly solved. Based on this, the coupling between the density and vorticity is clarified in a non-perturbative way. The relativistic correspondence to the Lagrangian perturbation theory in the Newtonian cosmology is also emphasized.Comment: 14 pages (RevTeX); accepted for publication in Phys. Rev.

How is the local-scale gravitational instability influenced by the surrounding large-scale structure formation?

We develop the formalism to investigate the relation between the evolution of the large-scale (quasi) linear structure and that of the small-scale nonlinear structure in Newtonian cosmology within the Lagrangian framework. In doing so, we first derive the standard Friedmann expansion law using the averaging procedure over the present horizon scale. Then the large-scale (quasi) linear flow is defined by averaging the full trajectory field over a large-scale domain, but much smaller than the horizon scale. The rest of the full trajectory field is supposed to describe small-scale nonlinear dynamics. We obtain the evolution equations for the large-scale and small-scale parts of the trajectory field. These are coupled to each other in most general situations. It is shown that if the shear deformation of fluid elements is ignored in the averaged large-scale dynamics, the small-scale dynamics is described by Newtonian dynamics in an effective Friedmann-Robertson-Walker (FRW) background with a local scale factor. The local scale factor is defined by the sum of the global scale factor and the expansion deformation of the averaged large-scale displacement field. This means that the evolution of small-scale fluctuations is influenced by the surrounding large-scale structure through the modification of FRW scale factor. The effect might play an important role in the structure formation scenario. Furthermore, it is argued that the so-called {\it optimized} or {\it truncated} Lagrangian perturbation theory is a good approximation in investigating the large-scale structure formation up to the quasi nonlinear regime, even when the small-scale fluctuations are in the non-linear regime.Comment: 15pages, Accepted for publication in Gravitation and General Relativit