Multipole Gravitational Lensing and High-order Perturbations on the Quadrupole Lens

An arbitrary surface mass density of gravitational lens can be decomposed into multipole components. We simulate the ray-tracing for the multipolar mass distribution of generalized SIS (Singular Isothermal Sphere) model, based on the deflection angles which are analytically calculated. The magnification patterns in the source plane are then derived from inverse shooting technique. As have been found, the caustics of odd mode lenses are composed of two overlapping layers for some lens models. When a point source traverses such kind of overlapping caustics, the image numbers change by \pm 4, rather than \pm 2. There are two kinds of images for the caustics. One is the critical curve and the other is the transition locus. It is found that the image number of the fold is exactly the average value of image numbers on two sides of the fold, while the image number of the cusp is equal to the smaller one. We also focus on the magnification patterns of the quadrupole (m = 2) lenses under the perturbations of m = 3, 4 and 5 mode components, and found that one, two, and three butterfly or swallowtail singularities can be produced respectively. With the increasing intensity of the high-order perturbations, the singularities grow up to bring sixfold image regions. If these perturbations are large enough to let two or three of the butterflies or swallowtails contact, eightfold or tenfold image regions can be produced as well. The possible astronomical applications are discussed.Comment: 24 pages, 6 figure

Chaotic Properties of Subshifts Generated by a Non-Periodic Recurrent Orbit

The chaotic properties of some subshift maps are investigated. These subshifts are the orbit closures of certain non-periodic recurrent points of a shift map. We first provide a review of basic concepts for dynamics of continuous maps in metric spaces. These concepts include nonwandering point, recurrent point, eventually periodic point, scrambled set, sensitive dependence on initial conditions, Robinson chaos, and topological entropy. Next we review the notion of shift maps and subshifts. Then we show that the one-sided subshifts generated by a non-periodic recurrent point are chaotic in the sense of Robinson. Moreover, we show that such a subshift has an infinite scrambled set if it has a periodic point. Finally, we give some examples and discuss the topological entropy of these subshifts, and present two open problems on the dynamics of subshifts

Scanning Tunneling Spectroscopic Studies of the Low-Energy Quasiparticle Excitations in Cuprate Superconductors

We report scanning tunneling spectroscopic (STS) studies of the low-energy quasiparticle excitations of cuprate superconductors as a function of magnetic field and doping level. Our studies suggest that the origin of the pseudogap (PG) is associated with competing orders (COs), and that the occurrence (absence) of PG above the superconducting (SC) transition T_c is associated with a CO energy Δ_(CO) larger (smaller) than the SC gap Δ_(SC). Moreover, the spatial homogeneity of Δ_(SC) and Δ_(CO) depends on the type of disorder in different cuprates: For optimally and under-doped YBa_2Cu_3O_(7−δ) (Y-123), we find that Δ_(SC) < Δ_(CO) and that both Δ_(SC) and Δ(CO) exhibit long-range spatial homogeneity, in contrast to the highly inhomogeneous STS in Bi_2Sr_2CaCu_2O_(8+x) (Bi-2212). We attribute this contrast to the stoichiometric cations and ordered apical oxygen in Y-123, which differs from the non-stoichiometric Bi-to-Sr ratio in Bi-2212 with disordered Sr and apical oxygen in the SrO planes. For Ca-doped Y-123, the substitution of Y by Ca contributes to excess holes and disorder in the CuO_2 planes, giving rise to increasing inhomogeneity, decreasing Δ_(SC) and Δ_(CO), and a suppressed vortex-solid phase. For electron-type cuprate Sr_(0.9)La_(0.1)CuO_2 (La-112), the homogeneous Δ_(SC) and Δ_(CO) distributions may be attributed to stoichiometric cations and the absence of apical oxygen, with Δ_(CO) < Δ_(SC) revealed only inside the vortex cores. Finally, the vortex-core radius (ξ_(halo)) in electron-type cuprates is comparable to the SC coherence length ξ_(SC), whereas ξ_(halo) ∼ 10ξ_(SC) in hole-type cuprates, suggesting that ξ_(halo) may be correlated with the CO strength. The vortex-state irreversibility line in the magnetic field versus temperature phase diagram also reveals doping dependence, indicating the relevance of competing orders to vortex pinning

Non-Commutative Gauge Theories and the Cosmological Constant

We discuss the issue of the cosmological constant in non-commutative non-supersymmetric gauge theories. In particular, in orbifold field theories non-commutativity acts as a UV cut-off. We suggest that in these theories quantum corrections give rise to a vacuum energy \rho, that is controlled by the non-commutativity parameter \theta, \rho ~ 1/theta^2 (only a soft logarithmic dependence on the Planck scale survives). We demonstrate our claim in a two-loop computation in field theory and by certain higher loop examples. Based on general expressions from string theory, we suggest that the vacuum energy is controlled by non-commutativity to all orders in perturbation theory.Comment: 11 pages, RevTex. 4 eps figures. v2: Typos corrected. To appear in Phys.Rev.