Cusp Summations and Cusp Relations of Simple Quad Lenses

We review five often used quad lens models, each of which has analytical solutions and can produce four images at most. Each lens model has two parameters, including one that describes the intensity of non-dimensional mass density, and the other one that describes the deviation from the circular lens. In our recent work, we have found that the cusp and the fold summations are not equal to 0, when a point source infinitely approaches a cusp or a fold from inner side of the caustic. Based on the magnification invariant theory, which states that the sum of signed magnifications of the total images of a given source is a constant, we calculate the cusp summations for the five lens models. We find that the cusp summations are always larger than 0 for source on the major cusps, while can be larger or smaller than 0 for source on the minor cusps. We also find that if these lenses tend to the circular lens, the major and minor cusp summations will have infinite values, and with positive and negative signs respectively. The cusp summations do not change significantly if the sources are slightly deviated from the cusps. In addition, through the magnification invariants, we also derive the analytical signed cusp relations on the axes for three lens models. We find that both on the major and the minor axes the larger the lenses deviated from the circular lens, the larger the signed cusp relations. The major cusp relations are usually larger than the absolute minor cusp relations, but for some lens models with very large deviation from circular lens, the minor cusp relations can be larger than the major cusp relations.Comment: 8 pages, 4 figures, accepted for publication in MNRA

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

Magnification relations of quad lenses and applications on Einstein crosses

In this work, we mainly study the magnification relations of quad lens models for cusp, fold and cross configurations. By dividing and ray-tracing in different image regions, we numerically derive the positions and magnifications of the four images for a point source lying inside of the astroid caustic. Then, based on the magnifications, we calculate the signed cusp and fold relations for the singular isothermal elliptical lenses. The signed fold relation map has positive and negative regions, and the positive region is usually larger than the negative region as has been confirmed before. It can also explain that for many observed fold image pairs, the fluxes of the Fermat minimum images are apt to be larger than those of the saddle images. We define a new quantity cross relation which describes the magnification discrepancy between two minimum images and two saddle images. Distance ratio is also defined as the ratio of the distance of two saddle images to that of two minimum images. We calculate the cross relations and distance ratios for nine observed Einstein crosses. In theory, for most of the quad lens models, the cross relations decrease as the distance ratios increase. In observation, the cross relations of the nine samples do not agree with the quad lens models very well, nevertheless, the cross relations of the nine samples do not give obvious evidence for anomalous flux ratio as the cusp and fold types do. Then, we discuss several reasons for the disagreement, and expect good consistencies for more precise observations and better lens models in the future.Comment: 12 pages, 11 figures, accepted for publication in MNRA

Theoretical studies of tone noise from a fan rotor

An analytical study was made of some possible rotor alone noise sources of dipole, quadrapole and monopole characters which generate discrete tone noise. Particular emphasis is given to the tone noise caused by fan inlet flow distortion and turbulence. Analytical models are developed to allow prediction of absolute levels. Experimental data measured on a small scale fan is presented which indicates inlet turbulence interaction with a fan rotor can be a source of tone noise. Predicted and measured tone noise for the small scale rotor are shown to be in reasonable agreement

Natural and anthropogenic forest fires recorded in the Holocene pollen record from a Jinchuan peat bog, northeastern China

Pollen and charcoal particles from a Jinchuan peat (northeastern China) were examined to investigate the fire origin and interaction between climate, vegetation, fire and human activity during the Holocene. Pollen results show that: (i) a broadleaved deciduous forest was dominant during the early Holocene; (ii) from ~5500 cal. yr B.P. there was a gradual increase in coniferous trees (mainly Pinus), and a decrease in broadleaved deciduous trees (e.g. Quercus, Juglans, and Ulmus–Zelkova); (iii) after ~4200 cal. yr B.P., the deciduous forest was replaced by a mixed forest of coniferous and deciduous trees; (iv) coniferous trees including Pinus, Abies and Picea further increased after ~2000 cal. yr B.P., reflecting a cooler and drier climate after ~5500–4200 cal. yr B.P. Two layers of abundant microfossil charcoal particles (250–10 μm) and the coexistence of macrofossil particles (N2 mm) suggest two local fires: fire event 1 (5120±66 cal. yr B.P.) and fire event 2 (1288±8 cal. yr B.P., AD 662±8). Charcoal layer 1, with a large amount of Monolete psilate spores, is superimposed on the long-term trend of vegetation changes, indicating a natural origin for fire event 1 that was probably facilitated by drying environmental conditions since the mid-Holocene. Cerealia-type pollen and a low percentage of Monolete psilate spores were observed in charcoal layer 2, indicating that fire event 2 was caused by clearing. We suggest that fire event 2 may be related to the spread of the Han farming culture accompanied by the territorial expansion of the Tang Dynasty to the studied area in AD 668

Effect of Hydrostatic Pressure on the Superconductivity in NaxCoO2.yH2O

The effect of hydrostatic pressure on the superconducting transition temperature of Na{0.35}CoO{2}.yH{2}O was investigated by ac susceptibility measurements up to 1.6 GPa. The pressure coefficient of T{c} is negative and the dependence T{c}(p) is nonlinear over the pressure range investigated. The magnitude of the average dlnT{c}/dp=-0.07 GPa^{-1} is comparable to the pressure coefficient of electron-doped high-T{c} copper oxide superconductors with a similar value of T{c}. Our results provide support to the assumption of two-dimensional superconductivity in Na{0.35}CoO{2}.yH{2}O, which is similar to the cuprate systems, and suggest that intercalation of larger molecules may lead to an enhancement of T{c}.Comment: Revised Manuscrip

Generalised Spin Projection for Fermion Actions

The majority of compute time doing lattice QCD is spent inverting the fermion matrix. The time that this takes increases with the condition number of the matrix. The FLIC(Fat Link Irrelevant Clover) action displays, among other properties, an improved condition number compared to standard actions and hence is of interest due to potential compute time savings. However, due to its two different link sets there is a factor of two cost in floating point multiplications compared to the Wilson action. An additional factor of two has been attributed due to the loss of the so-called spin projection trick. We show that any split-link action may be written in terms of spin projectors, reducing the additional cost to at most a factor of two. Also, we review an efficient means of evaluating the clover term, which is additional expense not present in the Wilson action.Comment: 4 page