Searching for a preferred direction with Union2.1 data

A cosmological preferred direction was reported from the type Ia supernovae (SNe Ia) data in recent years. We use the Union2.1 data to give a simple classification of such studies for the first time. Because the maximum anisotropic direction is independent of isotropic dark energy models, we adopt two cosmological models ($\Lambda$CDM, $w$CDM) for the hemisphere comparison analysis and $\Lambda$CDM model for dipole fit approach. In hemisphere comparison method, the matter density and the equation of state of dark energy are adopted as the diagnostic qualities in the $\Lambda$CDM model and $w$CDM model, respectively. In dipole fit approach, we fit the fluctuation of distance modulus. We find that there is a null signal for the hemisphere comparison method, while a preferred direction ($b=-14.3^\circ \pm 10.1^\circ, l=307.1^\circ \pm 16.2^\circ$) for the dipole fit method. This result indicates that the dipole fit is more sensitive than the hemisphere comparison method.Comment: 8 pages, 2 figures, accepted for publication in MNRA

Analytical Solutions of Singular Isothermal Quadrupole Lens

Using analytical method, we study the Singular Isothermal Quadrupole (SIQ) lens system, which is the simplest lens model that can produce four images. In this case, the radial mass distribution is in accord with the profile of the Singular Isothermal Sphere (SIS) lens, and the tangential distribution is given by adding a quadrupole on the monopole component. The basic properties of the SIQ lens have been studied in this paper, including deflection potential, deflection angle, magnification, critical curve, caustic, pseudo-caustic and transition locus. Analytical solutions of the image positions and magnifications for the source on axes are derived. As have been found, naked cusps will appear when the relative intensity $k$ of quadrupole to monopole is larger than 0.6. According to the magnification invariant theory of the SIQ lens, the sum of the signed magnifications of the four images should be equal to unity \citep{dal98}. However, if a source lies in the naked cusp, the summed magnification of the left three images is smaller than the invariant 1. With this simple lens system, we study the situations that a point source infinitely approaches a cusp or a fold. The sum of magnifications of cusp image triplet is usually not equal to 0, and it is usually positive for major cusp while negative for minor cusp. Similarly, the sum of magnifications of fold image pair is usually neither equal to 0. Nevertheless, the cusp and fold relations are still equal to 0, in that the sum values are divided by infinite absolute magnifications by definition.Comment: 12 pages, 2 figures, accepted for publication in ApJ

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

Thoughts on Jinan's Establishment of the National Central City of China's Yellow River Basin

The national central city is not a single existence, and it must be based on the national-level urban agglomeration. At the same time, it must have superior geographical advantages and rich natural resources as the basis for development. Urban agglomeration promotes national central cities; on the contrary, national central cities can also drive the common development of urban agglomerations. During the National People’s Congress and National Committee of the Chinese People’s Political Consultative Conference this year, Jinan Municipal Government proposed to create a national central city in the Yellow River Basin as the development goal. In this article, the measures taken by Jinan Municipal Government to create the national central city of the Yellow River Basin are put forward, and the impact of the surrounding urban agglomeration on the development of Jinan is pointed out. Meanwhile, the opportunities and challenges that Jinan will bring to the surrounding urban agglomeration by establishing the national central city are elaborated

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

An Improved Upper Bound for SAT

We show that the CNF satisfiability problem can be solved $O^*(1.2226^m)$ time, where $m$ is the number of clauses in the formula, improving the known upper bounds $O^*(1.234^m)$ given by Yamamoto 15 years ago and $O^*(1.239^m)$ given by Hirsch 22 years ago. By using an amortized technique and careful case analysis, we successfully avoid the bottlenecks in previous algorithms and get the improvement

The Magnification Invariant of Circularly-symmetric Lens Models

In the context of strong gravitational lensing, the magnification of image is of crucial importance to constrain various lens models. For several commonly used quadruple lens models, the magnification invariants, defined as the sum of the signed magnifications of images, have been analytically derived when the image multiplicity is a maximum. In this paper, we further study the magnification of several disk lens models, including (a) exponential disk lens, (b) Gaussian disk lens, (c) modified Hubble profile lens, and another two of the popular three-dimensional symmetrical lens model, (d) NFW lens and (e) Einasto lens. We find that magnification invariant does also exist for each lens model. Moreover, our results show that magnification invariants can be significantly changed by the characteristic surface mass density $\kappa_{\rm c}$.Comment: 14 pages, 6 figures. Accepted for publication in RA