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

Sandtime: A Tangible Interaction Featured Sensory Play Installation For Children To Increase Social Connection

From the study of social-interaction enhanced gaming design, aimed at providing a public environment which supports tangible & social interactions among children, we designed Sandtime. Sandtime is a public installation designed to encourage such interaction. Using the Tangible Interaction Design approach, this gaming installation features collaborative play and social interactions under public context, where children can collaboratively interact with the virtual onscreen characters by manipulating physical objects. This design is based on the study of how interactive gaming facilities can help to ease anxiety and enhance social interactions among children. In this paper, we want to continue this line of research by exploring further the elements that can enhance such interaction experience. This paper focuses specifically on the sensory play and how it can help to facilitate social interaction

A New Fast Monte Carlo Code for Solving Radiative Transfer Equations based on Neumann Solution

In this paper, we proposed a new Monte Carlo radiative transport (MCRT) scheme, which is based completely on the Neumann series solution of Fredholm integral equation. This scheme indicates that the essence of MCRT is the calculation of infinite terms of multiple integrals in Neumann solution simultaneously. Under this perspective we redescribed MCRT procedure systematically, in which the main work amounts to choose an associated probability distribution function (PDF) for a set of random variables and the corresponding unbiased estimation functions. We can select a relatively optimal estimation procedure that has a lower variance from an infinite possible choices, such as the term by term estimation. In this scheme, MCRT can be regarded as a pure problem of integral evaluation, rather than as the tracing of random walking photons. Keeping this in mind, one can avert some subtle intuitive mistakes. In addition the $\delta$-functions in these integrals can be eliminated in advance by integrating them out directly. This fact together with the optimal chosen random variables can remarkably improve the Monte Carlo (MC) computational efficiency and accuracy, especially in systems with axial or spherical symmetry. An MCRT code, Lemon (Linear Integral Equations' Monte Carlo Solver Based on the Neumann solution), has been developed completely based on this scheme. Finally, we intend to verify the validation of Lemon, a suite of test problems mainly restricted to flat spacetime have been reproduced and the corresponding results are illustrated in detail.Comment: 37 pages, 28 figures. The code can be download from: https://github.com/yangxiaolinyn/Lemon (or https://bitbucket.org/yangxiaolinsc/lemonsourcecode/src/main/) and https://doi.org/10.5281/zenodo.4686355. Comments are welcom