Assessing the effect of lens mass model in cosmological application with updated galaxy-scale strong gravitational lensing sample

By comparing the dynamical and lensing masses of early-type lens galaxies, one can constrain both the cosmological parameters and the density profiles of galaxies. We explore the constraining power on cosmological parameters and the effect of the lens mass model in this method with 161 galaxy-scale strong lensing systems, which is currently the largest sample with both high resolution imaging and stellar dynamical data. We assume a power-law mass model for the lenses, and consider three different parameterizations for $\gamma$ (i.e., the slope of the total mass density profile) to include the effect of the dependence of $\gamma$ on redshift and surface mass density. When treating $\delta$ (i.e., the slope of the luminosity density profile) as a universal parameter for all lens galaxies, we find the limits on the cosmological parameter $\Omega_m$ are quite weak and biased, and also heavily dependent on the lens mass model in the scenarios of parameterizing $\gamma$ with three different forms. When treating $\delta$ as an observable for each lens, the unbiased estimate of $\Omega_m$ can be obtained only in the scenario of including the dependence of $\gamma$ on both the redshift and the surface mass density, that is $\Omega_m = 0.381^{+0.185}_{-0.154}$ at 68\% confidence level in the framework of a flat $\Lambda$CDM model. We conclude that the significant dependencies of $\gamma$ on both the redshift and the surface mass density, as well as the intrinsic scatter of $\delta$ among the lenses, need to be properly taken into account in this method.Comment: Accepted for publication in MNRAS; 17 pages, 5 figures, 2 table

Hard thresholding hyperinterpolation over general regions

This paper proposes a novel variant of hyperinterpolation, called hard thresholding hyperinterpolation. This approximation scheme of degree $n$ leverages a hard thresholding operator to filter all hyperinterpolation coefficients which approximate the Fourier coefficients of a continuous function by a quadrature rule with algebraic exactness $2n$. We prove that hard thresholding hyperinterpolation is the unique solution to an $\ell_0$-regularized weighted discrete least squares approximation problem. Hard thresholding hyperinterpolation is not only idempotent and commutative with hyperinterpolation, but also satisfies the Pythagorean theorem. By estimating the reciprocal of the Christoffel function, we demonstrate that the upper bound of the uniform norm of hard thresholding hyperinterpolation operator is not greater than that of hyperinterpolation operator. Hard thresholding hyperinterpolation possesses denoising and basis selection abilities as Lasso hyperinterpolation. To judge the denoising effects of hard thresholding and Lasso hyperinterpolations, this paper yields a criterion that combines the regularization parameter and the product of noise coefficients and signs of hyperinterpolation coefficients. Numerical examples on the spherical triangle and the cube demonstrate the denoising performance of hard thresholding hyperinterpolation.Comment: 19 pages, 7 figure

Phase transitions and thermodynamics of the two-dimensional Ising model on a distorted Kagom\'{e} lattice

The two-dimensional Ising model on a distorted Kagom\'{e} lattice is studied by means of exact solutions and the tensor renormalisation group (TRG) method. The zero-field phase diagrams are obtained, where three phases such as ferromagnetic, ferrimagnetic and paramagnetic phases, along with the second-order phase transitions, have been identified. The TRG results are quite accurate and reliable in comparison to the exact solutions. In a magnetic field, the magnetization ($m$), susceptibility and specific heat are studied by the TRG algorithm, where the $m=1/3$ plateaux are observed in the magnetization curves for some couplings. The experimental data of susceptibility for the complex Co(N$_3$)$_2$(bpg)$\cdot$ DMF$_{4/3}$ are fitted with the TRG results, giving the couplings of the complex $J=22K$ and $J'=33K$