Theoretical consideration on the convergence of the fixed-point iteration method in the CIE mesopic photometry system MES2

Currently the fixed-point iteration method with initial guess 0=0.5 is officially recommended by the CIE MES2 system [CIE 191:2010] in order to compute the adaptation coefficient and the mesopic luminance . However, recently, Gao et al. [Opt. Express 25, 18365 (2017)] and Shpak et al. [Lighting Res. Technol. 49, 111 (2017)] have numerically found that the fixed-point iteration method could be not convergent for large values of / . Shpak et al. suspected that, to achieve convergence, the / ratio cannot be greater than 17. In this paper, a theoretical consideration for the CIE MES2 system is given. Namely, it is shown that the ratio / be smaller than a constant 2 (≈18.1834) is a sufficient condition for the convergence of the fixed-point iteration method. In addition, a new initial guess strategy, achieving faster convergence, is proposed

On the α-migrativity of multivariate semi-copulas

In this paper we introduce and study the α-migrative property for the class of multivariate semi-copulas. In particular, a characterization of α-migrative copulas is provided

Diagonal plane sections of trivariate copulas

We introduce the notion of diagonal plane section of a trivariate copula as an additional tool to describe its tail dependence behavior. This notion extends the concept of diagonal section of a bivariate copula. We provide existence results for trivariate copulas with a given diagonal plane function. Related results are discussed, especially about extensions to a higher dimensional setting

Some general results for moments in bivariate distributions

Integration by parts formula in n variables, Integration by parts in two variables, Bivariate survival function, Covariance between two functions, Generalized Gamma function,