A two point block backward difference method is established to solve Riccati differential equations directly. Based on a predictor-corrector two point block backward difference method (2PBBD), a code is developed using a set of integration coefficients that eliminates the need to be calculated at every step change. The method requires calculating the integration coefficients only once in the beginning. The 2PBBD has an added advantage of a recurrence relationship between coefficients of different orders which provides a more elegant algorithm. The recurrence relationship between coefficients also reduces the computational cost. � 2016 Author(s).

This article investigates a function f(x), constructed from the Nikol'skii class in S2. The estimation obtained will show that the Riesz mean of the spectral expansions is unable to be strengthened due to absence of localization caused by a singualrity at a definite point f(x), on the sphere. � 2017 Author(s).

We examine the comprehensive simultaneous relationship between countries, tertiary sector, financial leverage and firm financial performance. Most researchers routinely control for the industry or the country as a dummy variable in order to test how the firm financial leverage affect firm financial performance. However, the significant difference in the tertiary sector and, if there is, how the tertiary sector is different across countries, namely Malaysia and Indonesia have not extensively examined. Therefore, from the methodological standpoint, the PLS-SEM procedures and, as an extension, the PLS-MGA approach would make a valuable difference to current knowledge, especially from the perspectives of the capital structure theory. We test the equality of coefficient effects using Partial Least Square of Multi-group analysis (PLS-MGA) between country comparisons for tertiary sector of the economy. The data sample consists of 13 subsectors with the total final sample of 2720 for Malaysian and Indonesian tertiary sectors. The result reveals that there are few firm-country attributes to financial leverage and financial performance are differ across countries. However, we find that the Malaysian and Indonesian tertiary sector shows consistent equal effects in their financial leverage and firm financial performance.

In this paper we prove a precise equiconvergence relation between index of the Bochner-Riesz means of the expansions and power of the singularity of the distributions with compact support. � Published under licence by IOP Publishing Ltd.

An asymptotic formula for the Lesbegue constant of the Riesz means of Fourier-Laplace series on the sphere obtained in this paper. � Published under licence by IOP Publishing Ltd.

Convergence problems has been the focus of interest for researchers that are working in the fields of spectral theory. In the current research we investigate issues relating to the summability of the Fourier-Laplace series on the unit sphere. The necessary conditions which are required to obtain good estimation for summability of the Fourier-Laplace series investigated. This research will also provide new and sufficient conditions in the form of theorems and lemmas which will validate the uniform summability of the Fourier-Laplace series on the sphere. Published under licence by IOP Publishing Ltd.

This study examines the problem of uniform convergence for the functions from the Nikolskii class. The uniform convergences for the Riesz means of the Fourier-Laplace series in the Nikolskii class Hpa(SN) proved under certain conditions. � 2017 Author(s).