Error bounds for low-rank approximations of the first exponential integral kernel

A hierarchical matrix is an efficient data-sparse representation of a matrix, especially useful for large dimensional problems. It consists of low-rank subblocks leading to low memory requirements as well as inexpensive computational costs. In this work, we discuss the use of the hierarchical matrix technique in the numerical solution of a large scale eigenvalue problem arising from a finite rank discretization of an integral operator. The operator is of convolution type, it is defined through the first exponential-integral function and, hence, it is weakly singular. We develop analytical expressions for the approximate degenerate kernels and deduce error upper bounds for these approximations. Some computational results illustrating the efficiency and robustness of the approach are presented.

Stability, sustainability and viability of the Portuguese Football Federation

Purpose: to evaluate and study the viability, stability and the ability of the Portuguese Football Federation (PFF) to generate sustained profits. Methodology: Data were collected based on the Audit Reports of the institution during 2012-2014 and a financial and economic analysis was performed in order to establish some indicators of solvability, profitability and financial balance. Findings: It exists a lack of consistency in managing the profits obtained. We can also suggest that should be given a greater interest to the management of their own intangible assets, as brand management, for example. Practical implications: By making known to leaders and managers of this type of institutions that exists a link between participation in international championships and increase of their profitability may encourage them to better managing these cash inputs in order to decrease the dependence of Governmental financing. We also found that the management of their own intangible assets, as brand management, for example, could probably add more positive financial results.