Let M be a maximal subalgebra of the Lie algebra L. A subalgebra C of L is said to be a completion for M if C is not contained in M but every proper subalgebra of C that is an ideal of L is contained in M. The set I(M) of all completions of M is called the index complex of M in L. We use this concept to investigate the influence of the maximal subalgebras on the structure of a Lie algebra, in particular, finding new characterizations of solvable and supersolvable Lie algebras.

This paper aims at providing a mathematical foundation for the terms of the well spread supervisory rule “initial market value of assets must be at least equal to provision plus solvency capital”.

It starts with a risk-adjusted assessment — given by a set of test probabilities — of the future cash-flows coming from a company business plan and attempts to define terms of a supervisory accounting mode.

First, inspired by the idea of “representation” of obligations by “equivalent” assets, we define the supervisory provision (or “liability”) attached to existing obligations. This provision is market consistent according to the mathematical definition by Cheridito, Filipovic and Kupper and satisfies a property of equilibrium between supervision's wish for stress testing and management's possibility for appropriate choice of assets.

The comparison between the initial market price of assets and the supervisory provision defines “solvability” of existing obligations.

In a second step the paper defines a required solvency capital as related to the level of discrepancy between assets and obligations of a company. Solvency of a business plan is defined by requiring as initial market value an additional amount over the one needed for solvability: this is the required solvency capital. A business plan with zero required solvency capital is said to have an optimal replicating asset portfolio.

It is shown that — under a natural additional condition, that of a market prudent set of test probabilities — solvability of an obligation allows for solvency of a related business plan, by choice of the asset portfolio.

The paper emphasizes the distinction between supervisory and market oriented accounting hinted to in the CEIOPS CP 20 consultative paper.

The object of this paper is to show that every soluble right orderable group is locally indicable. The proof identifies an interesting connection between the theory of right orderable groups and the theory of amenable groups and bounded cohomology.

We obtain the following generalization of a well known result of Huppert. If p is the largest primer divisor of the order of a finite group G and q is any prime distinct from p, then G is supersolvable if and only if every maximal subgroup whose index is relatively prime to either p or q, has prime index.