Intersecting Lorenz curves and aversion to inverse downside inequality

This paper defines and characterizes the concept of an increase in inverse downside inequality and show that, when the Lorenz curves of two income distributions intersect, how the change from one distribution to the other is judged by an inequality index exhibiting inverse downside inequality aversion often depends on the relative strengths of its aversion to inverse downside inequality and inequality aversion. For the class of linear inequality indices, of which the Gini coefficient is a member, a measure characterizing the strength of an index’s aversion to inverse downside inequality against its own inequality aversion is shown to determine the ranking by the index of two distributions whose Lorenz curves cross once. The precise condition under which the same result generalizes to the case of multiple-crossing Lorenz curves is also identified.

Characters on C(X)

The precise condition on a completely regular space X for every character on C(X) to be an evaluation at some point in X is that X be realcompact. Usually, this classical result is obtained by relying heavily on involved (and even nonconstructive) extension arguments. This note provides a direct proof that is accessible to a large audience.

The module structure of Solomon's descent algebra

A close connection is uncovered between the lower central series of the free associative algebra of countable rank and the descending Loewy series of the direct sum of all Solomon descent algebras Δn, n ∈ ℕ0. Each irreducible Δn-module is shown to occur in at most one Loewy section of any principal indecomposable Δn-module.A precise condition for his occurence and formulae for the Cartan numbers are obtained.

MAXIMUM ENTROPY FOR REDUCED MOMENT PROBLEMS

The existence of maximum entropy solutions for a wide class of reduced moment problems on arbitrary open subsets of ℝd is considered. In particular, new results for the case of unbounded domains are obtained. A precise condition is presented under which solvability of the moment problem implies existence of a maximum entropy solution.

INSUFFICIENCY OF SCIENTIFIC REPORTING TO PRACTICE NEEDS

Physicians agree that the scientific literature ordinarily omits information which they consider essential to application: information on indications and clinical outcomes, risks and complications and, especially, the precise condition of the patients who did not do well. Scientists may deal in theories and statistics, but practitioners treat unique individuals.