STABLE RANK OF LEAVITT PATH ALGEBRAS OF ARBITRARY GRAPHS

The stable rank of Leavitt path algebras of row-finite graphs was computed by Ara and Pardo. In this paper we extend this to an arbitrary directed graph. In part our computation proceeds as for the row-finite case, but we also use knowledge of the row-finite setting by applying the desingularising method due to Drinen and Tomforde. In particular, we characterise purely infinite simple quotients of a Leavitt path algebra.

The Spectral Scale of a Self-Adjoint Operator in a Semifinite von Neumann Algebra

We extend Akemann, Anderson, and Weaver'sSpectral Scaledefinition to include selfadjoint operators fromsemifinitevon Neumann algebras. New illustrations of spectral scales in both the finite and semifinite von Neumann settings are presented. A counterexample to a conjecture made by Akemann concerning normal operators and the geometry of the their perspective spectral scales (in the finite setting) is offered.

ON THE POSSIBILISTIC DECISION MODEL: FROM DECISION UNDER UNCERTAINTY TO CASE-BASED DECISION

This paper improves a previously proposed axiomatic setting for qualitative decision under uncertainty in the von Neumann and Morgenstern' style, where only ordinal linear scales are required for assessing uncertainty and utility. Two qualitative criteria are axiomatized in a finite setting: a pessimistic one and an optimistic one, respectively obeying an uncertainty aversion axiom and an uncertainty-attraction axiom. These criteria generalize the well-known maximin and maximax criteria, making them more realistic. They are suited to one-shot decisions and they are not based on the notion of mean value, but take the form of medians. Elements for a qualitative case-based decision methodology are also proposed, with pessimistic and optimistic evaluations formally similar to the expressions which cope with uncertainty, up to modifying factors which cope with the lack of normalization of similarity evaluations. Finally two extensions of the model are analysed: (i) the case of generalized possibilistic mixtures, using a t-norm instead of min, and (ii) the case of evaluating either preferences or uncertainty on Cartesian products of ordinal scales.

EXTREMAL PROPERTIES OF BELIEF MEASURES IN THE THEORY OF EVIDENCE

An extension of Shannon entropy to the theory or belief and evidence is analyzed. Expressed as the weighed sum of logarithms of beliefs, it is termed an index of confusion. It is shown to exhibit a pathological behavior when reaching its maximum. The alternative, linear version of this index is shown to behave properly, reaching its extremum for a natural extension of the uniform distribution. Lastly, the consonant case of nested supports of evidence, is studied thoroughly, both in the finite setting, and when the frame of discernment is a real line.