We prove that if $$ is an idempotent reflexive left Goldie ring whose simple singular left $$-modules are GP-injective, then $$ is a finite product of simple left Goldie rings. As a byproduct of this result we are able to show that if $$ is semiprime, left Goldie and left weakly $$-regular, then $$ is a finite product of simple left Goldie rings.

We construct examples of finite complexes without odd cells which have non-trivial $$-term of the Atiyah-Hirzebruch spectral sequence of KO-theory for $$.

Take a simple arc $$ in an open Riemann suface $$ carrying a nonconstant harmonic function $$ with finite Dirichlet integral $$. Form a Riemann surface $$ with lump $$ by pasting $$ with $$ crosswise along $$, i.e. $$, and the transplant $$ of $$ on $$ to $$ characterized by its being harmonic on $$ with $$ and $$ at the ideal boundary of $$ and hence of $$ in a suitable sense. We are interested in the comparison of $$ with $$ when we take a variety of choices of pasting arcs $$ in $$, and we will prove that $$ for any $$ level arc $$ in $$, $$ for any $$ conjugate level arc $$ in $$, and as a consequence of these two facts there is a nondegenerate arc $$ (i.e. not a point arc $$) in $$ such that $$.

Spaces of probability measures with positive density function on a compact Riemannian manifold are endowed with a closed 2-form associated with the Fisher information metric by using a divergence-free vector field. In this note we give a necessary and sufficient condition on the vector field that this 2-form is a strong symplectic structure.