We study the global dimension function $${\text {gldim}}:{\text {Aut}}\backslash {\text {Stab}}{\mathcal {D}}/\mathbb {C}\rightarrow \mathbb {R}_{\ge 0}$$ on quotient of space Bridgeland stability conditions a triangulated category $${\mathcal {D}}$$ as well Toda’s Gepner equation $$\Phi (\sigma )=s\cdot \sigma $$ for some $$\sigma \in and $$(\Phi ,s)\in {Aut}}{\mathcal {D}}\times {C}$$ . For b...

We associate a contractible “outer space” to any free product of groups G = G1 ∗ · · · ∗ Gq. It equals Culler-Vogtmann space when G is free, McCulloughMiller space when no Gi is Z. Our proof of contractibility (given when G is not free) is based on Skora’s idea of deforming morphisms between trees. Using the action of Out(G) on this space, we show that Out(G) has finite virtual cohomological di...

An analytic families index is defined for (cusp) pseudodifferential operators on a fibration with fibres which are compact manifolds with boundaries. This provides an extension to the boundary case of the setting of the (pseudodifferential) Atiyah-Singer theorem and to the pseudodifferential case of the families Atiyah-Patodi-Singer index theorem for Dirac operators due to Bismut and Cheeger an...

Open forms of global constraints allow the addition of new variables to an argument during the execution of a constraint program. Such forms are needed for difficult constraint programming problems where problem construction and problem solving are interleaved. However, in general, filtering that is sound for a global constraint can be unsound when the constraint is open. This paper provides a ...