Separation of variables in the generalized 4th Appelrot class

We consider the analogue of the 4th Appelrot class of motions of the Kowalevski top for the case of two constant force fields. The trajectories of this family fill the four-dimensional surface O^4 in the six-dimensional phase space. The constants of three first integrals in involution restricted to this surface fill one of the sheets of the bifurcation diagram in R^3. We point out the pair of partial integrals to obtain the explicit parametric equations of this sheet. The induced system on O^4 is shown to be Hamiltonian with two degrees of freedom having the thin set of points where the induced symplectic structure degenerates. The region of existence of motions in terms of the integral constants is found. We provide the separation of variables on O^4 and the algebraic formulae for the initial phase variables.Comment: LaTex, 16 pages, 1 figur

Addendum to `Fake Projective Planes'

The addendum updates the results presented in the paper `Fake Projective Plane, Invent Math 168, 321-370 (2007)' and makes some additions and corrections. The fake projective planes are classified into twenty six classes. Together with a recent work of Donald Cartwright and Tim Steger, there is now a complete list of fake projective planes. There are precisely one hundred fake projective planes as complex surfaces classified up to biholomorphism.Comment: A more refined classification is given in the new versio

Updating DL-Lite ontologies through first-order queries

In this paper we study instance-level update in DL-LiteA, the description logic underlying the OWL 2 QL standard. In particular we focus on formula-based approaches to ABox insertion and deletion. We show that DL-LiteA, which is well-known for enjoying first-order rewritability of query answering, enjoys a first-order rewritability property also for updates. That is, every update can be reformulated into a set of insertion and deletion instructions computable through a nonrecursive datalog program. Such a program is readily translatable into a first-order query over the ABox considered as a database, and hence into SQL. By exploiting this result, we implement an update component for DLLiteA-based systems and perform some experiments showing that the approach works in practice.Peer ReviewedPostprint (author's final draft