The logic of justified belief,explicit knowledge,and conclusive evidence

We present a complete, decidable logic for reasoning about a notion of completely trustworthy (“conclusive”) evidence and its relations to justifiable (implicit) belief and knowledge, as well as to their explicit justifications. This logic makes use of a number of evidence-related notions such as availability, admissibility, and “goodness” of a piece of evidence, and is based on an innovative modification of the Fitting semantics for Artemov?s Justification Logic designed to preempt Gettier-type counterexamples. We combine this with ideas from belief revision and awareness logics to provide an account for explicitly justified (defeasible) knowledge based on conclusive evidence that addresses the problem of (logical) omniscience.     

Optimal Control of the Euler Equations via Relaxation Approaches

We discuss two linear relaxation approaches to the optimal control of nonlinear hyperbolic systems, in particular the control of Euler flows in gas dynamics. The first method is a relaxation system that is due to Jin and Xin [2], the second one is a Lattice-Boltzman approach [3], where we use one spatial dimension and five velocities (D1Q5 model). Both methods are incorporated in an adjoint based steepest-descent algorithm for the optimisation. Convincing numerical results are presented for both methods for an example with discontinuous solutions. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homotopy Type of Complexes of Graph Homomorphisms between Cycles

In this paper we study the homotopy type of Hom(C_m,C_n), where C_k is the cyclic graph with k vertices. We enumerate connected components of Hom(C_m,C_n) and show that each such component is either homeomorphic to a point or homotopy equivalent to S¹. Moreover, we prove that Hom(C_m,L_n) is either empty or is homotopy equivalent to the union of two points, where L_n is an n-string, i.e., a tree with n vertices and no branching points.