On the Logic of Common Belief

We investigate an axiomatization of the notion of common belief (knowledge) that makes use of no rules of inference (apart from Modus Ponens and Necessitation) and highlight the property of the set of accessibility relations that characterizes each axiom. Mathematics Subject Classification: 03B45, 68T25.     

Third-order tensor potentials for the Riemann and Weyl tensors

The representations of the Riemann and the Weyl tensors of a four-dimensional Riemannian manifold through covariant derivatives of third-order potentials are examined in detail. The Weyl tensor always admits a completely general representation whereas the Riemann tensor does not. Nevertheless there exists a class of Riemannian manifolds whose Riemann tensors may be calculated in terms of potentials; in this connection, specific examples are exhibited explicitly. The possibility of introducing gauges on the potentials is reexamined in connection with the previous result. New properties of the representations are also discussed.     

Solution phase synthesis of a library of tetrasubstituted pyrrole amides

An efficient strategy for the solution-phase parallel synthesis of a library of pyrrole-amides is described. Key reactions include functional homologation of beta-ketoesters with a set of aldehydes followed by oxidation to produce a series of differently substituted 1,4-dicarbonyl compounds. Rapid cyclization using a microwave-assisted Paal-Knorr reaction provided a set of 24 pyrrole esters that were further functionalized through a trimethylaluminum-mediated aminolysis to obtain a larger library of 288 diverse pyrrole-3-amides. The tetrasubstitution allows a good exploration of the chemical space around the central pyrrole core. The last step was entirely automated with a Bohdan Myriad personal synthesizer.