Unscaled spatial branging process with interaction: macrospic equation and local equilibrium

For some spatial branching processes with interaction considered as measure–valued processes, convergence to solutions of non–linear macroscopic equation and local equilibrium are proved, without scaling but providing each particle with a small mass ε and assuming convergence of the initial distribution when ε goes to 0     

Extendibility of bilinear forms on banach sequence spaces

We study Hahn-Banach extensions of multilinear forms defined on Banach sequence spaces. We characterize c ₀ in terms of extension of bilinear forms, and describe the Banach sequence spaces in which every bilinear form admits extensions to any superspace.     

In Situ Generation of ArCu from CuF2 Makes Coupling of Bulky Aryl Silanes Feasible and Highly Efficient

A bimetallic system of Pd/CuF₂, catalytic in Pd and stoichiometric in Cu, is very efficient and selective for the coupling of fairly hindered aryl silanes with aryl, anisyl, phenylaldehyde, p‐cyanophenyl, p‐nitrophenyl, or pyridyl iodides of conventional size. The reaction involves the activation of the silane by Cu^II, followed by disproportionation and transmetalation from the Cu^I(aryl) to Pd^II, upon which coupling takes place. Cu^III formed during disproportionation is reduced to Cu^I(aryl) by excess aryl silane, so that the CuF₂ system is fully converted into Cu^I(aryl) and used in the coupling. Moreover, no extra source of fluoride is needed. Interesting size selectivity towards coupling is found in competitive reactions of hindered aryl silanes. Easily accessible [PdCl₂(IDM)(AsPh₃)] (IDM = 1,3‐dimethylimidazol‐2‐ylidene) is by far the best catalyst, and the isolated products are essentially free from As or Pd (<1 ppm). The mechanistic aspects of the process have been experimentally examined and discussed.     

Identity, Leibniz's Law and Non-transitive Reasoning

Arguments based on Leibniz's Law seem to show that there is no room for either indefinite or contingent identity. The arguments seem to prove too much, but their conclusion is hard to resist if we want to keep Leibniz's Law. We present a novel approach to this issue, based on an appropriate modification of the notion of logical consequence.