The blocking probability of spider-web networks

We determine the limiting behavior of the blocking probability for spider-web networks, a class of crossbar switching networks proposed by Ikeno. We use a probabilistic model proposed by the author, in which the busy links always form disjoint routes through the network. We show that if the occupancy probability is below the threshold 2 - √2 = 0.5857…, then the blocking probability tends to zero, whereas above this threshold it tends to one. This provides a theoretical explanation for results observed empirically in simulations by Bassalygo, Neiman, and Vvedenskaya.     

An expedient route to diaza-spirocycles utilizing a sequential multicomponent α-aminoallylation/ring-closing metathesis strategy

A general method for the preparation of diaza-spirocycles is reported. This method used an olefin metathesis in order to construct the desired spirocyclic framework. Beginning with commercially available protected amino ketones, this strategy ultimately produced pharmacologically relevant diaza-scaffolds in an efficient and high-yielding process.     

Ionic liquids as reagents and solvents in conjunction with microwave heating: rapid synthesis of alkyl halides from alcohols and nitriles from aryl halides

We show that using ionic liquids as reagents in conjunction with microwave heating it is possible to prepare primary alkyl halides from the corresponding alcohols rapidly. Using ionic liquids as solvents in conjunction with microwave heating it is possible to prepare aryl nitriles from the corresponding aryl bromides or iodides. The scope and limitations of using microwave-promotion as a tool in these reactions is discussed.     

Zur Trennung des Eisens von Nickel und Kobalt

Ohne Zusammenfassung     

The synthesis of 6-trifluoroethyl-l-lysine: a method to introduce functionality at C-6 of l-lysine

Described is a method of introducing trifluoroalkyl groups at C-6 of lysine. This chemistry has the potential to introduce a variety of functionality at C-6 of lysine.     

Laguerre approximation of stable manifolds with application to connecting orbits

We present an algorithm, based on approximation by Laguerre polynomials, for computing a point on the stable manifold of a stationary solution of an autonomous system. A superconvergence phenomenon means that the accuracy of our results is much higher than the usual spectral accuracy. Both the theory and the implementation of the method are considered. Finally, as an application of the algorithm, we describe a fully spectral approximation of homo- and heteroclinic orbits.

     

How many black holes fit on the head of a pin?

The Bekenstein–Hawking entropy of certain black holes can be computed microscopically in string theory by mapping the elusive problem of counting microstates of a strongly gravitating black hole to the tractable problem of counting microstates of a weakly coupled D-brane system, which has no event horizon, and indeed comfortably fits on the head of a pin. We show here that, contrary to widely held beliefs, the entropy of spherically symmetric black holes can easily be dwarfed by that of stationary multi-black-hole “molecules” of the same total charge and energy. Thus, the corresponding pin-sized D-brane systems do not even approximately count the microstates of a single black hole, but rather those of a zoo of entropically dominant multicentered configurations. Fourth Award in the 2007 Essay Competition of the Gravity Research Foundation.