Eliminating graphs by means of parallel knock-out schemes

In 1997 Lampert and Slater introduced parallel knock-out schemes, an iterative process on graphs that goes through several rounds. In each round of this process, every vertex eliminates exactly one of its neighbors. The parallel knock-out number of a graph is the minimum number of rounds after which all vertices have been eliminated (if possible). The parallel knock-out number is related to well-known concepts like perfect matchings, hamiltonian cycles, and 2-factors.We derive a number of combinatorial and algorithmic results on parallel knock-out numbers: for families of sparse graphs (like planar graphs or graphs of bounded tree-width), the parallel knock-out number grows at most logarithmically with the number n of vertices; this bound is basically tight for trees. Furthermore, there is a family of bipartite graphs for which the parallel knock-out number grows proportionally to the square root of n. We characterize trees with parallel knock-out number at most 2, and we show that the parallel knock-out number for trees can be computed in polynomial time via a dynamic programming approach (whereas in general graphs this problem is known to be NP-hard). Finally, we prove that the parallel knock-out number of a claw-free graph is either infinite or less than or equal to 2.     

Stability of models with four-fermion interaction in the presence of vector fields

On the basis of an effective potential for quarks and gluons, for models with fourfermion interaction, stability is examined both with and without vector fields (gluons). The stability conditions relate the main characteristics of four-quark models with quark and gluon condensates.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 189, pp. 3–9, 1991.     

The impact of nonequilibrium gain in a spectral laser diode model

We describe the inclusion of nonequilibrium gain into a self-consistent 2.5D CW spectral laser diode model and report on the use of this model to investigate the origin of gain compression in a 975 nm high-brightness tapered QW laser diode. Nonequilibrium gain is calculated using a dynamic gain model, which simulates the dynamic relaxation of the quantum well carrier energy distributions under the influence of steady-state electrical and optical excitation. Calculated gain and spontaneous emission spectra are included in the laser model via parameterised look up tables. Both simulated and experimentally measured intracavity spontaneous emission spectra show an increased carrier density and a blue-shift of the gain maximum with increasing bias caused by carrier heating and spectral hole burning. The accurate incorporation of nonequilibrium gain compression is therefore vital for the accurate prediction of the operating characteristics of these devices and for the experimental determination of the active region temperature.     

1-Hydro-3-vinylhexamethylcyclotetrasiloxane and its polymerization

1,3-Dimethyl-3-vinyl-1,3-dichlorosiloxane is prepared by partial cohydrolysis of methylvinyldichlorosilane and methyldichlorosilane, and its cohydrolysis with sym-tetra-methyldichlorodisiloxane gives 1-hydro-3-vinylhexamethyl-cyclotetrasiloxane. A study is made of the kinetics of the polymerization of the latter in CCL₄ in the presence of H₂PtCl₆ · 6H₂O