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A gas network consists of pipes to transport the gas from the suppliers to the consumers. Due to friction with the pipe walls, gas pressure gets lost. Compressors compensate this pressure loss at the cost of consuming fuel gas. The aim of gas network optimization is to minimize the fuel gas consumption of the compressors in such a way that the contracts with consumers and suppliers are fulfilled. This problem leads to a very complex mixed integer nonlinear optimization problem. We present a linear approach which concentrates on time-dependent and discrete aspects. The nonlinearities are approximated by piece-wise linear functions using the concept of SOS constraints. We develop a branch-and-cut algorithm which has the potential to guarantee global optimality of the linearized problem where the nonlinearities are approximated within a given accuracy. We include an adequate handling of the SOS conditions and a separation algorithm for switching processes of compressors. A simulated annealing algorithm is included yielding feasible solutions. Our computational results show the success of our approach in this challenging field of gas network optimization.

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Hopf bifurcation for smooth systems is characterized by a crossing of a pair of complex conjugate eigenvalues of the linearized problem through the imaginary axis. Since this approach is not at hand for non-smooth systems, we use the geometrical characterization given by the change from an unstable to a stable focus through a centre for a basic (piecewise) linear system.In that way we find two mechanisms for the destabilizing of the basic stationary solution and for the generation of bifurcating periodic orbits: a generation switch of the stability properties or the influence of the unstable subsystem measured by the time of duration spent in the subsystem. The switch between stable and unstable subsystems seems to be a general source of destabilization observed in several mechanical systems. We expect that the features analysed for planar systems will help us to understand higher-dimensional systems as well.

Myelin oligodendrocyte glycoprotein (MaG), a putative autoantigen in multiple sclerosis (MS), is a quantitatively minor component of the CNS. In view of the difficulties associated with the purification of MOG from brain tissues, the extracellular domain of human MOG corresponding to the N-terminal 121 amino acids was expressed in Escherichia co/i as a glutathione sulfotransferase fusion protein. The expressed protein was localized to inclusion bodies, and varying the growth parameters resulted in the solubilization of small amounts of GST-MOG that could be affinity purified on glutathione agarose columns. The fusion protein found in the inclusion bodies could be solubilized with urea. The solubilized fusion protein was cleaved with thrombin, and the extracellular domain was purified by CM Sephadex 50 chromatography to homogeneity. Injection of recombinant human MOG into different strains of mice resulted in the induction of an MS-like disease, characterized by severe neurological impairment and extensive CNS demyelinated lesions. Recombinant MOG produced in E. co/i should prove to be useful as a highly purified biological reagent for immunological, pathological, functional, and structural studies.