A simplicial complex Δ is called$flag$if all minimal nonfaces of Δ have at most two elements. The following are proved: First, if Δ is a flag simplicial pseudomanifold of dimension$d$−1, then the graph of Δ (i) is (2$d$−2)-vertex-connected and (ii) has a subgraph which is a subdivision of the graph of the$d$-dimensional cross-polytope. Second, the$h$-vector of a flag simplicial homology sphere Δ of dimension$d$−1 is minimized when Δ is the boundary complex of the$d$-dimensional cross-polytope.

This paper deals with the derivation and analysis of the the Hall Magneto-Hydrodynamic equations. We first provide a derivation of this system from a two-fluids Euler-Maxwell system for electrons and ions, through a set of scaling limits. We also propose a kinetic formulation for the Hall-MHD equations which contains as fluid closure different variants of the Hall-MHD model. Then, we prove the existence of global weak solutions for the incompressible viscous resistive Hall-MHD model. We use the particular structure of the Hall term which has zero contribution to the energy identity. Finally, we discuss particular solutions in the form of axisymmetric purely swirling magnetic fields and propose some regularization of the Hall equation.

In this paper, we study the transonic shock problem for the full compressible Euler system in a general two-dimensional de Laval nozzle as proposed in Courant and Friedrichs (Supersonic flow and shock waves, Interscience, New York, 1948): given the appropriately large exit pressure pe(x), if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes pe(x). We solve this problem completely for a general class of de Laval nozzles whose divergent parts are small and arbitrary perturbations of divergent angular domains for the full steady compressible Euler system. The problem can be reduced to solve a nonlinear free boundary value problem for a mixed hyperbolic–elliptic system. One of the key ingredients in the analysis is to solve a nonlinear free boundary value problem in a weighted Hölder space with low regularities for a second order quasilinear elliptic equation with a free parameter (the position of the shock curve at one wall of the nozzle) and non-local terms involving the trace on the shock of the first order derivatives of the unknown function.

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In this paper, we prove the uniqueness of topological multivortex solutions for the self-dual Maxwell–Chern–Simons U(1)U(1) model if the Chern–Simons coupling parameter is sufficiently large and the charge of electron is sufficiently small or large. On the other hand, we also establish the sharp region of the flux for non-topological solutions and provide the classification of radial solutions of all types in the case of one vortex point.