For a fixed geometric configuration, hydrodynamic instabilities and bifurcation processes of laminar isothermal planar opposed jet flows with symmetric and slightly asymmetric inlet boundary conditions are investigated numerically using a high-resolution approach based on spectral element method. In current configuration, when inlet boundary conditions are symmetric, in the range of the Reynolds number considered (Re <= 200), multiple new symmetry-breaking bifurcations are observed and four new flow patterns are identified. Their hydrodynamic characteristics are analyzed, in particular their symmetries. In addition, the case that inlet boundary conditions are slightly asymmetric is investigated. It is found that bifurcation processes are extremely sensitive to this small symmetry-breaking imperfection and much different from those in the symmetric case. Furthermore, model equations are constructed by symmetry consideration to explain the numerical results based on hydrodynamic equations.

Let M be a compact Hamiltonian T−space, with finite fixed point set MT . An equivariant class is determined by its restriction to MT , and to each fixed point p ∈ MT and generic component of the moment map, there corresponds a canonical class τp. For a special class of Hamiltonian T−spaces, the value τp,q of τp at a fixed point q can be determined through an iterated interpolation procedure, and we obtained a formula for τp,q as a sum over ascending chains from p to q. In general the number of such chains is huge, and the main result of this paper is a procedure to reduce the number of relevant chains, through a systematic degeneration of the interpolation direction. The resulting formula for τp,q resembles, via the localization formula, an integral over a space of chains, and we prove that, for complex Grassmannians, τp,q can indeed be expressed as the integral of an equivariant form over a smooth Schubert variety.

We prove that the number of critical points of a Li–Tam Green’s function on a complete open Riemannian surface of finite type admits a topological upper bound, given by the first Betti number of the surface. In higher dimensions, we show that there are no topological upper bounds on the number of critical points by constructing, for each nonnegative integer N, a Riemannian manifold diffeomorphic to Rn (n > 3) whose minimal Green’s function has at least N non-degenerate critical points. Variations on the method of proof of the latter result yield contractible n-manifolds whose minimal Green’s functions have level sets diffeomorphic to any fixed codimension 1 compact submanifold of R^n.

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