We use the Atiyah-Bott-Segal-Singer Lefschetz fixed point formula to calculate the Hirzebruch ?_y genus X_y(S^[n]), where S^[n] is the Hilbert scheme of points of length n of a surface S. Combinatorial interpretation of the weights of the fixed points of the natural torus action on (²)^[n] is used. This is the first step to prove a conjectural formula about the elliptic genus of the Hilbert schemes.

We study the large time asymptotic speeds (turbulent flame speeds sT) of the simplified HamiltonJacobi (HJ) models arising in turbulent combustion. One HJ model is G-equation describing the front motion law in the form of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation, G-equations are HJ equations with convex (L1 type) but non-coercive Hamiltonians. The other is the quadratically nonlinear (L2 type) inviscid HJ model of MajdaSouganidis derived from the KolmogorovPetrovskyPiskunov reactive fronts. Motivated by a question posed by Embid, Majda and Souganidis (1995)[10], we compare the turbulent flame speeds sTs from these inviscid HJ models in two-dimensional cellular flows or a periodic array of steady vortices via sharp asymptotic estimates in the regime of large amplitude. The estimates are obtained by analyzing the action minimizing trajectories in the Lagrangian representation of solutions (Lax formula and its extension) in combination with delicate gradient bound of viscosity solutions to the associated cell problem of homogenization. Though the inviscid turbulent flame speeds share the same leading order asymptotics, their difference due to nonlinearities is identified as a subtle double logarithm in the large flow amplitude from the sharp growth laws. The turbulent flame speeds differ much more significantly in the corresponding viscous HJ models. 2013 Elsevier Masson SAS. All rights reserved.

In this paper we generalize a result in [J. An, Z. Wang, On the realization of Riemannian symmetric spaces in Lie groups, Topology Appl. 153 (7) (2005) 10081015, showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of fixed points of involutions are also proved.

Wave propagation in two-dimensional generalized honeycomb lattices is studied. By employing the tight-binding (TB) approximation, the linear dispersion relation and associated discrete envelope equations are derived for the lowest band. In the TB limit, the Bloch modes are localized at the minima of the potential wells and can analytically be constructed in terms of local orbitals. Bloch-mode relations are converted into integrals over orbitals. With this methodology, the linear dispersion relation is derived analytically in the TB limit. The nonlinear envelope dynamics are found to be governed by a unified nonlinear discrete wave system. The lowest Bloch band has two branches that touch at the Dirac points. In the neighborhood of these points, the unified system leads to a coupled nonlinear discrete Dirac system. In the continuous limit, the leading-order evolution is governed by a continuous nonlinear Dirac system. This system exhibits conical diffraction, a phenomenon observed in experiments. Coupled nonlinear Dirac systems are also obtained. Away from the Dirac points, the continuous limit of the discrete equation leads to coupled nonlinear Schrödinger equations when the underlying group velocities are nearly zero. With semiclassical approximations, all the parameters are estimated analytically.

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