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.

We study the properties of the n-volumic scalar curvature in this note. Lott-Sturm-Villani's curvature-dimension condition CD(κ,n) was showed to imply Gromov's n-volumic scalar curvature ≥nκ under an additional n-dimensional condition and we show the stability of n-volumic scalar curvature ≥κ with respect to smGH-convergence. Then we propose a new weighted scalar curvature on the weighted Riemannian manifold and show its properties.

This research, initiated by a problem excerpted from Baidu, one of the most famous internet search engines in China, explores two types of movements, “converging curve” and “tracing curve” named by this research group, which really act in a regular pattern. In the research, by applying scores of mathematical methods, we found that the key to converging curves is to spot the invariant elements from the changeful motions. For tracing curves, we adopt estimating equations to get the solution. Due to the limit of time and the relevant knowledge of the research group, many unexpected problems constantly appeared which are quite far beyond our ability. Yet we still tried to apply some promising mathematical thoughts, and have achieved some exciting results. In the process of discovering and rediscovering, we did not succeed in finding a perfect solution to the problem. After all, as a group of middle school students fascinated by mathematics, creating and discovering more are the greatest delight of us all.

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