We investigate the problem of counting 1/16 Bogomol’nyi-Prasad-Sommerfield (BPS) operators in N=4 super-Yang-Mills theory at weak coupling. We present the complete set of 1/16 BPS operators in the infinite-N limit, which agrees with the counting of free BPS multigraviton states in the gravity dual AdS_5×S^5. Further, we conjecture that all 1/16 BPS operators in N=4 super-Yang-Mills theory are of the multigraviton form, and give numerical evidence for this conjecture. We discuss the implication of our conjecture and the seeming failure in reproducing the entropy of large 1/16 BPS black holes in AdS_5.

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To study the non-linear stability of a non-trivial profile for a multi-dimensional systems of gas dynamics, the combination of the Green function on estimating the lower order derivatives and the energy method for the higher order derivatives is shown to be not only useful but sometimes maybe also essential. In this paper, we study the stability of a planar diffusion wave for the isentropic Euler equations with damping in two-dimensional space. By introducing an approximate Green function for the linearized equations around the planar diffusion wave and by applying the energy method, we prove the global existence and the L 2 convergence rate of the solution when the initial data is a small perturbation of the planar diffusion wave. The decay rates of the perturbation and its lower order spatial derivatives obtained are optimal in the L 2 norm. Furthermore, the constructed approximate Green function in this paper can be