In this paper, we develop, analyze and test the Fourier spectral methods for solving the Degasperis-Procesi(DP) equation which contains nonlinear high order derivatives, and possibly discontinuous or sharp transition solutions. The $L^2$ stability is obtained for general numerical solutions of the Fourier Galerkin method and Fourier collocation (pseudospectral) method.By applying the Gegenbauer reconstruction technique as a post-processing method to the Fourier spectral solution, we reduce the oscillations arising from the discontinuity successfully. The numerical simulation results for different types of solutions of the nonlinear Degasperis-Procesi equation are provided to illustrate the accuracy and capability of the methods.

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We construct relative and global Euler sequences of a module. We apply it to prove some acyclicity results of the Koszul complex of a module and to compute the cohomology of the sheaves of (relative and absolute) differential $p$ -forms of a projective bundle. In particular we generalize Bott’s formula for the projective space to a projective bundle over a scheme of characteristic zero.