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We present a simple proof on the formation of flocking in the F. Cucker and S. Smale system [Jpn. J. Math. (3) 2, No. 1, 197–227 (2007; Zbl 1166.92323)] based on the explicit construction of a Lyapunov functional. Our results also provide a unified condition on the initial states in which the exponential convergence to the flocking state will occur. For large particle systems, we give a rigorous justification for the mean-field limit from the many particle Cucker-Smale system to the Vlasov equation with flocking dissipation as the number of particles goes to infinity.

Let Mp(2n) be the metaplectic covering of Sp(2n) over a local field of characteristic zero. The core of the theory of endoscopy for Mp(2n) is the geometric transfer of orbital integrals to its elliptic endoscopic groups. The dual of this map, called the spectral transfer, is expected to yield endoscopic character relations which should reveal the internal structure of L-packets. As a first step, we characterize the image of the collective geometric transfer in the non-archimedean case, then reduce the spectral transfer to the case of cuspidal test functions by using a simple stable trace formula. In the archimedean case, we establish the character relations and determine the spectral transfer factors by rephrasing the works by Adams and Renard.

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