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This paper is concerned with a partially observed optimal control problem described by mean-field forward and backward stochastic differential equations. Moreover, the control variable enters the diffusion coefficient and the control domain is non-convex. Utilising Girsanov's theorem as well as extended Ekeland's variational principle, a maximum principle is established in the form of Pontryagin's type. As an application, a linear-quadratic control problem is studied in terms of the stochastic filtering. [ABSTRACT FROM AUTHOR]

The authors consider a mixed super quasi-Einstein manifold, which is conducive to understanding deeply the global characteristics of the universe including its topology. Firstly, the existence of such manifolds is ensured by an interesting example. Then, the authors determine the curvature characterizations of a class of mixed super quasi-Einstein manifolds. Finally, the present paper depicts a W 2 -conservative characteristic of mixed super quasi-Einstein manifolds associated with a recurrent generator. [ABSTRACT FROM AUTHOR]