Abstract. In this paper we prove a mass-capacity inequality and a volumetric Penrose inequality for conformally flat manifolds, in arbitrary dimensions. As a by-product of the proofs, Pólya-Szegö and Aleksandrov-Fenchel inequalities for mean-convex Euclidean domains are obtained. For each inequality, the case of equality is characterized.

We consider asymptotically flat Riemannian manifolds with non-negative scalar curvature that are conformal to R n \Ω, n ≥ 3, and so that their boundary is a minimal hypersurface. (Here, Ω ⊂ R n is open bounded with smooth mean-convex boundary.) We prove that the ADM mass of any such manifold is bounded below by 1 2 (V /βn) (n−2)/n , where V is the Euclidean volume of Ω and βn is the volume of the Euclidean unit n-ball. This gives a partial proof to a conjecture of Bray and Iga (Commun. Anal. Geom. 10:999-1016, 2002). Surprisingly, we do not require the boundary to be outermost.

In this paper we construct the first examples of (n + m + 2)-dimensional asymptotically flat Riemannian manifolds with non-negative scalar curvature that have outermost minimal hypersurfaces with non-spherical topology for n, m ≥ 1.The outermost minimal hypersurfaces are, topologically, S n × S m+1 . In the context of general relativity these hypersurfaces correspond to outermost apparent horizons of black holes.

This paper makes two distinctive contributions to the field of biosignal analysis, including performing signal processing in the topological domain and handling extremely small dataset. Currently, there have been no related works that can efficiently tackle the dilemma between avoiding electrochemical reaction and accelerating assay process using ACEK.

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