Pearl’s well-known d-separation criterion for an acylic directed graph (ADG) is a pathwise separation criterion that can be used to efficiently identify all valid conditional independence relations in the Markov model determined by the graph. This paper introduces p-separation, a pathwise separation criterion that efficiently identifies all valid conditional independences under the Andersson-Ma...

We establish, in innnite dimensional Banach space, a nonconvex separation property for general closed sets that is an extension of Hahn-Banach separation theorem. We provide some consequences in optimization, in particular the existence of singular multipliers and show the relation of our principle with the extremal principle of Mordukhovich.

In this paper we proved a nonconvex separation property for general sets which coincides with the Hahn-Banach separation theorem when sets are convexes. Properties derived from the main result are used to compute the subgradient set to the distance function in special cases and they are also applied to extending the Second Welfare Theorem in economics and proving the existence of singular multi...

Pixels in an image are essentially constituted of two properties, position and colour. Pixel Property Separation, a radically different approach for Symmetric-key image encryption, separates these properties to disturb the semantics of the image. The scheme operates in two orthogonal stages each requiring an encryption key. The first stage creates the Position Vector, an ordered set of Pixel Po...

Consider the following seemingly rhetorical question: Is it crucial for a property-tester to know the error parameter 2 in advance? Previous papers dealing with various testing problems, suggest that the answer may be no, as in these papers there was no loss of generality in assuming that 2 is given as part of the input, and is not known in advance. Our main result in this paper, however, is th...