A convex subset B of a real locally convex space X is said to have the separation property if it can be separated from any closed convex subset A of X, which is disjoint from B, by a closed hyperplane. The strong separation theorem says that if B is weakly compact then it has the separation property. In this paper, we present several versions for the converse and discuss some applications. For ...

We consider acyclic directed mixed graphs, in which directed edges (x → y) and bi-directed edges (x ↔ y) may occur. A simple extension of Pearl’s d-separation criterion, called m-separation, is applied to these graphs. We introduce a local Markov property which is equivalent to the global property resulting from the m-separation criterion.

In this article, we report the synthesis of single-crystalline nickel silicide nanowires (NWs) via chemical vapor deposition method using NiCl2·6H2O as a single-source precursor. Various morphologies of δ-Ni2Si NWs were successfully acquired by controlling the growth conditions. The growth mechanism of the δ-Ni2Si NWs was thoroughly discussed and identified with microscopy studies. Field emissi...

The separation of liquid mixtures has been studied for a long time. Separation is based on the difference in physical properties including pore size and electrostatic interaction. Therefore, there are many difficulties materials having similar such as ethanol–water 1,4-dioxane–water mixtures. While pervaporation transport rates by permeability through membrane suggested, it still high energy co...

We investigate the separation property for hierarchies of Schrödinger operators for identical particles. We show that such hierarchies of translation invariant second order differential operators are necessarily linear. A weakened form of the separation property, related to a strong form of cluster decomposition, allows for homogeneous hierarchies of nonlinear differential operators. Some conne...

The famous R. James’ Theorem (see [3–7] and [8]) asserts that a Banach space E is reflexive if and only if the closed unit ball UE has the James’ property, i.e. every continuous linear functional f of E attains its supremum in UE. James’ Theorem does not hold, however, for general normed spaces [6]. We prove in this talk that a normed space X is reflexive if and only if UX has the separation pr...