In this paper, we introduce the concept of strongly binary g^⋆-closed sets in topological space and investigate group structure set all

We show that if a real x ∈ 2 is Hausdorff H-random, where h is an order function, then it is also random for a continuous probability measure μ such that the μ-measure of the basic open cylinders shrinks according to h. We use this result to derive a characterization of effective Hausdorff dimension similar to Frostman’s Theorem.

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In this note we prove that an extended-real-valued lower semi-continuous convex function Φ defined on a reflexive Banach space X achieves its supremum on every nonempty bounded and closed convex set of its effective domain Dom Φ, if and only if the restriction of Φ to Dom Φ is sequentially continuous with respect to the weak topology on the underlying space X .

We provide a specific representation of convex polynomials nonnegative on a convex (not necessarily compact) basic closed semi-algebraic set K ⊂ Rn. Namely, we prove that almost all of them belong to a specific subset of the quadratic module generated by the concave polynomials that define K. In particular, almost all nonnegative convex polynomials are sums of squares. Mathematics Subject Class...