and Applied Analysis 3 Definition 1.4 see 18, 20–23 . 1 The intersection of all fuzzy α-closed resp., semiclosed, preclosed, γ-closed, semi-preclosed, β-closed sets containing a fuzzy set A is called a fuzzy α-closure resp., semiclosure, preclosure, γ-closure, semi-preclosure, β-closure of A. 2 The union of all fuzzy α-open resp., semiopen, preopen, γ-open, semi-preopen, β-open sets contained i...

Given Ω,Λ ⊂ R two bounded open sets, and f and g two probability densities concentrated on Ω and Λ respectively, we investigate the regularity of the optimal map ∇φ (the optimality referring to the Euclidean quadratic cost) sending f onto g. We show that if f and g are both bounded away from zero and infinity, we can find two open sets Ω′ ⊂ Ω and Λ′ ⊂ Λ such that f and g are concentrated on Ω′ ...

We provide a Hilbert-Mumford Criterion for actions of reductive groups G on Q-factorial complex varieties. The result allows to construct open subsets admitting a good quotient by G from certain maximal open subsets admitting a good quotient by a maximal torus of G. As an application, we indicate how to obtain all invariant open subsets with good quotient for a given G-action on a complete Q-fa...

The Hölder continuity of the truncated moment map a shade function in Euclidean space is established vicinity principal semi-algebraic set. proof combines volume bounds sets and convex optimization methods. main estimate applied to potential type transform specific two real variables, for perturbations quadrature domains.

Many algorithms for determining properties of real algebraic or semi-algebraic sets rely upon the ability to compute smooth points. In this paper, we present a procedure based on computing critical points some well-chosen function that guarantees computation in each bounded connected component (real) atomic set. Our technique is intuitive principal, performs well previously difficult examples, ...