This paper proposes three enlargements of the conventional Moreau–Rockafellar subdifferential: sup-, sup⋆- and symmetric subdifferentials. They satisfy most fundamental properties convexity, weak∗-closedness and, if function is bounded, weak∗-compactness. Moreover, they are nonempty for Rainwater function, possess certain calculus rules, provide necessary sufficient optimality conditions not ne...

In this paper, we discuss two variants of the generalized nonlinear vector variational-like inequality problem. We provide their solutions by adopting topological approach. Topological properties such as compactness, closedness, and net theory are used in proof. The admissibility function space topology KKM-Theorem have played important role proving results.

Tensor network states (TNS) are a powerful approach for the study of strongly correlated quantum matter. The curse dimensionality is addressed by parametrizing many-body state in terms partially contracted tensors. These tensors form substantially reduced set effective degrees freedom. In practical algorithms, functionals like energy expectation values or overlaps optimized over certain sets TN...

The objective of this paper is to present a comprehensive study of dual representations of risk measures and convex functionals defined on an Orlicz space L or an Orlicz heart H. The first part of our study is devoted to the Orlicz pair (L, L). In this setting, we present a thorough analysis of the relationship between order closedness of a convex set C in L and the closedness of C with respect...

Let X be a topological space and let Φ ⊂ C(X). Then there exists a topological space Y containing X as a subspace and such that Φ = C(Y )|X, if and only if Φ is weakly composition closed, i.e., for any index set I, any fi ∈ Φ (i ∈ I) and any continuous map k : R → R we have k ◦ 〈fi〉 ∈ Φ, where 〈fi〉 : X → R is the map with i-th coordinate fi. The analogous statement is valid for functions to any...

where Ω is an ordinal (or the class of ordinals) and F is a family of (unary or finitary) operations on a topological space. The natural semantics is as follows: the modality 〈α〉 is interpreted by the αth iteration of the Cantor derivative, while the modalities 〈F 〉 and (F−1) are interpreted by the image and the preimage of the operation F . We describe logics in this language that are sound w....

We apply the existence and special properties of Gauduchon metrics to give several applications. The first one is concerned with implications algebro-geometric nature under a Hermitian metric nonnegative holomorphic sectional curvature. second show non-existence sections on vector bundles certain conditions. third restriction $\partial\bar{\partial}$-closedness some real $(n-1,n-1)$-forms compa...