We prove some unconditional cases of the Existential Closedness problem for modular $j$-function. For this, we show that any finitely generated field can find a "convenient" set generators. This is done by showing in equipped with functions replicating algebraic behaviour $j$-function and its derivatives, one define natural closure operator three equivalent different ways.

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.