Research proof index

The potential is stable if it is strongly stable, i.e. its quartic term is always positive. The proof of this result relies on the compactness of the closed unit ball in EuclideanSpace ℝ (Fin 3), and the extreme value theorem.

The potential stabilityCounterExample is not stable.

Let R₀ be a topological ring, topologically of finite type (over ℤ). Consider a family of (cardinality) finite rings R i with the discrete topology whose cardinalites are unifomly bounded. Given a family of continuous ring homs f i : R →+* R i, the lift R →+* 𝒰(Rᵢ) is also continuous.

Let R₀ be a topological ring, topologically of finite type (over ℤ). Consider a family of (cardinality) finite continuous R₀-algebras R i with the discrete topology whose cardinalites are unifomly bounded. Then 𝒰(Rᵢ) ≃ₐ[R] R i for F-many i.

Let R₀ be a topological ring, topologically of finite type (over ℤ). Consider a family of (cardinality) finite rings R i with the discrete topology whose cardinalites are unifomly bounded. Given a family of continuous ring homs f i : R →+* R i, there exists F-many i such that 𝒰(Rᵢ) ≃+* R i and this map is compatible with f.

Let R₀ be a topological ring, topologically of finite type (over ℤ). Consider a family of (cardinality) finite rings R i with the discrete topology whose cardinalites are unifomly bounded. Given a family of continuous surjective ring homs f i : R →+* R i, the lift R →+* 𝒰(Rᵢ) is also surjective.