The unfolding and determinacy theorems for subgroups of 𝒜 and 𝒦

Volume preserving subgroups of $${{\mathcal A}}$$ and $${{\mathcal K}}$$ and singularities in unimodular geometry

“…These subgroups G q of G fail to be geometric subgroups of A and K in the sense of Damon [11,12], hence the usual determinacy and unfolding theorems do not hold for G q . In this situation moduli and even functional moduli often appear already in codimension zero, and, e.g., for R n this is indeed the case: a Morse function has a functional modulus (and hence infinite modality) in the volume preserving case [49].…”

On pairs of regular foliations in $${\mathbb{R}^3}$$ and singularities of map-germs

“…However, Damon showed that these results are also valid for a larger class of subgroups of K and A, which he called geometric subgroups of K and A ( [7]). These are subgroups that satisfy some algebraic properties that ensure that all the results on finite determinacy and versal unfoldings are valid for the action of such subgroups on m n · E(n, p).…”

“…We can then classify germs of functions in m 3 up to A * = G × L-equivalence. However, the group A * is not a Damon geometric subgroup [7], so the classification is in the formal category. (Basically, we show that there exists an integer k such that m l…”

“…The group K V is a Damon geometric subgroup (see [7]). A result of Damon [7] states that F is r − K V -determined if and only if there exists an integer l such that…”

Stability of equivariant bifurcation problems with two types of state variables and their unfoldings in presence of parameter symmetry

“…Some relevant concepts that we do not state can be found in Ref. [6]. {(R x,y,λ,α , J x,y,λ,α )} ≡ {(ε α , μ α ), (ε λ,α (Δ), μ α ε λ,α (Δ)), (ε y,λ,α (Γ, Δ), μ α ε y,λ,α (Γ, Δ)), (ε x,y,λ,α (Γ, Δ), μ α ε x,y,λ,α (Γ, Δ))} , which is a system of DA-algebras and Jacobson ideals.…”