On the zero set of <i>G</i>-equivariant maps
Bibliographic record
Abstract
Abstract Let G be a finite group acting on vector spaces V and W and consider a smooth G-equivariant mapping f: V → W. This paper addresses the question of the zero set of f near a zero x with isotropy subgroup G. It is known from results of Bierstone and Field on G-transversality theory that the zero set in a neighbourhood of x is a stratified set. The purpose of this paper is to partially determine the structure of the stratified set near x using only information from the representations V and W. We define an index s(Σ) for isotropy subgroups Σ of G which is the difference of the dimension of the fixed point subspace of Σ in V and W. Our main result states that if V contains a subspace G-isomorphic to W, then for every maximal isotropy subgroup Σ satisfying s(Σ) > s(G), the zero set of f near x contains a smooth manifold of zeros with isotropy subgroup Σ of dimension s(Σ). We also present partial results in the case of group representations V and W which do not satisfy the conditions of our main theorem. The paper contains many examples and raises several questions concerning the computation of zero sets of equivariant maps. These results have application to the bifurcation theory of G-reversible equivariant vector fields.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".