NON-ABELIAN AFFINE GROUPS WITH SCALAR LINEAR PART ACTING ON R^n
Bibliographic record
Abstract
In this paper, we explore the action of a non-Abelian group G on R^n , where G consists of affine transformations with linear parts being scalar multiples of the identity matrix. We provide a complete criterion for when G admits a dense orbit, thus characterizing its hypercyclic behavior. Additionally, we present an explicit and unified description of the closure of any orbit under the group action. Specifically, we express the closure of an orbit passing through any point \( x \in \mathbb{R}^n \) as:G(x) = \overline{\Lambda} \cdot T_x(\overline{K})$, where \Lambda is a multiplicative subgroup of R^-{0} , and K is an additive subgroup invariant under \Lambda . This form reveals a structure that partially mirrors the symmetry seen in Abelian group actions. The results offer a framework that can be extended to a broader class of non-Abelian affine groups.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.002 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.003 | 0.006 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.004 |
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; both teacher heads agree on what is shown here.
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".