Freie Ringe und ihre geometrische Realisierung als planare Zykluskomplexe.Verlängerung von J.J. Satz der freien Gruppe von Rotman.
Bibliographic record
Abstract
This paper introduces free rings, free path groups and free path rings. A free ring is a Herstein ring on a free group. A free path group G which is a J.H.C. Whitehead homotopy group free on a collection of continuous maps h : [0, 1] → X (called paths) in a path cycle on a space X. Each path cycle is a sequence of paths with no end path. A free path ring is a Herstein ring on a free path group. The main results in this paper are (1) Every path cycle is realizable as a free path ring (Extension of J.J. Rotman's free group theorem), (2) Every 1-cycle cell complex is realizable as a path cycle and (3) Every 1cycle is realizable as a free ring (this reverbates back to W. Dyke's 1882 work on the derivation of free groups from circuits on simply connected polygons).
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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.008 | 0.002 |
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".