Transversals with Residue in Moderately Overlapping <i>T</i>(<i>k</i>)-Families of Translates
Bibliographic record
Abstract
Abstract Let K denote an oval, a centrally symmetric compact convex domain with non-empty interior. A family of translates of K is said to have property T(k) if for every subset of at most k translates there exists a common line transversal intersecting all of them. The integer k is the stabbing level of the family. Two translates Ki = K + ci and Kj = K + cj are said to be σ-disjoint if σK + ci and σK + cj are disjoint. A recent Helly-type result claims that for every σ > 0 there exists an integer k(σ) such that if a family of σ-disjoint unit diameter discs has property T(k)|k ≥ k(σ), then there exists a straight line meeting all members of the family. In the first part of the paper we give the extension of this theorem to translates of an oval K. The asymptotic behavior of k(σ) for σ → 0 is considered as well. Katchalski and Lewis proved the existence of a constant r such that for every pairwise disjoint family of translates of an oval K with property T(3) a straight line can be found meeting all but at most r members of the family. In the second part of the paper σ-disjoint families of translates of K are considered and the relation of σ and the residue r is investigated. The asymptotic behavior of r(σ) for σ → 0 is also discussed.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".