Area-Proportional Drawings of Intersecting Families of Simple Closed Curves.
Bibliographic record
Abstract
A FISC, or family of intersecting simple closed curves, is a collection of simple closed curves in the plane with the properties that there is some open region common to the interiors of all the curves, and that every two curves intersect in finitely many points or arcs. Let F be a FISC with a set of open regions R. F is said to be area-proportional with respect to weight function ω : R → R if there is a positive constant α such that for any two finite regions, r1 and r2, area(r1)/area(r2) = αω(r1)/ω(r2). We consider F as a directed plane graph, ~ G(F), where the curve intersections are vertices and the curve arcs between vertices are edges. Edges are directed so that each of F ’s curves is traversed in a clockwise fashion. The directed plane dual of ~ G(F), denoted ~ D(F), has edges oriented to indicate inclusion in fewer interiors of the curves. The graph ~ G(F) has an area-proportional drawing with respect to ω if there is some FISC C that is area-proportional to ω and where F can be transformed into C by a continuous transformation of the plane. We describe an O(n|V |) algorithm for creating an area-proportional drawing of ~ G(F) = (V,E) where F is a FISC with n curves and ~ D(F) has only one source and only one sink. For the case of n-Venn diagrams, since |V | ≤ 2 − 2, this yields an O(|V |lg|V |) drawing algorithm.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.012 | 0.001 |
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".