Bibliographic record
Abstract
Abstract Let fi be a polynomial of degree nj, having all its zeros in the interval [aj,bj], where j 1, 2 and a1 < b1 < a2 < b2. If f1 (b1) h(a2) = 0, then, by Rolle’s theorem, the product f : z ↦ f1 (z) f2 (z) has exactly n1 ™1 critical points in [a1,b1], one critical point in (b1,a2), and n2 -1 critical points in [a2,b2]. When the zeros of f1 and f2 do not lie on two disjoint segments of a straight line, we may apply the Gauss Lucas theorem to conclude that, if fi is the set of the zeros of fj, where j = 1, 2, then the critical points of f lie in the convex hull of ε1 Uε2. The following theorem of Walsh shows that, in this general case, we can say more in terms of circular domains 𝒦1, 𝒦2 containing the sets ε1, ε2, respectively.
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 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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.004 | 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; 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".