Maximally monotone operators with ranges whose closures are not convex and an answer to a recent question by Stephen Simons
Bibliographic record
Abstract
In his recent Proceedings of the AMS paper, “Gossez’s skew linear map and its pathological maximally monotone multifunctions”, Stephen Simons proved that the closure of the range of the sum of the Gossez operator and a multiple of the duality map is not convex whenever the scalar is between 0 0 and 4 4 . The problem of the convexity of that range when the scalar is equal to 4 4 was explicitly stated. In this paper, we answer this question in the negative for any scalar greater than or equal to 4 4 . We derive this result from an abstract framework that allows us to also obtain a corresponding result for the Fitzpatrick-Phelps integral operator.
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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 | 0.004 | 0.016 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.007 |
| Scholarly communication | 0.002 | 0.014 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.004 | 0.006 |
| Insufficient payload (model declined to judge) | 0.004 | 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".