Corrigenda to “Codomain rigidity of the Dirichlet to Neumann operator for the Riemannian wave equation”
Bibliographic record
Abstract
The proof of Lemma 4.4 in our article, which appeared in Trans. Amer. Math. Soc. 371 (2019), 8781–8810, contains a flaw. In proving the existence of a minimizer of the map A ↦ I ϵ [ A ] \mathbf {A} \mapsto I_\epsilon [\mathbf {A}] defined therein, we stated that this map is a convex function of A \mathbf {A} . This is incorrect, as I ϵ I_\epsilon is a composition of two convex functions, a quadratic form and an absolute value, and since the absolute value function is not monotonic, there is no guarantee that the resulting functional is convex. This short article corrects this flaw by showing that there is a continuous convex functional J ϵ J_\epsilon such that I ϵ [ A ] = J ϵ [ A 2 ] I_\epsilon [\mathbf {A}] = J_\epsilon [\mathbf {A}^2] , and then employing weak lower semi-continuity of J ϵ J_\epsilon to demonstrate the existence of a minimizer of I ϵ I_\epsilon .
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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 | 0.002 | 0.013 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.133 | 0.033 |
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".