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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper A right-arrow from bar upper I Subscript epsilon Baseline left-bracket bold upper A right-bracket"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">A</mml:mi> </mml:mrow> <mml:mo stretchy="false"> ↦ </mml:mo> <mml:msub> <mml:mi>I</mml:mi> <mml:mi> ϵ </mml:mi> </mml:msub> <mml:mo stretchy="false">[</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">A</mml:mi> </mml:mrow> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {A} \mapsto I_\epsilon [\mathbf {A}]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defined therein, we stated that this map is a convex function of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This is incorrect, as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript epsilon"> <mml:semantics> <mml:msub> <mml:mi>I</mml:mi> <mml:mi> ϵ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">I_\epsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J Subscript epsilon"> <mml:semantics> <mml:msub> <mml:mi>J</mml:mi> <mml:mi> ϵ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">J_\epsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript epsilon Baseline left-bracket bold upper A right-bracket equals upper J Subscript epsilon Baseline left-bracket bold upper A squared right-bracket"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>I</mml:mi> <mml:mi> ϵ </mml:mi> </mml:msub> <mml:mo stretchy="false">[</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">A</mml:mi> </mml:mrow> <mml:mo stretchy="false">]</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>J</mml:mi> <mml:mi> ϵ </mml:mi> </mml:msub> <mml:mo stretchy="false">[</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">A</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">I_\epsilon [\mathbf {A}] = J_\epsilon [\mathbf {A}^2]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and then employing weak lower semi-continuity of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J Subscript epsilon"> <mml:semantics> <mml:msub> <mml:mi>J</mml:mi> <mml:mi> ϵ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">J_\epsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to demonstrate the existence of a minimizer of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript epsilon"> <mml:semantics> <mml:msub> <mml:mi>I</mml:mi> <mml:mi> ϵ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">I_\epsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .
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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) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".