Form estimates for the đ(đ„)-Laplacean
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
We consider the problem of establishing conditions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that ensure that the form associated with the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Laplacean is positive bounded below. It was shown recently by Fan, Zhang and Zhao thatâunlike the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p=</mml:annotation> </mml:semantics> </mml:math> </inline-formula> constant caseâthis is not possible if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a strict extrema in the domain. They also considered the closely related problem of eigenvalue existence and estimates. Our main tool is the adaptation of a technique, employed by Protter for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals 2 comma"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p=2,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> involving arbitrary vector fields. We also examine related results obtained by a variant of Picone Identity arguments. We directly consider problems in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega subset-of upper R Superscript n"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mo> â </mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\Omega \subset R^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-equal-to 1 comma"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> â„ </mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">n\ge 1,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and while we focus on Dirichlet boundary conditions we also indicate how our approach can be used in cases of mixed boundary conditions, of unbounded domains and of discontinuous <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p left-parenthesis x right-parenthesis period"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p(x).</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Our basic criteria involve restrictions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and its gradient.
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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 | 0.002 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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 it