Spectral asymptotics for Sturm-Liouville equations with indefinite weight
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Bibliographic record
Abstract
The Sturm-Liouville equation <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus left-parenthesis p y Superscript prime Baseline right-parenthesis Superscript prime Baseline plus q y equals lamda r y on left-bracket 0 comma l right-bracket"> <mml:semantics> <mml:mrow> <mml:mo> − </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:msup> <mml:mi>y</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>q</mml:mi> <mml:mi>y</mml:mi> <mml:mo>=</mml:mo> <mml:mi> λ </mml:mi> <mml:mi>r</mml:mi> <mml:mi>y</mml:mi> <mml:mspace width="thickmathspace"/> <mml:mspace width="thickmathspace"/> <mml:mtext>on</mml:mtext> <mml:mspace width="thickmathspace"/> <mml:mspace width="thickmathspace"/> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>l</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} -(py’)’ + qy =\lambda ry \;\; \text {on}\;\; [0,l] \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> is considered subject to the boundary conditions <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column y left-parenthesis 0 right-parenthesis cosine alpha 2nd Column a m p semicolon equals left-parenthesis p y Superscript prime Baseline right-parenthesis left-parenthesis 0 right-parenthesis sine alpha comma 2nd Row 1st Column y left-parenthesis l right-parenthesis cosine beta 2nd Column a m p semicolon equals left-parenthesis p y Superscript prime Baseline right-parenthesis left-parenthesis l right-parenthesis sine beta period EndLayout"> <mml:semantics> <mml:mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" side="left" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:mi>y</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mi>cos</mml:mi> <mml:mo> </mml:mo> <mml:mi> α </mml:mi> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:msup> <mml:mi>y</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mi>sin</mml:mi> <mml:mo> </mml:mo> <mml:mi> α </mml:mi> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi>y</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>l</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>cos</mml:mi> <mml:mo> </mml:mo> <mml:mi> β </mml:mi> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:msup> <mml:mi>y</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>l</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>sin</mml:mi> <mml:mo> </mml:mo> <mml:mi> β </mml:mi> <mml:mo>.</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:annotation encoding="application/x-tex">\begin{align*} y(0)\cos \alpha &= (py’)(0)\sin \alpha ,\\ y(l)\cos \beta &= (py’)(l)\sin \beta . \end{align*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> We assume that <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> is positive and that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p r"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mi>r</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">pr</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is piecewise continuous and changes sign at its discontinuities. We give asymptotic approximations up to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis 1 slash StartRoot n EndRoot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msqrt> <mml:mi>n</mml:mi> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(1/\sqrt {n})</
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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