Convergence of cascade algorithms associated with nonhomogeneous refinement equations
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Bibliographic record
Abstract
This paper is devoted to a study of multivariate nonhomogeneous refinement equations of the form <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi left-parenthesis x right-parenthesis equals g left-parenthesis x right-parenthesis plus sigma-summation Underscript alpha element-of double-struck upper Z Superscript s Baseline Endscripts a left-parenthesis alpha right-parenthesis phi left-parenthesis upper M x minus alpha right-parenthesis comma x element-of double-struck upper R Superscript s Baseline comma"> <mml:semantics> <mml:mrow> <mml:mi> ϕ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:munder> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> α </mml:mi> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msup> </mml:mrow> </mml:munder> <mml:mi>a</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi> α </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi> ϕ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mi>x</mml:mi> <mml:mo> − </mml:mo> <mml:mi> α </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mspace width="2em"/> <mml:mi>x</mml:mi> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} \phi (x) = g(x) + \sum _{\alpha \in \mathbb {Z}^s} a(\alpha ) \phi (Mx-\alpha ), \qquad x \in \mathbb {R}^s, \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi equals left-parenthesis phi 1 comma ellipsis comma phi Subscript r Baseline right-parenthesis Superscript upper T"> <mml:semantics> <mml:mrow> <mml:mi> ϕ </mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi> ϕ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi> ϕ </mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>T</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\phi = (\phi _1,\ldots ,\phi _r)^T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the unknown, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g equals left-parenthesis g 1 comma ellipsis comma g Subscript r Baseline right-parenthesis Superscript upper T"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>T</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">g = (g_1,\ldots ,g_r)^T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a given vector of functions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript s"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s times s"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo> × </mml:mo> <mml:mi>s</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">s \times s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> dilation matrix, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.001 |
| 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 it