Extension of a generalized Pexider equation
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
The equations <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k left-parenthesis s plus t right-parenthesis equals script l left-parenthesis s right-parenthesis plus n left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>+</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi> ℓ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">k(s+t)=\ell (s)+n(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k left-parenthesis s plus t right-parenthesis equals m left-parenthesis s right-parenthesis n left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>+</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">k(s+t)=m(s)n(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , called Pexider equations, have been completely solved on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R squared period"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}^2.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> If they are assumed to hold only on an open region, they can be extended to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (the second when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is nowhere 0) and solved that way. In this paper their common generalization <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k left-parenthesis s plus t right-parenthesis equals script l left-parenthesis s right-parenthesis plus m left-parenthesis s right-parenthesis n left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>+</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi> ℓ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">k(s+t)=\ell (s)+m(s)n(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is extended from an open region to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and then completely solved if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not constant on any proper interval. This equation has further interesting particular cases, such as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k left-parenthesis s plus t right-parenthesis equals script l left-parenthesis s right-parenthesis plus m left-parenthesis s right-parenthesis k left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>+</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi> ℓ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi>
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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