Two functional equations preserving functional forms
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Bibliographic record
Abstract
Two functional equations are considered that are motivated by three considerations: work in utility theory and psychophysics, questions concerning when pairs of degree 1 homogeneous functions can be homomorphic and calculating their homomorphisms, and the link of the latter questions to quasilinear mean values. The first equation is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}\boldsymbol{{\mathit{h}}}(\boldsymbol{{\mathrm{{\sigma}}}}(\boldsymbol{{\mathit{y}}})\boldsymbol{{\mathit{x}}{\mathrm{\hspace{.167em}+\hspace{.167em}}}}[\boldsymbol{{\mathrm{1\hspace{.167em}-\hspace{.167em}{\sigma}}}}(\boldsymbol{{\mathit{y}}})]\boldsymbol{{\mathit{y}}})\boldsymbol{{\mathrm{\hspace{.167em}=\hspace{.167em}{\tau}}}}(\boldsymbol{{\mathit{y}}})\boldsymbol{{\mathit{h}}}(\boldsymbol{{\mathit{x}}})\boldsymbol{\hspace{.167em}}\end{equation*}\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}\boldsymbol{{\mathrm{+\hspace{.167em}}}}[\boldsymbol{{\mathrm{1\hspace{.167em}-\hspace{.167em}{\tau}}}}(\boldsymbol{{\mathit{y}}})]\boldsymbol{{\mathit{h}}}(\boldsymbol{{\mathit{y}}})\boldsymbol{\hspace{1em}}(\boldsymbol{{\mathit{x}}{\mathrm{\hspace{.167em}{\geq}\hspace{.167em}}}{\mathit{y}}{\mathrm{\hspace{.167em}{\geq}\hspace{.167em}0}}})\boldsymbol{{\mathrm{,}}}\end{equation*}\end{document} where h maps [0, ∞[ into a subset of [0, ∞[ and is strictly increasing and continuously differentiable; the functions σ and τ map [0, ∞[ continuously into [0, 1], σ( y ) > 0 for y > 0 but σ is not 1 on ]0, ∞[. The solutions are fully determined. (Recently Zsolt Páles has eliminated the differentiability assumption.) The second equation is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}\boldsymbol{{\mathit{h}}}[\boldsymbol{{\mathit{y}}{\mathrm{\hspace{.167em}+\hspace{.167em}}}{\mathit{f}}}(\boldsymbol{{\mathit{x}}{\mathrm{\hspace{.167em}-\hspace{.167em}}}{\mathit{y}}})]\boldsymbol{{\mathrm{\hspace{.167em}=\hspace{.167em}}}{\mathit{h}}}(\boldsymbol{{\mathit{y}}})\boldsymbol{{\mathrm{\hspace{.167em}+\hspace{.167em}}}{\mathit{g}}}[\boldsymbol{{\mathit{h}}}(\boldsymbol{{\mathit{x}}})\boldsymbol{{\mathrm{\hspace{.167em}-\hspace{.167em}}}{\mathit{h}}}(\boldsymbol{{\mathit{y}}})]\boldsymbol{\hspace{1em}}(\boldsymbol{{\mathit{x}}{\mathrm{\hspace{.167em}{\geq}\hspace{.167em}}}{\mathit{y}}{\mathrm{\hspace{.167em}{\geq}\hspace{.167em}0}}})\boldsymbol{{\mathrm{,}}}\end{equation*}\end{document} where h maps [0, ∞[ onto a subinterval of positive length of [0, ∞[ and is strictly increasing and twice continuously differentiable, f and g map [0, ∞[ onto [0, ∞[ and are twice differentiable, and either f "(0) ≠ 0 or g "(0) ≠ 0. The solutions are fully determined under these conditions. When f "(0) = g "(0) = 0 and h " is not identically zero, we determine the solutions under the added assumption of analyticity. It remains an open problem to find the solutions in the latter case under the assumption of only second order differentiability. A more general open problem is to eliminate all differentiability conditions for the second equation.
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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 | 0.007 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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