Jensen's inequality for nonconvex functions ∗
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Bibliographic record
Abstract
Abstract . Jensen’sinequalityisformulatedforconvexifiable(gen-erallynonconvex)functions. Keywords: Jensen’sinequality, convexifiable function, arithmeticmeantheorem AMSsubjectclassifications:26B25, 52A40Received June 23, 2004 Accepted November 2, 2004 1. Introduction Jensen’s inequality is 100 years old, e.g., [1, 2, 3] . It says that the value of a convexfunction at a point, which is a convex combination of finitely many points, is lessthan or equalto the convex combination of values of the function at these points.Using symbols: If : R n → R is convex then f pi =1 λ i x i ≤ pi =1 λ i f (x i )(1)for every set of p points x i ,i =1 ,...,p, in the Euclidean space R n and for all realscalars λ i ≥ 0, i =1 ,...,p , such that pi =1 λ i =1.In this note the inequality (1) is extended from convex to convexifiable func-tions, e.g., [4, 5]. These include all twice continuously differentiable functions, allonce continuously differentiable functions with Lipschitz derivative and all analyticfunctions. As a specialcase we obtain a new form of the arithmetic mean theorem.
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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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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