CHARACTERIZATIONS OF LINEARLY INDEPENDENT FUNCTIONS
Why this work is in the frame
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Bibliographic record
Abstract
For functions f1, . . . , fn on a set D, we characterize their linear independence with an invertible matrix from their values at n distinct points in D. With the matrix, the pointwise convergence of a sequence {gk} of functions in the span{f1, · · · , fn} is shown to be equivalent to those of the sequences of the coordinates of gks in the span. When fis are bounded, a pointwise convergent sequence {gk} must uniformly converge to a function in the span. It turns out that the limit of a convergent sequence {gk} inherits the continuity, differentiability, and integrability of fis. Furthermore the (pointwise or uniform) convergence of a sequence of solutions of an n-th order constant coefficients linear differential equation is completely determined by that of the sequence of relevant initial conditions.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 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) | 0.002 | 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