The theorem of the complement for nested sub-Pfaffian sets
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Bibliographic record
Abstract
Let R be an o-minimal expansion of the real field, and let Lnest(R) be the language consisting of all nested Rolle leaves over R. We call a set nested sub-Pfaffian over R if it is the projection of a positive Boolean combination of definable sets and nested Rolle leaves over R. Assuming that R admits analytic cell decomposition, we prove that the complement of a nested sub-Pfaffian set over R is again a nested sub-Pfaffian set over R. As a corollary, we obtain that if R admits analytic cell decomposition, then the Pfaffian closure P(R) of R is obtained by adding to R all nested Rolle leaves over R, a one-stage process, and that P(R) is model complete in the language Lnest(R).
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.002 |
| 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 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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