On the analyticity of WLUD<sup>∞</sup> functions of one variable and WLUD<sup>∞</sup> functions of several variables in a complete non-Archimedean valued field
Bibliographic record
Abstract
Abstract Let $\mathcal {N}$ be a non-Archimedean-ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order, and whose Hahn group is Archimedean. In this paper, we first review the properties of weakly locally uniformly differentiable (WLUD) functions, $k$ times weakly locally uniformly differentiable (WLUD $^{k}$ ) functions and WLUD $^{\infty }$ functions at a point or on an open subset of $\mathcal {N}$ . Then, we show under what conditions a WLUD $^{\infty }$ function at a point $x_0\in \mathcal {N}$ is analytic in an interval around $x_0$ , that is, it has a convergent Taylor series at any point in that interval. We generalize the concepts of WLUD $^{k}$ and WLUD $^{\infty }$ to functions from $\mathcal {N}^{n}$ to $\mathcal {N}$ , for any $n\in \mathbb {N}$ . Then, we formulate conditions under which a WLUD $^{\infty }$ function at a point $\boldsymbol {x_0} \in \mathcal {N}^{n}$ is analytic at that point.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.006 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".