Complex Analytic Functions with Natural Boundary
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Bibliographic record
Abstract
The analytic functions with natural boundaries have been only occasionally mentioned in literature. They were defined mainly by lacunary power series of Hadamard type, except for the modular function which is the result of a laborious construction. The case of infinite Blaschke products which cannot be analytically continued over the unit circle is also known, yet the authors have no knowledge about any study devoted to these functions. The purpose of this article is to take a closer look upon these functions, to find new techniques of generating them and to bring this topic into the mainstream study of analytic functions. A special attention is devoted to the theory of Blaschke products, which is completed with new results related to their boundary behavior, making possible the study of the Blaschke products with natural boundary. We apply to them the same method of study as for ordinary infinite Blaschke products obtaining mirror functions with respect to the unit circle. The working tool is that of the fundamental domains, which are easily revealed by the technique of continuation over a curve, or lifting of a curve, having its origins in the differential geometry. Graphic illustrations contribute to a better understanding of the theoretical endeavors.
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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 | 0.001 | 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 | 0.001 |
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