The graph of a family of functions over quadratic extensions of finite fields
Bibliographic record
Abstract
Brochero and Teixeira (2023) [4] showed the behavior of the functional graph of f a ( X ) = X q + 1 + a X 2 over quadratic extensions of finite fields explicitly for a ∈ { 1 , − 1 } . In this article, we create a family of functions using repeated iterations of the function f a ( X ) and taking values of a ∈ { 1 , − 1 } in each iteration. Let α be an n -sequence of values for a , taken over { 1 , − 1 } , and f α ( X ) be the resulting function. We present the form of f α ( X ) and use it to derive a closed formula for the number and length of cycles present in the functional graph of f α ( X ) . We then determine the shape of the trees hanging from each cycle and gather all the results in our main theorem.
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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| 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 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".