Polynomial expansions via embedded Pascal’s triangles
Bibliographic record
Abstract
An expansion is given for polynomials of the form (ω + λ1) · · ·(ω +λn). The coefficients of the resulting polynomials are related to their roots, and a system of equations that enables one to numericallydetermine the roots in terms of the coefficients is specified. The case where all the roots λi are equal is considered as well. A multinomial extension to polynomials of the form (x1+ · · · + xI )n is then provided. As it turns out, the coefficients of the monomials contained in the resulting polynomial expansion can be determined in terms of the coefficients of the monomials included in the expansion of (x1+ · · · + xI-1 )n and the rows of embedded Pascal’s triangles of successive orders. An algorithm is provided for generating and concatenating these rows, with the particulars of its implementation by means of the symbolic computation software Mathematica being discussed as well. Potential applications of such expansions to combinatorics and genomics are also suggested.
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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 | 0.001 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.015 | 0.004 |
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".