Bibliographic record
Abstract
Performing calculations modulo a set of relations is a basic technique in algebra. For instance, computing the inverse of an integer modulo a prime integer or computing the inverse of the complex number 3 + 2t modulo the relation ℓ2 + 1 = 0. Computing modulo a set S containing more than one relation requires from S to have some mathematical structure. For instance, computing the inverse of p = x + y modulo S = {x2 + y + 1,y2 + x + 1} is difficult unless one realizes that this question is equivalent to computing the inverse of p modulo C = {x4 + 2x2 + x + 2,y + x2 + 1}. Indeed, from there one can simplify p using y = -x2 - 1 leading to q = -x2 + x - 1 and compute the inverse of q modulo x4 + 2x2 + x + 2 (using the extended Euclidean algorithm) leading to -1/2x3 - 1/2x. One commonly used mathematical structure for a set of algebraic relations is that of a Gröbner basis. It is particularly well suited for deciding whether a quantity is null or not modulo a set of relations. For inverse computations, the notion of a regular chain is more adequate. For instance, computing the inverse of p = x + y modulo the set C = {y2 - 2x + 1,x2 - 3x + 2}, which is both a Gröbner basis and a regular chain, is easily answered in this latter point of view. Indeed, it naturally leads to consider the GCD of p and Cy = y2 - 2x + 1 modulo the relation Cx = x2 - 3x + 2 = 0, which is [EQUATION] This shows that p has no inverse if x = 1 and has an inverse (which can be computed and which is -y + 2) if x = 2.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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) | 0.002 | 0.002 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.004 | 0.005 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.355 | 0.253 |
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".