Bibliographic record
Abstract
The space of rational functions has natural Poisson structure discovered by Atiyah and Hitchin. We con-sider its derivation by means of quadratic r-matrix structure which naturally arises for the corresponding scattering problem and provide a trigonometric analogue. Mathematical Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20 1 Poisson brackets on space of rational maps In the work by Atiayh and Hitchin [1] it was introduced a symplectic structure on the space RN of rational functions of the form S(λ) = N−1∑ i=0 ρi λ − βi (1.1) To describe this symplectic structure we represent function S(λ) as a ratio of two polynomials S(λ) = p(λ) q(λ) (1.2) where q(λ) = (λ − β1)... (λ − βN). Then the Atiyah-Hitchin symplectic form looks like follows: ω = N∑ i=1 dp(βi) ∧ dβi p(βi) (1.3) The symplectic structure implies the following Poisson brackets {p(βm), p(βn)} = 0, {βm, βn} = 0 (1.4) {p(βm), βn} = p(βm)δmn (1.5) These Poisson brackets in turn imply brackets between polynomials p(λ) and q(λ): {q(λ), q(µ)} = 0, {p(λ), p(µ)} = 0 (1.6) {p(λ), q(µ)} = p(λ)q(µ) − q(λ)p(µ) λ − µ (1.7) It was shown by Gekhtman and Faybusovich in [3] that the relations (1.6) and (1.7) are equivalent to the following brackets on the space of rational functions RN (1.3): {S(λ), S(µ)} = (S(λ) − S(µ)) 2 λ − µ (1.8) In fact, the bracket (1.8) can be obviously extended to the space of all rational functions being the ratio of two polynomials of arbitrary degree. However the requirement of distinct roots remains essential. As it was mentioned by K.Takasaki [4] , the Gekhtman-Faybusovich bracket (1.8) is naturally related to the rational quadratic Sklyanin bracket
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How this classification was reachedexpand
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 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| 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) | 0.000 | 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 teacher head, 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".