Generating functions, polynomials and vortices with alternating signs in Bose–Einstein condensates
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Bibliographic record
Abstract
In this work, we construct suitable generating functions for vortices of alternating signs in the realm of quasi-two-dimensional Bose–Einstein condensates in the large density (so-called Thomas–Fermi) limit, where the vortices can be treated as effective particles. In addition to the vortex–vortex interaction included in earlier fluid dynamics constructions of such functions, the vortices here precess around the center of the trap. This results in the generating functions of the vortices of positive charge and of negative charge satisfying a modified, so-called, Tkachenko differential equation. From that equation, we reconstruct collinear few-vortex equilibria obtained in earlier work, as well as extend them to larger numbers of vortices. Moreover, particular moment conditions can be derived e.g. about the sum of the squared locations of the vortices for arbitrary vortex numbers. Furthermore, the relevant differential equation can be generalized appropriately in the two-dimensional complex plane and allows for the construction e.g. of polygonal vortex ring and multi-ring configurations, as well as ones with rings surrounding a central vortex that are again connected to earlier bibliography and associated numerical computations.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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)
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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