Magnetic Vortices, Abrikosov Lattices, and Automorphic Functions
Bibliographic record
Abstract
This chapter presents some recent results on the Ginzburg-Landau equations of superconductivity, and reviews appropriate background. The Ginzburg–Landau equations describe the key mesoscopic and macroscopic properties of superconductors and form the basis of the phenomenological theory of superconductivity. One of the most interesting mathematical and physical phenomena connected with Ginzburg–Landau equations is the presence of vortices in their solutions. Vortex represents a localized defect where the normal state intrudes and magnetic flux penetrates. For vortex lattices the energy is infinite, but the flux quantization still holds for each lattice cell because of gauge-periodic boundary conditions. The chapter describes the existence of Abrikosov solutions at low magnetic fields near the first critical magnetic field. Configurations containing several vortices are not static solutions. Heuristically, this is due to an effective inter-vortex interaction, which causes the vortex centers to move.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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".