Order Reconstruction for Nematics on Squares with Isotropic Inclusions: A Landau--De Gennes Study
Bibliographic record
Abstract
We prove the existence of a well order reconstruction solution (WORS)-type Landau--de Gennes critical point on a square domain with an isotropic concentric square inclusion, with tangent boundary conditions on the outer square edges. There are two geometrical parameters---the outer square edge length $\lambda$, and the aspect ratio $\rho$, which is the ratio of the inner and outer square edge lengths. The WORS exists for all geometrical parameters and all temperatures, and is globally stable for either $\lambda$ small enough or for $\rho$ sufficiently close to unity. We study three different types of Landau--de Gennes critical points in this setting: critical points with the minimal two degrees of freedom consistent with the imposed boundary conditions, critical points with three degrees of freedom, and critical points with five degrees of freedom. We identify the competitors for the WORS in the two- and three-dimensional settings. In the three-dimensional setting, we numerically find up to $28$ critical points for moderately large values of $\rho$, of which diagonal solutions are global energy minimizers when they exist. We find two nonenergy minimizing critical points with five degrees of freedom for very small values of $\rho$, with an escaped profile around the isotropic square inclusion.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".