An Embedding Technique for the Solution of Reaction-Diffusion Equations\n on Algebraic Surfaces with Isolated Singularities
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Bibliographic record
Abstract
In this paper we construct a parametrization-free embedding technique for\nnumerically evolving reaction-diffusion PDEs defined on algebraic curves that\npossess an isolated singularity. In our approach, we first desingularize the\ncurve by appealing to techniques from algebraic geometry. We create a family of\nsmooth curves in higher dimensional space that correspond to the original curve\nby projection. Following this, we pose the analogous reaction-diffusion PDE on\neach member of this family and show that the solutions (their projection onto\nthe original domain) approximate the solution of the original problem. Finally,\nwe compute these approximants numerically by applying the Closest Point Method\nwhich is an embedding technique for solving PDEs on smooth surfaces of\narbitrary dimension or codimension, and is thus suitable for our situation. In\naddition, we discuss the potential to generalize the techniques presented for\nhigher-dimensional surfaces with multiple singularities.\n
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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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| 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