Globally optimal volume-trap arrangements for the narrow-capture problem inside a unit sphere
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Bibliographic record
Abstract
The determination of statistical characteristics for particles undergoing Brownian motion in constrained domains has multiple applications in various areas of research. This work presents an attempt to systematically compute globally optimal configurations of traps inside a three-dimensional domain that minimize the average of the mean first passage time (MFPT) for the narrow capture problem, the average time it takes a particle to be captured by any trap. For a given domain, the mean first passage time satisfies a linear Poisson problem with Dirichlet-Neumann boundary conditions. While no closed-form general solution of such problems is known, approximate asymptotic MFPT expressions for small traps in a unit sphere have been found. These solutions explicitly depend on trap parameters, including locations, through a pairwise potential function. After probing the applicability limits of asymptotic formulas through comparisons with numerical and available exact solutions of the narrow capture problem, full three-dimensional global optimization was performed to find optimal trap positions in the unit sphere for 2≤N≤100 identical traps. The interaction energy values and geometrical features of the putative optimal trap arrangements are presented.
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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 | 0.000 | 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)
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 it