Discrete harmonic measure, Green’s functions and symmetrization: a unified probabilistic approach
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Bibliographic record
Abstract
We use probabilistic methods based on the work of Haliste (1965) to obtain various new versions of theorems of Baernstein (1974) on the effects of circular symmetrizations on harmonic measure and Green's functions.Our results are quite general and include a number of cases of symmetrization in discrete settings.For instance, in the setting of the discrete cylinder ZZ m we obtain complete generalized analogues of Baernstein's results on harmonic measure and Green's functions, and even get a discrete version of Beurling's (1933) shove theorem.1. Introduction.Suppose Z m be the integers modulo m, let S n = {x R n+1 : |x| = 1} be the unit sphere of dimension n, use H n to denote ndimensional hyperbolic space, and put T = S 1 .
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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) | 0.001 | 0.002 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.004 | 0.008 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 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 it