A practical method to experimentally evaluate the Hausdorff dimension: An alternative phase-transition-based methodology
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Bibliographic record
Abstract
We introduce a methodology to estimate numerically the Hausdorff dimension of a geometric set. This practical method has been conceived as a subsequent tool of another context study, associated to our concern to distinguish between various fractal sets. Its conception is natural since it can be related to the original idea involved in the definitions of Hausdorff measure and Hausdorff dimension. It is based on the critical behavior of the measure spectrum functions of the set around its Hausdorff dimension value. We illustrate on several well-known examples, the ability of this method to accurately estimate the Hausdorff dimension. Also, we show how the transition property, exhibited by the quantities used as substitutes of the Hausdorff measure in the corresponding fractal dimension relationships, can be used to accurately estimate the fractal dimension. To show the potential of our method, we also report the results of Hausdorff dimension measurements on some typical examples, compared to a direct application of the scaling relation involved in the box-counting dimension definition.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.009 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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.
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