Multi-scaling limits for relativistic diffusion equations with random initial data
Bibliographic record
Abstract
Let u ( t , x ) , t > 0 , x ∈ R n , u(t,\mathbf {x}),\ t>0,\ \mathbf {x}\in \mathbb {R}^{n}, be the spatial-temporal random field arising from the solution of a relativistic diffusion equation with the spatial-fractional parameter α ∈ ( 0 , 2 ) \alpha \in (0,2) and the mass parameter m > 0 \mathfrak {m}> 0 , subject to a random initial condition u ( 0 , x ) u(0,\mathbf {x}) which is characterized as a subordinated Gaussian field. In this article, we study the large-scale and the small-scale limits for the suitable space-time re-scalings of the solution field u ( t , x ) u(t,\mathbf {x}) . Both the Gaussian and the non-Gaussian limit theorems are discussed. The small-scale scaling involves not only scaling on u ( t , x ) u(t,\mathbf {x}) but also re-scaling the initial data; this is a new type result for the literature. Moreover, in the two scalings the parameter α ∈ ( 0 , 2 ) \alpha \in (0,2) and the parameter m > 0 \mathfrak {m}> 0 play distinct roles for the scaling and the limiting procedures.
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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 | 0.007 | 0.038 |
| Meta-epidemiology (narrow) | 0.002 | 0.002 |
| Meta-epidemiology (broad) | 0.003 | 0.003 |
| Bibliometrics | 0.005 | 0.001 |
| Science and technology studies | 0.003 | 0.005 |
| Scholarly communication | 0.005 | 0.009 |
| Open science | 0.004 | 0.007 |
| Research integrity | 0.003 | 0.005 |
| Insufficient payload (model declined to judge) | 0.013 | 0.002 |
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".