Estimates for Fundamental Solutions of Second-Order Parabolic Equations
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Bibliographic record
Abstract
In this paper we study the second-order parabolic equation ċċċ∂tu(t,x)=∇ċa(t,x)ċ∇u(t,x)-b(t,x)ċ∇u(t,x)+∇b∩(t,x)u(t,x)u(t,x)+V(t,x)(1) in a domain [0,T]×Rd ⊂ Rd+1, where a=(aij)i,j=1d is matrix of bounded measurable coefficients, b=(bj)j=1d, and b^=(b^j)j=1d are measurable (in general, singular) vector fields, V is a measurable potential, T is a fixed positive number, and ∂tu = ∂u/∂t, and we employ the notation ċċċ∇ċaċ∇u=∑i,j=1d∂xiaij∂xju,bċ∇u=∑j=1dbj∂xju,∇b∩u=∑j=1d∂xj(b∩ju). We introduce a new class of coefficients in the lower-order terms for which we prove the existence and the uniqueness of the weak fundamental solution, and for this we derive Gaussian upper and lower bounds.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| 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) | 0.003 | 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