Heat kernel estimates for inner uniform subsets of Harnack-type Dirichlet spaces
Bibliographic record
Abstract
The main result of this thesis is the two-sided heat kernel estimates for both Dirich-let and Neumann problem in any inner uniform domain of the Euclidean space Rn. The results of this thesis are shown to hold more generally for any inner uniform domain in many other spaces with Gaussian-type heat kernel estimates. We as-sume that the heat equation is associated with a local divergence form differential operator, or more generally with a strictly local Dirichlet form on a complete locally compact metric space. Other results include the (parabolic) Harnack inequality and the boundary Harnack principle. BIOGRAPHICAL SKETCH Pavel Gyrya was born in Kharkiv, Ukraine in 1980. He studied in Kharkiv lyceum number 27 until 1997 and spent his school years focusing on mathematical com-petitions. He then went on to study for two years in Moscow State university and then transferred to the University of Toronto. After graduation from University of Toronto he entered the graduate program in Cornell university, and after two years of studying all the major subjects in mathematics, began his work with Prof. Saloff-Coste with the concentration on Analysis and Probability. After gradua-tion from Cornell University Pavel will switch gears and join American Express company to focus on business decisions based on modelling of financial risk. iii ACKNOWLEDGEMENTS
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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 | 0.003 | 0.014 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.004 | 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 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".