Ultracontractive Properties for Directed Graph Semigroups with Applications to Coupled Oscillators
Bibliographic record
Abstract
Abstract It is now well known that ultracontractive properties of semigroups with infinitesimal generator given by an undirected graph Laplacian operator can be obtained through an understanding of the geometry of the underlying infinite weighted graph. The aim of this work is to extend these results to semigroups with infinitesimal generators given by a directed graph Laplacian operator through an analogous inspection of the geometry of the underlying directed graph. In particular, we introduce appropriate nomenclature to discuss the geometry of an infinite directed graph, as well as provide sufficient conditions to extend ultracontractive properties of undirected graph Laplacians to those of the directed variety. Such directed graph Laplacians can often be observed in the study of coupled oscillators, where recent work made explicit the link between synchronous patterns to systems of identically coupled oscillators and ultracontractive properties of undirected graph semigroups. Therefore, in this work we demonstrate the applicability of our results on directed graph semigroups by extending the aforementioned investigation beyond the idealized case of identically coupled oscillators.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
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".