Simplicial Approximation of the Hodge Laplacian Using Cauchy Sequences of Hilbert Complexes
Bibliographic record
Abstract
Discrete differential geometry arises from the use of discrete spaces such as graphs, simplicial, cubical, or polyhedral complexes for modeling geometric structures on manifolds. A common practice in this work is to transport structures on smooth manifolds to discrete counterparts in a process referred to as discretization. Discretizations often appear as elements of a sequence that approximates the smooth structure on the manifold through some measure of convergence. Algorithms which produce such sequences are highly sought after for computational applications but frequently ignore deeper structural relationships between successive discrete models. This thesis makes contributions to the discretization of Hodge theory through the construction of a framework that serves to axiomatize a foundational set of results in the field. The salient feature of this framework is the ability to directly measure the difference in approximation accuracy between discretizations without reference to the overarching smooth structure. This provides a Cauchy-type characterization of sequences of discretizations while opening the scope of inquiry to a much larger class of problems involving the analysis of Hodge Theory through Cauchy sequences.
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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 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.002 |
| 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".