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.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".