Construction of New Continuous K-g-Frames within Hilbert C∗-Modules
Bibliographic record
Abstract
In this paper, we investigate the construction of new continuous c-K-g-frames in Hilbert C*-modules, extending and generalizing existing frame theory results. Our main theorem establishes sufficient positivity conditions on an auxiliary operator T to ensure that the transformed family \(\{\Lambda_\omega T\}_{\omega \in \Omega}\) forms a continuous c-K-g-frame. Through examples, we illustrate the necessity of these positivity conditions. We also present a method for combining multiple continuous c-K-g-frames into a single continuous c-\(\sum_i K_i T_i\)-g-frame. We prove associativity of continuous c-K-g-frames under product measure spaces, and explore exactness and stability under restriction to some measurable subsets for non-decreasing continuous K-g-frames with respect to an ordered measurable space. Additionally, we characterize dual frames in this setting, providing insight into their existence and construction. Our results extend and unify various notions of continuous frames in both Hilbert spaces and Hilbert C*-modules.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 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".