Riesz Sequences and Frames of Exponentials associated with non-full rank lattices.
Bibliographic record
Abstract
Let Rd be a measurable set of nite positive measure (not necessarily bounded). Let (cj)kj =1 be a given collection of vectors in Rd, and let H be the dual lattice of a full rank lattice K Rd. For 2 Rd, let e denote the exponential e (x) := e2 ih ;xi; x 2 Rd: It is known that, the collection E( ) := fe : 2 g; where = f(cj + h) 2 Rd : h 2 H; j 2 f1; :::; kgg; forms Riesz basis on Rd if the domain is a k-tile domain and if, in addition, it satis es an extra arithmetic property, called the admissibility condition. The theory of shift invariant spaces generated by the full rank lattice K plays an important role to analyze and solve the above problem. The main goal of this thesis is to study a variant of the problem above where the dual lattice H is replaced by a non-full rank lattice in Rd. In particular, given an at most countable index set J and a collection of vectors (cj)j2J Rd, we examine the existence of Riesz sequences, frames and Riesz bases of the form E( ) := fe : 2 g; where = f(cj +h) 2 Rd : h 2 H; j 2 Jg; on Rd as above, and H, a non-full rank lattice in Rd. Our results are obtained using an extention of the theory of shift invariant subspaces of L2(Rd), where the shifts are now generated by a non-full rank lattice in Rd.
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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 | 0.002 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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".