On singular limits of finite Hilbert transform operators on multi‐intervals
Bibliographic record
Abstract
Abstract In this paper, we study the small‐ spectral asymptotics of an integral operator defined on two multi‐intervals and , when the multi‐intervals touch each other (but their interiors are disjoint). The operator is closely related to the multi‐interval finite Hilbert transform (FHT). This case can be viewed as a singular limit of self‐adjoint Hilbert–Schmidt integral operators with so‐called integrable kernels, where the limiting operator is still bounded, but has a continuous spectral component. The regular case when , and is of the Hilbert–Schmidt class, was studied in an earlier paper by the authors. The main assumption in this paper is that is a single interval (although part of our analysis is valid in a more general situation). We show that the eigenvalues of , if they exist, do not accumulate at . Combined with the results in an earlier paper by the authors, this implies that , the subspace of discontinuity (the span of all eigenfunctions) of , is finite dimensional and consists of functions that are smooth in the interiors of and . We also obtain an approximation to the kernel of the unitary transformation that diagonalizes , and obtain a precise estimate of the exponential instability of inverting . Our work is based on the method of Riemann–Hilbert problem and the nonlinear steepest descent method of Deift and Zhou.
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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 | 0.001 | 0.003 |
| 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 | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.002 |
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; both teacher heads agree on what is shown here.
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".