Singular value decomposition of a finite Hilbert transform defined on\n several intervals and the interior problem of tomography: the Riemann-Hilbert\n problem approach
Bibliographic record
Abstract
We study the asymptotics of singular values and singular functions of a\nFinite Hilbert transform (FHT), which is defined on several intervals.\nTransforms of this kind arise in the study of the interior problem of\ntomography. We suggest a novel approach based on the technique of the matrix\nRiemann-Hilbert problem and the steepest descent method of Deift-Zhou. We\nobtain a family of matrix RHPs depending on the spectral parameter $\\lambda$\nand show that the singular values of the FHT coincide with the values of\n$\\lambda$ for which the RHP is not solvable. Expressing the leading order\nsolution as $\\lambda\\to 0$ of the RHP in terms of the Riemann Theta functions,\nwe prove that the asymptotics of the singular values can be obtained by\nstudying the intersections of the locus of zeroes of a certain Theta function\nwith a straight line. This line can be calculated explicitly, and it depends on\nthe geometry of the intervals that define the FHT. The leading order\nasymptotics of the singular functions and singular values are explicitly\nexpressed in terms of the Riemann Theta functions and of the period matrix of\nthe corresponding normalized differentials, respectively. We also obtain the\nerror estimates for our asymptotic results.\n
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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 | 0.000 |
| 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 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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".