A fast and well-conditioned spectral method for singular integral\n equations
Bibliographic record
Abstract
We develop a spectral method for solving univariate singular integral\nequations over unions of intervals by utilizing Chebyshev and ultraspherical\npolynomials to reformulate the equations as almost-banded infinite-dimensional\nsystems. This is accomplished by utilizing low rank approximations for sparse\nrepresentations of the bivariate kernels. The resulting system can be solved in\n${\\cal O}(m^2n)$ operations using an adaptive QR factorization, where $m$ is\nthe bandwidth and $n$ is the optimal number of unknowns needed to resolve the\ntrue solution. The complexity is reduced to ${\\cal O}(m n)$ operations by\npre-caching the QR factorization when the same operator is used for multiple\nright-hand sides. Stability is proved by showing that the resulting linear\noperator can be diagonally preconditioned to be a compact perturbation of the\nidentity. Applications considered include the Faraday cage, and acoustic\nscattering for the Helmholtz and gravity Helmholtz equations, including\nspectrally accurate numerical evaluation of the far- and near-field solution.\nThe Julia software package SingularIntegralEquations.jl implements our method\nwith a convenient, user-friendly interface.\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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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) | 0.000 | 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".