Spectral theory of partial differential equations
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
These lectures present highlights of spectral theory for selfadjoint partial differential operators, emphasizing problems with discrete spectrum. Spectral methods permeate the theory of partial differential equations. Linear PDEs are often solved by separation of variables, getting eigenvalues when the spectrum is discrete and continuous spectrum when it is not. Further, linearized stability of a steady state or traveling wave of a nonlinear PDE depends on the sign of the first eigenvalue, or more generally on the location of the spectrum in the complex plane. We define eigenvalues in terms of quadratic forms on a general Hilbert space. Particular applications include the eigenvalues of the Laplacian under Dirichlet and Neumann boundary conditions. Rayleigh-type principles characterize the first and higher eigenvalues, and lead to a number of comparison and domain monotonicity properties. Lastly, the role of eigenvalues in stability analysis is investigated for a reaction-diffusion equation in one spatial dimension. Computable examples are presented before the general theory. Some ideas are used before being properly defined, but overall students gain a better understanding of the purpose of the theory by gaining first a solid grounding in specific examples.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.003 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.014 | 0.001 |
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 it