Bibliographic record
Abstract
Any complex sound is a combination of simple ‘pure tones’ of different frequencies. For example, a musical chord is a superposition of three (or more) musical notes, each with a different frequency. In fact, a musical note itself is not really a single frequency at all; a note consists of a ‘fundamental’ frequency, plus a cascade of higher-frequency ‘harmonics’. The energy distribution of these harmonics is part of what gives each musical instrument its distinctive sound. The decomposition of a sound into separate frequencies is sometimes called its power spectrum . A crude graphical representation of this power spectrum is visible on most modern stereo systems (the little jiggling red bars). Fourier theory is based on the idea that a real-valued function is like a sound, which can be represented as a superposition of ‘pure tones’ (i.e. sine waves and/or cosine waves) of distinct frequencies. This provides a ‘coordinate system’ for expressing functions, and within this coordinate system we can express the solutions for many partial differential equations in a simple and elegant way. Fourier theory is also an essential tool in probability theory and signal analysis (although we will not discuss these applications in this book). The idea of Fourier theory is simple, but to make this idea rigorous enough to be useful, we must deploy some formidable mathematical machinery. So we will begin by developing the necessary background concerning inner products, orthogonality, and the convergence of functions.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.011 | 0.004 |
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".