Why this work is in the frame
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Bibliographic record
Abstract
Parseval’s theorem states that the energy of a signal is preserved by the discrete Fourier transform (DFT). Parseval’s formula shows that there is a nonlinear <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">invariant</i> function for the DFT, so the total energy of a signal can be computed from the signal or its DFT using the same nonlinear function. In this paper, we try to answer the question of whether there are linear invariant functions for the DFT, and how they can be found, along with their potential applications in digital signal processing. In order to answer this question, we first prove that the only linear equations that are preserved by the DFT are its orthogonal projections. Then, using Hilbert spaces and adjoint operators, we propose an algorithm that computes all linear invariant functions for the DFT. These linear invariant functions are also shown to be useful and important in a variety of signal-processing applications, particularly for finding some boundaries for transformed signals without explicitly evaluating the DFT, and vice versa. Additionally, using the proposed identities, we demonstrate that the average of a circular auto-correlation function for a large class of signals is preserved by the DFT. Finally, the results reported in this paper are verified for several short-length and long-length DFTs, including a 256-point DFT.
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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) | 0.000 | 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) | 0.003 | 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 it