Discrete and continuous cosine transform generalized to Lie groups SU(2)×SU(2) and O(5)
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Bibliographic record
Abstract
We develop and describe continuous and discrete transforms of class functions on compact semisimple Lie group G as their expansions into series of uncommon special functions, called here C-functions in recognition of the fact that the functions generalize cosine to any dimension n<∞. A uniform discretization of the problem on lattices of any density is described. Continuous and discrete orthogonality of C-functions is shown. Discrete transform is known in the case n=1 as the cosine transform. Continuous extension of the discrete transform is described. In general, C-functions are the contributions to irreducible characters from just one orbit of the Weyl group of G. Their products are fully decomposable to the sums of C-functions, so are the reductions to subgroups of the Lie group. They are eigenfunctions of Laplace operator, satisfying Neumann conditions at the boundary of the fundamental region of G, etc. A ready-to-use presentation is made of two of the four variants of the two-dimensional transforms. Both variants have in common exploitation of square lattices for the discrete version of the transforms. They are based on the compact Lie groups SU(2)×SU(2) and O(5), or, equivalently, Sp(4). Remaining two groups, SU(3) and G(2), involve triangular lattices. They are considered separately. Processing digital data, sampled on square lattices, is our motivating application.
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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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 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 it