Discrete and continuous cosine transform generalized to Lie groups SU(2)×SU(2) and O(5)
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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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".