Discrete and continuous cosine transform generalized to Lie groups SU(3) and G(2)
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Bibliographic record
Abstract
In the paper we complete the development and description of the four variants of two-dimensional generalization of the cosine transform started in [Patera and Zaratsyan, J. Math Phys. 46, 053514 (2005)]. Each variant is based on a compact semisimple Lie group G of rank 2. Here, the groups are SU(3) and G(2). The cosines are generalized as the corresponding C-functions of the Lie group. A C-function is the contribution to an irreducible character from one orbit of the appropriate Weyl group. An explicit description is provided for expansions of functions given on the fundamental region F of the two compact simple Lie groups into series of C-functions. The fundamental region F is an equilateral triangle for SU(3) and half of such a triangle for G(2). Expansion coefficients are calculated using orthogonality of C-functions on F. Discrete expansions are set up on a grid FM⊂F. The grid is defined group theoretically for all positive integers M. It consists of points in F that represent conjugacy classes of elements of the finite maximal Abelian subgroup of G generated by its elements of order M. The C-functions are orthogonal on such a grid; hence, coefficients of discrete expansions are calculated independently of the continuous expansions. Processing digital data, sampled on triangular lattices, is the motivating application here.
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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) | 0.001 | 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