Partial derivatives over hypercomplex variables for non-analytic functions in eight-dimensional algebra
Bibliographic record
Abstract
This paper provides the formulation of partial derivatives over hypercomplex variables for functions of octonions or split-biquaternions. The proposed partial derivatives are an extension to the case of octonions and split-biquaternions algebras of the well-known Wirtinger/CR-calculus for complex-numbers algebra, and HR-calculus or generalized HR-calculus for quaternion algebra. The octonionic functions considered in this paper are not constrained by analyticity, holomorphy, harmonicity, conformality, regularity or monogenic property constraints. Instead, the only constraint for the octonionic functions considered in this paper is that each element of an octonionic function be real-differentiable over the (real-valued) variable of each element in an octonion or split-biquaternion variable. We first show that the approach based on rotations over basis components used in the development of HR-calculus or generalized-HR calculus is not suitable for the two considered 8-dimensional hypercomplex algebras. Instead, we justify the use of transformations based on Hadamard matrices, which allows a common framework for the development of partial derivatives for complex-numbers, quaternion, octonion and split-biquaternion algebras. The results of this paper can be an important step towards the development of optimization algorithms and adaptive filtering algorithms of hypercomplex variables from 8-dimensional hypercomplex algebras.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".