[ARTICLE] Gamma Derivatives of the Extended Sine and Cosine Functions for the Upgraded Mass-Spring Oscillatory System (Canadian Journal of Mathematics Research / 2023) - GAMMA TRIGONOMETRY
Bibliographic record
Abstract
ABSTRACT Derivatives in trigonometry have always been defined in orthogonal contexts (i.e., where the y-axis is set perpendicular to the x-axis). Within the context of trigonometric, the present work expands the concept of derivative (operating by the principle of 90 degrees phase shift when applicable to sine and cosine functions) to the realm where the y-axis is at a variable angle $\gamma$ to the x-axis (i.e., non-orthogonal systems). This gives rise to the concept of the \emph{gamma derivative} --- which expands the classical derivative to impart phase shifts of $\gamma$ degrees. Hence, the ordinary derivative (with respect to $\alpha$) or $d/d \alpha$ is a particular case of the more general \emph{gamma derivative} or $d_\gamma/d_\gamma \alpha$. Formula for the $n^{th}$ gamma derivative of the extended sine and cosine functions are defined. For applied mathematics, the gamma derivatives of the extended sine function $\sin^*(\alpha,\gamma)$ and cosine function $\cos^*(\alpha,\gamma)$ determine the extended governing equation of the energy-coupled mass-spring oscillatory system, and by extended analogy that of the electrical LC (Inductance-Capacitance) circuit.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.007 | 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".