Reflections Over the Dual Ring—Applications to Kinematic Synthesis
Bibliographic record
Abstract
Least-square problems arise in multiple application areas. The numerical algorithm intended to compute offline the minimum (Euclidian)-norm approximation to an overdetermined system of linear equations, the core of least squares, is based on Householder reflections. It is self-understood, in the application of this algorithm, that the coefficient matrix is dimensionally homogeneous, i.e., all its entries bear the same physical units. Not all applications lead to such matrices, a case in point being parameter identification in mechanical systems involving rigid bodies, whereby both rotation and translation occur; the former being dimensionless and the latter bearing units of length. Because of this feature, dual numbers have found extensive applications in these fields, as they allow the analyst to include translations within the same relations applicable to rotations, on dualization2 of the rotation equations, as occurring in the geometric, kinematic, or dynamic analyses of mechanical systems. After recalling the basic background on dual numbers and introducing reflection matrices defined over the dual ring, we obtain the dual version of Householder reflections applicable to the offline implementation of parameter identification. For the online parameter identification, recursive least squares are to be applied. We provide also the dual version of recursive least squares. Numerical examples are included to illustrate the underlying principles and algorithms.
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How this classification was reachedexpand
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) | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".