Bibliographic record
Abstract
In this thesis two problems concerning linear transformations on Mn, the algebra of n-square matrices over the complex numbers, are considered. The first is the determination of the structure of those transformations which map non-singular matrices to non-singular matrices; the second is the determination of the structure of those transformations which, for some positive integer r, preserve the sum of the r x r principal subdeterminants of each matrix. In what follows, we use E to denote this sum, and the phrase "direct product" to refer to transformations of the form T(A) = cUAV for all A in Mn or T(A) = cUA'V for all A in Mn where U, V are fixed members of Mn and c is a complex number. The main result of the thesis is that both non-singularity preservers and Er-preservers, if r ≥ 4, are direct products. The cases r=1,2,3 are discussed separately. If r=1, it is shown that E₁ preservers have no significant structure. If r=2, it is shown that there are two types of linear transformations which preserve E₂, and which are not direct products. Finally, it is shown that these counter examples do not generalize to the case r=3. These results and their proofs will also be found in a forthcoming paper by M. Marcus and JR. Purves in the Canadian Journal of Mathematics, entitled Linear Transformations of Algebras of Matrices: Invariance of the Elementary Symmetric Functions.
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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 | 0.002 |
| 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 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.010 | 0.003 |
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".