Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
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.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.000 | 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 | 0.001 |
| Open science | 0.001 | 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