Bibliographic record
Abstract
In this article, we present some recent results related to the calculation of the induced p-norm of n times n circulant matrices A(n, a, b) with diagonal entries equal to a in R and off-diagonal entries equal to b in R. For circulant matrices with nonnegative entries, an explicit formula for the induced p-norm (p between 1 and infinity) is given, whereas for A(n, a, b), a > 0 the situation is no longer so simple and calls for a more subtle analysis. As a matter of fact, while \nthe 2-norm of A(n, a, b) is precisely determined, the exact value of the induced p-norm for 1 < p < infinity and p not equal to 2, still remains elusive. Nevertheless, we provide a lower bound as well as two different categories of upper bounds. As an indication of not being far from the exact values, our estimates coincide at both ends points (i.e., p = 1 and p = infinity) as well as at p = 2 with the precise values. As an abstract approach, we also introduce the *-algebra generated by a normal matrix A accompanied by an axis-oriented norm, and obtain some estimations of the norm of elements of the *-algebra. We then exhibit the connection between the new generalized estimates and the previously obtained estimates in the special case where A is a circulant matrix. Finally, using an optimization-oriented approach, we gain insights on the nature of the maximizing vectors for the p-norm of Ax divided by the p-norm of x. This leads us to formulate a conjecture that, if proven valid, would make it possible to derive an exact formula for the induced p-norm of \nA(n, a, b) in a particular case of special interest, namely when a = 1-n/n and b = 1/n.
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.
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 | 0.003 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.005 | 0.002 |
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".