Counting Simplexes in R³
Bibliographic record
Abstract
A finite set of vectors S#R n is called a simplex i# S is linearly dependent but all its proper subsets are independent. This concept arises in particular from stoichiometry. We are interested in this paper in the number of simplexes contained in some H#R n , which we denote by simp(H). This investigation is particularly interesting for H spanning R n and containing no collinear vectors. Our main result shows that for any H#R 3 of fixed size not equal to 3, 4 or 7 and such that H spans R 3 and contains no collinear vectors, simp(H)is minimal if and only if H is contained in two planes intersecting in H,andone of which is of size exactly 3. The minimal configurations for |H| =3,4,7are also completely described. The general problem for R n remains open. # The research of the first author was partially supported by NSERC of Canada. + The research of the second author was partially supported by the Fund "Peregrinatio I" of MOL Co. Hungary, Grant no. 3/1994. 1 the electro...
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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 | 0.002 | 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) | 0.015 | 0.004 |
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; both teacher heads agree on what is shown here.
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".