PARTITIONS INTO SUMMANDS OF THE FORM [ma]
Bibliographic record
Abstract
§1. Asymptotic estimates for the number of partitions of the integer n into summands chosen from an arithmetic progression have been derived by several authors, see for example Meinardus [3]. In this note we investigate a natural extension which has not previosusly appeared in the literature. We shall study the asymptotic behaviour of the numbers p a (n) and q (n), the number a of partitions of n into summands and distinct summands, respectively, chosen from the sequence [ma], m = 1,2,... where a> 1 is an irrational number and [x] denotes the largest integer 5 x. If y = a- [a] then for almost all y c (0,1) in the Lebesgue sense we shall obtain asymptotic formulae (given in Theorem 2 below) for pa (n) and q a (n). However, when y is of finite class, that is, there does not exist a number y such that as i 3-(1.1) f1+a+'Isin tyul}-for every positive e, we can only deduce log pa (n) _ 7T 3 + 0(n ő) log q a (n) = r 3a + 0(n ő) for every positive 6. This is closely connected with the well known fact the larger the class of y, defined to be the largest y satisfying equation (1.1), the less evenly distributed is ma- [me]. It is worth noting that if a = [a] + p/q + e with E very small, then for a long stretch the sequence 4 q [ma] is the union of arithmetic progressions whose difference is small compared to their length. Finally we point out that no significant difference arises if we consider partitions into the sequence [ma+S]. §2. In this section we first apply the results of Roth and Szekeres [4]. Their results hold subject to the conditions: PROC. SEVENTH MANITOBA CONFERENCE ON
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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 | 0.002 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.003 | 0.002 |
| Scholarly communication | 0.003 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.016 | 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".