Bibliographic record
Abstract
Two definitions of division by an interval containing zero are compared: a functional one and a relational one. We show that the relational definition also provides interval inverses for other functions that do not have point inverses, such as max and the absolute-value function. Applying the same approach to the ≤ relation introduces the “companion functions ” of relations. By regarding the arithmetic operations +, −, ∗, and / as ternary relations, we obtain the interval versions of the operations. This opens the way for regarding arithmetic problems such as evaluating expressions and solving equations as Constraint Satisfaction Problems (csps). These have a useful computational theory, which is, however, influenced by their predominantly discrete applications. We generalize the conventional formulation to better accommodate real-valued variables, and state the main results. When these results are applied to numerical csps we relate the interval evaluation of an arithmetic expression to the family of solving algorithms of csps. The key to our method of bringing interval arithmetic and interval constraints under a common denominator are companion functions. These functions form an alternative characterization of n-ary relations and appear to be a new contribution to the mathematical theory of relations.
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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 | 0.012 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.006 |
| Science and technology studies | 0.001 | 0.006 |
| Scholarly communication | 0.005 | 0.012 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.001 | 0.006 |
| Insufficient payload (model declined to judge) | 0.012 | 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".