On the foundations for a measure theory and integration in two and three dimensions and a theory of delta functions over the Levi-Civita field
Bibliographic record
Abstract
The field of real numbers does not permit a direct representation of the (improper) delta functions used for the description of impulsive (instantaneous) or concentrated (localized) sources. Of course, within the framework of distributions, these concepts can be accounted for in a rigorous fashion, but at the expense of the intuitive interpretation. The existence of infinitely small numbers and infinitely large numbers in the Levi-Civita field allows us to have well-behaved delta functions which, when restricted to the real numbers, reduce to the Dirac delta function. Here we develop the foundations for a mathematically rigorous theory of localized and instantaneous sources that has a clear and unambiguous way of specifying a mathematically concentrated source. We use the already existent one variable measure and integration theory on Levi-Civita field to construct the foundations of a measure and integration theory in two and three dimensions. First we construct measurable sets using sets with boundaries that can be expressed as analytic functions and we show the the resulting measure is Lebesgue-like. In particular we prove the measurability of countable sets, the countable union of measurable sets, and the finite intersection of measurable sets. Following that we use analytic functions to construct a larger class of measurable functions, we then define the integral of a measurable function over a measurable set. We prove several propositions regarding measurable functions and the associated integration theory including that the set of measurable functions is closed under multiplication and addition, and that integration is linear. This allows for a wide range of applications for the delta function in one, two, and three dimensions and sets the course for a more extensive study of this topic in the future.
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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 | 0.003 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.002 | 0.011 |
| Scholarly communication | 0.005 | 0.006 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.003 | 0.005 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".