Generalized Quadratic Functional Equation and Its Stability over Non-Archimedean Normed Space
Bibliographic record
Abstract
A functional equation is one of the most important and fascinating areas of mathematics, which involves simple algebraic manipulations and can lead to a variety of interesting results. In recent decades, numerous authors have studied different types of functional equation and its stability, such as Hyers-Ulam Stability, Hyers-Ulam-Rassias Stability, and generalized Hyers-Ulam Stability. The stability of functional equations and mixedtype functional equations has been extensively explored by numerous researchers across various spaces, yielding intriguing results primarily in the classical (Archimedean) setting. In recent years, attention has shifted towards investigating the Hyers-Ulam stability (HUS) of generalized Quadratic functional equations in non-Archimedean normed spaces. This article demonstrates the Hyers-Ulam Stability (HUS) of Quadratic functional equations. (3 -) + ( + 3) = 10() + 10(), ( -) + ( + ) = ( 2 + 1)() + ( 2 + 1)(), for any integer 0, in NAN space by using the direct method. Also, we have given some suitable counterexamples.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
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".