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Bibliographic record
Abstract
The aim of this paper is to study smoothness, approximate continuity, and approximate derivative in a nonstandard manner with respect to infinitesimal parameters. The new nonstandard introduced definitions are combined with standard and nonstandard intermediate value property. Particularly, we show that the existence of continuous and smooth function has the infinitesimal intermediate value property. Moreover, for the same result, we reduce the continuity condition to the infinitesimal intermediate value condition References Abdeljalil, S. (2018). A new approach to nonstandard analysis. Sahand Communications in Mathematical Analysis, 12(1), 195-254. Arkhangel'skii, A. V. (2001). Fundamentals of general topology: problems and exercises (Vol. 13). (D. Translated from the Russian. Reidel, Trans.) Springer Netherlands. Benyamini, Y. & Lindenstrauss, J. (2000). Geometric Nonlinear Analysis (Vol. 48). Amer. Math. Soc. Colloq. Publ. Borwein, J. M., & Preiss, D. (1987). A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions. Transactions of the American Mathematical Society, 303(2), 517-527. Bottazzi, E. (2018). A transfer principle for the continuations of real Functions to the Levi-Civita field p-adic numbers. Ultrametric Analysis and Applications, 10(3), 179-191. Ciesielski, K. (1997). Set theory for the working mathematician (Vol. 39). Londan: Cambridge University Press. Ciurea, G. (2018). A nonstandard approach of helly’ selection principle in complete metric spaces. Proceedings of the Romanian Academy Series A Mathematics Physics Technical Sciences Information Science, 19(1), 11-17. Deville, R., Godefroy, G., & Zizler, V. (1993). A smooth variational principle with applications to Hamilton–Jacobi equations in infinite dimensions. Journal of functional analysis, 111(1), 197-212. Diener, F., & Diener, M. (1995). Nonstandard analysis in practice. Berlin, Heidelberg: Springer Verlag. Duanmu, H. (2018). Applications of nonstandard analysis to Markov processes and statistical decision theory. Doctoral Dissertation, University of Toronto. Goldblatt, R. (1998). Lectures on the hyperreals: an introduction to nonstandard analysis (Vol. 188). Springer Science & Business Media. Goldbring, I. ((2014). Lecture notes on nonstandard analysis. UCLA Summer School in Logic. Hamad, I. O. & Haasan, A. O. (2021). Nonstandard Completion of a non-complete metric spaces. Zanko Journal of Pure and Applied Science, 33(4), 129-135. Hamad, I. O. (2011). Generalized curvature and torsion in nonstandard analysis. Berlin, Germany: Lambert Academic Publishing. Hamad, I. O. (2016). On some nonstandard development of intermediate value property. Assiut University Journal of Mathematics and Computer Science, 45(2), 35-45. Jwahir, H. G. (2021). Nonstandard Analysis for some convergence sequences theories via a transfer principle. American Journal of Applied Sciences, 18(1), 33-5-. Kiro, A. (2020). Taylor coefficients of smooth functions. Journal d'Analyse Mathématique, 14(1), 193-269. Narici, L., & Beckenstein, E. (2011). Topological Vector Spaces (Second ed.). (P. a. mathematics, Ed.) Chapman and Hall/CRC. Nelson, E. (1977). Internal set theory: a new approach to nonstandard analysis. Bulletin of the American Mathematical Society, 1165- 1198. O’Malley, R. J. (1976). Baire* 1, Darboux functions. Proceedings of the American Mathematical Society, 60(1), 187-192. Robert, A. (1988). Nonstandard analysis. John Wiley & Sons Lid. Robinson, A. (1961). Non-standard analysis. Proceedings of the American Mathematical Society, 64(23), 432-440. Robinson, A. (1996). Non-standard analysis (3 ed.). Princeton University Press. Sun, Y. (2015). Nonstandard analysis in mathematical economics. In Nonstandard Analysis for the Working Mathematician (pp. 349-399). Springer, Dordrecht. Vanderwerff, J. (1992). Smooth approximations in Banach space. Proceedings of the American Mathematical Society, 115(1), 113-120.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.201 | 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 it