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Non–Standard Analysis Revisited: An Easy Axiomatic Presentation Oriented Towards Numerical Applications

Open Access
|Mar 2022

References

  1. Amodio, P., Iavernaro, F., Mazzia, F., Mukhametzhanov, M. and Sergeyev, Y. (2017). A generalized Taylor method of order three for the solution of initial value problems in standard and infinity floating-point arithmetic, Mathematics and Computers in Simulation 141: 24–39, DOI: 10.1016/j.matcom.2016.03.007.10.1016/j.matcom.2016.03.007
  2. Arora, J.S. (2004). Introduction to Optimum Design, Elsevier, San Diego.10.1016/B978-012064155-0/50012-4
  3. Astorino, A. and Fuduli, A. (2020). Spherical separation with infinitely far center, Soft Computing 24(23): 17751–17759.10.1007/s00500-020-05352-2
  4. Benci, V. and Cococcioni, M. (2020). The algorithmic numbers in non-Archimedean numerical computing environments, Discrete and Continuous Dynamical Systems S 14(5): 1673–1692, DOI: 10.3934/dcdss.2020449.10.3934/dcdss.2020449
  5. Benci, V. and Di Nasso, M. (2003). Numerosities of labelled sets: A new way of counting, Advances in Mathematics 173(1): 50–67.10.1016/S0001-8708(02)00012-9
  6. Benci, V. and Di Nasso, M. (2018). How to Measure the Infinite: Mathematics with Infinite and Infinitesimal Numbers, World Scientific, Singapore.
  7. Benci, V., Di Nasso, M. and Forti, M. (2006). The eightfold path to nonstandard analysis, in N. J. Cutland et al. (Eds), Nonstandard Methods and Applications in Mathematics, AK Peters, Wellesley, pp. 3–44.
  8. Benci, V., Horsten, L. and Wenmackers, S. (2018). Infinitesimal probabilities, British Journal for the Philosophy of Science 69(2): 509–552.10.1093/bjps/axw013601260429977092
  9. Bierman, G.J. (2006). Factorization Methods for Discrete Sequential Estimation, Courier Corporation, North Chelmsford.
  10. Cococcioni, M., Cudazzo, A., Pappalardo, M. and Sergeyev, Y. (2020). Solving the lexicographic multi-objective mixed-integer linear programming problem using branch-and-bound and grossone methodology, Communications in Nonlinear Science and Numerical Simulation 84: 105177, DOI: 10.1016/j.cnsns.2020.105177.10.1016/j.cnsns.2020.105177
  11. Cococcioni, M. and Fiaschi, L. (2020). The Big-M method with the numerical infinite M, Optimization Letters 15: 2455–2468, DOI: 10.1007/s11590-020-01644-6.10.1007/s11590-020-01644-6
  12. Cococcioni, M., Fiaschi, L. and Lambertini, L. (2021). Non-Achimedean zero-sum games, Journal of Applied and Computational Mathematics 113483: 1–17, DOI: 10.1016/j.cam.2021.113483.10.1016/j.cam.2021.113483
  13. Conway, J.H. (2000). On Numbers and Games, CRC Press, New York.10.1201/9781439864159
  14. De Leone, R. (2018). Nonlinear programming and grossone: Quadratic programming and the role of constraint qualifications, Applied Mathematics and Computation 318: 290–297, DOI: 10.1016/j.amc.2017.03.029.10.1016/j.amc.2017.03.029
  15. De Leone, R., Egidi, N. and Fatone, L. (2020a). The use of grossone in elastic net regularization and sparse support vector machines, Soft Computing 24(23): 17669–17677.10.1007/s00500-020-05185-z
  16. De Leone, R., Fasano, G., Roma, M. and Sergeyev, Y.D. (2020b). Iterative grossone-based computation of negative curvature directions in large-scale optimization, Journal of Optimization Theory and Applications 186(2): 554–589.10.1007/s10957-020-01717-7
  17. Dehn, M. (1900). Die Legendre’schen S¨atzeüber die Winkelsumme im Dreieck, Mathematische Annalen 53(1900): 404–439, DOI: 10.1007/BF01448980.10.1007/BF01448980
  18. Deveau, M. and Teismann, H. (2014). 72+ 42: Characterizations of the completeness and Archimedean properties of ordered fields, Real Analysis Exchange 39(2): 261–304.10.14321/realanalexch.39.2.0261
  19. Falcone, A., Garro, A., Mukhametzhanov, M.S. and Sergeyev, Y.D. (2020a). Representation of grossone-based arithmetic in simulink for scientific computing, Soft Computing 24(23): 17525–17539.10.1007/s00500-020-05221-y
  20. Falcone, A., Garro, A., Mukhametzhanov, M.S. and Sergeyev, Y.D. (2020b). A Simulink-based software solution using the infinity computer methodology for higher order differentiation, Applied Mathematics and Computation 409: 125606, DOI: 10.1016/j.amc.2020.125606.10.1016/j.amc.2020.125606
  21. Fiaschi, L. and Cococcioni, M. (2018). Numerical asymptotic results in game theory using Sergeyev’s infinity computing, International Journal of Unconventional Computing 14: 1–25.
  22. Fiaschi, L. and Cococcioni, M. (2020). Non-Archimedean game theory: A numerical approach, Applied Mathematics and Computation 409: 125356, DOI: 10.1016/j.amc.2020.125356.10.1016/j.amc.2020.125356
  23. Fiaschi, L. and Cococcioni, M. (2021). A non-Archimedean interior point method for solving lexicographic multi-objective quadratic programming problems, EURO Journal on Computational Optimization, (submitted).10.3390/math10234536
  24. Gagniuc, P.A. (2017). Markov Chains: From Theory to Implementation and Experimentation, Wiley, Hoboken.10.1002/9781119387596
  25. Golub, G.H. and Van Loan, C.F. (2013). Matrix Computations, JHU Press, Baltimore.10.56021/9781421407944
  26. Iavernaro, F., Mazzia, F., Mukhametzhanov, M. and Sergeyev, Y. (2020). Conjugate-symplecticity properties of Euler–Maclaurin methods and their implementation on the infinity computer, Applied Numerical Mathematics 155: 58–72, DOI: 10.1016/j.apnum.2019.06.011.10.1016/j.apnum.2019.06.011
  27. Keisler, H.J. (1976). Foundations of Infinitesimal Calculus, Prindle, Weber & Schmidt, Boston.
  28. Krishnamoorthy, A. and Menon, D. (2013). Matrix inversion using Cholesky decomposition, 2013 IEEE Conference on Signal Processing: Algorithms, Architectures, Arrangements, and Applications (SPA’13), Poznan, Poland, pp. 70–72.
  29. Lai, L., Fiaschi, L., Cococcioni, M. and Deb, K. (2021a). Handling priority levels in mixed Pareto-lexicographic many-objective optimization problems, Evolutionary Multi-Criterion Optimization, Shenzhen, China, pp. 362–374, DOI: 10.1007/978-3-030-72062-9_29.10.1007/978-3-030-72062-9_29
  30. Lai, L., Fiaschi, L., Cococcioni, M. and Deb, K. (2021b). Solving mixed pareto-lexicographic multi-objective optimization problems: The case of priority levels, IEEE Transactions on Evolutionary Computation 25(5): 971–985, DOI: 10.1109/TEVC.2021.3068816.10.1109/TEVC.2021.3068816
  31. Levi-Civita, T. (1892). Sugli infiniti ed infinitesimi attuali quali elementi analitici, Atti del R. Istituto Veneto di Scienze Lettere ed Arti Series 7: 1892–1893.
  32. Mises, R. and Pollaczek-Geiringer, H. (1929). Praktische verfahren der gleichungsauflösung, Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 9(1): 58–77.10.1002/zamm.19290090105
  33. Pohlhausen, E. (1921). Berechnung der eigenschwingungen statisch-bestimmter fachwerke, Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 1(1): 28–42.10.1002/zamm.19210010104
  34. Robinson, A. (1996). Non-Standard Analysis, 2nd Edn, Princeton University Press, Princeton.
  35. Sergeyev, Y. (2017). Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems, EMS Surveys in Mathematical Sciences 4(2): 219––320.10.4171/EMSS/4-2-3
  36. Sergeyev, Y.D., Mukhametzhanov, M., Mazzia, F., Iavernaro, F. and Amodio, P. (2016). Numerical methods for solving initial value problems on the infinity computer, Journal of Unconventional Computing 12(1): 3–23.
  37. Thompson, G.L. and Weil, Jr, R.L. (1969). Further relations between game theory and eigensystems, SIAM Review 11(4): 597–602.10.1137/1011091
  38. Thompson, G.L. and Weil, R.L. (1972). The roots of matrix pencils (Ay = λBy): Existence, calculations, and relations to game theory, Linear Algebra and Its Applications 5(3): 207–226.10.1016/0024-3795(72)90003-1
  39. Weil, Jr, R.L. (1968). Game theory and eigensystems, SIAM Review 10(3): 360–367.10.1137/1010061
DOI: https://doi.org/10.34768/amcs-2022-0006 | Journal eISSN: 2083-8492 | Journal ISSN: 1641-876X
Language: English
Page range: 65 - 80
Submitted on: Feb 12, 2021
Accepted on: Sep 21, 2021
Published on: Mar 31, 2022
Published by: Sciendo
In partnership with: Paradigm Publishing Services
Publication frequency: 4 times per year

© 2022 Vieri Benci, Marco Cococcioni, Lorenzo Fiaschi, published by Sciendo
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