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


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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 Search in Google Scholar

Arora, J.S. (2004). Introduction to Optimum Design, Elsevier, San Diego.10.1016/B978-012064155-0/50012-4 Search in Google Scholar

Astorino, A. and Fuduli, A. (2020). Spherical separation with infinitely far center, Soft Computing 24(23): 17751–17759.10.1007/s00500-020-05352-2 Search in Google Scholar

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 Search in Google Scholar

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 Search in Google Scholar

Benci, V. and Di Nasso, M. (2018). How to Measure the Infinite: Mathematics with Infinite and Infinitesimal Numbers, World Scientific, Singapore. Search in Google Scholar

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. Search in Google Scholar

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 Search in Google Scholar

Bierman, G.J. (2006). Factorization Methods for Discrete Sequential Estimation, Courier Corporation, North Chelmsford. Search in Google Scholar

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 Search in Google Scholar

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 Search in Google Scholar

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 Search in Google Scholar

Conway, J.H. (2000). On Numbers and Games, CRC Press, New York.10.1201/9781439864159 Search in Google Scholar

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 Search in Google Scholar

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 Search in Google Scholar

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 Search in Google Scholar

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 Search in Google Scholar

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 Search in Google Scholar

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 Search in Google Scholar

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 Search in Google Scholar

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. Search in Google Scholar

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 Search in Google Scholar

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 Search in Google Scholar

Gagniuc, P.A. (2017). Markov Chains: From Theory to Implementation and Experimentation, Wiley, Hoboken.10.1002/9781119387596 Search in Google Scholar

Golub, G.H. and Van Loan, C.F. (2013). Matrix Computations, JHU Press, Baltimore.10.56021/9781421407944 Search in Google Scholar

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 Search in Google Scholar

Keisler, H.J. (1976). Foundations of Infinitesimal Calculus, Prindle, Weber & Schmidt, Boston. Search in Google Scholar

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. Search in Google Scholar

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 Search in Google Scholar

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 Search in Google Scholar

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. Search in Google Scholar

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 Search in Google Scholar

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 Search in Google Scholar

Robinson, A. (1996). Non-Standard Analysis, 2nd Edn, Princeton University Press, Princeton. Search in Google Scholar

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 Search in Google Scholar

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. Search in Google Scholar

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 Search in Google Scholar

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 Search in Google Scholar

Weil, Jr, R.L. (1968). Game theory and eigensystems, SIAM Review 10(3): 360–367.10.1137/1010061 Search in Google Scholar

eISSN:
2083-8492
Language:
English
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Journal Subjects:
Mathematics, Applied Mathematics