Abstract
Ultrafunctions are a particular class of functions defined on a non-Archimedean field \({\mathbb{R}^{\ast } \supset \mathbb{R}}\). They have been introduced and studied in some previous works (Benci, Adv Nonlinear Stud 13:461–486, 2013; Benci and Luperi Baglini, EJDE, Conf 21:11–21, 2014; Benci, Basic Properties of ultrafunctions, to appear in the WNDE2012 Conference Proceedings, arXiv:1302.7156, 2014). In this paper we introduce a modified notion of ultrafunction and discuss systematically the properties that this modification allows. In particular, we will concentrate on the definition and the properties of the operators of derivation and integration of ultrafunctions.
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Benci, V.: Ultrafunctions and generalized solutions. Adv. Nonlinear Stud. 13, 461–486, arXiv:1206.2257 (2013)
Benci, V.: An algebraic approach to nonstandard analysis. In: Buttazzo, G. (ed.) Calculus of variations and partial differential equations. Springer, Berlin, 285–326 (1999)
Benci V., Di Nasso M.: Numerosities of labelled sets: a new way of counting. Adv. Math. 173, 50–67 (2003)
Benci V., Di Nasso M.: Alpha-theory: an elementary axiomatic for nonstandard analysis. Expo. Math. 21, 355–386 (2003)
Benci V., Di Nasso M., Forti M.: An Aristotelean notion of size. Ann. Pure Appl. Logic 143, 43–53 (2006)
Benci, V.; Galatolo, S.; Ghimenti, M.: An elementary approach to Stochastic Differential Equations using the infinitesimals. In: Contemporary mathematics, 530, pp. 1–22, Ultrafilters across Mathematics, American Mathematical Society (2010)
Benci, V.; Horsten, H.; Wenmackers, S.: Non-Archimedean probability. Milan J. Math. arXiv:1106.1524 (2012)
Benci V., Luperi Baglini L.: A model problem for ultrafunctions. EJDE, Conf. 21, 11–21 (2014)
Benci, V.; Luperi Baglini, L.: Basic properties of ultrafunctions. To appear in the WNDE2012 Conference Proceedings. arXiv:1302.7156 (2014)
Benci, V.; Luperi Baglini, L.: A non-Archimedean algebra and the Schwartz impossibility theorem. To appear in Monatshefte für Mathematik, arXiv:1401.0475
Du Bois-Reymond P.: Über die Paradoxen des Infinit är-Calcüls, Math. Ann. 11, 150–167 (1877)
Ehrlich Ph.: The Rise of non-Archimedean mathematics and the roots of a misconception I: the emergence of non-Archimedean systems of magnitudes. Arch. Hist. Exact Sci. 60, 1–121 (2006)
Levi-Civita, T.: Sugli infiniti ed infinitesimi attuali quali elementi analitici. Atti del R. Istituto Veneto di Scienze Lettere ed Arti, Venezia (Serie 7), 1765–1815 (1892–93)
Hilbert, D.: Grundlagen der Geometrie. Festschrift zur Feier der Enthüllung des Gauss–Weber Denkmals in Göttingen, Teubner, Leipzig (1899)
Keisler, H.J.: Foundations of infinitesimal calculus, prindle, Weber & Schmidt, Boston (1976). [This book is now freely downloadable at: http://www.math.wisc.edu/~keisler/foundations.html]
Robinson, A.: Non-standard analysis. In: Proceedings of the Royal Academy of Sciences, Amsterdam (Series A) 64, 432–440 (1961)
Veronese, G.: Il continuo rettilineo e l’assioma V di Archimede. Memorie della Reale Accademia dei Lincei, Atti della Classe di scienze naturali, fisiche e matematiche 4, 603–624 (1889)
Veronese G.: Intorno ad alcune osservazioni sui segmenti infiniti o infinitesimi attuali. Math. Ann. 47, 423–432 (1896)
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Benci, V., Luperi Baglini, L. Generalized functions beyond distributions. Arab. J. Math. 4, 231–253 (2015). https://doi.org/10.1007/s40065-014-0114-5
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DOI: https://doi.org/10.1007/s40065-014-0114-5