Research

The common theme in my research is the use of non Archimedean mathematics, (NAM) especially of nonstandard analysis.

My research can be divided in two main directions: applications of NAM in analysis and related subjects, and applications of NAM in combinatorial number theory.

Applications in analysis

  1. R. Scandone, L. Luperi Baglini, K. Simonov, A characterization of singular Schrödinger operators on the half-line, Canad. Math. Bull. (2020), 1-19, doi:10.4153/S0008439520000958.
  2. V. Benci, L. Luperi Baglini, K. Simonov, Infinitesimal and infinite numbers as an approach to quantum
    mechanics, Quantum 3, 137 (2019), https://doi.org/10.22331/q-2019-05-03-137.
  3. V. Benci, L. Luperi Baglini, M. Squassina, Generalized solutions of variational problems and applications, Adv. Nonlinear Anal., https://www.degruyter.com/view/j/anona.2020.9.issue-1/anona-2018-0146/anona-2018-0146.xml.
  4. A. Lecke, L. Luperi Baglini, P. Giordano, The classical theory of calculus of variations for generalized smooth functions, Adv. Nonlinear Anal., https://doi.org/10.1515/anona-2017-0150.
  5. V. Benci, L. Luperi Baglini, Generalized solutions in PDE’s and the Burgers’ equation, J. Diff. Eq., Vol. 263 (2017), Issue 10, 6916-6952, DOI: https://doi.org/10.1016/j.jde.2017.07.034.
  6. L. Luperi Baglini, P. Giordano, The category of Colombeau algebras, Monatsh. Math., Vol. 182 (2017), Issue 3, 649-674, https://link.springer.com/article/10.1007/s00605-016-0990-1
  7. P. Giordano, L. Luperi Baglini, Asymptotic gauges: generalization of Colombeau type algebras, Math. Nachr., Vol. 289 (2016), Issue 2-3, 247-274, https://onlinelibrary.wiley.com/doi/abs/10.1002/mana.201400278 .
  8. V. Benci, L. Luperi Baglini, A non-archimedean algebra and the Schwartz impossibility theorem, Monatsh. Math., Vol. 176 (2015), 503-520, https://link.springer.com/article/10.1007/s00605-014-0647-x .
  9. V. Benci, L. Luperi Baglini, Generalized functions beyond distributions, Arab. J. Math, Vol. 4 (2014), Issue 4, 231-253, https://link.springer.com/article/10.1007/s40065-014-0114-5 .
  10. V. Benci, L. Luperi Baglini, Ultrafunctions and applications, DCDS-S, Vol. 7 (2014), Issue 4, 593-616, https://aimsciences.org/article/doi/10.3934/dcdss.2014.7.593 .
  11. V. Benci, L. Luperi Baglini, A topological approach to non-Archimedean mathematics, in: “Geometric Properties for Parabolic and Elliptic PDE” (F. Gazzola, K. Ishige, C. Nitsch, P. Salani eds.), Springer Proceedings in Mathematics & Statistics, Vol. 176 (2016), 17-40, https://www.springerprofessional.de/en/a-topological-approach-to-non-archimedean-mathematics/10565586 .
  12. V. Benci, L. Luperi Baglini, A generalization of Gauss’ divergence theorem, in: “Recent Advances in Partial Differential Equations and Applications” (V. D. Radulescu, A. Sequeira, V. A. Solonnikov eds.),
    Contemporary Mathematics 666 (2016), 69-84, http://www.ams.org/books/conm/666/ .
  13. V. Benci, L. Luperi Baglini, A model problem for ultrafunctions, in: “Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems”, EJDE, Conference 21
    (2014), 11-21, ISSN: 1072-6691.
  14. V. Benci, L. Luperi Baglini, Basic properties of ultrafunctions, in: “Analysis and Topology in Nonlinear Differential Equations”, Progress in Nonlinear Differential Equations and Their Applications, Vol. 85
    (2014), 61-86, 10.1007/978-3-319-04214-5_4.

Applications in combinatorial number theory

  1. P. Arruda, L. Luperi Baglini, Rado equations solved by linear combinations of idempotent ultrafilters, submitted.
  2. L. Luperi Baglini, Partition regularity of polynomial systems near 0, Semigr. Forum, in publication.
  3. L. Luperi Baglini, Ultrafilters, monads and combinatorial properties, accepted for publication in Mathematical Logic Quarterly, arXiv:
    1712.06073    .
  4. M. Di Nasso, L. Luperi Baglini, Ramsey properties of nonlinear Diophantine equations, Adv. Math., Vol. 324 (2018), 84-117, https://www.sciencedirect.com/science/article/pii/S0001870816309100.
  5. L. Luperi Baglini, F-finite embeddabilities of sets and ultrafilters, Arch. Math. Logic, Vol. 55 (2016), Issue 5, 705-734, https://link.springer.com/article/10.1007/s00153-016-0489-4 .
  6. L. Luperi Baglini, Partition regularity of nonlinear polynomials: a nonstandard approach, Integers, Vol. 14 (2014), A-30, http://math.colgate.edu/~integers/vol14.html.
  7. L. Luperi Baglini, Ultrafilters maximal for finite embeddability, J. Log. Anal., Vol. 6 (2014), A-6, 1-16, http://logicandanalysis.org/index.php/jla/article/view/226/98 .
  8. L. Luperi Baglini, A nonstandard technique in combinatorial number theory, in: “Selected Papers of EuroComb’13” (J. Nesetril, M. Pellegrini eds.), European J. Combin., Vol. 48 (2015), 71-80, https://www.sciencedirect.com/science/article/pii/S0195669815000323 .
  9. L. Luperi Baglini, “Partition regularity of nonlinear polynomials: a nonstandard approach”, in: “The Seventh European Conference on Combinatorics, Graph Theory and Applications” (J. Nesetril, M. Pellegrini, eds.), CRM Series, Vol. 16 (2013), 407-412, https://link.springer.com/chapter/10.1007/978-88-7642-475-5_65 .

Preprints

  1. L. Luperi Baglini, P. Giordano, Fixed point iteration methods for arbitrary generalized ODE, in preparation,
    https://www.mat.univie.ac.at/~giordap7/preprint_ODE.pdf .
  2. P. Giordano, L. Luperi Baglini, Picard-Lindelöf theorem for PDE, in preparation, https://www.mat.univie.ac.at/~giordap7/PL-PDE.pdf .
  3. M. Di Nasso, L. Luperi Baglini, Finite embeddabilities as group actions on N, in preparation.
  4. L. Luperi Baglini, Y. Puig de Dios, Finite embeddabilities in linear dynamics, in preparation.

Altre Pubblicazioni

L. Luperi Baglini, Hyperintegers and Nonstandard Techniques in Combinatorics of Numbers, PhD Dissertation (2012), University of Siena, arXiv: 1212.2049.