Mathematical and Statistical Theory
What are the theoretical foundations of the quantiative tools motivated by the applications?
Background
Theoretical results are at the foundation of the methodologies used to study natural phenomena. The possibility to rely on mathematical abstractions enables research to extract general principles of broad use and to extend the development of new methodologies to areas that are connected more strongly if modeled using mathematical concepts.
Goals
The mathematical and statistical foundations developed in this line of research are motivated by both their applications and also by their independent theoretical interest. The contributions are focused on problems emerging in functional analysis, partial differential equations, statistics on manifolds, and theoretical machine learning.
Broader Impacts
The spirit that permeates this group of results aims at interdisciplinarity and inclusiveness and wants to provide a structure of ideas that impact the whole range from mathematical and statistical theories to applied settings.
Relevant Publications
Selvitella, A.M. (2024). On the metastability of learning algorithms in physics-informed neural networks: a case study on Schr\"{o}dinger operators. ICML 2024 Workshop High-dimensional Learning Dynamics 2024: The Emergence of Structure and Reasoning. https://openreview.net/pdf?id=N6eW99aNlg
A.M. Selvitella and J.J. Valdes. (2021). An extension of the gamma test to binary variables and its use as a machine learning tool. International Journal of Pattern Recognition and Artificial Intelligence, 35 (10), 2151010. https://doi.org/10.1142/S0218001421510101
A.M. Selvitella. (2020). Uniqueness of the ground state of the NLS on Hd via analytical and topological methods. Rocky Mountain Journal of Mathematics, 50 (5), 1817-1832. https://doi.org/10.1216/rmj.2020.50.1817
A.M. Selvitella. (2019). A characterization of elliptical distributions through stretched orthogonal matrices. Journal of Applied Probability and Statistics, 14 (3), 1-22.
A.M. Selvitella. (2019). On geometric probability distributions on the torus and applications to molecular biology. Electronic Journal of Statistics, 13 (2), 2717-2763. https://doi.org/10.1214/19-EJS1579
A.M. Selvitella. (2019). Qualitative properties of stationary solutions of the NLS on the hyperbolic space without and with external potentials. Communications in Pure and Applied Analysis, 18 (5), 2663-2677. https://doi.org/10.3934/cpaa.2019118
A.M. Selvitella. (2018). A Rosetta Stone for information theory and differential equations. Communications in Advanced Mathematical Sciences, 1 (1), 45-64. https://doi.org/10.33434/cams.448407
A.M. Selvitella. (2017). The Monge-Ampère equation in transformation theory and an application to 1/α - probabilities. Communications in Statistics - Theory and Methods, 46 (4), 2037-2054. https://doi.org/10.1080/03610926.2015.1040509
A.M. Selvitella. (2017). On 1/α - characteristic functions and applications to asymptotic statistical inference, Communications in Statistics - Theory and Methods, 46 (4), 1941-1958. https://doi.org/10.1080/03610926.2015.1030427
N. Balakrishnan and A.M. Selvitella. (2017). Symmetry of a distribution via symmetry of order statistics. Statistics and Probability Letters, 129, 367-372. https://doi.org/10.1016/j.spl.2017.06.023
A.M. Selvitella. (2017). The Simpson’s paradox in quantum mechanics. Journal of Mathematical Physics, 58 (3), 032101. https://doi.org/10.1063/1.4977784
A.M. Selvitella. (2017). The ubiquity of the Simpson’s paradox. Journal of Statistical Distributions and Applications, 4, 1-16. https://doi.org/10.1186/s40488-017-0056-5
10. B. Franke, J.-F. Plante, R. Roscher, E.A. Lee, C. Smyth, A. Hatefi, F. Chen, E. Gil, A. Schwing, A.M. Selvitella, M.M. Hoffman, R. Grosse, D. Hendricks, and N. Reid. (2016). Statistical inference, learning and models in big data. International Statistical Review, 84 (3), 371-289. https://doi.org/10.1111/insr.12176
A.M. Selvitella. (2016). The maximal Strichartz family of Gaussian distributions. International Journal of Differential Equations, 2016, 2343975. https://doi.org/10.1155/2016/2343975
A.M. Selvitella. (2016). The dual approach to stationary and evolution quasilinear PDEs. Nonlinear Differential Equations and Applications, 23, 1-22. https://doi.org/10.1007/s00030-016-0367-0
A.M. Selvitella. (2015). Nondegeneracy of the Ground State for Quasilinear Schrödinger Equations. Calculus of Variations and Partial Differential Equations, 53, 349-364. http://dx.doi.org/10.1007/s00526-014-0751-8
A.M. Selvitella. (2015). Remarks on the sharp constant for the Schrödinger Strichartz estimate and applications. Electronic Journal of Differential Equations, 2015 (270), 1-19. https://ejde.math.txstate.edu/Volumes/2015/270/selvitella.pdf
W. Craig, A.M. Selvitella, and Y. Wang. (2013). Birkhoff Normal Form for the Nonlinear Schrödinger Equation. Rendiconti Lincei. Matematica e Applicazioni, 24, 215-228. http://dx.doi.org/10.4171/RLM/653
A.M. Selvitella and Y. Wang. (2012). Morawetz and Interaction Morawetz estimates for a Quasilinear Schrödinger Equation. Journal of Hyperbolic Differential Equations, 09 (04), 613-639. https://doi.org/10.1142/S0219891612500208
A.M. Selvitella. (2011). Uniqueness and Nondegeneracy of the Ground State for a class of Quasilinear Schrödinger Equations with a small parameter. Nonlinear Analysis: Theory, Methods and Applications, 74 (5), 1731-1737. https://doi.org/10.1016/j.na.2010.10.045
A.M. Selvitella. (2010). Semiclassical evolution of two rotating solitons for the Nonlinear Schrödinger equation with electric potential. Advances in Differential Equations, 15 (3-4), 315-348. https://projecteuclid.org/journals/advances-in-differential-equations/volume-15/issue-3_2f_4/Semiclassical-evolution-of-two-rotating-solitons-for-the-Nonlinear-Schr%C3%B6dinger/ade/1355854752.full
A.M. Selvitella. (2008). Asymptotic Evolution for the Semiclassical Nonlinear Schrödinger Equation in presence of electric and magnetic fields. Journal of Differential Equations, 245 (9), 2566-2584. https://doi.org/10.1016/j.jde.2008.05.012
Major Events
Data Science and Machine Learning Seminar Series. 2019/2020, 2020/2021, and 2021/2022.
The Data Science Show Episode 1, Episode 2 [Guest: Russell Greiner - AMII & University of Alberta], and Episode 3 [Guest: Mayor Tom Henry - City of Fort Wayne].
AAAI Symposium on Survival Prediction: Algorithms, Challenges and Applications 2021. Organizers: M. van der Schaar, R. Greiner, T.A. Gerds, and N. Kumar. Thought Leader of the Discussion Group on “Counterfactual Reasoning and Causality").
Data Science Week. 2019-2020-2021-2022. Co-organizer: K.L. Foster, Ball State University.
Main Collaborators
Narayanaswamy Balakrishnan - McMaster University Homepage
Julio J. Valdes - National Research Council - Canada Google Scholar