Mary Vaughan

Mary Vaughan

Ph.D.

Department of Mathematics and Statistics
The University of Western Australia


IMA Research Interests: Analysis and Partial Differential Equations

List of Publications

You can find all my papers and preprints on ArXiv.
  1. One-sided fractional derivatives, fractional Laplacians, and weighted Sobolev spaces,
    Nonlinear Anal. (2020) (with P. R. Stinga).
  2. The dual Kaczmarz algorithm,
    Acta Appl Math. (2020) (with A. Aboud, E. Curl, S. N. Harding, and E. S. Weber).
  3. Fractional elliptic equations in nondivergence form: definition, applications and Harnack inequality,
    J. Math. Pures Appl. (2021) (with P. R. Stinga).
  4. A convergence result for the derivation of front propagation in nonlocal phase field models,
    to appear in Interfaces Free Bound. (2024) (with S. Patrizi).
  5. Continuous symmetrizations and uniqueness of solutions to nonlocal equations,
    to appear in Analysis & PDE (2024) (with M. G. Delgadino).
  6. The discrete dislocation dynamics of multiple dislocation loops,
    submitted (2024) (with S. Patrizi).
  7. Interior Schauder estimates for fractional elliptic equations in nondivergence form,
    submitted (2024) (with P. R. Stinga).
  8. Asymptotic expansion of a nonlocal phase transition energy,
    submitted (2024) (with S. Dipierro, S. Patrizi, and E. Valdinoci).
  9. A threshold for higher-order asymptotic development of genuinely nonlocal phase transition energies,
    submitted (2024) (with S. Dipierro and E. Valdinoci).
  10. Bond-breaking criteria in peridynamics: Critical stretch vs. critical energy density,
    in preparation (2024) (with P. Seleson and P. Stinga).
  11. A new graph-directed construction of nonlocal energies on the unit interval,
    in preparation (2024) (with A. Aboud, P. Alonso Ruiz, and T. Das).

Ph.D. Thesis: Here is a link to my dissertation.


Video Presentations


Research Training:
In 2018, I was one of twenty students selected to participate in the NSF Mathematical Sciences Graduate Internship. I interned at Oak Ridge National Laboratory under the direction of Pablo Seleson.
When referring to the nth order derivative dnu(t)/dtn, L'Höpital asked Leibniz in his 1965 letter
"What if n is 1/2?" Leibniz replied "It will lead to a paradox. From this apparent paradox,
one day useful consequences will be drawn."