Notes & PDFs

The Sturm-Liouville problem on the half-line. This document is my Master’s project: I spent one semester working on it full-time. The original project was to generalize the Sturm-Liouville problem on $[a,b]$, which can be solved through an eigenfunction expansion, to the real half-line $[0,+\infty)$ by following H. Weyl’s original geometric proof, written in German in 1910. H. Weyl’s proof is rigorous in the “limit-circle” case, but incomplete in the “limit-point” case. I then turned to Levitan and Sargsjan’s book Sturm-Liouville and Dirac Operators, but the proof of Lemma 2.4.3 in the book does not justify a very important and delicate limiting argument. My supervisor and I were unable to find a simple way to bridge this gap at the time of the project. In my opinion, one must first prove the surjectivity of the generalized Fourier transform in order to obtain this result (cf. Corollary 4.57 in my project). To do so, I followed the argument in Titchmarsh’s book Eigenfunction Expansions Associated with Second-Order Differential Equations, Part One. I therefore hope to help others who become interested in this beautiful subject.

The Spectral Theorem for unbounded self-adjoint operators. This document is a semester project completed during my Master’s degree. It presents a proof of the spectral theorem for unbounded self-adjoint operators, following Weidmann’s proof. I then present applications of this theorem to the Hamburger moment problem and to the characterization of so-called “operator-monotone” functions.

Transformation Theory and an Application to the Kepler Problem. This document is a semester project completed during the final year of my Bachelor’s degree. It is an introduction to Hamiltonian systems, including a proof of the (simplified) Noether theorem that connects the symmetries of a Hamiltonian system with its conserved quantities. The theorem is then applied to recover conservation laws from physics (conservation of energy, mass or momenta) and to solve the geodesic problem on the sphere. Finally, the geodesic problem on the sphere and the negative-energy Kepler problem (i.e. the two-body problem) are shown to be equivalent. This immediately shows that two moving celestial bodies either follow a periodic trajectory or collide in finite time.

Predicting the number of satellites of female horseshoe crabs. This document is my final project for an Applied Biostatistics course. The aim was to analyze Brockmann’s (1996) “Horseshoe crab satellite” dataset using a generalized linear model.

Data Analysis: Prostate Cancer. This document is my final project for a Linear Models course. The aim was to analyze a prostate cancer dataset by T.A. Stamey et al. (1989) using a linear model.