A related problem of formation of Stokes waves from a generic plane wave is considered. It is determined that over long time a plane wave tends to a solution that is effectively described by a Stokes wave with a perturbation moving in the opposite direction to the Stokes wave. This perturbation to the Stokes wave may be described by an effective Hamiltonian, that has quadratic and cubic terms with respect to the perturbations.

A train of Stokes waves can be studied assuming a slowly-varying envelope, with dynamics of the envelope subject to the nonlinear Schroedinger equation (NLSE). In the second part of the present work we provide comparison of two numerical methods to solve NLSE. The first one is the standard second order split-step method based on an operator splitting approach. The second method is the Hamiltonian-conserving method referred to as the Hamiltonian integration method (HIM). HIM allows exact conservation of the Hamiltonian and wave action but requires implicit time stepping. We find that the NLSE can benefit from the Hamiltonian-conserving method compared to the split step method in particular for such solutions as the Akmediev and the Kuznetsov-Ma solitons as well as multisoliton solutions. We find that numerical error for HIM is systematically smaller than for the split-step scheme for the same timestep. At the same time, one can take orders of magnitude larger time steps in HIM, compared to split step, while still ensuring numerical stability. We propose the Hamiltonian-conserving method for the Majda-Maclaughlin-Tabak (MMT) model, which is a generalization of NLSE.

]]>In this dissertation, we develop the first maximum likelihood (ML) method that infers a species tree from the ranked gene trees. We introduce the software PRANC, which can compute the probabilities of ranked gene trees under the coalescent process and infer an ML species tree. We propose methods to estimate a starting tree to be able to locate the ML species tree quickly. To illustrate the methods proposed, we analyze two experimental studies of skinks and gibbons.

]]>analysis of magnetic resonance spectroscopy (MRS) data can be done. This

analysis compares five methods of quality classification; three of these are

legacy methods, Maudsley et al. (2006), Zhang et al. (2018), and

Bustillo et al. (2020), and two newly created methods that used a random forests

classifier (RFC) to inform their classifications. We found that the random forest

classifier was the most accurate at predicting spectra quality (balanced

accuracy for RF of 88% vs legacy of 70%, 72%, or 72%). A

Random-Forests-Informed Filtering method (RFIFM) for quality classification was

created by bounding four of the highest ranking features in the RFC to mimic

the classification methods of the legacy methodologies. The RFIFM had only

slightly decreased accuracy compared to the RFC (85% vs 88%), but still

outclassed the legacy methods. Overall, the top features in the RFC show that

the best measures of quality relate to the frequency of the metabolite peaks in

the spectra.

]]>II. Collocation with piecewise polynomial functions is developed as a method for solving two-point bour:rlary value problems. Convergence is shown for a general class of linear problems and a rather broad class of nonlinear problems. Some computational examples are presented to illustrate the wide applicability and efficiency of the procedure.

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