ChapterZero

Qualification exams are over! I can do something other than study and worry now.

There were three exams. The first was on a boatload of applied complex variables stuff: the first term of the associated course covered basic complex analy, bilinear and conformal mappings, Schwarz-Christofel transformations, series solutions to ODEs with analytic and nonanalytic coefficients, integration by parts for asymptotic evaluation of integrals. The second term continued asymptotics with watson’s lemma and laplace’s method, fourier method, stationary phase, steepest descent, then we covered the basics of qualitative analysis of nonlinear ODEs– existence and uniqueness through Floquet theory– and regular perturbation theory. The third term covered singular perturbation theory, boundary layer expansions, WKB, fourier series, completeness of eigenfunctions of sturm-liouville problems, separation of variables, green’s functions, and basic integral equations. The actual exam had a question on asymptotic evaluation of an integral, a boundary layer problem, and a completeness of SL eigenfunctions problem.

The second covered three one term courses: the linear algebra course (pretty standard stuff there, I’d guess: invariant subspaces, jordan decomposition, special classes of operators like normal/self-adjoint, SVD, spectral theory, basic Hilbert space theory like projections, riesz representation, adjoints, square roots of positive operators), the functional/real analysis course (Banach, Hilbert space, spectral theory, and lebesgue integration in R and Lp spaces), and the probability course (non-measure theoretic applied probability: basic counting arguments, poisson spatial processes, guassian random variables, law of large numbers, branching processes, brownian motion). The functional analysis question was to show that a closed operator from one Hilbert space into another is bounded beneath iff it is invertible and has a closed range. The linear algebra question was to show that if T is a normal operator and W a T-invariant subspace, then W can be written as the direct sum of the intersections of itself with the eigenspaces of T. There were three probability questions: the first was an application of Bayes rule, the second was a multi-part question on Poisson processes and their conditioning properties, and the third was on a Galton-Watson process.

The final exam covered the numerical mathematics course: in the first term we talked about implementing various matrix decompositions like QR, LU, SVD, Cholesky using various transformations like Householder, Givens rotations, etc. and the stability/backward stability of these methods, their cost in terms of operations, and then iterative methods for solving systems and finding eigenvalues. The second term focused on various tools: splines, least squares, interpolation, etc. before getting to the numerical solution of IVPs and analysis of the convergence of the various methods, shooting and relaxing for solving BVPs. The third term covered the numerical solution of PDEs with finite difference methods, analysis of stability and consistency, shock capturing schemes, finite element methods, multigrid, and boundary element methods. The questions from the first term were to come up with an algorithm using Householder transformations to zero out the bottom rows in a rectangular lower triangular matrix, making it essentially a square lower triangular matrix; also, to use Gaussian elimination to show that at some point during the process of reducing a given matrix, the Schur complement replaces a part of the matrix, and given positive/negative definite conditions on two subblocks of the matrix, to show its nonsingularity, the existence and uniqueness of the LU decomposition, and that without pivoting the algorithm would be unstable. The second term questions were to prove the order of convergence of the RK2 method, then do some stuff with Hermite interpolation– which I don’t remember, because I skipped it– to improve the method, I think, and to approximate an exponentially decaying integral over all R using the trapezoid rule with unit step size over a small interval, while justifying the use of a small interval and explaining the fact that the resulting error is very small. The third term question was to find a backward in time, centered in space FD scheme for the solution of a linear advection equation, prove its consistency, prove unconditional stability via Neumann analysis, explain the advantages and disadvantages of implicit schemes such as this, and come up with a modified equation which the solution to the difference scheme satisfies to a higher order in time.