\documentstyle{article} \oddsidemargin=0in \headheight=-1.1in \textwidth=6.5in \textheight=9.5in \baselineskip=18pt \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % MACROS % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\eq{\begin{equation}} \def\endeq{\end{equation}} \def\reals{{{\rm I}\kern - .15em{\rm R}}} \noindent \large \centerline{{\bf Ph.D. COMPREHENSIVE EXAMINATION TOPICS IN}} \centerline{{\bf Applied Mathematics - August 1999}} \vspace{0.1in} \noindent What follows is intended as a guide to the topics and terminology from Applied Mathematics for which the student is responsible. This list reflects the material covered in class, from the text, additional handouts and homework assignments. \normalsize \noindent \rule{6.5in}{0.01in}\\ \noindent Material covered in M560-61 \begin{itemize} \item[1.] Linear algebra on $\reals^n, A\in \reals^{n\times n}$. \begin{itemize} \item[$\bullet$] Eigenvalues, eigenvectors, diagonalization \item[$\bullet$] Adjoint, self-adjoint, orthogonal diagonalization \item[$\bullet$] Fredholm alternative \item[$\bullet$] Least Squares \item[$\bullet$] Moore-Penrose Inverse \item[$\bullet$] Singular Value decomposition \end{itemize} \item[2.] Linear algebra on vector spaces \begin{itemize} \item[$\bullet$] Change of bases \item[$\bullet$] Gram-Schmidt orthogonalization \item[$\bullet$] Similarity transformations \end{itemize} \item[3.] Manifolds \begin{itemize} \item[$\bullet$] Linear manifolds \item[$\bullet$] Invariant manifolds for $A\in \reals^{n \times n}$ \item[$\bullet$] Closed linear manifolds \end{itemize} \item[3.] Hilbert Spaces \begin{itemize} \item[$\bullet$] Approximation in Hilbert spaces \item[$\bullet$] Complete (orthonormal) sets \item[$\bullet$] Bessel's inequaility \item[$\bullet$] Parseval's equality \item[$\bullet$] Orthogonal complements (closed) \item[$\bullet$] Projection theorem \end{itemize} \item[4.] Sturm Liouville Theory \begin{itemize} \item[$\bullet$] Regular S-L problems (defn) \item[$\bullet$] Basic existence of complete set theorem \item[$\bullet$] Eigenvalue problems and equations \item[$\bullet$] Separated and nonseparated boundary conditions \item[$\bullet$] Regular SL $\Rightarrow$ orthogonality of eigenfunctions \item[$\bullet$] Regular SL $\Rightarrow$ real eigenvalues \end{itemize} \item[5.] Operators \begin{itemize} \item[$\bullet$] Nullspace, range \item[$\bullet$] Bounded, continuous \item[$\bullet$] Adjoint of a bounded operator, Riesz Representation Theorem \item[$\bullet$] Fredholm alternative for bounded operators with closed range \item[$\bullet$] Compact operators and properties \item[$\bullet$] Compact operators spectral theory \end{itemize} \newpage \item[6.] Integral Equations \begin{itemize} \item[$\bullet$] Fredholm (separable and nonseparable-defns) \item[$\bullet$] Volterra \item[$\bullet$] Solution of Volterra eqns. by Laplace transform \item[$\bullet$] Solution of separable Fredholm eqns. \item[$\bullet$] Eigenvalue problems \item[$\bullet$] Conversion of BVP's to Fredholm integral equations \item[$\bullet$] Hilbert-Schmidt operators \item[$\bullet$] separable and nonseparable kernels \item[$\bullet$] Resolvent and pseudo resolvent kernels and operators \item[$\bullet$] Fredholm alternative for $(I-\lambda K)u=f$ with $K$ compact \end{itemize} \item[7.] Distributions and Green's functions \begin{itemize} \item[$\bullet$] Regular and singular distributions \item[$\bullet$] Differentiation, algebraic manipulations, properties \item[$\bullet$] distributional solutions of distributional equations \item[$\bullet$] $\delta(x), Pf(\frac{1}{x})$ \item[$\bullet$] $\delta$ sequences \item[$\bullet$] Green's functions for ordinary differential equations \item[$\bullet$] Solutions of BVP using Green's functions \end{itemize} \item[8.] Differential Operators \begin{itemize} \item[$\bullet$] Formal Adjoint of a differential operator \item[$\bullet$] Domain of a differential operator \item[$\bullet$] Adjoints, Green's functions and distributions \item[$\bullet$] Symmetry of Green's functions \item[$\bullet$] Matrix systems and formal adjoints \item[$\bullet$] Green's functions for Matrix systems \item[$\bullet$] Homogenization of boundary conditions, extended domains \item[$\bullet$] existence of adjoints and Green's functions: 2nd order \item[$\bullet$] Conversion of BVP to integral equations \item[$\bullet$] Fredholm alternative for invertible self-adjoint operators \end{itemize} \item[9.] Calculus of variations \begin{itemize} \item[$\bullet$] Admissible sets, functional minimization \item[$\bullet$] First variations for functionals \item[$\bullet$] Euler-Lagrange equations for $y=y(x), x\in[a,b]$ and special cases \item[$\bullet$] Nonexistence of minimizers \item[$\bullet$] Constrained problems: Lagrange multipliers \item[$\bullet$] Geodesics \item[$\bullet$] Higher-order derivatives \item[$\bullet$] Natural boundary conditions \item[$\bullet$] Several independent variables \item[$\bullet$] Minimum area problems \item[$\bullet$] Variational forms for BVP: Sturm-Liouville, Elliptic PDE \item[$\bullet$] Hamilton's Principle, Lagrangians, Hamiltonians \item[$\bullet$] Lagrangian Mechanics \item[$\bullet$] Hamiltonian Mechanics, Hamilton's equations \item[$\bullet$] NOT COVERED: Hamilton-Jacobi theory and canonical transformations \end{itemize} \item[10.] Spectral theory for operators \begin{itemize} \item[$\bullet$] Resolvent set, point spectrum, continuous spectrum, residual spectrum \item[$\bullet$] Spectral theorem for symmetric operators (complex field) \item[$\bullet$] Existence and boundedness of inverse of resolvent operator \item[$\bullet$] Functions of matrix and integral operators \item[$\bullet$] Functions of operators with complete sets (Hilbert space) \item[$\bullet$] Spectral representation of Operators (basic theorems) \item[$\bullet$] Spectral representation of the identity/delta function \item[$\bullet$] Representation of delta function and transform pairs \item[$\bullet$] Fourier Transformation and complex inversion \item[$\bullet$] Application of Fourier transforms to PDE, ODE, int. eqns, differential delay equations \end{itemize} \end{itemize} \noindent \rule{6.5in}{0.01in}\\ \noindent {\bf References} \begin{itemize} \item[1.] Principles of Applied Mathematics: Transformation and Approximation, J. P. Keener, Addison-Wesley Pub., 1995. \item[2.] Green's Functions and Boundary Value Problems, 2nd ed., Stakgold, 1998. (Alternate source for 2. above) \item[3.] Class notes \end{itemize} \end{document}