Preliminary Examinations

Those students who wish to continue toward a Ph.D. degree must take two written examinations called the Preliminary Examinations. One written Preliminary Examination covers material from Math600 and Math602 (Analysis) and another examination covers material from Math672 (Linear Algebra) or from Math 611 (Numerical Linear Algebra).

These examinations will be given twice each year, once before the start of the Fall Semester, and once before the start of the Spring Semester. For students entering with a Bachelor's degree, it is required that the Preliminary Examinations be passed by the beginning of the fourth semester of study. Providing it is before the beginning of the fourth semester, a student may take these examinations several times. Only the part not passed needs to be repeated.

Students who failed to pass both subject areas of the Preliminary Examinations by the beginning of the fourth semester will be asked to leave the graduate program.

Analysis

Topics covered on the exam include the following. Many of these topics have been discussed in Math 600 and Math 602, some of you students have had as an undergraduate and others you may not have seen. References are given for each topic.

1. Metric Spaces: open and closed sets, compactness, connected sets, complete sets, continuous functions on metric spaces
   ([1], Chapters 3 and 4).
2. Continuity and Differentiation: mean value theorem, Rolle’s theorem, Taylor’s formula, derivatives of vector valued functions, uniform continuity, monotonic functions, functions of bounded variation
   ([1], Chapters 5 and 6).
3. Integration: The Riemann-Stieltjes integral, fundamental theorem of calculus, sufficient conditions for existence of Riemann-Stieltjes integral, differention under the integral sign, interchange of order of integration, mean value theorems ([1], Chapter 7).
4. Infinite Sequences and Series: Limit superior and limit inferior, monotonic sequences, alternating series, absolute and conditional convergence, power series, tests for convergence of series, rearrangement of series ([1], Chapter 8).
5. Sequences of Functions: Pointwise convergence, uniform convergence, uniform convergence and continuity, differentiability and integration ([1], Chapter 9).
6. Functions of Several Variables: Directional derivatives, the total derivative, Jacobians, inverse function theorem, implicit function theorem, extrema problems ([1], Chapters 12 and 13).
7. Vector Calculus: Line integrals, Green’s theorem, surface integrals, Stokes theorem, the divergence theorem
   ([2], Chapters 10, 11 and 12).
8. Analytic Function Theory: Analytic functions, Cauchy’s theorem, Cauchy’s integral theorem, the maximum principle, the identity theorem, Taylor and Laurent series, the residue theorem, elementary conformal mappings ([1], Chapter 16).

References:

1. Tom Apostol, Mathematical Analysis 2nd edition, Addison Wesley, 1974.
2. Tom Apostol, Calculus, Vol 2, 2nd edition, John Wiley, 1969.

Linear Algebra

Subspaces, bases, dimension, Linear transformations and matrix representations, Linear functionals, adjoints, and dual spaces, Scalar products and orthogonality, Symmetric, hermitian, and unitary operators, Eigenvectors and eigenvalues (spectral theorem), Jordan canonical form.

References:

• Sh. Axler, Linear Algebra Done Right, Springer-Verlag
• M. L. Curtis, Abstract Linear Algebra,(skip Chapter 2(B) and Chapter 5), (Springer-Verlag).
• K. Huffman and R. Kunze, Linear Algebra, (skip Chapters 6.5, 7.2, 7.6, 9.4, 9.6, 10.6), (Prentice-Hall).
• S. Lang, Linear Algebra, Third Edition, (Springer-Verlag).

Note: Material skipped in one of these references might be found in other references, but presented in a more elementary form.

Numerical Linear Algebra

Direct and iterative methods for the solution of linear systems, LU factorization, row pivoting, stable QR factorization, solution of linear least squares problems by normal equations and QR, stability and conditioning issues, power and inverse iterations, QR iteration, singular value decomposition, simple iterations for sparse matrices, conjugate gradients and other Krylov subspace iterations.

Suggested References:

• L.N. Trefethen and D. Bau, III, Numerical Linear AlgebraSIAM (see e.g., I, II, III.)
• J.W. Demmel, Applied Numerical Linear Algebra, SIAM (Sections 2.1, 2.2, 2.3, 2.4, 3.1-3.3, 4.4, 6.5, 6.6)
• A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, Springer-Verlag (Sections 3.1, 3.3, 4.1, 4.2, 4.3, 5.2, 5.3, 5.4, 5.5, 5.8)