Every student planning to earn the PhD in Applied Mathematics must pass the two Joint Program Examinations. One exam covers real analysis (MA 653 and MA 654).  The other exam covers linear algebra and numerical linear algebra (MA 544 and MA 614).   Each exam is three and one half hours long

The exams are administered twice a year.  During each administration, a student may take one or both of the exams.  A single exam may be attempted at most twice, with a maximum of three attempts allowed for passing both exams.

Any student considering taking this examination should meet as soon as possible with the Department Chair and Graduate Program Director, Dr. Toka Diagana, in Room 258A Shelby Center for helpful advice and information.

Plan II (non-thesis option) masters degree students who have passed the Joint Program Exam will not be required to take the final oral examination for the masters degree.

## Topics in Real Analysis

• Lebesgue measure on R1: outer measure, measurable sets and Lebesgue measure, non-measurable sets, measurable functions.
• The Lebesgue integral in R1: positive functions and general functions, comparison with the proper and improper Riemann integral.
• Differentiation and integration: monotone functions, functions of bounded variation, absolute continuity, the fundamental theorem of calculus.
• Definition of a positive measure, measure spaces, measurable functions, the integral with respect to a positive measure.
• Convergence theorems for positive measures: monotone and dominated convergence.
• Lp spaces for positive measures with p=1,2,...,∞, definition, completeness.
• Product measure, Lebesgue measure on Rk, Fubini's theorem.

## Topics in Linear Algebra

• Vector spaces over a field: subspaces
• quotient spaces
• complementary subspaces
• bases as maximal linearly independent subsets
• finite dimensional vector spaces
• linear transformations
• null spaces
• ranges
• invariant subspaces
• vector space isomorphisms
• matrix of a linear transformations
• rank and nullity of linear transformations and matrices
• change of basis
• equivalence and similarity of matrices
• dual spaces and bases
• diagonalization of linear operators and matrices
• Cayley-Hamilton theorem and minimal polynomials
• Jordon canonical forms
• real and complex normed and inner product spaces
• Cauchy-Schwarz and triangle inequalities
• orthogonal complements, orthonormal sets
• Fourier coefficients and the Bessel inequality
• adjoint of a linear operator
• positive definite operators and matrices
• unitary diagonalization of normal operators and matrices
• orthogonal diagonalization of real, symmetric matrices