UAH > Math > Joint Program Exam

Joint Program Examination

The Joint Program Examination is required of all students for the PhD in Applied Mathematics. The exam covers two core areas

1. Real analysis (MA 653, MA 654)
2. Linear and numerical linear algebra (MA 544, MA 614)

The test is administered in two parts, corresponding to the two core areas. Each part is three and one half hours long. The exam may be taken at most twice.

Any student considering taking this examination should meet as soon as possible with the Graduate Program Director, Dr. Kenneth Howell, in 201B 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.
• L ^p spaces for positive measures with p = 1, 2, ..., ∞, definition, completeness.
• Product measure, Lebesgue measure on R^k, 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

Schedule

• Fall 2009: Tuesday, September 15 and Thursday, September 17.

Previous Exams

Links to some of the previous exams are given below in Portable Document Format (PDF). You will need Adobe Acrobat Reader to view these files.

Date	Real Analysis	Linear Algebra
Spring 2009
Fall 2008
Spring 2008
Fall 2007
Spring 2007
Fall 2006
Spring 2006
Fall 2005
Spring 2005
Fall 2004
Spring 2004
Spring 2003
Fall 2002
Fall 2001
Spring 2001
Spring 2000
Fall 1999
Spring 1999
Fall 1998
Spring 1998
Fall 1997
Spring 1997
Fall 1996
Spring 1996
Fall 1995