Topics for Qualifying Exam in Analysis
r-algebras
Measures, outer measures, Borel measures
Measurable Functions
Lebergee integration in abstract measure spaces and in R , Lebesgue measure
Riesz repres
Real analysis qualifying exam
Jan. 13, 2010
1. (a) Let f be a continuous map of a metric space X into a metric space Y .
True or False. If false either give a counterexample, or make the statement tru
Analysis Qualifying Exam
August 2010
You must justify your answers in full detail, and
explicitly check all the assumptions of any theorem you use.
1. Assume that f, f1 , f2 , L1 (R) (Lebesgue measure
PhD Qualifying Exam, January 2010
Probability
You may use the following facts.
(i) Y is Poisson with parameter > 0 if P (Y = k ) = e k /k ! for k = 0, 1, . . . , in which case E (Y ) = and
V ar(Y ) =
Qualifying Exam
Probability
August, 2010
1. Let Xi , i 1 be IID and set N = inf cfw_n 1 : Xn > X1 (inf = ). Prove that
P (N > n) 1/n for n 1, and use this to show EN = .
2. Let Xi , i 1 be independen
Topics for Qualifying Exam in Complex Analysis
I
Complex Plane and Elementary Function.
a) Complex Numbers
b) Polar Representation
c) Stereographic Projection
d) The Square and Square Root Functions
e
Complex Part
1. Show that the function f (z ) = 1/z has no a holomorphic antiderivative on cfw_< 1 < |z | < 2.
2. Suppose that f is an entire function and f 2 is a holomorphic
polynomial. Show that f
Qualifying Exam, Complex Analysis, August 2010
1. Let n > 0 be an integer. How many solutions does the equation 3z n = ez have in the
open unit disk? Justify your answer in full detail.
2. Let f (z )
QUALIFYING EXAM COMPLEX ANALYSIS
Thursday, January 8, 2009
Show ALL your work. Write all your solutions in clear, logical steps. Good luck!
Your Name:
Problem Score Max
1
20
2
20
3
30
4
30
Total
100
P
Topics for Qualifying Exam in Combinatorics
I. Enumeration
A. Selections with and without repetitions (combinations &
permutations)
B. Partitions
1. Stirling numbers of the first and second kind
C. Pr
Combinatorics Qualifying Exam
Practice Questions
July 21, 2008
1. (a) Dene (v , k , )-design. Derive some identities involving these three parameters.
(b) How are these designs related to nite pro jec
Qualifying Exam in Combinatorics
19 August, 2008
1. Let be a 3-connected planar graph with planar dual . Prove or
disprove:
If is a Cayley graph and is is vertex-transitive, then is a Cayley
graph.
If
Qualifying Exam in Combinatorics
12 January, 2007
1. Suppose that G is a graph with no vertex of valence < 5, exactly 13
vertices of valence 5, no vertex of valence 7, and possibly an assortment
of ve