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 representation theorem for positive Borel measures and for co
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 true by
either adding a hypothesis or modifying the conclu
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), and that as n (i) fn f pointwise on R
and (ii) fn
1
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 ) = .
(ii) Y is exponential with parameter > 0 if Y has den
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 independent with
P (Xi = i) = P (Xi = i) =
1
2i
and
P (Xi = 0) =
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) The Exponential Function
f) The Logarithm Function
g)
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 is also a holomorphic polynomial.
3. Suppose that a fun
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 ) =
n0
an z n be holomorphic in the unit disk U such that
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
Problem 1. Let f = f (z ) be analytic in the unit disk,
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. Principle of Inclusion-Exclusion
1. Surjections
2. Derang
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 jective planes? Give a picture of the simplest nite
pro je
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 you believe the statement to be true, give a proof, or
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 vertices of other valences.
(a) What is the least number