MATH 553 Qualifying Exam
August 2012
Note. There are 6 questions in 14 parts. Each part worths 9 points, except Problem 6 worths 18 points. But
the maxmal points you may receive is 100 points. You may do them in any order but label your solutions
clearly.
Qualifying examination
MA544, Summer 2012
Name:
1. (30 points) Let (X, M, ) be a measure space. Let f and g be real-valued integrable
functions on X with
f d =
gd
X
X
Show that either
(a) f = g a.e. on X , or
(b) there exists a set E M such that
E
f d >
E
MA 54400 - Qualifying Exam
August 8, 2011
Prof. Donatella Danielli
Problem Score Max. pts.
1
25
2
20
3
30
4
25
Total
100
1
2
1. Let f L1 (R), and let F (t) = R f (x) cos (tx) dx.
(a) Prove that F (t) is continuous for t R.
(b) Prove the following Riemann-
Name
MATH 544 Qualifying Examination
August 2010
L. Brown
There are 6 numbered problems, each of which is worth 20 points. All answers must be
justied. Use the space provided or the backs of the pages. Please ask if you do not
understand some of the termi
QUALIFYING EXAMINATION
August 2009
MATH 544Antnio S Barreto
o
a
Identier (PLEASE PRINT CLEARLY)
Instructions:
I) This exam booklet contains 6 problems and 13 pages. The value of each question
is indicated next to its statement. Write the solutions on this
QUALIFYING EXAMINATION
August 2013
MATH 544R. Ba uelos
n
Instructions: There are a total of 6 problems. A problem appears on each of the following
pages. Problems are worth 20 points each. Use the space provided for the solutions, using
back pages as need
Math 544
Qualifying Examination
January 2008
Prof. N. Garofalo
Name.
I. D. no. .
Problem Score Max. pts.
1
20
2
20
3
25
4
20
5
20
6
25
Total
130
Problem 1. (20 points) Let f L1 (Rn ) L2 (Rn ) be a function such that for some s > n/2 its
Fourier transform
MA 54400 - Qualifying Exam
January 3, 2012
Prof. Donatella Danielli
Problem Score Max. pts.
1
20
2
25
3
30
4
25
Total
100
In order to receive full credit, you need to show your work and justify your arguments.
1
2
1. Let f (x, y ), 0 x, y 1, satisfy the f
MA 544 QUALIFYING EXAMINATION
January, 2010
Antnio S Barreto
o
a
Identier:
(Please print clearly in case the front cover gets lost)
PLEASE FOLLOW THESE INSTRUCTIONS:
1) This exam booklet contains 7 problems and 15 pages. The value of each question
is indi
Math 544 Qualifying Exam,
January 2009,
L. Lempert
Each problem is worth 5 points.
1. Let a < b be real numbers, gi : R [a, b] arbitrary functions, i N, and h : R R
continuous. Supposing that the gi converge uniformly, prove that h gi also converge
unifor
QUALIFYING EXAMINATION
January 2013
MATH 544R. Ba uelos
n
Student ID:
(PLEASE PRINT CLEARLY)
Instructions: There are a total of 6 problems in this exam. A problem appears on each of
the following pages. Problems are worth 20 points each. Use the space pro
MA 553
GOINS
QUALIFYING EXAM SPRING 2008
This exam is to be done in two hours in one continuous sitting. Begin each question on a new sheet
of paper. In answering any part of a question, you may assume the results in previous parts, even if
you have not s
ABSTRACT ALGEBRA COMPREHENSIVE EXAM AUG, 2013
Attempt all questions. Time 2 hrs
(1) (20 pts) Let G be a group such that for a xed integer n > 1, (xy )n = xn y n
for all x, y G. Let G(n) = cfw_xn |x G and G(n) = cfw_x G|xn = e.
(a) Prove that G(n) and G(n)
Qualifying Examination
MA 553
Time: 2 hours
August 13, 2010
Instructor: F. Shahidi
PUID:
1
2
3
4
5
6
7
8
Total
1
MA 553
Qualifying Exam
August 13, 2010
Inst: F. Shahidi
PUID
(20 pts)
1. Show that every group of order 143 is cyclic. You are only allowed to
QUALIFYING EXAMINATION
AUGUST 2009
MA 553
1. (13 points) Let G be a group such that G/Z (G) is Abelian, and let H = cfw_e
be a normal subgroup of G. Show that H Z (G) = cfw_e. (Hint: Consider the
commutator subgroup G of G.)
2. (15 points) Let G be a grou
Math 553
Qualifying Exam
January 2 2008
1. (20 pts) Let G be a nontrivial nite group.
(a) What is meant by a composition series for G?
(b) State the Jordan-Hlder theorem.
o
(c) What does it mean for G to be simple?
(d) What does it mean for G to be solvab
MATH 553 QUALIFYING EXAMINATION, JANUARY 2013
READ THIS = : Please begin each question (IV) on a new sheet of paper.
In answering any part of a question, you may assume the results in previous parts, even if
you havent done them.
[Bold numbers] indicate p
MATH 553 QUALIFYING EXAMINATION
January 6, 2012
PLEASE BEGIN YOUR ANSWER TO EACH PROBLEM IIV ON A NEW SHEET.
WHEN ANSWERING ANY PART OF A PROBLEM, YOU MAY USE PRECEDING PARTS, EVEN IF
YOU HAVENT SOLVED THEM.
I. [15 points] Let G be a nite group, let p be
MATH 553 Qualifying Exam
January 2011
Note. There are 5 questions in 15 parts. Each part worths 8 points. But the maxmal points you may receive
is 100 points. You may do them in any order but label your solutions clearly. You may do any problem/part
by as
MATH 553 QUALIFYING EXAMINATION
January 2010
Please begin each question IV on a new sheet.
In doing any part of a multipart problem, you may assume youve done the preceding
parts, even if you havent.
I. [32 points] Let p and q be (positive) integer primes
Math 544
Qualifying Examination
August, 2008 Prof. Davis
(15) 1. Show that if E R and if | |e stands for outer measure then
|E |e =
|E [n, n + 1]|e .
n=
Also prove that if Oi , i 1, are open subsets of R satisfying
Oi = R then
i=1
k
lim
k
Oi
i=1
= |E |e .