FALL 2005 PRELIMINARY EXAMINATION
1A. Let M be a compact metric space and let (Ui )iI be an open cover of M . Show that
there exists > 0 such that, for all x, y M , if d(x, y) < then there is some j with both
x and y in Uj .
2A. Prove that, if f (z) = P (
Preliminary Exam - Spring 1983
Problem 1 Let f : R + R + be a monotone decreasing function, dened
on the positive real numbers with
f (x) dx < .
0
Show that
lim xf (x) = 0.
x
Problem 2 Let A = (aij ) be an nn real matrix satisfying the conditions:
aii > 0
Preliminary Exam - Spring 1981
Problem 1 Let , , and k be the usual unit vectors in R3 . Let F denote the
vector eld
(x2 + y 4) + 3xy + (2xz + z 2 )k.
1. Compute
F (the curl of F ).
2. Compute the integral of
F over the surface x2 +y 2 +z 2 = 16, z
0.
P
Preliminary Exam - Spring 1982
Problem 1 Prove the Fundamental Theorem of Algebra: Every nonconstant
polynomial with complex coecients has a complex root.
Problem 2 Let S Rn be a subset which is uncountable. Prove that there
is a sequence of distinct poin
Preliminary Exam - Spring 1980
Problem 1 Let f : R R be the unique function such thatf (x) = x if
x < and f (x + 2n) = f (x) for all n Z.
1. Prove that the Fourier series of f is
n=1
(1)n+1 2 sin nx
n
2. Prove that the series does not converge uniformly.
Preliminary Exam - Spring 1979
Problem 1 Let f : Rn \ cfw_0 R be dierentiable. Suppose
f
(x)
x0 xj
lim
exists for each j = 1, . . . , n.
1. Can f be extended to a continuous map from Rn to R?
2. Assuming continuity at the origin, is f dierentiable from Rn
Preliminary Exam - Spring 1978
Problem 1 Let k
0 be an integer and dene a sequence of maps
fn : R R,
fn (x) =
xk
,
x2 + n
n = 1, 2, . . . .
For which values of k does the sequence converge uniformly on R? On every
bounded subset of R?
Problem 2 Prove that
Preliminary Exam - Spring 1977
Problem 1 Suppose f is a dierentiable function from the reals into the
reals. Suppose f (x) > f (x) for all x R, and f (x0 ) = 0. Prove that
f (x) > 0 for all x > x0 .
Problem 2 Suppose that f is a real valued function of on
Problem 1A.
Score:
Suppose that X is a compact metric space. If Y is another metric space (possibly noncompact), let p : X Y Y be the map p(x, y) = y. Show that if Z is a closed subset of X Y
then p(Z) is closed in Y .
Solution:
Suppose that cfw_yi is a
Preliminary Exam - Spring 1987
Problem 1 A standard theorem states that a continuous real valued function
on a compact set is bounded. Prove the converse: If K is a subset of Rn and
if every continuous real valued function on K is bounded, then K is compa
Preliminary Exam - Spring 1988
Problem 1 Suppose that f (x), < x < , is a continuous real valued
function, that f (x) exists for x = 0, and that lim f (x) exists. Prove that
x0
f (0) exists.
Problem 2 Determine the last digit of
2323
2323
in the decimal s
Preliminary Exam - Summer 1983
Problem 1 The number 21982145917308330487013369 is the thirteenth power
of a positive integer. Which positive integer?
Problem 2 Let f : C C be an analytic function such that
k 1
1 + |z|
dm f
dz m
is bounded for some k and m
Preliminary Exam - Summer 1980
Problem 1 Exhibit a real 33 matrix having minimal polynomial (t2 +1)(t
10), which, as a linear transformation of R3 , leaves invariant the line L
through (0, 0, 0) and (1, 1, 1) and the plane through (0, 0, 0) perpendicular
Preliminary Exam - Summer 1978
Problem 1 For each of the following either give an example or else prove
that no such example is possible.
1. A nonabelian group.
2. A nite abelian group that is not cyclic.
3. An innite group with a subgroup of index 5.
4.
Preliminary Exam - Summer 1977
Problem 1 Prove the following statements about the polynomial ring F[x],
where F is any eld.
1. F[x] is a vector space over F.
2. The subset Fn [x] of polynomials of degree
sion n + 1 in F[x].
n is a subspace of dimen-
3. Th
Preliminary Exam - Spring 1995
Problem 1 For each positive integer n, dene fn : R R by fn (x) =
cos nx. Prove that the sequence of functions cfw_fn has no uniformly convergent subsequence.
Problem 2 Let A be the 33 matrix
1 1
0
1
2 1
0 1
1
Determine al
Preliminary Exam - Spring 1994
Problem 1 Let the collection U of open subsets of R cover the interval [0, 1].
Prove that there is a positive number such that any two points x and y of
[0, 1] satisfying |x y| < belong together to some member of the cover U
Preliminary Exam - Spring 1998
Problem 1 Prove that the polynomial z 4 + z 3 + 1 has exactly one root in the
quadrant cfw_z = x + iy | x, y > 0.
Problem 2 Let f be analytic in an open set containing the closed unit disc.
Suppose that |f (z)| > m for |z| =
Preliminary Exam - Spring 1989
Problem 1 Let a1 , a2 , . . . be positive numbers such that
an < .
n=1
Prove that there are positive numbers c1 , c2 , . . . such that
lim cn =
n
cn an < .
and
n=1
Problem 2 Let F be a eld, n and m positive integers, and A
Preliminary Exam - Spring 2001
Problem 1 Let F be a nite eld of order q, and let V be a two dimensional
vector space over F. Find the number of endomorphisms of V that x at least
one nonzero vector.
Problem 2 Let the continuous function f : R R be periodi
Preliminary Exam - Spring 2000
Problem 1 Are the 44
1 0 0
0 1 0
A=
0 0 0
0 0 0
matrices
0
0
and
1
0
1
1
B=
1
1
0
1
0
0
0 0
1 1
0 0
1 0
similar?
Problem 2 Let cfw_fn be a uniformly bounded equicontinuous sequence of
n=1
real-valued functions on the c
Preliminary Exam - Fall 1980
Problem 1 Dene
cos x
2 +xt)
e(t
F (x) =
dt.
sin x
Compute F (0).
Problem 2 Are the matrices give below similar ?
1 0 0
1 1 0
A = 1 1 1
and
B = 0 1 0
1 0 2
0 0 2
Problem 3 Do there exist functions f (z) and g(z) that are analy
Preliminary Exam - Fall 1983
Problem 1 Evaluate
(sech x)2 cos x dx
0
where is a real constant and
sech x =
ex
2
+ ex
Problem 2 Let Mnn (F) be the ring of nn matrices over a eld F. For
n
1 does there exist a ring homomorphism from M(n+1)(n+1) (F) onto
Mnn
Preliminary Exam - Fall 1981
Problem 1 Evaluate the integral
cos x
dx .
1 + x4
Problem 2 Consider an autonomous system of dierential equations
dxi
= Fi (x1 , . . . , xn ),
dt
where F = (F1 , . . . , Fn ) : Rn Rn is a C 1 vector eld.
1. Let U and V be two
1A
Find a sequence rn of positive rational numbers such that, rn
n=0
converges and for any prime p and any positive integer m, pm divides
the numerator of sk sj (written in lowest terms) for k and j suciently
large, where sk = k rn .
n=0
Solution. Take
n!
Problem 1A.
Find the volume of the solid given by x2 + z 2 1, y 2 + z 2 1. (Hint:
Solution: The volume is
1
1
4xydz where x = y =
1
(something)dz.)
1
1 z 2 . This integral has value 16/3.
Problem 2A.
Let f (x) be a irreducible polynomial over the rational
Problem 1A.
Score:
Let
. . . X 2 X1
be a nested sequence of closed nonempty connected subsets of a compact metric space X.
Prove that Xi is nonempty and connected.
i=1
Solution:
Since Xi is closed in X, it is compact. The intersection of a nested sequence
Preliminary Exam - Fall 1977
Problem 1 Let
A=
7 15
.
2 4
Find a real matrix B such that B 1 AB is diagonal.
Problem 2
1. Using only the axioms for a eld F, prove that a system
of m homogeneous linear equations in n unknowns withm < n and
coecients in F ha
PRELIM EXAM. Fall2009
Problem 1A.
Let p(k) be a degree n polynomial with complex coecients dened by
p(k) = p0 +
k
k
k
p1 +
p2 + +
pn .
1
2
n
Dene
p(k)z k .
f (z) =
k=0
Find the radius of convergence of the power series, prove that f (z) is a rational func
Preliminary Exam - Fall 1987
Problem 1 Prove that (cos )p
cos(p ) for 0
/2 and 0 < p < 1.
Problem 2 Suppose that cfw_fn is a sequence of nondecreasing functions which
map the unit interval into itself. Suppose that
lim fn (x) = f (x)
n
pointwise and that