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Preliminary Exam - Fall 1992
Problem 1
Are the matrices
and
similar?
Problem 2
Let
denote the ideal in
, the ring of
polynomials with coefficients in
, generated by
and
.
Is
a prime ideal?
Problem 3
- How many roots does the polynomial
have in the disc ?
- How many roots does the same polynomial have on the real axis?
Problem 4
Let the real valued function
solve the initial value problem
Prove that there is exactly one
such that
.
Problem 5
Let
be a group and
and
subgroups such that
has a finite index in
. Prove that
has a finite index in
.
Problem 6
Let
and
be metric spaces and
a continuous surjective map such that
for every pair of points
in
.
- If is complete, must be complete? Give a proof or a
counterexample.
- If is complete, must be complete? Give a proof or a
counterexample.
Problem 7
Evaluate
where
is the unit circle with counterclockwise orientation.
Problem 9
Let the function
be analytic in the region
of the complex plane. Prove that if
is real valued on the interval
of the real axis, then
is also real valued on the interval
.
Problem 10
How many Sylow
-subgroups does the dihedral group
of
order
have, when
is odd?
Problem 11
Let
be a finite-dimensional vector space over a field
,
and let
be a bilinear map (not necessarily symmetric).
Define the subspaces
and
by
Prove that
.
Problem 12
Let
be a sequence of real valued
functions on
such that, for all n,
Prove that the sequence has a subsequence that converges uniformly on .
Problem 14
Let
and
be positive continuous functions on
,
with
everywhere. Assume the initial value problem
has a solution defined on all of
. Prove that the initial value problem
also has a solution defined on all of
.
Problem 15
Let
be the group of all real 2
2 matrices of the
form
with
. Let
be the subgroup of those matrices in
having
.
(a) Prove that is a normal subgroup of and that is isomorphic to
.
(b) Find a proper normal subgroup of that contains properly.
Problem 16
Let f be a
function of
into
such that
has rank
everywhere. Assume
is proper (which, by
definition, means
is compact whenever
is compact). Prove that
.
Problem 17
Let
be a real number, and let the function
be
defined in
by
Prove that
is a harmonic function.
Problem 18
Let
be a positive integer. Determine those real numbers
for which every sequence
of real numbers satisfying the
recurrence relation
has period k (i.e.,
for all n).
Previous: Spring92
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Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10