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Preliminary Exam - Spring 1997
Problem 1
For which values of the exponents
does the
following series converge?
Problem 2
Let
be a metric space with metric
. Let
be a
nonempty closed subset of
. Define
by
Show that
is continuous, and that
if and
only if
Problem 3
Suppose that
is continuous, nonnegative for
,
with
Prove that
Problem 4
Let
and
be two entire functions such that, for all
,
for some real constant
(independent of
). Show that there are constants
such that
Problem 5
Prove that
is independent of the real parameter
.
Problem 6
Suppose that
is a topological space and
is a finite-dimensional
subspace of the vector space of continuous real valued functions on
.
Prove that there exist a basis
for
and points
in
such that
.
Problem 8
Classify all abelian groups of order
up to isomorphism.
Problem 9
Let
be the ring of
matrices over a field. Suppose
is a ring and
is a homomorphism. Show that
is either
injective or zero.
Problem 10
Let
be a bounded function (i.e., there is a constant
such that
for all
). Suppose the graph
of
is a closed subset of
. Prove that
is continuous.
Problem 11
Suppose that
for all
, and that
. Show that
as
.
Problem 12
Evaluate the integral
Problem 13
Suppose that
is injective and everywhere holomorphic.
Prove that there exist
with
such that
for all
Problem 14
Show that
for any complex
matrix
, where
is defined
as in Problem
.
Problem 15
Suppose that
and
are
matrices such that
,
, and
is invertible. Show that
and
have the same rank.
Problem 16
Suppose that
is a commutative algebra with identity over
(i.e.,
is a commutative ring containing
as a subring with
identity). Suppose further that
for all nonzero
elements
. Show that if the dimension of
as a vector space
over
is finite and at least two, then the equations
is
satisfied by at least three distinct elements
.
Problem 17
Let
denote the multiplicative group of invertible
matrices over the ring of integers modulo
. Find the order of
for each prime
and positive integer
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Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10