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Preliminary Exam - Fall 1981
Problem 1
Evaluate the integral
Problem 2
Consider an autonomous system of differential equations
where
is a
vector field.
- Let and be two solutions on . Assuming that
for all x, z in
, show that
is a decreasing function of .
- Let be a solution defined for . Assuming that
show that there exists
such that
Problem 3
Let
be the group of all permutations of
objects
and let
be a subgroup of
of order
, where
is
a prime not dividing
. Show that
has a
fixed point; that is,
one of the objects is left fixed by every element of
.
_group,>fixed point
Problem 5
The Fibonacci numbers
are defined recursively by
,
_Fibonacci numbers
, and
for
. Show that
exists, and evaluate the limit.
Note: See also Problem .
Problem 6
Let
and
be continuous functions on
such that
,
, for all
.
Prove that
Problem 7
Find a specific polynomial with rational coefficients having
as a root.
Problem 8
- How many zeros does the function
have inside
the unit circle (counting multiplicities)?
- Are the zeros distinct?
Problem 9
Let
be the vector space of all real 2
2 matrices. Let
and define a linear transformation
by
L(X) = AXB. Compute the trace and the determinant of
.
Problem 10
Let
be an
matrix whose entries
are
real valued differentiable functions defined on
. Assume that the
determinant
of
is everywhere positive. Let
be the inverse matrix of
. Prove the formula
Problem 11
Consider the complex 3
3 matrix
where
.
- Show that
, where
- Use Part 1 to find the complex eigenvalues of .
- Generalize Parts 1 and 2 to nn matrices.
Problem 12
Let a, b be real constants and let
Show that u is harmonic and find an entire function f(z)
whose real part is u.
Correction: cannot be the real part of an entire function.
Why? Change slightly and do the problem.
Problem 13
Let
be a real valued function on
of class
. A point
is a critical point of f if all the partial
derivatives of f vanish at x; a critical point is nondegenerate
if the
matrix
is nonsingular.
Let x be a nondegenerate critical point of f. Prove that there
is an open neighborhood of x which contains no other critical
points (i.e., the nondegenerate critical points are isolated).
Problem 14
Let
be a
function and consider the system
of second order differential equations
where
Let
be a solution of this system
on a finite interval
.
- Show that the function
is constant for .
- Assuming that
for all
,
show that , , and are bounded on ,
and then prove all three limits
exist.
Problem 15
Let
be a holomorphic map of the unit disc
into itself, which is not the identity map
. Show that
can
have, at most, one fixed point.
Problem 16
Let
be a group with three normal subgroups
. Suppose
and
for all
with
. Show that
is abelian
and
is isomorphic to
for all
.
Problem 17
Let
be a continuous function on
. Evaluate the following
limits.
-
-
Problem 18
Let
and
be two real
matrices. Suppose there is
a complex invertible
matrix
such that
.
Show that there is a real invertible
matrix
such
that
. (In other words, if two real matrices are
similar over
, then they are similar over
.)
Problem 19
Either prove or disprove (by a counterexample) each of the
following statements:
- Let
,
be such that
Then
- If
is continuous and is an open
set in , then is an open set in .
- Let f be of class on the interval
. Suppose that
for all
and all x in the interval. Then f is real analytic;
that is, it has a convergent power series expansion in a neighborhood
of each point of the interval. _function,>real analytic
Problem 20
Let
be a group of order
which has a normal subgroup
of order
. Prove that
is abelian.
Previous: Summer81
Next: Spring82
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10