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Preliminary Exam - Spring 1983
Problem 1
Let
be a monotone decreasing function, defined
on the positive real numbers with
Show that
Problem 2
Let
be an n
n real matrix satisfying the
conditions:
Show that
.
Problem 3
A fractional linear transformation maps the annulus
(where
) onto the domain bounded by the two circles
and
. Find
.
Problem 4
In the triangular network in
depicted below, the
points
,
,
, and
are respectively
,
,
, and
. Describe the structure of the group of all
Euclidean transformations of
which leave this network invariant.
file=../Fig/Pr/Sp83-4,width=2.7in
Problem 5
Find all solutions
to
Problem 6
Suppose that
is a continuous function on
which is periodic
with period
, i.e.,
. Show:
- The function is bounded above and below and achieves
its maximum and minimum.
- The function is uniformly continuous on .
- There exists a real number such that
Problem 7
Let
be the group of integers
, under addition, where
is a prime
number. Suppose that
is an integer satisfying
,
and let
be the group
(
factors).
Show that
has no automorphism of order
.
Problem 8
Suppose that
is an integer. Prove that the sum
is not an integer.
Problem 9
Suppose that
is continuous and satisfies
for all
and some
.
Prove that
is one-to-one, onto, and has a continuous inverse.
Note: See also Problem .
Problem 10
Evaluate
and justify your calculations.
Problem 11
Let
be an invertible real
matrix. Show that there is
a decomposition
in which
is an
real orthogonal
matrix and
is upper-triangular with positive diagonal entries.
Is this decomposition unique?
Problem 12
Determine all the complex analytic functions
defined on the unit
disc
which satisfy
for
.
Problem 13
Let
be real numbers. Show
that the infinite series
converges uniformly over
to a continuous limit function
. Show, further, that the limit
exists.
Problem 14
Let
be an abelian group which is generated by, at most,
elements.
Show that each subgroup of
is again generated by, at most,
elements.
Problem 15
Let
and
be complex polynomials with the degree of
at least
two more than the degree of
. Show that there is a positive number
such that
for each simple closed curve
which does not intersect
.
Problem 16
Let
be a real vector space of dimension
, and let
be a nondegenerate bilinear form. Suppose that
is a linear subspace of
such that the restriction of
to
is identically
. Show that
we have
.
Problem 18
Let
be the field with seven elements. How many 3
3 matrices
with coefficients in
have determinant
? How many have determinant
?
Problem 19
Show that the initial value problem
has a solution defined on all of
.
Problem 20
Show that the interval
cannot be written as a countably
infinite disjoint union of closed subintervals of
.
Previous: Fall82
Next: Summer83
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10