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Preliminary Exam - Spring 1978
Problem 1
Let
be an integer and define a sequence of maps
For which values of
does the sequence converge uniformly on
? On every bounded subset of
?
Problem 2
Prove that a map
is continuous only if its graph
is closed in
. Is the converse true?
Note: See also Problem .
Problem 3
Let
be a nonconstant entire function.
Prove that
is dense in
.
Problem 5
let
denote the ring of integers modulo n. Let
be the ring of polynomials with coefficients in
. Let
denote the ideal in
generated by
.
- For which values of n,
, is the quotient
ring
a field?
- Give the multiplication table for
.
Problem 6
Prove that the sum of two algebraic numbers is algebraic. (An
algebraic number is a complex number which is a root of a polynomial
with rational coefficients.)
Problem 7
What is the volume enclosed by the ellipsoid
Problem 9
Determine the Jordan Canonical Form of the matrix
Problem 10
Suppose
is a real
matrix.
- Is it true that must commute with its transpose?
- Suppose the columns of (considered as vectors) form
an orthonormal set; is it true that the rows of must also
form an orthonormal set?
Problem 11
Show that there is a complex analytic function defined on the set
whose derivative is
Is there a complex analytic function on
whose derivative is
Problem 12
Prove that the uniform limit of a sequence of complex analytic
functions is complex analytic. Is the analogous theorem true for
real analytic functions?
Problem 13
- For which real numbers does the differential
equation
have a solution on some interval ?
- For which values of are there intervals on which
two solutions of are defined?
Problem 14
Let
be a group of order
which has a normal subgroup of order
.
Prove that
is abelian.
Problem 15
Is
irreducible over the field of real numbers? The field
of rational numbers? A field with
elements?
Problem 16
Let
and
denote real n
n symmetric matrices such that
.
Prove that
and
have a common eigenvector in
.
Problem 17
Evaluate
where
.
Problem 18
Let
be a matrix with entries in a field
. The
row rank of
over
is the maximal number of rows which are linearly independent
(as vectors) over
. The
column rank is similarly defined using
columns instead of rows. _matrix,>row rank _matrix,>column rank
- Prove row rank = column rank.
- Find a maximal linearly independent set of columns of
taking
.
- If
is a subfield of
, and has entries
in
, how is the row rank of over
related to the
row rank of over
?
Problem 19
Let
be Riemann integrable over
for
all b such that
.
- If f is bounded, prove that f is Riemann integrable over
.
- What if f is not bounded?
Previous: Fall77
Next: Summer78
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10