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Preliminary Exam - Spring 1979
Problem 1
Let
![% latex2html id marker 690
$f:\mbox{$\mathbb{R}^{n}$}\setminus \{0\} \to \mbox{$\mathbb{R}^{}$}$](img1-6.gif)
be differentiable. Suppose
exists for each
![$j = 1, \ldots, n$](img3-6.gif)
.
- Can
be extended to a continuous map from
to
?
- Assuming continuity at the origin, is
differentiable
from
to
?
Problem 3
Let
![$S_7$](img19-6.gif)
be the group of permutations of a set of seven elements.
Find all
![$n$](img20-6.gif)
such that some element of
![$S_7$](img21-6.gif)
has order
![$n$](img22-6.gif)
.
Problem 4
Prove that every compact metric space has a countable dense subset.
Problem 5
Find all solutions to the differential equation
Problem 6
Prove that if
![$1 < \lambda < \infty$](img24-6.gif)
, the function
has only one zero in the half-plane
![$\Re z<0$](img26-6.gif)
, and that this zero is real.
Problem 8
Let
![$M$](img28-6.gif)
be a real nonsingular 3
![$\times$](img29-6.gif)
3 matrix. Prove there are
real matrices
![$S$](img30-6.gif)
and
![$U$](img31-6.gif)
such that
![$M = SU = US$](img32-6.gif)
, all the eigenvalues
of
![$U$](img33-6.gif)
equal
![$1$](img34-6.gif)
, and
![$S$](img35-6.gif)
is diagonalizable over
![% latex2html id marker 825
$\mathbb{C}\,^{}$](img36-6.gif)
.
Problem 9
Let
![$M$](img37-6.gif)
be an
![$n\times n$](img38-6.gif)
complex matrix. Let
![$G_M$](img39-6.gif)
be the set of complex
numbers
![$\lambda$](img40-6.gif)
such that the matrix
![$\lambda M$](img41-6.gif)
is similar to
![$M$](img42-6.gif)
.
- What is
if
- Assume
is not nilpotent. Prove
is finite.
Problem 10
Let
![$f(x)$](img47-6.gif)
be a polynomial over
![% latex2html id marker 878
$\mbox{$\mathbb{Z}^{}$}_p$](img48-6.gif)
, the field of integers
![$\bmod p$](img49-6.gif)
. Let
![$g(x)=x^p-x$](img50-6.gif)
. Show that the greatest
common divisor of
![$f(x)$](img51-6.gif)
and
![$g(x)$](img52-6.gif)
is the product of the distinct
linear factors of
![$f(x)$](img53-6.gif)
.
Problem 11
Let
![$S$](img54-6.gif)
be a collection of abelian groups, each of order
![$720$](img55-6.gif)
, no two
of which are isomorphic. What is the maximum cardinality
![$S$](img56-6.gif)
can have?
Problem 12
Let
![$G$](img57-6.gif)
be a group with three normal subgroups
![$N_1$](img58-6.gif)
,
![$N_2$](img59-6.gif)
, and
![$N_3$](img60-6.gif)
. Suppose
![$N_i \cap N_j$](img61-6.gif)
= {e} and
![$N_i N_j = G$](img62-6.gif)
for all
![$i,j$](img63-6.gif)
with
![$i\neq j$](img64-6.gif)
. Show that
![$G$](img65-6.gif)
is abelian and
![$N_i$](img66-6.gif)
is isomorphic to
![$N_j$](img67-6.gif)
for all
![$i,j$](img68-6.gif)
.
Problem 13
Consider the system of differential equations:
Prove there exists a solution defined for all
![$t \in [0,1]$](img70-6.gif)
,
such that
and also
Problem 15
Which of the following matrices are similar as matrices over
![% latex2html id marker 1059
$\mathbb{R}^{}$](img83-6.gif)
?
Problem 16
For which
![% latex2html id marker 1081
$z \in \mbox{$\mathbb{C}\,^{}$}$](img86-6.gif)
does
converge?
Problem 17
Let
![$P$](img88-5.gif)
and
![$Q$](img89-5.gif)
be complex polynomials with the degree of
![$Q$](img90-5.gif)
at least two
more than the degree of
![$P$](img91-5.gif)
. Prove there is an
![$r>0$](img92-5.gif)
such that if
![$C$](img93-5.gif)
is a closed curve outside
![$\vert z\vert=r$](img94-5.gif)
, then
Problem 18
Show that for any continuous function
![% latex2html id marker 1130
$f:[0,1] \to
\mbox{$\mathbb{R}^{}$}$](img96-5.gif)
and
![$\varepsilon > 0$](img97-5.gif)
, there is a function of the form
for some
![% latex2html id marker 1136
$n \in \mbox{$\mathbb{Z}^{}$}$](img99-5.gif)
, where
![% latex2html id marker 1140
$C_0,\ldots, C_n \in \mbox{$\mathbb{Q}\,^{}$}$](img100-5.gif)
and
![$\vert g(x) - f(x)\vert<\varepsilon$](img101-5.gif)
for all x in
![$[0,1]$](img102-5.gif)
.
Problem 19
Let
![$P$](img103-3.gif)
be a 9
![$\times$](img104-3.gif)
9 real matrix such that
![$x^{t}Py = - y^{t}Px$](img105-2.gif)
for all column vectors
![$x,y$](img106-2.gif)
in
![% latex2html id marker 1170
$\mathbb{R}^{9}$](img107-2.gif)
.
Prove that
![$P$](img108-2.gif)
is singular.
Previous: Fall78
Next: Summer79
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10