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Preliminary Exam - Spring 1989
Problem 1
Let
![$a_1, a_2, \ldots$](img1-32.gif)
be positive numbers such that
Prove that there are positive numbers
![$c_1, c_2, \ldots$](img3-32.gif)
such that
Problem 2
Let
![% latex2html id marker 679
$\mbox{\bf {F}}$](img5-32.gif)
be a field,
![$n$](img6-32.gif)
and
![$m$](img7-32.gif)
positive integers, and
![$A$](img8-32.gif)
an
![$n\times n$](img9-32.gif)
matrix with entries in
![% latex2html id marker 689
$\mbox{\bf {F}}$](img10-32.gif)
such that
![$A^m = 0$](img11-32.gif)
.
Prove that
![$A^n = 0$](img12-32.gif)
.
Problem 3
Let f be a continuous real valued function on
![$[0,1]\times[0,1]$](img13-32.gif)
.
Let the function g on
![$[0,1]$](img14-32.gif)
be defined by
Prove that g is continuous.
Problem 4
Prove that if
![$1 < \lambda < \infty$](img16-32.gif)
, the function
has only one zero in the half-plane
![$\Re z<0$](img18-32.gif)
, and this zero is on the real axis.
Problem 6
Let
![$S$](img34-32.gif)
be the subspace of
![$M_{n\times n}$](img35-32.gif)
(the vector space of all real
![$n\times n$](img36-32.gif)
matrices) generated by all matrices of the form
![$AB - BA$](img37-32.gif)
with
![$A$](img38-32.gif)
and
![$B$](img39-32.gif)
in
![$M_{n\times n}$](img40-32.gif)
. Prove that
![$\dim(S) = n^2-1$](img41-32.gif)
.
Problem 7
Let
Find the general solution of the matrix differential equation
![$dX/dt=AXB$](img43-32.gif)
for the unknown
4
![$\times$](img44-32.gif)
4 matrix function
![$X(t)$](img45-32.gif)
.
Problem 8
Let
![$f$](img46-32.gif)
be an analytic function mapping the open unit disc,
![$D$](img47-32.gif)
,
into itself. Assume that there are two points
![$a$](img48-32.gif)
and
![$b$](img49-32.gif)
in
![$D$](img50-32.gif)
,
with
![$a \neq b$](img51-32.gif)
, such that
![$f(a) = a$](img52-32.gif)
and
![$f(b) = b$](img53-32.gif)
. Prove
that
![$f$](img54-32.gif)
is the identity function (
![$f(z)=z$](img55-32.gif)
).
Problem 9
For
![$G$](img56-32.gif)
a group and
![$H$](img57-32.gif)
a subgroup, let
![$C(G,H)$](img58-32.gif)
denote the collection
of left cosets of
![$H$](img59-32.gif)
in
![$G$](img60-32.gif)
. Prove that if
![$H$](img61-32.gif)
and
![$K$](img62-32.gif)
are two subgroups
of
![$G$](img63-32.gif)
of infinite index, then
![$G$](img64-32.gif)
is not a finite union of cosets
from
![$C(G,H) \cup C(G,K)$](img65-32.gif)
.
Problem 10
Let the real
![$2n\times 2n$](img66-32.gif)
matrix
![$X$](img67-32.gif)
have the form
where
![$A$](img69-32.gif)
,
![$B$](img70-32.gif)
,
![$C$](img71-32.gif)
, and
![$D$](img72-32.gif)
are
![$n\times n$](img73-32.gif)
matrices that commute with one
another. Prove that
![$X$](img74-31.gif)
is invertible if and only if
![$AD - BC$](img75-31.gif)
is invertible.
Problem 11
Let
![$R$](img76-31.gif)
be a ring with at least two elements. Suppose that for each nonzero
![$a$](img77-31.gif)
in
![$R$](img78-31.gif)
there is a unique
![$b$](img79-31.gif)
in
![$R$](img80-31.gif)
(depending on
![$a$](img81-31.gif)
) with
![$aba = a$](img82-31.gif)
.
Show that
![$R$](img83-31.gif)
is a division ring.
Problem 12
Evaluate
where
![$C$](img85-31.gif)
is the circle
![$\vert z\vert = 2$](img86-31.gif)
with counterclockwise orientation.
Problem 13
Let
![$D_n$](img87-31.gif)
be the dihedral group, the group of rigid motions of a regular
n-gon
![$(n\geq 3)$](img88-30.gif)
. (It is a noncommutative group of order
![$2n$](img89-30.gif)
.) Determine
its center
![$C=\{c \in D_n \;\vert\; cx = xc \; for\; all\; x \in D_n\}$](img90-30.gif)
.
Problem 14
Suppose
![$f$](img91-30.gif)
is a continuously differentiable map of
![% latex2html id marker 953
$\mathbb{R}^{2}$](img92-30.gif)
into
![% latex2html id marker 955
$\mathbb{R}^{2}$](img93-30.gif)
.
Assume that
![$f$](img94-30.gif)
has only finitely many singular points, and that for
each positive number
![$M$](img95-30.gif)
, the set
![% latex2html id marker 963
$\{z \in \mbox{$\mathbb{R}^{2}$} \;\vert\; \vert f(z)\vert \leq M\}$](img96-30.gif)
is
bounded. Prove that
![$f$](img97-30.gif)
maps
![% latex2html id marker 967
$\mathbb{R}^{2}$](img98-30.gif)
onto
![% latex2html id marker 969
$\mathbb{R}^{2}$](img99-29.gif)
.
Problem 15
Let
![${B} =(b_{ij})_{i,j=1}^{20}$](img100-28.gif)
be a real
![$20\times 20$](img101-28.gif)
matrix such that
Prove that
![${B}$](img104-25.gif)
is nonsingular.
Problem 16
Let
![$f$](img105-24.gif)
and
![$g$](img106-23.gif)
be entire functions such that
![${\displaystyle{\lim_{z\to\infty}}} f(g(z))=\infty$](img107-22.gif)
.
Prove that
![$f$](img108-22.gif)
and
![$g$](img109-22.gif)
are polynomials.
Problem 18
Let
![$f$](img126-9.gif)
be a real valued function on
![% latex2html id marker 1092
$\mathbb{R}^{2}$](img127-9.gif)
with the following properties:
- For each
in
, the function
is
continuous.
- For each
in
, the function
is
continuous.
is compact whenever
is a compact subset of
.
Prove that
![$f$](img137-7.gif)
is continuous.
Previous: Fall88
Next: Fall89
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10