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Preliminary Exam - Fall 1993
Problem 1
Let
![$X$](img1-42.gif)
be a metric space and
![$(x_{n})$](img2-42.gif)
a convergent sequence in
![$X$](img3-42.gif)
with limit
![$x_{0}$](img4-42.gif)
. Prove
that the set
![$C = \{x_0, x_1, x_2,...\}$](img5-42.gif)
is compact.
Problem 2
Let
![$A$](img6-42.gif)
be the additive group of rational
numbers, and let
![$M$](img7-42.gif)
be the multiplicative group
of positive rational numbers. Determine all homomorphisms
of
![$A$](img8-42.gif)
into
![$M$](img9-42.gif)
.
Problem 3
Describe the region in the complex plane where the
infinite series
converges. Draw a sketch of the region.
Problem 4
Let
![% latex2html id marker 708
$\mbox{\bf {F}}$](img11-42.gif)
be a field. For
![$m$](img12-42.gif)
and
![$n$](img13-42.gif)
positive integers, let
![$M_{m\times n}$](img14-42.gif)
be the vector space of
![$m \times n$](img15-42.gif)
matrices over
![% latex2html id marker 718
$\mbox{\bf {F}}$](img16-42.gif)
. Fix
![$m$](img17-42.gif)
and
![$n$](img18-42.gif)
, and fix
matrices
![$A$](img19-42.gif)
and
![$B$](img20-42.gif)
in
![$M_{m\times n}$](img21-42.gif)
. Define the
linear transformation
![$T$](img22-42.gif)
from
![$M_{n\times m}$](img23-42.gif)
to
![$M_{m\times n}$](img24-42.gif)
by
Prove that if
, then
is not invertible.
Problem 5
Let the function
![$x(t) \; (-\infty < t < \infty)$](img28-42.gif)
be a
solution of the differential equation
such that
![$x(0) = x(1) = 0$](img30-42.gif)
. (Here,
![$b$](img31-42.gif)
and
![$c$](img32-42.gif)
are
real constants.) Prove that
![$x(n) =0$](img33-42.gif)
for every integer
![$n$](img34-42.gif)
.
Problem 6
Let
![$A$](img35-42.gif)
,
![$B$](img36-42.gif)
, and
![$C$](img37-42.gif)
be finite abelian groups.
Prove that if
![$A \times B$](img38-42.gif)
is isomorphic to
![$A \times C$](img39-42.gif)
,
then
![$B$](img40-42.gif)
is isomorphic to
![$C$](img41-42.gif)
.
Problem 7
Let
![$(n \geq 2)$](img43-42.gif)
be the space of real
n
![$\times$](img44-42.gif)
n matrices, identified in the usual way with
the Euclidean space
![% latex2html id marker 812
$\mathbb{R}^{n^{2}}$](img45-42.gif)
. Let
![$F$](img46-42.gif)
be the
determinant map of
![$M_{n \times n}$](img47-42.gif)
into
![% latex2html id marker 818
$\mathbb{R}$](img48-42.gif)
:
![$F(X)=\det (X)$](img49-42.gif)
.
Find all of the
critical points of
![$F$](img50-42.gif)
; that is,
all matrices
![$X$](img51-42.gif)
such that
![$DF(X) = 0$](img52-42.gif)
. _function,>critical points
Problem 8
Prove that if
![$A$](img53-42.gif)
is an
![$n \times n$](img54-42.gif)
matrix
over
![% latex2html id marker 845
$\mathbb{C}$](img55-42.gif)
, and if
![$A^k = I$](img56-42.gif)
for some positive
integer
![$k$](img57-42.gif)
, then
![$A$](img58-42.gif)
is diagonalizable.
Problem 10
Let
![$f$](img60-42.gif)
be a continuous real valued function on
![$[0, \infty)$](img61-42.gif)
. Let
![$A$](img62-42.gif)
be the set of real numbers
![$a$](img63-42.gif)
that can be expressed as
![$a = \lim_{n \to \infty} f(x_{n})$](img64-42.gif)
for some
sequence
![$(x_{n})$](img65-42.gif)
in
![$[0, \infty)$](img66-42.gif)
such that
![$\lim_{n \to \infty} x_n = \infty$](img67-42.gif)
.
Prove that if
![$A$](img68-42.gif)
contains the two numbers
![$a$](img69-42.gif)
and
![$b$](img70-42.gif)
,
then it contains the entire interval with endpoints
![$a$](img71-42.gif)
and
![$b$](img72-42.gif)
.
Problem 11
Let
![$R$](img73-42.gif)
be a commutative ring with identity.
Let
![$G$](img74-41.gif)
be a finite subgroup of
![$R^*$](img75-41.gif)
, the group of
units of
![$R$](img76-41.gif)
. Prove that if
![$R$](img77-41.gif)
is an integral domain,
then
![$G$](img78-41.gif)
is cyclic.
Problem 12
Evaluate
![$\frac{1}{2\pi i} \int_{\gamma} f(z) \ dz$](img79-41.gif)
for the function
![$f(z) = z^{-2}(1 - z^{2})^{-1}e^z$](img80-41.gif)
and the curve
![$\gamma$](img81-40.gif)
depicted by
file=../Fig/Pr/Fa93-12,width=3.7in
Problem 13
Show that there are at least two nonisomorphic
nonabelian groups of order
![$40$](img82-40.gif)
.
Problem 14
Let
![$n$](img83-40.gif)
be an integer larger than
![$1$](img84-40.gif)
. Is
there a differentiable function on
![$[0,\infty)$](img85-39.gif)
whose
derivative equals its
![$n^{th}$](img86-39.gif)
power and whose value
at the origin is positive?
Problem 15
Prove that the matrix
has two positive and two negative eigenvalues (taking
into account multiplicities).
Problem 16
Let
![$K$](img88-38.gif)
be a continuous real valued function
on
![$[0,1] \times [0,1]$](img89-38.gif)
. Let
![$F$](img90-38.gif)
be the family of
functions
![$f$](img91-38.gif)
on
![$[0,1]$](img92-37.gif)
of the form
with
![$g$](img94-37.gif)
a real valued continuous function on
![$[0,1]$](img95-36.gif)
satisfying
![$\vert g\vert \leq 1$](img96-36.gif)
everywhere. Prove that the
family
![$F$](img97-36.gif)
is equicontinuous.
Problem 18
Let
![$f$](img117-15.gif)
be a continuous real valued function
on
![$[0,1]$](img118-13.gif)
, and let the function
![$h$](img119-13.gif)
in the complex
plane be defined by
- Prove that
is analytic in the entire plane.
- Prove that
is the zero function only if
is the zero function.
Previous: Spring93
Next: Spring94
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10