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Preliminary Exam - Spring 1994
Problem 1
Let the collection
![$\cal U$](img1-43.gif)
of open subsets of
![% latex2html id marker 658
$\mathbb{R}$](img2-43.gif)
cover the
interval
![$[0,1]$](img3-43.gif)
. Prove that there is a positive number
![$\delta$](img4-43.gif)
such that any two points
![$x$](img5-43.gif)
and
![$y$](img6-43.gif)
of
![$[0,1]$](img7-43.gif)
satisfying
![$\vert x-y\vert < \delta$](img8-43.gif)
belong together to some member of the cover
![$\cal U$](img9-43.gif)
.
Problem 4
Let
![$G$](img21-43.gif)
be a group having a subgroup
![$A$](img22-43.gif)
of finite
index. Prove that there is a normal subgroup
![$N$](img23-43.gif)
of
![$G$](img24-43.gif)
contained in
![$A$](img25-43.gif)
such that
![$N$](img26-43.gif)
is of finite index in
![$G$](img27-43.gif)
.
Problem 6
Prove or disprove: A square complex matrix,
![$A$](img33-43.gif)
, is similar to its
transpose,
![$A^{t}$](img34-43.gif)
.
Problem 7
Let
![$a_{1}, a_{2}, \ldots , a_n$](img35-43.gif)
be complex numbers. Prove that
there is a point
![$x$](img36-43.gif)
in
![$[0,1]$](img37-43.gif)
such that
Problem 8
Find all automorphisms of
![% latex2html id marker 789
$\mathbb{Z}[x]$](img39-43.gif)
, the ring of
polynomials over
![% latex2html id marker 791
$\mathbb{Z}$](img40-43.gif)
.
Problem 9
- Let
and
be open connected subsets of
the complex plane, and let
be an analytic function
in
such that
. Assume
is
compact whenever
is a compact subset of
. Prove
that
.
- Prove that the last equality can fail if analytic
is replaced by continuous in the preceding statement.
Problem 10
Let f be a continuous real valued function on
![% latex2html id marker 840
$\mathbb{R}$](img50-43.gif)
such that the improper Riemann integral
![$\int^{\infty}_{-\infty} \, \vert f(x)\vert \, dx$](img51-43.gif)
converges. Define the function g on
![% latex2html id marker 844
$\mathbb{R}$](img52-43.gif)
by
Prove that g is continuous.
Problem 11
Let
![% latex2html id marker 858
$T: \mathbb{R}^{n} \to \mathbb{R}^n$](img54-43.gif)
be a diagonalizable
linear transformation. Prove that there is an orthonormal basis
for
![% latex2html id marker 860
$\mathbb{R}^n$](img55-43.gif)
with respect to which
![$T$](img56-43.gif)
has an upper-triangular
matrix.
Problem 12
Let
![$f = u + iv$](img57-43.gif)
and
![$g = p + iq$](img58-43.gif)
be analytic functions
defined in a neighborhood of the origin in the complex
plane. Assume
![$\vert g^{\prime}(0)\vert \, < \, \vert f^{\prime}(0)\vert$](img59-43.gif)
. Prove that there is
a neighborhood of the origin in which the function
![$h = f + \overline{g}$](img60-43.gif)
is one-to-one.
Problem 13
Let
![% latex2html id marker 913
$\mbox{\bf {F}}$](img61-43.gif)
be a finite field with q elements.
Say that a function
![% latex2html id marker 915
$f: \mbox{\bf {F}} \to \mbox{\bf {F}}$](img62-43.gif)
is a
polynomial
function if there are elements
![$a_{0},a_{1}, \ldots ,a_n$](img63-43.gif)
of
![% latex2html id marker 919
$\mbox{\bf {F}}$](img64-43.gif)
such
that
![$f(x) = a_{0} + a_{1}x + \cdots + a_{n}x^n$](img65-43.gif)
for all
![% latex2html id marker 923
$x \in \mbox{\bf {F}}$](img66-43.gif)
.
How many polynomial functions are there? _function,>polynomial
Problem 14
Let
![$W$](img67-43.gif)
be a real 3
![$\times$](img68-43.gif)
3 antisymmetric matrix, i.e.,
![$W^{t} = - W$](img69-43.gif)
. Let the function
be a real solution of the vector differential equation
![${dX}/{dt} = WX$](img71-43.gif)
.
- Prove that
, the Euclidean norm of
, is
independent of
.
- Prove that if
is a vector in the null space of
,
then
is independent of
.
- Prove that the values
all lie on a fixed circle in
.
Problem 15
Let
![$\alpha$](img81-41.gif)
be a number in
![$(0,1)$](img82-41.gif)
. Prove that any
sequence
![$(x_{n})^{\infty}_{0}$](img83-41.gif)
of real
numbers satisfying the recurrence relation
has a limit, and find an expression for the limit in
terms of
![$\alpha$](img85-40.gif)
,
![$x_0$](img86-40.gif)
, and
![$x_1$](img87-40.gif)
.
Problem 16
For which numbers
![$a$](img88-39.gif)
in
![$(1, \infty)$](img89-39.gif)
is it true that
![$x^{a} \leq a^x$](img90-39.gif)
for all
![$x$](img91-39.gif)
in
![$(1, \infty)$](img92-38.gif)
?
Problem 17
Prove that there are at least two nonisomorphic
nonabelian groups of order
![$30$](img93-38.gif)
.
Problem 18
Let u be a real valued harmonic function
in the complex plane such that
for all z, where a and b are positive constants.
Prove that u is constant.
Previous: Fall93
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Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10