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Preliminary Exam - Fall 1991
Problem 1
Prove that every finite group of order at least
![$3$](img1-38.gif)
has a
nontrivial automorphism.
Problem 2
Let
![$f$](img2-38.gif)
be a continuous function from
![% latex2html id marker 676
$\mathbb{R}^{}$](img3-38.gif)
to
![% latex2html id marker 678
$\mathbb{R}^{}$](img4-38.gif)
such that
![$\vert f(x)-f(y)\vert\geq \vert x-y\vert$](img5-38.gif)
for all
![$x$](img6-38.gif)
and
![$y$](img7-38.gif)
. Prove that the
range of
![$f$](img8-38.gif)
is all of
![% latex2html id marker 688
$\mathbb{R}^{}$](img9-38.gif)
.
Note: See also Problem
.
Problem 3
Evaluate the integral
where n is a positive integer, and
![$C$](img11-38.gif)
is the circle
![$\vert z\vert=1$](img12-38.gif)
, with
counterclockwise orientation.
Problem 5
Let
![$f$](img22-38.gif)
be an infinitely differentiable function from
![% latex2html id marker 768
$\mathbb{R}^{}$](img23-38.gif)
to
![% latex2html id marker 770
$\mathbb{R}^{}$](img24-38.gif)
. Suppose that, for some positive integer
![$n$](img25-38.gif)
,
Prove that
for some
in
.
Problem 6
Let the function f be analytic in the disc
![$\vert z\vert<1$](img30-38.gif)
of the complex
plane. Assume that there is a positive constant
![$M$](img31-38.gif)
such that
Prove that
Problem 7
Consider the vector differential equation
where
![$A$](img35-38.gif)
is a smooth
![$n\times n$](img36-38.gif)
function on
![% latex2html id marker 830
$\mathbb{R}^{}$](img37-38.gif)
. Assume
![$A$](img38-38.gif)
has
the property that
![% latex2html id marker 834
$\langle {\rm A}(t)y, y \rangle \leq c\Vert y\Vert^2$](img39-38.gif)
for all
![$y$](img40-38.gif)
in
![% latex2html id marker 838
$\mathbb{R}^{n}$](img41-38.gif)
and all
![$t$](img42-38.gif)
, where
![$c$](img43-38.gif)
is a fixed real
number. Prove that any solution
![$x(t)$](img44-38.gif)
of the equation satisfies
![$\Vert x(t)\Vert\leq e^{ct}\Vert x(0)\Vert$](img45-38.gif)
for all
![$t>0$](img46-38.gif)
.
Problem 8
Let
![$a_1,a_2,a_3,\ldots$](img47-38.gif)
be positive numbers.
- Prove that
implies
.
- Prove that the converse of the above statement is false.
Problem 9
Let
![$G$](img50-38.gif)
be a group of order
![$2p$](img51-38.gif)
, where
![$p$](img52-38.gif)
is an odd prime. Assume
that
![$G$](img53-38.gif)
has a normal subgroup of order
![$2$](img54-38.gif)
. Prove that
![$G$](img55-38.gif)
is cyclic.
Problem 10
Let
![% latex2html id marker 907
$\mbox{\bf {F}}$](img56-38.gif)
be a finite field of order
![$p$](img57-38.gif)
. Compute the order of
![% latex2html id marker 911
$SL_3(\mbox{\bf {F}})$](img58-38.gif)
,
the group of 3
![$\times $](img59-38.gif)
3 matrices over
![% latex2html id marker 915
$\mbox{\bf {F}}$](img60-38.gif)
of determinant
![$1$](img61-38.gif)
.
Problem 11
Let
![$X$](img62-38.gif)
and
![$Y$](img63-38.gif)
be metric spaces and
![$f$](img64-38.gif)
a continuous map of
![$X$](img65-38.gif)
into
![$Y$](img66-38.gif)
. Let
![$K_1,K_2,\ldots$](img67-38.gif)
be nonempty compact subsets of
![$X$](img68-38.gif)
such that
![$K_{n+1} \subset K_n$](img69-38.gif)
for all
![$n$](img70-38.gif)
, and let
![$K=\bigcap K_n$](img71-38.gif)
. Prove that
![$f(K)= \bigcap f(K_n)$](img72-38.gif)
.
Problem 12
Let
![$p$](img73-38.gif)
be a nonconstant complex polynomial whose zeros are all in the
half-plane
![$\Im z>0$](img74-37.gif)
.
- Prove that
on the real axis.
- Find a relation between
and
Problem 13
Let
![$A = (a_{ij})_{i,j=1}^n$](img78-37.gif)
be a real
![$n\times n$](img79-37.gif)
matrix with
nonnegative entries such that
Prove that no eigenvalue of
![$A$](img81-36.gif)
has absolute value greater
than
![$1$](img82-36.gif)
.
Problem 14
Let
![${\cal B}$](img83-36.gif)
denote the unit ball of
![% latex2html id marker 1023
$\mathbb{R}^{3}$](img84-36.gif)
,
![% latex2html id marker 1027
${\cal B}=\{r\in\mbox{$\mathbb{R}^{3}$} \;\vert\; \Vert r\Vert\leq 1\}$](img85-36.gif)
.
Let
![$J=(J_1,J_2,J_3)$](img86-36.gif)
be a smooth vector field on
![% latex2html id marker 1031
$\mathbb{R}^{3}$](img87-36.gif)
that vanishes
outside of
![${\cal B}$](img88-35.gif)
and satisfies
![$\nabla \cdot \vec{J} = 0$](img89-35.gif)
.
- For
a smooth, scalar-valued function defined on a
neighborhood of
, prove that
- Prove that
Problem 15
Let
![$\mathfrak{I}$](img94-35.gif)
be the ideal in the ring
![% latex2html id marker 1070
$\mbox{$\mathbb{Z}^{}$}[x]$](img95-34.gif)
generated by
![$x-7$](img96-34.gif)
and
![$15$](img97-34.gif)
. Prove that the quotient ring
![% latex2html id marker 1078
$\mbox{$\mathbb{Z}^{}$}[x]/\mathfrak{I}$](img98-34.gif)
is isomorphic to
![% latex2html id marker 1082
$\mbox{$\mathbb{Z}^{}$}_{15}$](img99-33.gif)
.
Problem 16
Let
![$M_{n \times n}$](img100-32.gif)
be the space of real n
![$\times $](img101-32.gif)
n matrices. Regard it
as a metric space with the distance function
Prove that the set of nilpotent matrices in
is a closed set.
Problem 17
Let
![$f$](img104-28.gif)
be a
![$C^1$](img105-27.gif)
function from the interval
![$(-1,1)$](img106-25.gif)
into
![% latex2html id marker 1132
$\mathbb{R}^{2}$](img107-24.gif)
such that
![$f(0)=0$](img108-24.gif)
and
![$f'(0)\neq 0$](img109-24.gif)
. Prove that there is
a number
![$\varepsilon$](img110-24.gif)
in
![$(0,1)$](img111-23.gif)
such that
![$\Vert f(t)\Vert$](img112-22.gif)
is an
increasing function of
![$t$](img113-22.gif)
on
![$(0,\varepsilon)$](img114-19.gif)
.
Problem 18
Let the function
![$f$](img115-16.gif)
be analytic in the entire complex plane and
satisfy the inequality
![$\vert f(z)\vert\leq\vert\Re z\vert^{-1/2}$](img116-15.gif)
off
the imaginary axis. Prove that
![$f$](img117-14.gif)
is constant.
Previous: Spring91
Next: Spring92
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10