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Preliminary Exam - Fall 1989
Problem 1
Let
![$A$](img1-33.gif)
be a finite abelian group, and
![$m$](img2-33.gif)
the maximum of the orders of
the elements of
![$A$](img3-33.gif)
. Put
![$S=\{a \in A \;\vert\; \vert a\vert = m\}$](img4-33.gif)
. Prove
that
![$A$](img5-33.gif)
is generated by
![$S$](img6-33.gif)
.
Problem 2
Let
![% latex2html id marker 679
$f:[0,1]\to\mbox{$\mathbb{R}^{}$}$](img7-33.gif)
be a real valued continuously differentiable
function with
![$f(0)=0$](img8-33.gif)
. Suppose also that there is a constant
![$M>0$](img9-33.gif)
such that, for
![$0\leq x\leq 1$](img10-33.gif)
,
Prove that
![$f(x)=0$](img12-33.gif)
for
![$0\leq x \leq 1$](img13-33.gif)
.
Problem 3
Let
![$A$](img14-33.gif)
be a real, upper-triangular,
![$n\times n$](img15-33.gif)
matrix that commutes with
its transpose. Prove that
![$A$](img16-33.gif)
is diagonal.
Problem 4
Let
![$f(z)$](img17-33.gif)
be analytic for
![$\vert z\vert<1$](img18-33.gif)
and suppose that
Show that
![$\vert f'(0)\vert\leq 4$](img20-33.gif)
.
Problem 5
Let
![$G$](img21-33.gif)
be a group,
![$G'$](img22-33.gif)
its commutator subgroup, and
![$N$](img23-33.gif)
a normal subgroup
of
![$G$](img24-33.gif)
. Suppose that
![$N$](img25-33.gif)
is cyclic. Prove that
![$gn = ng$](img26-33.gif)
for all
![$g\in G'$](img27-33.gif)
and all
![$n\in N$](img28-33.gif)
.
Problem 6
Let
![% latex2html id marker 751
$X \subset \mbox{$\mathbb{R}^{n}$}$](img29-33.gif)
be a closed set and r a fixed positive real number.
Let
![% latex2html id marker 755
$Y = \{y \in \mbox{$\mathbb{R}^{n}$} \,\vert\, \vert x-y\vert=r \; for \,\, some \; x \in X\}$](img30-33.gif)
.
Show that
![$Y$](img31-33.gif)
is closed.
Problem 7
Let
![$A$](img32-33.gif)
and
![$B$](img33-33.gif)
be diagonalizable linear transformations of
![% latex2html id marker 775
$\mathbb{R}^{n}$](img34-33.gif)
into
itself such that
![$AB = BA$](img35-33.gif)
. Let
![$E$](img36-33.gif)
be an eigenspace of
![$A$](img37-33.gif)
. Prove that
the restriction of
![$B$](img38-33.gif)
to
![$E$](img39-33.gif)
is diagonalizable.
Problem 8
Evaluate the integral
Problem 9
Let
![% latex2html id marker 823
$\mbox{\bf {F}}$](img41-33.gif)
be a field,
![% latex2html id marker 825
$\mbox{\bf {F}}[x]$](img42-33.gif)
the polynomial ring in one variable
over
![% latex2html id marker 827
$\mbox{\bf {F}}$](img43-33.gif)
, and
![$R$](img44-33.gif)
a subring of
![% latex2html id marker 831
$\mbox{\bf {F}}[x]$](img45-33.gif)
with
![% latex2html id marker 833
$\mbox{\bf {F}} \subset R$](img46-33.gif)
.
Prove that there
exists a finite set
![$\{f_1,f_2,\ldots,f_n\}$](img47-33.gif)
of elements of
![% latex2html id marker 837
$\mbox{\bf {F}}[x]$](img48-33.gif)
such that
![% latex2html id marker 839
$R = \mbox{\bf {F}}[f_1,f_2,\ldots,f_n]$](img49-33.gif)
.
Problem 10
Let
![$\alpha$](img50-33.gif)
be a number in
![$(0,1)$](img51-33.gif)
. Prove that any
sequence
![$(x_{n})$](img52-33.gif)
of real
numbers satisfying the recurrence relation
has a limit, and find an expression for the limit in
terms of
![$\alpha$](img54-33.gif)
,
![$x_0$](img55-33.gif)
and
![$x_1$](img56-33.gif)
.
Problem 11
Let
![$\varphi$](img57-33.gif)
be Euler's totient function; so if
![$n$](img58-33.gif)
is a positive
_Euler>totient function
integer, then
![$\varphi(n)$](img59-33.gif)
is the number of integers
![$m$](img60-33.gif)
for which
![$1\leq
m \leq n$](img61-33.gif)
and
![$\gcd \{n,m\} = 1$](img62-33.gif)
. Let
![$a$](img63-33.gif)
and
![$k$](img64-33.gif)
be two integers, with
![$a>1$](img65-33.gif)
,
![$k>0$](img66-33.gif)
. Prove that
![$k$](img67-33.gif)
divides
![$\varphi(a^k-1)$](img68-33.gif)
.
Problem 12
Let
![$f(z)$](img69-33.gif)
be analytic in the annulus
![$\Omega = \{1<\vert z\vert<2\}$](img70-33.gif)
.
Assume that
![$f$](img71-33.gif)
has no zeros in
![$\Omega$](img72-33.gif)
. Show that there exists an
integer
![$n$](img73-33.gif)
and an analytic function
![$g$](img74-32.gif)
in
![$\Omega$](img75-32.gif)
such that, for
all
![$z \in \Omega$](img76-32.gif)
,
![$f(z)=z^ne^{g(z)}$](img77-32.gif)
.
Problem 13
Let
![${A}$](img78-32.gif)
be an
![$n\times n$](img79-32.gif)
real matrix,
![${A}^t$](img80-32.gif)
its
transpose. Show that
![${A}^t{A}$](img81-32.gif)
and
![${A}^t$](img82-32.gif)
have the same
range. In other words, given
![$y$](img83-32.gif)
, show that the equation
![$y =
{A}^t{A}x$](img84-32.gif)
has a solution
![$x$](img85-32.gif)
if and only if the equation
![$y = {A}^tz$](img86-32.gif)
has a solution
![$z$](img87-32.gif)
.
Problem 14
Let
![% latex2html id marker 985
$X \subset\mbox{$\mathbb{R}^{n}$}$](img88-31.gif)
be compact and let
![% latex2html id marker 989
$f:X \to \mbox{$\mathbb{R}^{ }$}$](img89-31.gif)
be continuous.
Given
![$\varepsilon > 0$](img90-31.gif)
, show there is an
![$M$](img91-31.gif)
such that for all
![$x, y\in X$](img92-31.gif)
,
Problem 15
Let
![$G_n$](img94-31.gif)
be the free group on
![$n$](img95-31.gif)
generators. Show that
![$G_2$](img96-31.gif)
and
![$G_3$](img97-31.gif)
are not isomorphic.
Problem 16
Prove that the polynomial
has at least one root in the disc
![$\vert z\vert<1$](img99-30.gif)
.
Problem 17
Let
![$V$](img100-29.gif)
be a vector space of finite-dimension
![$n$](img101-29.gif)
over a field of characteristic
![$0$](img102-29.gif)
. Prove that
![$V$](img103-26.gif)
is not the union of finitely many subspaces of dimension
![$n-1$](img104-26.gif)
.
Problem 18
Let the function
![$f$](img105-25.gif)
from
![$[0,1]$](img106-24.gif)
to
![$[0,1]$](img107-23.gif)
have the following properties:
Prove that the arclength of the graph of
![$f$](img113-21.gif)
does not exceed
![$3$](img114-18.gif)
.
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Next: Spring90
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10