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Preliminary Exam - Fall 1995
Problem 1
Let
![$G$](img1-45.gif)
be a group generated by
![$n$](img2-45.gif)
elements. Find an upper bound
![$N(n,k)$](img3-45.gif)
for the number of subgroups
![$H$](img4-45.gif)
of
![$G$](img5-45.gif)
with the index
![$[G:H]=k$](img6-45.gif)
.
Problem 2
Let
![$A$](img7-45.gif)
be a finite subset of the unit disc in the plane,
and let
![$N(A,r)$](img8-45.gif)
be the set of points at distance
![$\leq r$](img9-45.gif)
from
![$A$](img10-45.gif)
, where
![$0 < r < 1$](img11-45.gif)
. Show that the length of the
boundary
![$N(A,r)$](img12-45.gif)
is, at most,
![$C/r$](img13-45.gif)
for some constant
![$C$](img14-45.gif)
independent of
![$A$](img15-45.gif)
.
Problem 3
Find the radius of convergence
![$R$](img16-45.gif)
of the Taylor series about
![$z=1$](img17-45.gif)
of the function
![$f(z)=1/(1+z^2+z^4+z^6+z^8+z^{10})$](img18-45.gif)
.
Express your answer in terms of real numbers and square roots only.
Problem 4
Suppose
![$A$](img19-45.gif)
and
![$B$](img20-45.gif)
are real
![$n\times n$](img21-45.gif)
matrices and
![$C$](img22-45.gif)
is a complex
![$n\times n$](img23-45.gif)
matrix such that
Find a real
![$n\times n$](img25-45.gif)
matrix
![$D$](img26-45.gif)
such that
![$DAD^{-1}=B$](img27-45.gif)
.
Problem 5
Prove that
![% latex2html id marker 738
$\mbox{$\mathbb{Q}\,^{}$}[x,y]/\langle x^2+y^2-1 \rangle$](img28-45.gif)
is an integral domain and
that its field of fractions is isomorphic to the field of
rational functions
![% latex2html id marker 742
$\mbox{$\mathbb{Q}\,^{}$}(t)$](img29-45.gif)
.
Problem 6
Determine all real numbers
![$L > 1$](img30-45.gif)
so that the boundary value problem
has a nonzero solution.
Problem 7
Let
![$g(z)=\sum^\infty_{n=0} g_nz^n$](img33-45.gif)
and
![$h(z)=\sum^\infty_{n=0}h_nz^n$](img34-45.gif)
be entire functions.
Find a formula for the coefficients
![$f_n$](img35-45.gif)
in the Taylor
expansion about
![$z=0$](img36-45.gif)
of
Problem 8
Show that an
![$n\times n$](img38-45.gif)
matrix of complex numbers
![$A$](img39-45.gif)
satisfying
for
![$1\leq i\leq n$](img41-45.gif)
must be invertible.
Problem 9
Let
![$x_1$](img42-45.gif)
be a real number,
![$0 < x_1 < 1$](img43-45.gif)
, and define a sequence by
![$x_{n+1}=x_n-x_n^{n+1}$](img44-45.gif)
. Show that
![$\liminf_{n\to\infty} x_n > 0$](img45-45.gif)
.
Problem 10
Let
![% latex2html id marker 822
$\mbox{\bf {F}}$](img46-45.gif)
be a field and
![% latex2html id marker 824
$\mbox{\bf {F}}^*$](img47-45.gif)
be the multiplicative group of
nonzero elements. Let
![$G$](img48-45.gif)
be a subgroup of
![% latex2html id marker 828
$\mbox{\bf {F}}^*$](img49-45.gif)
of
finite order
![$n$](img50-45.gif)
. Show that
![$G$](img51-45.gif)
is cyclic.
Problem 11
Let
![$f(z)=u(z)+iv(z)$](img52-45.gif)
be holomorphic in
![$\vert z\vert < 1$](img53-45.gif)
,
![$u$](img54-45.gif)
and
![$v$](img55-45.gif)
real. Show that
for
![$0 < r < 1$](img57-45.gif)
if
![$u(0)^2=v(0)^2$](img58-45.gif)
.
Problem 12
Let
![$f$](img59-45.gif)
and
![$f'$](img60-45.gif)
be continuous on
![$[0,\infty)$](img61-45.gif)
and
![$f(x)=0$](img62-45.gif)
for
![$x\geq 10^{10}$](img63-45.gif)
. Show that
Problem 13
Show that
for
![$\vert z\vert < 1$](img66-45.gif)
.
Problem 14
Let
![% latex2html id marker 911
$f(x)\in \mbox{$\mathbb{Q}\,^{}$}[x]$](img67-45.gif)
be a polynomial with rational coefficients.
Show that there is a
![% latex2html id marker 915
$g(x)\in \mbox{$\mathbb{Q}\,^{}$}[x]$](img68-45.gif)
,
![$g\neq 0$](img69-45.gif)
, such that
![$f(x)g(x)=a_2x^2+a_3x^3+a_5x^5+\cdots +a_px^p$](img70-45.gif)
is a polynomial in which only prime exponents appear.
Problem 15
Let
![% latex2html id marker 941
$f:\mbox{$\mathbb{R}^{}$}\to \mbox{$\mathbb{R}^{}$}$](img71-45.gif)
be a
![$C^\infty$](img72-45.gif)
function. Assume that
![$f(x)$](img73-45.gif)
has a
local minimum at
![$x=0$](img74-44.gif)
. Prove there is a disc centered on the
![$y$](img75-44.gif)
axis which lies above the graph of
![$f$](img76-44.gif)
and touches the graph at
![$(0,f(0))$](img77-44.gif)
.
Problem 16
Let
![$A$](img78-44.gif)
and
![$B$](img79-44.gif)
be nonsimilar
![$n\times n$](img80-44.gif)
complex matrices with
the same minimal and the same characteristic polynomial.
Show that
![$n\geq 4$](img81-43.gif)
and the minimal polynomial is not equal to
the characteristic polynomial.
Problem 17
Let
![$f_1,f_2,\dots ,f_n$](img82-43.gif)
be continuous real valued functions
on
![$[a,b]$](img83-43.gif)
. Show that the set
![$\{ f_1,\dots ,f_n\}$](img84-43.gif)
is
linearly dependent on
![$[a,b]$](img85-42.gif)
if and only if
Problem 18
Let
![% latex2html id marker 1002
$f:\mbox{$\mathbb{R}^{}$}\to \mbox{$\mathbb{R}^{}$}$](img87-41.gif)
be a nonzero
![$C^\infty$](img88-40.gif)
function such that
![$f(x)f(y)=f\left(\sqrt{x^2+y^2}\right)$](img89-40.gif)
for all
![$x$](img90-40.gif)
and
![$y$](img91-40.gif)
and
that
![$f(x)\to 0$](img92-39.gif)
as
![$\vert x\vert\to\infty$](img93-39.gif)
.
- Prove that
is an even function and that
is
.
- Prove that
satisfies the differential equation
, and find the most general function satisfying
the given conditions.
Previous: Spring95
Next: Spring96
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10