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Preliminary Exam - Summer 1984
Problem 1
Show that if a subgroup
![$H$](img1-22.gif)
of a group
![$G$](img2-22.gif)
has just one left coset different
from itself, then it is a normal subgroup of
![$G$](img3-22.gif)
.
Problem 2
Let
![% latex2html id marker 683
$\mathbb{Z}^{}$](img4-22.gif)
be the ring of integers and
![% latex2html id marker 687
$\mbox{$\mathbb{Z}^{}$}[x]$](img5-22.gif)
the polynomial ring over
![% latex2html id marker 689
$\mathbb{Z}^{}$](img6-22.gif)
.
Show that
is irreducible in
![% latex2html id marker 693
$\mbox{$\mathbb{Z}^{}$}[x]$](img8-22.gif)
.
Problem 3
Let
![% latex2html id marker 726
$f: \mbox{$\mathbb{R}^{m}$} \to \mbox{$\mathbb{R}^{n}$}, \; n \geq 2$](img9-22.gif)
, be a linear transformation of
rank
![$n-1$](img10-22.gif)
. Let
![$f(v) = (f_1(v),f_2(v),\ldots,f_n(v))$](img11-22.gif)
for
![% latex2html id marker 734
$v \in\mbox{$\mathbb{R}^{m}$}$](img12-22.gif)
. Show that a necessary and sufficient condition for
the system of inequalities
![$f_i(v)>0$](img13-22.gif)
,
![$i=1,\ldots,n$](img14-22.gif)
, to have no
solution is that there exist real numbers
![$\lambda_i\geq 0$](img15-22.gif)
, not all
zero, such that
Problem 4
Let
be a real matrix with
![$a, b, c, d > 0$](img18-22.gif)
. Show that
![$A$](img19-22.gif)
has an eigenvector
with
![$x, y > 0$](img21-22.gif)
.
Problem 6
Let
![$\rho > 0$](img29-22.gif)
. Show that for
![$n$](img30-22.gif)
large enough, all the zeros of
lie in the circle
![$\vert z\vert<\rho$](img32-22.gif)
.
Problem 7
Let
![% latex2html id marker 857
$f:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$](img33-22.gif)
be
![$C^1$](img34-22.gif)
and let
If
![$f'(x_0)\neq 0$](img36-22.gif)
, show that this transformation is locally invertible
near
![$(x_0,y_0)$](img37-22.gif)
and the inverse has the form
Problem 8
Let
![$\varphi(s)$](img39-22.gif)
be a
![$C^2$](img40-22.gif)
function on
![$[1,2]$](img41-22.gif)
with
![$\varphi$](img42-22.gif)
and
![$\varphi'$](img43-22.gif)
vanishing at
![$s=1,2$](img44-22.gif)
. Prove that there is a constant
![$C>0$](img45-22.gif)
such
that for any
![$\lambda>1$](img46-22.gif)
,
Problem 9
Consider the solution curve
![$\left( x(t),y(t) \right)$](img48-22.gif)
to the equations
with initial conditions
![$x(0)=0$](img50-22.gif)
and
![$y(0)=0$](img51-22.gif)
. Prove that the solution
must cross the line
![$x=1$](img52-22.gif)
in the
![$xy$](img53-22.gif)
plane by the time
![$t=2$](img54-22.gif)
.
Problem 10
Let
![$C^{1/3}$](img55-22.gif)
be the set of real valued functions f on the closed
interval
![$[0,1]$](img56-22.gif)
such that
;
is finite, where by definition
Verify that
![$\Vert\cdot\Vert$](img60-22.gif)
is a norm for the space
![$C^{1/3}$](img61-22.gif)
, and prove that
![$C^{1/3}$](img62-22.gif)
is complete with respect to this norm.
Problem 11
Let
![$S_n$](img63-22.gif)
denote the group of permutations of
![$n$](img64-22.gif)
letters. Find
four different subgroups of
![$S_4$](img65-22.gif)
isomorphic to
![$S_3$](img66-22.gif)
and nine
isomorphic to
![$S_2$](img67-22.gif)
.
Problem 13
Let
![$A$](img83-21.gif)
be a 2
![$\times$](img84-21.gif)
2 matrix over
![% latex2html id marker 1063
$\mathbb{C}\,^{}$](img85-21.gif)
which is not a scalar
multiple of the identity matrix
![$I$](img86-21.gif)
. Show that any 2
![$\times$](img87-21.gif)
2
matrix
![$X$](img88-20.gif)
over
![% latex2html id marker 1071
$\mathbb{C}\,^{}$](img89-20.gif)
commuting with
![$A$](img90-20.gif)
has the form
![$X=\alpha I+\beta A$](img91-20.gif)
,
where
![% latex2html id marker 1079
$\alpha, \beta \in \mbox{$\mathbb{C}\,^{}$}$](img92-20.gif)
.
Problem 14
Suppose
![$V$](img93-20.gif)
is an
![$n$](img94-20.gif)
-dimensional vector space over the field
![% latex2html id marker 1100
$\mbox{\bf {F}}$](img95-20.gif)
. Let
![$W\subset V$](img96-20.gif)
be a subspace of dimension
![$r<n$](img97-20.gif)
. Show that
Problem 15
Let
![% latex2html id marker 1126
$\mbox{$\mathbb{Z}^{}$}_3$](img99-20.gif)
be the field of integers
![$\bmod 3$](img100-19.gif)
and
![% latex2html id marker 1132
$\mbox{$\mathbb{Z}^{}$}_3[x]$](img101-19.gif)
the
corresponding polynomial ring. Decompose
![$x^3+x+2$](img102-19.gif)
into irreducible
factors in
![% latex2html id marker 1138
$\mbox{$\mathbb{Z}^{}$}_3[x]$](img103-16.gif)
.
Problem 16
Let
![$p(z)$](img104-16.gif)
be a nonconstant polynomial with real coefficients such
that for some real number
![$a$](img105-15.gif)
,
![$p(a)\neq 0$](img106-14.gif)
but
![$p'(a)=
p''(a)=0$](img107-14.gif)
. Prove that the equation
![$p(z)=0$](img108-14.gif)
has a nonreal root.
Problem 17
Suppose
has radius of convergence
![$R>0$](img110-14.gif)
. Show that
is entire and that for
![$0<r<R$](img112-13.gif)
, there is a constant
![$M$](img113-13.gif)
such that
Problem 18
Show that there is a unique continuous function
![% latex2html id marker 1198
$f:[0,1]\to\mbox{$\mathbb{R}^{}$}$](img115-11.gif)
such that
Problem 19
Let
![$x(t)$](img117-9.gif)
be the solution of the differential equation
with initial conditions
![$x(0)=0$](img119-8.gif)
and
![$x'(0)=0$](img120-8.gif)
. Show that for
suitable constants
![$\alpha$](img121-8.gif)
and
![$\delta$](img122-8.gif)
,
Previous: Spring84
Next: Fall84
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10