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Preliminary Exam - Fall 1994
Problem 1
For which values of the real number
![$a$](img1-44.gif)
does the series
converge?
Problem 2
Prove that the matrix
has one positive eigenvalue and one negative eigenvalue.
Problem 3
Evaluate the integrals
Problem 4
Suppose the group
![$G$](img5-44.gif)
has a nontrivial subgroup
![$H$](img6-44.gif)
which is
contained in every nontrivial subgroup of
![$G$](img7-44.gif)
. Prove that
![$H$](img8-44.gif)
is contained in the center of
![$G$](img9-44.gif)
.
Problem 5
- Find a basis for the space of real solutions
of the differential equation
- Find a basis for the subspace of real solutions of
that satisfy
Problem 6
Let
![$A= \left( a_{ij} \right)_{i,j=1}^{n}$](img13-44.gif)
be a real
![$n \times n$](img14-44.gif)
matrix such that
![$a_{ii} \geq 1$](img15-44.gif)
for all
![$i$](img16-44.gif)
, and
Prove that
![$A$](img18-44.gif)
is invertible.
Problem 7
Let
![$f$](img19-44.gif)
be a continuously differentiable function
from
![% latex2html id marker 754
$\mathbb{R}^{2}$](img20-44.gif)
into
![% latex2html id marker 756
$\mathbb{R}$](img21-44.gif)
. Prove that
there is a continuous one-to-one function
![$g$](img22-44.gif)
from
![$[0,1]$](img23-44.gif)
into
![% latex2html id marker 762
$\mathbb{R}^{2}$](img24-44.gif)
such that the composite function
![$f \circ g$](img25-44.gif)
is constant.
Problem 8
Let
![% latex2html id marker 805
$\mathbb{Q}$](img26-44.gif)
be the field of rational numbers. For
![$\theta$](img27-44.gif)
a real number, let
![% latex2html id marker 809
$\mbox{\bf {F}}_{\theta}=\mathbb{Q}(\sin \theta)$](img28-44.gif)
and
![% latex2html id marker 811
$\mbox{\bf {E}}_{\theta}=\mathbb{Q}\left(\sin \frac{\theta}{3}\right)$](img29-44.gif)
.
Show that
![% latex2html id marker 813
$\mbox{\bf {E}}_{\theta}$](img30-44.gif)
is an extension field of
![% latex2html id marker 815
$\mbox{\bf {F}}_{\theta}$](img31-44.gif)
, and determine all possibilities
for
![% latex2html id marker 817
${\dim}_{\mbox{\scriptsize\bf F}_{\theta}} \mbox{\bf {E}}_{\theta}$](img32-44.gif)
.
Problem 10
Let the function
![% latex2html id marker 873
$f: \mathbb{R}^{n} \to \mathbb{R}^{n}$](img34-44.gif)
satisfy
the following two conditions:
- (i)
is compact whenever
is a compact subset of
.
- (ii)
- If
is a decreasing
sequence of compact subsets of
, then
Prove that
![$f$](img41-44.gif)
is continuous.
Problem 11
Write down a list of 5
![$\times $](img42-44.gif)
5 complex matrices, as long as possible,
with the following properties:
- The characteristic polynomial of each matrix in the list is
;
- The minimal polynomial of each matrix in the list is
;
- No two matrices in the list are similar.
Problem 12
Suppose the coefficients of the power series
are given by the recurrence relation
Find the radius of convergence of the series and the function to which
it converges in its disc of convergence.
Problem 13
Let
![$p$](img47-44.gif)
be an odd prime and
![% latex2html id marker 948
$\mbox{\bf {F}}_{p}$](img48-44.gif)
the field of
![$p$](img49-44.gif)
elements.
How many elements of
![% latex2html id marker 952
$\mbox{\bf {F}}_{p}$](img50-44.gif)
have square roots in
![% latex2html id marker 954
$\mbox{\bf {F}}_{p}$](img51-44.gif)
?
How many have cube roots in
![% latex2html id marker 956
$\mbox{\bf {F}}_{p}$](img52-44.gif)
?
Problem 14
Find the maximum area of all triangles that can be inscribed in an ellipse
with semiaxes
![$a$](img53-44.gif)
and
![$b$](img54-44.gif)
, and describe the triangles that have maximum area.
Note: See also Problem
.
Problem 15
Let
![$M_{7\times 7}$](img55-44.gif)
denote the vector space of real 7
![$\times $](img56-44.gif)
7 matrices. Let
![$A$](img57-44.gif)
be a diagonal matrix in
![$M_{7\times 7}$](img58-44.gif)
that has
![$+1$](img59-44.gif)
in four diagonal positions and
![$-1$](img60-44.gif)
in three diagonal positions. Define the linear transformation
![$T$](img61-44.gif)
on
![$M_{7\times 7}$](img62-44.gif)
by
![$T(X)=AX-XA$](img63-44.gif)
. What is the dimension of the range of
![$T$](img64-44.gif)
?
Problem 16
Let
![${\cal D}$](img65-44.gif)
denote the open unit disc in
![% latex2html id marker 1036
$\mathbb{R}^{2}$](img66-44.gif)
. Let
![$u$](img67-44.gif)
be an
eigenfunction for the Laplacian in
![${\cal D}$](img68-44.gif)
; that is,
a real valued function of class
![$C^{2}$](img69-44.gif)
defined in
![$\overline{\cal D}$](img70-44.gif)
,
zero on the boundary of
![${\cal D}$](img71-44.gif)
but not identically zero, and
satisfying the differential equation _Laplacian,>eigenfunction
where
![$\lambda$](img73-44.gif)
is a constant. Prove that
and hence that
![$\lambda < 0$](img75-43.gif)
.
Problem 17
Let
![$R$](img76-43.gif)
be a ring with identity, and let
![$u$](img77-43.gif)
be an element of
![$R$](img78-43.gif)
with a right inverse. Prove that the following conditions on
![$u$](img79-43.gif)
are
equivalent:
has more than one right inverse;
is a zero divisor;
is not a unit.
Problem 18
Let the function
![$f$](img83-42.gif)
be analytic in the complex plane, real on the real
axis,
![$0$](img84-42.gif)
at the origin, and not identically
![$0$](img85-41.gif)
. Prove that if
![$f$](img86-41.gif)
maps the imaginary axis into a straight line, then that straight line
must be either the real axis or the imaginary axis.
Previous: Spring94
Next: Spring95
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10