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Preliminary Exam - Fall 1996
Problem 1
Let
![$M$](img1-47.gif)
be the set of real valued continuous functions
![$f$](img2-47.gif)
on
![$[0,1]$](img3-47.gif)
such that
![$f'$](img4-47.gif)
is continuous on
![$[0,1]$](img5-47.gif)
, with the norm
Which subsets of
![$M$](img7-47.gif)
are compact?
Problem 2
A real-valued function
![$f$](img8-47.gif)
on a closed bounded interval
![$[a,b]$](img9-47.gif)
is said to be upper semicontinuous provided that for every
![$\varepsilon > 0$](img10-47.gif)
and
![$p\in [a,b]$](img11-47.gif)
, there is a
![$\delta=\delta (\varepsilon,p)> 0$](img12-47.gif)
such that if
![$x\in [a,b]$](img13-47.gif)
and
![$\vert x-p\vert < \delta$](img14-47.gif)
then
![$f(x) <
f(p)+\varepsilon$](img15-47.gif)
. Prove that an upper semicontinuous function is bounded
above on
![$[a,b]$](img16-47.gif)
.
Problem 3
Evaluate the integral
Problem 4
Does there exist a function
![$f$](img18-47.gif)
, analytic in the punctured plane
![% latex2html id marker 719
$\mbox{$\mathbb{C}\,^{}$} \setminus \{0\}$](img19-47.gif)
, such that
for all nonzero
![$z$](img21-47.gif)
?
Problem 5
Prove that any linear transformation
![% latex2html id marker 743
$T:\mbox{$\mathbb{R}^{3}$} \to \mbox{$\mathbb{R}^{3}$}$](img22-47.gif)
has
- a one-dimensional invariant subspace
- a two-dimensional invariant subspace.
Problem 6
Let
![$A$](img23-47.gif)
and
![$B$](img24-47.gif)
be real 2
![$\times $](img25-47.gif)
2 matrices such that
Show that there exists a real 2
![$\times $](img27-47.gif)
2 matrix
![$T$](img28-47.gif)
such that
Problem 7
Suppose
![$p$](img30-47.gif)
is a prime. Show that every element of
![% latex2html id marker 788
$GL_2(\mbox{\bf {F}}_p)$](img31-47.gif)
has order
dividing either
![$p^2-1$](img32-47.gif)
or
![$p(p-1)$](img33-47.gif)
.
Problem 8
Show the denominator of
![$\left(\!\!\begin{array}{c} 1/2 \\ n\end{array}\!\!
\right)$](img34-47.gif)
is a power of
![$2$](img35-47.gif)
for all integers
![$n$](img36-47.gif)
.
Problem 9
For positive integers
![$a$](img37-47.gif)
,
![$b$](img38-47.gif)
and
![$c$](img39-47.gif)
show that
Problem 10
If
![$f$](img41-47.gif)
is a
![$C^2$](img42-47.gif)
function on an open interval, prove that
Problem 11
Let
![$f$](img44-47.gif)
be continuous and nonnegative on
![$[0,1]$](img45-47.gif)
and suppose that
Prove that
![$f(t)\leq 1+t$](img47-47.gif)
for
![$0\leq t\leq 1$](img48-47.gif)
.
Problem 12
Find the number of roots, counted with their multiplicities, of
which lie between the circles
![$\vert z\vert=1$](img50-47.gif)
and
![$\vert z\vert=2$](img51-47.gif)
.
Problem 13
Define
![% latex2html id marker 878
$F:\mbox{$\mathbb{C}\,^{3}$} \to \mbox{$\mathbb{C}\,^{3}$}$](img52-47.gif)
by
Show that
![$F$](img54-47.gif)
is onto but not one-to-one.
Problem 14
Let
![$f$](img55-47.gif)
be holomorphic on and inside the unit circle
![$C$](img56-47.gif)
.
Let
![$L$](img57-47.gif)
be the length of the image of
![$C$](img58-47.gif)
under
![$f$](img59-47.gif)
. Show that
Problem 15
Is there a real 2
![$\times $](img61-47.gif)
2 matrix
![$A$](img62-47.gif)
such that
Exhibit such an
![$A$](img64-47.gif)
or prove there is none.
Problem 16
Let
Show that every real matrix
![$B$](img66-47.gif)
such that
![$AB=BA$](img67-47.gif)
has the form
for some real numbers
![$a$](img69-47.gif)
,
![$b$](img70-47.gif)
, and
![$c$](img71-47.gif)
.
Problem 17
Let
![% latex2html id marker 963
$\mbox{$\mathbb{Z}^{}$}[x]$](img72-47.gif)
be the ring of polynomials in the indeterminate
![$x$](img73-47.gif)
with
coefficients in the ring
![% latex2html id marker 967
$\mathbb{Z}^{}$](img74-46.gif)
of integers. Let
![% latex2html id marker 971
$\mathfrak{I}\subset \mbox{$\mathbb{Z}^{}$}[x]$](img75-46.gif)
be the ideal generated by
![$13$](img76-46.gif)
and
![$x-4$](img77-46.gif)
. Find an integer
![$m$](img78-46.gif)
such
that
![$0\leq m\leq 12$](img79-46.gif)
and
Problem 18
Prove that any finite group is isomorphic to
- a subgroup of the group of permutations of
objects
- a subgroup of the group of permutations of
objects which consists only of even permutations.
Previous: Spring96
Next: Spring97
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10