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Preliminary Exam - Fall 1987
Problem 1
Prove that
![$(\cos \theta )^p \leq \cos (p\,\theta )$](img1-28.gif)
for
![$0 \leq \theta \leq \pi /2$](img2-28.gif)
and
![$0 < p < 1$](img3-28.gif)
.
Problem 2
Suppose that
![$\{f_n\}$](img4-28.gif)
is a sequence of
nondecreasing functions which map the unit interval into
itself. Suppose that
pointwise and that
![$f$](img6-28.gif)
is a continuous function. Prove
that
![$f_n(x) \to f(x)$](img7-28.gif)
uniformly as
![$n \to \infty$](img8-28.gif)
,
![$0 \leq x \leq 1$](img9-28.gif)
. Note that the functions
![$f_n$](img10-28.gif)
are not
necessarily continuous.
Problem 3
Show that the following limit exists
and is finite:
Problem 4
Let u and v be two real valued
![$C^1$](img12-28.gif)
functions on
![% latex2html id marker 706
$\mathbb{R}^{2}$](img13-28.gif)
such that
the gradient
![$\nabla u$](img14-28.gif)
is never
![$0$](img15-28.gif)
, and such that, at each point,
![$\nabla v$](img16-28.gif)
and
![$\nabla u$](img17-28.gif)
are linearly dependent vectors. Given
![% latex2html id marker 718
$p_0 = (x_0,y_0) \in \mbox{$\mathbb{R}^{2}$}$](img18-28.gif)
, show that there is a
![$C^1$](img19-28.gif)
function
![$F$](img20-28.gif)
of one variable such that
![$v(x,y) = F\left( u(x,y)\right) $](img21-28.gif)
in
some neighborhood of
![$p_0$](img22-28.gif)
.
Problem 5
Calculate
![$A^{100}$](img23-28.gif)
and
![$A^{-7}$](img24-28.gif)
, where
Problem 6
Let
![$G$](img26-28.gif)
and
![$H$](img27-28.gif)
be finite groups of relatively prime order. Show that
![$\mathrm{Aut}(G \times H)$](img28-28.gif)
, the group of automorphisms of
![$G \times H$](img29-28.gif)
, is
isomorphic to the direct product of
![$\mathrm{Aut}(G)$](img30-28.gif)
and
![$\mathrm{Aut}(H)$](img31-28.gif)
.
Problem 7
Let
![$A$](img32-28.gif)
and
![$B$](img33-28.gif)
be real n
![$\times$](img34-28.gif)
n symmetric matrices with
![$B$](img35-28.gif)
positive
definite. Consider the function defined for
![$x\neq 0$](img36-28.gif)
by
- Show that
attains its maximum value.
- Show that any maximum point
for
is an eigenvector
for a certain matrix related to
and
and show which matrix.
Problem 8
Let
![$R$](img43-28.gif)
be the set of 2
![$\times$](img44-28.gif)
2 matrices of the form
where a, b are elements of a given field
![% latex2html id marker 841
$\mbox{\bf {F}}$](img46-28.gif)
. Show that with
the usual matrix operations,
![$R$](img47-28.gif)
is a commutative ring
with identity. For which of the following fields
![% latex2html id marker 845
$\mbox{\bf {F}}$](img48-28.gif)
is
![$R$](img49-28.gif)
a field:
![% latex2html id marker 857
$F = \mbox{$\mathbb{Q}\,^{}$},\, \mbox{$\mathbb{C}\,^{}$},\, \mbox{$\mathbb{Z}^{}$}_5,\, \mbox{$\mathbb{Z}^{}$}_7$](img50-28.gif)
?
Problem 9
Evaluate the integral
Problem 10
If
![$f(z)$](img52-28.gif)
is analytic in the open disc
![$\vert z\vert<1$](img53-28.gif)
, and
![$\vert f(z)\vert\leq 1/(1-\vert z\vert)$](img54-28.gif)
, show that
Problem 11
Let
![$V$](img56-28.gif)
be a finite-dimensional vector space and
![$T : V \to V$](img57-28.gif)
a diagonalizable linear transformation.
Let
![$W \subset V$](img58-28.gif)
be a
linear subspace which is mapped into itself by
![$T$](img59-28.gif)
. Show that the
restriction of
![$T$](img60-28.gif)
to
![$W$](img61-28.gif)
is diagonalizable.
Problem 12
Given two
real ![$n\times n$](img62-28.gif)
matrices
![$A$](img63-28.gif)
and
![$B$](img64-28.gif)
, suppose that there is
a nonsingular
complex matrix
![$C$](img65-28.gif)
such that
![$CAC^{-1} = B$](img66-28.gif)
.
Show
that there exists a
real nonsingular
![$n\times n$](img67-28.gif)
matrix
![$C$](img68-28.gif)
with
this property.
Problem 13
Let
![$A$](img69-28.gif)
be the group of rational numbers under addition, and let
![$M$](img70-28.gif)
be the group of positive rational numbers under multiplication.
Determine all homomorphisms
![$\varphi : A \to M$](img71-28.gif)
.
Problem 14
Show that
![% latex2html id marker 963
$\mbox{$M_{n \times n}$}(\mbox{\bf {F}})$](img72-28.gif)
, the ring of all
![$n\times n$](img73-28.gif)
matrices over the
field
![% latex2html id marker 967
$\mbox{\bf {F}}$](img74-27.gif)
, has no proper two sided ideals.
Problem 15
Let
![$f(z)$](img75-27.gif)
be analytic for
![$z \neq 0$](img76-27.gif)
, and suppose that
![$f(1/z) = f(z)$](img77-27.gif)
.
Suppose also that
![$f(z)$](img78-27.gif)
is real for all
![$z$](img79-27.gif)
on the unit
circle
![$\vert z\vert=1$](img80-27.gif)
. Prove that
![$f(z)$](img81-27.gif)
is real for all real
![$z \neq 0$](img82-27.gif)
.
Problem 16
How many zeros (counting multiplicities) does the polynomial
have in the annular region
![$1 \leq \vert z\vert \leq 2$](img84-27.gif)
?
Problem 17
Let
![$u$](img85-27.gif)
be a positive
harmonic function on
![% latex2html id marker 1021
$\mathbb{R}^{2}$](img86-27.gif)
; that is,
Show that
![$u$](img88-26.gif)
is constant. _function,>harmonic
Problem 18
Find a curve
![$C$](img89-26.gif)
in
![% latex2html id marker 1042
$\mathbb{R}^{2}$](img90-26.gif)
, passing through the point
![$(3,2)$](img91-26.gif)
,
with the following property: Let
![$L(x_0,y_0)$](img92-26.gif)
be the segment of
the tangent line to
![$C$](img93-26.gif)
at
![$(x_0,y_0)$](img94-26.gif)
which lies in the first
quadrant. Then each point
![$(x_0,y_0)$](img95-26.gif)
of
![$C$](img96-26.gif)
is the midpoint of
![$L(x_0,y_0)$](img97-26.gif)
.
file=../Fig/Pr/Fa87-18,width=2.7in
Problem 19
Define a sequence of positive numbers as follows. Let
![$x_0 > 0$](img98-26.gif)
be
any positive number, and let
![$x_{n+1} = (1 + x_n)^{-1}$](img99-26.gif)
. Prove that
this sequence converges, and find its limit.
Problem 20
Let
![$S$](img100-25.gif)
be the set of all real
![$C^1$](img101-25.gif)
functions
![$f$](img102-25.gif)
on
![$[0,1]$](img103-22.gif)
such
that
![$f(0) = 0$](img104-22.gif)
and
Define
Show that the function
![$J$](img107-20.gif)
is bounded on
![$S$](img108-20.gif)
, and compute its
supremum. Is there a function
![$f_0 \in S$](img109-20.gif)
at which
![$J$](img110-20.gif)
attains
its maximum value? If so, what is
![$f_0$](img111-19.gif)
?
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Next: Spring88
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10