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Preliminary Exam - Spring 1980
Problem 1
Let
![% latex2html id marker 684
$f:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$](img1-9.gif)
be the unique function such that
![$f(x)=x$](img2-9.gif)
if
![$-\pi \leq x < \pi$](img3-9.gif)
and
![$f(x+2n\pi )=f(x)$](img4-9.gif)
for all
![% latex2html id marker 694
$n\in \mbox{$\mathbb{Z}^{}$}$](img5-9.gif)
.
- Prove that the Fourier series of
is
- Prove that the series does not converge uniformly.
- For each
, find the sum of the series.
Problem 2
Let
![% latex2html id marker 739
$f_n:\mbox{$\mathbb{R}^{}$} \to \mbox{$\mathbb{R}^{}$}$](img9-9.gif)
be differentiable for each n = 1, 2, ...
with
![$\vert f_n'(x)\vert\leq 1$](img10-9.gif)
for all n, x. Assume
for all x. Prove that
![% latex2html id marker 747
$g:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$](img12-9.gif)
is continuous.
Problem 3
Let
![$P_2$](img13-9.gif)
denote the set of real polynomials of degree
![$\leq 2$](img14-9.gif)
.
Define the map
![% latex2html id marker 773
$J : P_2 \to \mbox{$\mathbb{R}^{}$}$](img15-9.gif)
by
Let
![$Q = \{f \in P_2 \;\vert\; f(1) = 1\}$](img17-9.gif)
. Show that
![$J$](img18-9.gif)
attains
a minimum value on
![$Q$](img19-9.gif)
and determine where the minimum occurs.
Problem 4
Let
![$a>0$](img20-9.gif)
be a constant
![$\neq 2$](img21-9.gif)
. Let
![$C_a$](img22-9.gif)
denote the positively
oriented circle of radius
![$a$](img23-9.gif)
centered at the origin. Evaluate
Problem 5
Let
be an analytic function in the open unit disc
![% latex2html id marker 817
$\mathbb{D}^{}$](img26-9.gif)
.
Assume that
Prove that
![$f$](img28-9.gif)
is injective.
Problem 6
![$G$](img29-9.gif)
is a group of order
![$n$](img30-9.gif)
,
![$H$](img31-9.gif)
a proper subgroup of order
![$m$](img32-9.gif)
,
and
![$(n/m)!<2n$](img33-9.gif)
. Prove
![$G$](img34-9.gif)
has a proper normal subgroup different
from the identity.
Problem 7
Let
![$n\geq 2$](img35-9.gif)
be an integer such that
![$2^n+n^2$](img36-9.gif)
is prime. Prove
that
Problem 9
Let
Show that every real matrix
![$B$](img53-9.gif)
such that
![$AB = BA$](img54-9.gif)
has the form
![$sI+tA$](img55-9.gif)
, where
![% latex2html id marker 908
$s, t \in \mbox{$\mathbb{R}^{}$}$](img56-9.gif)
.
Problem 10
Consider the differential equation
- Find all its constant solutions.
- Discuss
, where
is
the solution such that
Problem 11
Let
![% latex2html id marker 951
${\cal S} = \{(x,y,z) \in \mbox{$\mathbb{R}^{3}$} \;\vert\; x^2+y^2+z^2=1\}$](img61-9.gif)
denote the
unit sphere in
![% latex2html id marker 953
$\mathbb{R}^{3}$](img62-9.gif)
. Evaluate the surface integral over S:
Problem 12
Let
![$M_{3\times 3}$](img64-9.gif)
denote the vector space of real 3
![$\times$](img65-9.gif)
3 matrices.
For any matrix
![$A \in M_{3\times 3}$](img66-9.gif)
, define the linear operator
![$L_A:M_{3\times 3} \to M_{3\times 3}$](img67-9.gif)
,
![$L_A(B)=AB$](img68-9.gif)
.
Suppose that the determinant of
![$A$](img69-9.gif)
is
![$32$](img70-9.gif)
and the minimal polynomial is
![$(t-4)(t-2)$](img71-9.gif)
.
What is the trace of
![$L_A$](img72-9.gif)
?
Problem 13
Let
![$G$](img73-9.gif)
be a group of permutations of a set
![$S$](img74-9.gif)
of
![$n$](img75-9.gif)
elements. Assume
![$G$](img76-9.gif)
is
transitive; that is, for any
![$x$](img77-9.gif)
and
![$y$](img78-9.gif)
in
![$S$](img79-9.gif)
, there is some
![$\sigma \in G$](img80-9.gif)
with
![$\sigma(x) = y$](img81-9.gif)
. _group of permutations,>transitive
- Prove that
divides the order of
.
- Suppose
. For which integers
can such a
have
order
?
Problem 14
Find a real matrix
![$B$](img88-8.gif)
such that
Problem 15
Show that a vector space over an infinite field cannot be the
union of a finite number of proper subspaces.
Problem 16
Let
![% latex2html id marker 1076
$f : \mbox{$\mathbb{R}^{n}$} \to \mbox{$\mathbb{R}^{n}$}$](img90-8.gif)
be continuously differentiable. Assume the
Jacobian matrix
![$\left(\partial f_i/\partial x_j\right)$](img91-8.gif)
has rank n everywhere.
Suppose
![$f$](img92-8.gif)
is
proper; that is,
![$f^{-1}(K)$](img93-8.gif)
is compact whenever
![$K$](img94-8.gif)
is compact. Prove
![% latex2html id marker 1090
$f(\mbox{$\mathbb{R}^{n}$}) = \mbox{$\mathbb{R}^{n}$}$](img95-8.gif)
. _function,>proper
Problem 17
![$S_9$](img96-8.gif)
is the group of permutations of the set of integers from
![$1$](img97-8.gif)
to
![$9$](img98-8.gif)
.
- Exhibit an element of
of order
.
- Prove that no element of
has order
.
Problem 18
For each
![% latex2html id marker 1145
$t\in \mbox{$\mathbb{R}^{}$}$](img103-6.gif)
, let
![$P(t)$](img104-6.gif)
be a symmetric real
![$n\times n$](img105-5.gif)
matrix whose entries are continuous functions of
![$t$](img106-4.gif)
. Suppose for
all
![$t$](img107-4.gif)
that the eigenvalues of
![$P(t)$](img108-4.gif)
are all
![$\leq -1$](img109-4.gif)
. Let
![$x(t)=\left( x_1(t),\ldots,x_n(t) \right)$](img110-4.gif)
be a solution of the vector
differential equation
Prove that
Problem 19
Let
be analytic in the disc
![% latex2html id marker 1191
$\mbox{$\mathbb{D}^{}$} = \{z \in \mbox{$\mathbb{C}\,^{}$} \;\vert\; \vert z\vert<1\}$](img114-2.gif)
.
Assume
![$f$](img115-2.gif)
maps
![% latex2html id marker 1197
$\mbox{$\mathbb{D}^{}$}$](img116-2.gif)
one-to-one onto a domain
![$G$](img117-2.gif)
having area
![$A$](img118-2.gif)
. Prove
Problem 20
Does there exist an analytic function mapping the annulus
onto the annulus
and taking
![$C_1 \to C_1, C_4 \to C_2$](img122-2.gif)
, where
![$C_r$](img123-2.gif)
is the
circle of radius r?
Previous: Fall79
Next: Summer80
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10