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Preliminary Exam - Summer 1981
Problem 1
Let
Justify the statement
where
Problem 3
Prove or disprove: The set
![% latex2html id marker 714
$\mathbb{Q}\,^{}$](img16-13.gif)
of rational numbers is the intersection of
a countable family of open subsets of
![% latex2html id marker 716
$\mathbb{R}^{}$](img17-13.gif)
.
Problem 4
Let
![% latex2html id marker 742
$f:\mbox{$\mathbb{R}^{}$} \to \mbox{$\mathbb{R}^{}$}$](img18-13.gif)
be continuous, with
Show that there is a sequence
![$(x_n)$](img20-13.gif)
such that
![$x_n \to \infty$](img21-13.gif)
,
![$x_nf(x_n) \to 0$](img22-13.gif)
, and
![$x_nf(-x_n) \to 0$](img23-13.gif)
as
![$n \to \infty$](img24-13.gif)
.
Problem 5
Let
![$S$](img25-13.gif)
denote the vector space of real
![$n \times n$](img26-13.gif)
skew-symmetric matrices.
For a nonsingular matrix
![$A$](img27-13.gif)
, compute the determinant of the linear map
![$T_A:S\to S,\, T_A(X)=AXA^{-1}$](img28-13.gif)
.
Problem 6
Let
![% latex2html id marker 788
$S \mathbb{O}(3)$](img29-13.gif)
denote the group of orthogonal transformations of
![% latex2html id marker 790
$\mathbb{R}^{3}$](img30-13.gif)
of
determinant
![$1$](img31-13.gif)
. Let
![% latex2html id marker 794
$Q \subset S \mathbb{O}(3)$](img32-13.gif)
be the subset of symmetric
transformations
![$\neq I$](img33-13.gif)
. Let
![$P^2$](img34-13.gif)
denote the space of lines through
the origin in
![% latex2html id marker 800
$\mathbb{R}^{3}$](img35-13.gif)
.
- Show that
and
are compact metric spaces (in their
usual topologies).
- Show that
and
are homeomorphic.
Problem 7
Compute
where
![$C$](img41-13.gif)
is the circle
![$\vert z\vert = 1/5$](img42-13.gif)
, positively oriented.
Problem 8
Show that
![$x^{10}+x^9+x^8+\cdots+x+1$](img43-13.gif)
is irreducible over
![% latex2html id marker 855
$\mathbb{Q}\,^{}$](img44-13.gif)
. How about
Problem 9
Let
![% latex2html id marker 886
$f:\mbox{$\mathbb{R}^{}$}\to \mbox{$\mathbb{R}^{}$}$](img46-13.gif)
be the function of period
![$2\pi$](img47-13.gif)
such that
![$f(x)=x^3$](img48-13.gif)
for
![$-\pi \leq x < \pi$](img49-13.gif)
.
- Prove that the Fourier series for
has the form
and write an integral formula for
(do
not evaluate it).
- Prove that the Fourier series converges for all
.
- Prove
Problem 11
Show that the equation
has, for each sufficiently small
![$\varepsilon>0$](img70-13.gif)
, exactly two solutions. Let
![$x(\varepsilon)$](img71-13.gif)
be the smaller one. Show that
- 1.
-
as
;
yet for any
,
- 2.
-
as
.
Problem 12
Show that no commutative ring with identity has additive group isomorphic
to
![% latex2html id marker 1014
$\mbox{$\mathbb{Q}\,^{}$}/\mbox{$\mathbb{Z}^{}$}$](img77-13.gif)
.
Problem 13
Let
![$G$](img78-13.gif)
be an additive group, and
![$u,v: G \to G$](img79-13.gif)
homomorphisms. Show that
the map
![$f: G \to G$](img80-13.gif)
,
![$f(x) = x-v \left( u(x) \right)$](img81-13.gif)
is surjective
if the map
![$h:G\to G$](img82-13.gif)
,
![$h(x)=x-u\left(v(x)\right)$](img83-13.gif)
is surjective.
Problem 14
Let
![% latex2html id marker 1051
$I \subset \mbox{$\mathbb{R}^{}$}$](img84-13.gif)
be the open interval from
![$0$](img85-13.gif)
to
![$1$](img86-13.gif)
.
Let
![% latex2html id marker 1059
$f:I \to \mbox{$\mathbb{C}\,^{ }$}$](img87-13.gif)
be
![$C^1$](img88-12.gif)
(i.e., the real and imaginary parts are continuously differentiable).
Suppose that
![$f(t) \to 0,\; f'(t) \to C \neq 0$](img89-12.gif)
as
![$t \to 0+$](img90-12.gif)
. Show
that the function
![$g(t)=\vert f(t)\vert$](img91-12.gif)
is
![$C^1$](img92-12.gif)
for sufficiently small
![$t>0$](img93-12.gif)
and
that
![$\lim_{t\to 0+}g'(t)$](img94-12.gif)
exists, and evaluate the limit.
Problem 15
Let
![$V$](img95-12.gif)
be a finite-dimensional vector space over the rationals
![% latex2html id marker 1091
$\mathbb{Q}\,^{}$](img96-12.gif)
and
let
![$M$](img97-12.gif)
be an automorphism of
![$V$](img98-12.gif)
such that
![$M$](img99-12.gif)
fixes no
nonzero vector in
![$V$](img100-12.gif)
. Suppose that
![$M^p$](img101-12.gif)
is the identity map on
![$V$](img102-12.gif)
,
where
![$p$](img103-10.gif)
is a prime number. Show that the dimension of
![$V$](img104-10.gif)
is
divisible by
![$p-1$](img105-9.gif)
.
Problem 16
Let
![$\{f_n\}$](img106-8.gif)
be a sequence of continuous functions
![% latex2html id marker 1137
$[0,1] \to \mbox{$\mathbb{R}^{}$}$](img107-8.gif)
such that
Let
![% latex2html id marker 1141
$K:[0,1] \times [0,1] \to \mbox{$\mathbb{R}^{}$}$](img109-8.gif)
be continuous. Define
![% latex2html id marker 1145
$g_n:[0,1] \to \mbox{$\mathbb{R}^{}$}$](img110-8.gif)
by
Prove that the sequence
![$\{g_n\}$](img112-8.gif)
converges uniformly.
Problem 17
Suppose that
![$f(z)$](img113-8.gif)
and
![$g(z)$](img114-6.gif)
are entire functions such that
![$\vert f(z)\vert\leq \vert g(z)\vert$](img115-6.gif)
for all
![$z$](img116-6.gif)
. Show that
![$f(z)=cg(z)$](img117-5.gif)
for some
constant
![% latex2html id marker 1175
$c \in \mbox{$\mathbb{C}\,^{}$}$](img118-5.gif)
.
Problem 18
Let
![$A$](img119-5.gif)
and
![$B$](img120-5.gif)
be square matrices of rational numbers such that
![$CAC^{-1}=B$](img121-5.gif)
for some real matrix
![$C$](img122-5.gif)
. Prove that such a
![$C$](img123-5.gif)
can be chosen to have
rational entries.
Problem 19
Prove that the number of roots of the equation
![$z^{2n}+\alpha^2z^{2n-1}+\beta^2=0$](img124-4.gif)
(
![$n$](img125-4.gif)
a natural number,
![$\alpha$](img126-4.gif)
and
![$\beta$](img127-4.gif)
real, nonzero) that have
positive real part is
if
is even, and
if
is odd.
Problem 20
Let
![$y=y(x)$](img132-4.gif)
be a solution of the differential equation
![$y''=-\vert y\vert$](img133-4.gif)
with
![$-\infty<x<\infty$](img134-3.gif)
,
![$y(0)=1$](img135-3.gif)
, and
![$y'(0)=0$](img136-3.gif)
.
- Show that
is an even function.
- Show that
has exactly one zero on the positive real axis.
Previous: Spring81
Next: Fall81
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10