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Preliminary Exam - Fall 1998
Problem 1
Let
![$M$](img1-52.gif)
be a metric space with metric
![$d$](img2-52.gif)
.
Let
![$C$](img3-52.gif)
be a compact subset of
![$M$](img4-52.gif)
,
and let
![$\left(U_\alpha\right)_{\alpha \in I}$](img5-52.gif)
be an open cover of
![$C$](img6-52.gif)
.
Show that there exists
![$\epsilon > 0$](img7-52.gif)
such that,
for every
![$p \in C$](img8-52.gif)
,
the open ball
![$B(p, \epsilon)$](img9-52.gif)
is contained in
at least one of the sets
![$U_\alpha$](img10-52.gif)
.
Problem 2
Find a function
![$y(x)$](img11-52.gif)
such that
![$y^{(4)} + y = 0$](img12-52.gif)
for
![$x \geq 0$](img13-52.gif)
,
![$y(0) = 0$](img14-52.gif)
,
![$y'(0) = 1$](img15-52.gif)
and
![$\displaystyle{\lim_{x\rightarrow\infty} y(x) = \lim_{x\rightarrow\infty} y'(x) = 0}$](img16-52.gif)
.
Problem 3
Prove that for any real
![$\alpha > 1$](img17-52.gif)
,
Problem 4
Let
![$f$](img19-52.gif)
be analytic in the closed unit disc, with
![$f(-\log 2) = 0$](img20-52.gif)
and
![$\vert f(z)\vert \leq \vert e^z\vert$](img21-52.gif)
for all
![$z$](img22-52.gif)
with
![$\vert z\vert = 1$](img23-52.gif)
. How large can
![$\vert f(\log 2)\vert$](img24-52.gif)
be?
Problem 5
Let
![% latex2html id marker 745
$f : \mathbb{R}^n \rightarrow \mathbb{R}$](img25-52.gif)
be a function such that
- the function
defined by
is bilinear,
- for all
and
,
.
Show that there is a linear transformation
![% latex2html id marker 757
$A : \mathbb{R}^n \rightarrow \mathbb{R}^n$](img31-52.gif)
such that
![$f(x) = \langle x, Ax \rangle $](img32-52.gif)
where
![$\langle \cdot, \cdot \rangle $](img33-52.gif)
is the usual inner product on
![% latex2html id marker 763
$\mathbb{R}^n$](img34-52.gif)
(in other words,
![$f$](img35-52.gif)
is a quadratic form).
Problem 6
Let
![$A$](img36-52.gif)
and
![$B$](img37-52.gif)
be linear transformations
on a finite dimensional vector space
![$V$](img38-52.gif)
.
Prove that
![$\dim \ker (AB) \leq \dim \ker A + \dim \ker B$](img39-52.gif)
.
Problem 7
A real symmetric
![$n \times n$](img40-52.gif)
matrix
![$A$](img41-52.gif)
is called
positive semi-definite if
![$x^tAx \geq 0$](img42-52.gif)
for all
![% latex2html id marker 808
$x \in \mathbb{R}^n$](img43-52.gif)
.
Prove that
![$A$](img44-52.gif)
is positive semi-definite if and only if
![$\mathrm{tr}\,AB \geq 0$](img45-52.gif)
for every real symmetric positive semi-definite n
![$\times$](img46-52.gif)
n matrix
![$B$](img47-52.gif)
.
Problem 8
Let
![$R$](img48-52.gif)
be a finite ring with identity.
Let
![$a$](img49-52.gif)
be an element of
![$R$](img50-52.gif)
which is not a zero divisor.
Show that
![$a$](img51-52.gif)
is invertible.
Problem 9
Suppose that
![$G$](img52-52.gif)
is a finite group such that
every Sylow subgroup is normal and abelian.
Show that
![$G$](img53-52.gif)
is abelian.
Problem 10
Let
![$f$](img54-52.gif)
be a real function on
![$[a, b]$](img55-52.gif)
.
Assume that
![$f$](img56-52.gif)
is differentiable and that
![$f'$](img57-52.gif)
is Riemann integrable.
Prove that
Problem 11
Find the minimal value of the areas of hexagons circumscribing
the unit circle in
![% latex2html id marker 862
$\mathbb{R}^2$](img59-52.gif)
.
Note: See also Problem
.
Problem 12
Let
![$\varphi(x, y)$](img60-52.gif)
be a function with
continuous second order partial derivatives such that
-
in the punctured plane
,
-
and
as
.
Let
![$C_R$](img66-52.gif)
be the circle
![$x^2 + y^2 = R^2$](img67-52.gif)
.
Show that the line integral
is independent of
![$R$](img69-52.gif)
, and evaluate it.
Problem 13
Let
![$f$](img70-52.gif)
be an entire function.
Define
![% latex2html id marker 914
$\Omega = \mathbb{C}\setminus (-\infty, 0]$](img71-52.gif)
,
the complex plane with the ray
![$(-\infty, 0]$](img72-52.gif)
removed.
Suppose that for all
![$z \in \Omega$](img73-52.gif)
,
![$\vert f(z)\vert \leq \vert\log z\vert$](img74-51.gif)
,
where
![$\log z$](img75-51.gif)
is the principal branch of the logarithm.
What can you conclude about the function
![$f$](img76-51.gif)
?
Problem 14
Let
![$z_1, \dots, z_n$](img77-51.gif)
be distinct complex numbers, and let
![$a_1, \dots, a_n$](img78-51.gif)
be nonzero complex numbers such that
![$S_p = \sum_{j = 1}^n a_j z_j^p = 0$](img79-51.gif)
for
![$p = 0, 1, \dots, m - 1$](img80-51.gif)
but
![$S_m \neq 0$](img81-50.gif)
.
Here
![$1 \leq m \leq n - 1$](img82-50.gif)
.
How many zeros does the rational function
![$f(z) = \sum_{j = 1}^n \frac{a_j}{z - z_j}$](img83-49.gif)
have in
![% latex2html id marker 951
$\mathbb{C}$](img84-48.gif)
?
Explain why
![$m \geq n$](img85-47.gif)
is impossible.
Problem 15
Let
![$A$](img86-47.gif)
and
![$B$](img87-45.gif)
be n
![$\times$](img88-44.gif)
n matrices.
Show that the eigenvalues of
![$AB$](img89-44.gif)
are
the same as the eigenvalues of
![$BA$](img90-44.gif)
.
Problem 17
Let
![% latex2html id marker 1010
$\mbox{\bf {F}}$](img100-37.gif)
be a finite field with
![$q$](img101-37.gif)
elements.
Denote by
![% latex2html id marker 1014
$GL_n(\mbox{\bf {F}})$](img102-37.gif)
the group of
invertible n
![$\times$](img103-34.gif)
n matrices with entries if
![% latex2html id marker 1018
$\mbox{\bf {F}}$](img104-33.gif)
.
What is the order of this group?
Problem 18
Show that the field
![% latex2html id marker 1030
$\mathbb{Q}(t_1, \dots, t_n)$](img105-32.gif)
of rational functions in
![$n$](img106-29.gif)
variables
over the rational numbers is isomorphic to a subfield of
![% latex2html id marker 1034
$\mathbb{R}$](img107-28.gif)
.
Previous: Spring98
Next: Spring99
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10