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Preliminary Exam - Spring 2000
Problem 1
Are the 4
![$\times$](img1-55.gif)
4 matrices
similar? Explain your reasoning.
Problem 2
Let
![$(f_n)^{\infty}_{n=1}$](img3-55.gif)
be a uniformly bounded equicontinuous sequence
of real-valued functions on the compact metric space
![$(X,d)$](img4-55.gif)
.
Define the functions
![% latex2html id marker 688
$g_n:X\to\mbox{$\mathbb{R}^{}$}$](img5-55.gif)
, for
![% latex2html id marker 690
$n\in \mathbb{N}$](img6-55.gif)
by
Prove that the sequence
![$(g_n)^{\infty}_{n=1}$](img8-55.gif)
converges uniformly.
Problem 3
Prove that the group
![% latex2html id marker 704
$G=\mathbb{Q}/\mathbb{Z}$](img9-55.gif)
has no proper
subgroup of finite index.
Problem 4
Let
![$f$](img10-55.gif)
and
![$g$](img11-55.gif)
be entire functions such that
![${\displaystyle{\lim_{z\to\infty}}} f(g(z))=\infty$](img12-55.gif)
.
Prove that
![$f$](img13-55.gif)
and
![$g$](img14-55.gif)
are polynomials.
Problem 5
Let
![$a$](img15-55.gif)
and
![$x_0$](img16-55.gif)
be positive numbers, and define the sequence
![$(x_n)^{\infty}_{n=1}$](img17-55.gif)
recursively by
Prove that this sequence converges, and find its limit.
Problem 7
Let
![$f$](img35-55.gif)
be a positive function of class
![$C^2$](img36-55.gif)
on
![$(0,\infty)$](img37-55.gif)
such that
![$f'\leq 0$](img38-55.gif)
and
![$f''$](img39-55.gif)
is bounded. Prove that
![$\displaystyle{\lim_{t\to\infty}}f'(t)=0$](img40-55.gif)
.
Problem 8
Find the cardinality of the set of all subrings of
![% latex2html id marker 829
$\mathbb{Q}$](img41-55.gif)
,
the field of rational numbers.
Problem 9
Evaluate
where the direction of integration is counterclockwise.
Problem 10
Let
![$S$](img43-55.gif)
be an uncountable subset of
![% latex2html id marker 849
$\mathbb{R}$](img44-55.gif)
. Prove that there exists
a real number
![$t$](img45-55.gif)
such that both sets
![$S\cap (-\infty,t)$](img46-55.gif)
and
![$S\cap (t,\infty)$](img47-55.gif)
are uncountable.
Problem 11
Let
![$A_n$](img48-55.gif)
be the
![$n\times n$](img49-55.gif)
matrix whose entries
![$a_{jk}$](img50-55.gif)
are given by
Prove that the eigenvalues of
![$A$](img52-55.gif)
are symmetric with respect
to the origin.
Problem 12
Suppose that
![$H_1$](img53-55.gif)
and
![$H_2$](img54-55.gif)
are distinct subgroups of a group
![$G$](img55-55.gif)
such that
![$[G:H_1]=[G:H_2]=3$](img56-55.gif)
. What are the possible values of
![$[G:H_1\cap H_2]$](img57-55.gif)
? Explain.
Problem 13
Let
![$f$](img58-55.gif)
be a nonconstant entire function whose values on the
real axis are real and nonnegative. Prove that all real zeros of
![$f$](img59-55.gif)
have even order.
Problem 14
Let
![$I_1,\dots ,I_n$](img60-55.gif)
be disjoint closed nonempty subintervals of
![% latex2html id marker 915
$\mathbb{R}$](img61-55.gif)
.
- Prove that if
is a real polynomial of degree less than
such that
then
.
- Prove that there is a nonzero real polynomial
of
degree
that satisfies
.
Problem 15
Let
![$F$](img69-55.gif)
, with components
![$F_1,\dots ,F_n$](img70-55.gif)
, be a differentiable map
of
![% latex2html id marker 962
$\mbox{$\mathbb{R}^{n}$}$](img71-55.gif)
into
![% latex2html id marker 966
$\mbox{$\mathbb{R}^{n}$}$](img72-55.gif)
such that
![$F(0)=0$](img73-55.gif)
. Assume that
Prove that there is a ball
![$B$](img75-54.gif)
in
![% latex2html id marker 974
$\mbox{$\mathbb{R}^{n}$}$](img76-54.gif)
with center
![$0$](img77-54.gif)
such that
![$F(B)\subset B$](img78-54.gif)
.
Problem 16
Let
![$A$](img79-54.gif)
be a complex
![$n\times n$](img80-54.gif)
matrix such that the sequence
![$(A^n)^{\infty}_{n=1}$](img81-53.gif)
converges to a matrix
![$B$](img82-53.gif)
. Prove that
![$B$](img83-52.gif)
is similar to a diagonal matrix with zeros and ones along the main diagonal.
Problem 17
Evaluate the integrals
Problem 18
Let
![$G$](img85-50.gif)
be a finite group and
![$p$](img86-50.gif)
a prime number. Suppose
![$a$](img87-48.gif)
and
![$b$](img88-47.gif)
are elements of
![$G$](img89-47.gif)
of order
![$p$](img90-47.gif)
such that
![$b$](img91-46.gif)
is not in the
subgroup generated by
![$a$](img92-45.gif)
. Prove that
![$G$](img93-45.gif)
contains at least
![$p^2-1$](img94-45.gif)
elements of order
![$p$](img95-43.gif)
.
Previous: Fall99
Next: Contents
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10