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Preliminary Exam - Fall 1997
Problem 1
Define a sequence of real numbers
![$(x_n)$](img1-49.gif)
by
Show that
![$(x_n)$](img3-49.gif)
converges, and evaluate its limit.
Problem 2
Let
![$f$](img4-49.gif)
be a real valued function that is differentiable
on an open interval containing
![$[a, b]$](img5-49.gif)
.
Prove that if
![$f'(a) < 0$](img6-49.gif)
and
![$f'(b) > 0$](img7-49.gif)
then there is
a point
![$c \in (a, b)$](img8-49.gif)
such that
![$f'(c) = 0$](img9-49.gif)
.
Problem 3
Let
![$f$](img10-49.gif)
be an entire function such that, for all
![$z$](img11-49.gif)
,
![$\vert f(z)\vert = \vert\sin z \vert$](img12-49.gif)
. Prove that there is a constant
![$C$](img13-49.gif)
of modulus
![$1$](img14-49.gif)
such that
![$f(z) = C \sin z $](img15-49.gif)
.
Problem 4
Evaluate the integral
where
![$n > 0$](img17-49.gif)
is an integer.
Problem 5
Let
![% latex2html id marker 719
$\mathbb{D}= \{z \;\vert\; \vert z\vert < 1\}$](img18-49.gif)
, the open unit disc in the complex plane.
Suppose that
![% latex2html id marker 721
$f : \mathbb{D}\rightarrow \mathbb{D}$](img19-49.gif)
is analytic,
and that there exist two distinct points
![% latex2html id marker 723
$a, b \in \mathbb{D}$](img20-49.gif)
with
![$f(a) = a$](img21-49.gif)
,
![$f(b) = b$](img22-49.gif)
. Prove that
![$f(z) = z$](img23-49.gif)
for all
![% latex2html id marker 731
$z \in \mathbb{D}$](img24-49.gif)
.
Problem 6
Let
![$\alpha_1, \alpha_2, \dots ,\alpha_n$](img25-49.gif)
be distinct real numbers.
Show that the
![$n$](img26-49.gif)
exponential functions
![$e^{\alpha_1t}, e^{\alpha_2t}, \dots ,e^{\alpha_nt}$](img27-49.gif)
are linearly independent over the real numbers.
Problem 7
Define the
index of a real symmetric matrix
![$A$](img28-49.gif)
_matrix,>index
to be the number of strictly positive eigenvalues of
![$A$](img29-49.gif)
minus the number of strictly negative eigenvalues.
Suppose
![$A$](img30-49.gif)
, and
![$B$](img31-49.gif)
are real symmetric
![$n \times n$](img32-49.gif)
matrices
such that
![$x^tAx \leq x^tBx$](img33-49.gif)
for all
![$n \times 1$](img34-49.gif)
matrices
![$x$](img35-49.gif)
.
Prove the the index of
![$A$](img36-49.gif)
is less than or equal to the index of
![$B$](img37-49.gif)
.
Problem 8
Suppose
![$H_i$](img38-49.gif)
is a normal subgroup of a group
![$G$](img39-49.gif)
for
![$1 \leq i \leq k$](img40-49.gif)
, such that
![$H_i \cap H_j = \{1\}$](img41-49.gif)
for
![$i \neq j$](img42-49.gif)
.
Prove that
![$G$](img43-49.gif)
contains a subgroup isomorphic to
![$H_1 \times H_2 \times \cdots \times H_k$](img44-49.gif)
if
![$k = 2$](img45-49.gif)
,
but not necessarily if
![$k \geq 3$](img46-49.gif)
.
Problem 9
Prove that if
![$p$](img47-49.gif)
is prime then every group of order
![$p^2$](img48-49.gif)
is abelian.
Problem 10
Prove that for all
![$x > 0$](img49-49.gif)
,
![$\sin x > x - x^3/6$](img50-49.gif)
.
Problem 11
Let
![% latex2html id marker 848
$f : \mbox{$\mathbb{R}^{}$} \rightarrow \mbox{$\mathbb{R}^{}$}$](img51-49.gif)
be twice differentiable,
and suppose that for all
![% latex2html id marker 852
$x \in \mbox{$\mathbb{R}^{}$}$](img52-49.gif)
,
![$\vert f(x)\vert \leq 1$](img53-49.gif)
and
![$\vert f''(x)\vert \leq 1$](img54-49.gif)
.
Prove that
![$\vert f'(x)\vert \leq 2$](img55-49.gif)
for all
![% latex2html id marker 862
$x \in \mbox{$\mathbb{R}^{}$}$](img56-49.gif)
.
Problem 12
A map
![% latex2html id marker 919
$f : \mbox{$\mathbb{R}^{m}$} \rightarrow \mbox{$\mathbb{R}^{n}$}$](img57-49.gif)
_map>compact _map>closed
is
proper if it is continuous and
![$f^{-1}(B)$](img58-49.gif)
is compact for each compact subset
![$B$](img59-49.gif)
of
![% latex2html id marker 927
$\mbox{$\mathbb{R}^{n}$}$](img60-49.gif)
;
![$f$](img61-49.gif)
is
closed if it is continuous and
![$f(A)$](img62-49.gif)
is closed
for each closed subset
![$A$](img63-49.gif)
of
![% latex2html id marker 937
$\mbox{$\mathbb{R}^{m}$}$](img64-49.gif)
.
- Prove that every proper map
is closed.
- Prove that every one-to-one closed map
is proper.
Problem 13
Conformally map the region inside the disc
![% latex2html id marker 989
$\{z \in \mbox{$\mathbb{C}\,^{}$} \;\vert\; \vert z - 1\vert \leq 1\}$](img67-49.gif)
and outside the
disc
![% latex2html id marker 993
$\{z \in \mbox{$\mathbb{C}\,^{}$} \;\vert\; \vert z - \frac{1}{2}\vert \leq \frac{1}{2}\}$](img68-49.gif)
onto the upper half-plane.
Problem 14
Evaluate the integral
where
![$k \geq 0$](img70-49.gif)
.
Problem 15
Let
![% latex2html id marker 1030
$M_{n\times n}(\mbox{\bf {K}})$](img71-49.gif)
be the vector space of
![$n \times n$](img72-49.gif)
matrices over a field
![% latex2html id marker 1034
$\mbox{\bf {K}}$](img73-49.gif)
.
Find the dimension of the subspace of
![% latex2html id marker 1036
$M_{n\times n}(\mbox{\bf {K}})$](img74-48.gif)
spanned by
![% latex2html id marker 1038
$\{XY - YX\,\vert\, X, Y \in M_{n\times n}(\mbox{\bf {K}})\}$](img75-48.gif)
.
Problem 16
Prove that if
![$A$](img76-48.gif)
is a 2
![$\times $](img77-48.gif)
2 matrix over the integers
such that
![$A^n = I$](img78-48.gif)
for some strictly positive integer
![$n$](img79-48.gif)
,
then
![$A^{12} = I$](img80-48.gif)
.
Problem 17
A group
![$G$](img81-47.gif)
is generated by two elements
![$a, b$](img82-47.gif)
, each of order
![$2$](img83-46.gif)
.
Prove that
![$G$](img84-45.gif)
has a cyclic subgroup of index
![$2$](img85-44.gif)
.
Problem 18
A finite abelian group
![$G$](img86-44.gif)
has the property that
for each positive integer
![$n$](img87-43.gif)
the set
![$\{x \in G \,\vert\, x^n = 1\}$](img88-42.gif)
has at most
![$n$](img89-42.gif)
elements.
Prove that
![$G$](img90-42.gif)
is cyclic,
and deduce that every finite field has cyclic multiplicative group.
Previous: Spring97
Next: Spring98
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10