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Preliminary Exam - Spring 1999
Problem 2
Suppose that a sequence of functions
![% latex2html id marker 722
$f_n : \mbox{$\mathbb{R}^{}$} \rightarrow \mbox{$\mathbb{R}^{}$}$](img16-53.gif)
converges
uniformly on
![% latex2html id marker 726
$\mbox{$\mathbb{R}^{}$}$](img17-53.gif)
to a function
![% latex2html id marker 732
$f : \mbox{$\mathbb{R}^{}$} \rightarrow \mbox{$\mathbb{R}^{}$}$](img18-53.gif)
, and that
![$\displaystyle{c_n = \lim_{x \rightarrow \infty} f_n(x)}$](img19-53.gif)
exists for each positive integer
![$n$](img20-53.gif)
.
Prove that
![$\displaystyle{\lim_{n \rightarrow \infty} c_n}$](img21-53.gif)
and
![$\displaystyle{\lim_{x \rightarrow \infty} f(x)}$](img22-53.gif)
both exist and are equal.
Problem 3
Suppose that
![$f$](img23-53.gif)
is a twice differentiable real-valued function on
![% latex2html id marker 764
$\mbox{$\mathbb{R}^{}$}$](img24-53.gif)
such that
![$f(0) = 0$](img25-53.gif)
,
![$f'(0) > 0$](img26-53.gif)
, and
![$f''(x) \geq f(x)$](img27-53.gif)
for all
![$x \geq 0$](img28-53.gif)
.
Prove that
![$f(x) > 0$](img29-53.gif)
for all
![$x > 0$](img30-53.gif)
.
Problem 4
Evaluate
![$\displaystyle{\int_0^{\infty}\frac{dx}{x^c(x + 1)}}$](img31-53.gif)
for each real number
![$c \in (0, 1)$](img32-53.gif)
.
Problem 6
Let
![$p$](img43-53.gif)
,
![$q$](img44-53.gif)
,
![$r$](img45-53.gif)
and
![$s$](img46-53.gif)
be polynomials of degree at most 3.
Which, if any, of the following two conditions is sufficient
for the conclusion that the polynomials are linearly dependent?
- At 1 each of the polynomials has the value 0.
- At 0 each of the polynomials has the value 1.
Problem 7
Suppose that the minimal polynomial of a linear operator
![$T$](img47-53.gif)
on a
seven-dimensional vector space is
![$x^2$](img48-53.gif)
. What are the possible
values of the dimension of the kernel of T?
Problem 8
Let
![$M$](img49-53.gif)
be a
![$3 \times 3$](img50-53.gif)
matrix
with entries in the polynomial ring
![% latex2html id marker 890
$\mbox{$\mathbb{R}^{}$}[t]$](img51-53.gif)
such that
![$\displaystyle{ M^3 =
\left( \begin{array}{ccc}
t & 0 & 0 \\
0 & t & 0 \\
0 & 0 & t \\
\end{array} \right)}$](img52-53.gif)
.
Let
![$N$](img53-53.gif)
be the matrix with real entries
obtained by substituting
![$t = 0$](img54-53.gif)
in
![$M$](img55-53.gif)
.
Prove that
![$N$](img56-53.gif)
is similar to
![$\displaystyle{\left( \begin{array}{ccc}
0 & 1 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0 \\
\end{array} \right)}$](img57-53.gif)
.
Problem 9
Let
![$G$](img58-53.gif)
be a finite group, with identity
![$e$](img59-53.gif)
.
Suppose that for every
![$a, b \in G$](img60-53.gif)
distinct from
![$e$](img61-53.gif)
,
there is an automorphism
![$\sigma$](img62-53.gif)
of
![$G$](img63-53.gif)
such that
![$\sigma(a) = b$](img64-53.gif)
. Prove that
![$G$](img65-53.gif)
is abelian.
Problem 10
Suppose that
![$f$](img66-53.gif)
is a twice differentiable real function
such that
![$f''(x) > 0$](img67-53.gif)
for all
![$x \in [a, b]$](img68-53.gif)
.
Find all numbers
![$c \in [a, b]$](img69-53.gif)
at which the area between the graph
![$y = f(x)$](img70-53.gif)
,
the tangent to the graph at
![$(c, f(c))$](img71-53.gif)
,
and the lines
![$x = a$](img72-53.gif)
,
![$x = b$](img73-53.gif)
, attains its minimum value.
Problem 11
Prove that if
![$n$](img74-52.gif)
is a positive integer and
![$\alpha$](img75-52.gif)
,
![$\varepsilon$](img76-52.gif)
are real numbers with
![$\varepsilon > 0$](img77-52.gif)
,
then there is a real function
![$f$](img78-52.gif)
with derivatives of all orders such that
-
for
and all
,
-
for
,
-
.
Problem 12
Suppose that
![$f$](img85-48.gif)
is holomorphic
in some neighborhood of
![$a$](img86-48.gif)
in the complex plane.
Prove that either
![$f$](img87-46.gif)
is constant on some neighborhood of
![$a$](img88-45.gif)
,
or there exist an integer
![$n > 0$](img89-45.gif)
and
real numbers
![$\delta, \varepsilon > 0$](img90-45.gif)
such that
for each complex number
![$b$](img91-44.gif)
satisfying
![$0 < \vert b - f(a)\vert < \varepsilon$](img92-43.gif)
,
the equation
![$f(z) = b$](img93-43.gif)
has exactly
![$n$](img94-43.gif)
roots
in
![% latex2html id marker 1021
$\{z \in \mbox{$\mathbb{C}\,^{}$} \vert \;\vert z - a\vert < \delta\}$](img95-41.gif)
.
Problem 13
Let
![$b_1$](img96-41.gif)
,
![$b_2$](img97-41.gif)
, ...be a sequence of real numbers such that
![$b_k \geq b_{k + 1}$](img98-41.gif)
for all
![$k$](img99-39.gif)
and
![$\displaystyle{\lim_{k \rightarrow \infty} b_k = 0}$](img100-38.gif)
.
Prove that the power series
![$\displaystyle{\sum_{k = 1}^{\infty} b_k z^k}$](img101-38.gif)
converges for all
complex numbers
![$z$](img102-38.gif)
such that
![$\vert z\vert \leq 1$](img103-35.gif)
and
![$z \neq 1$](img104-34.gif)
.
Problem 14
Let
![$A = \left(a_{i j}\right)$](img105-33.gif)
be a
![$n \times n$](img106-30.gif)
complex matrix
such that
![$a_{i j} \neq 0$](img107-29.gif)
if
![$i = j + 1$](img108-28.gif)
but
![$a_{i j} = 0$](img109-28.gif)
if
![$i \geq j + 2$](img110-28.gif)
.
Prove that
![$A$](img111-27.gif)
cannot have more than one Jordan block for any eigenvalue.
Problem 15
Let
![$M$](img112-26.gif)
be a square complex matrix,
and let
![$S = \{XMX^{-1} \vert \; X \mbox{is non-singular}\}$](img113-26.gif)
be the set of all matrices similar to
![$M$](img114-23.gif)
.
Show that
![$M$](img115-19.gif)
is a nonzero multiple of the identity matrix
if and only if no matrix in
![$S$](img116-18.gif)
has a zero anywhere on its diagonal.
Problem 17
Let
![% latex2html id marker 1172
$f(x) \in \mbox{$\mathbb{Q}\,^{}$}[x]$](img130-10.gif)
be an irreducible polynomial of degree
![$n \geq 3$](img131-10.gif)
.
Let
![$L$](img132-10.gif)
be the splitting field of
![$f$](img133-10.gif)
,
and let
![$\alpha \in L$](img134-9.gif)
be a zero of
![$f$](img135-9.gif)
.
Given that
![% latex2html id marker 1186
$[L : \mbox{$\mathbb{Q}\,^{}$}] = n!$](img136-8.gif)
,
prove that
![% latex2html id marker 1192
$\mbox{$\mathbb{Q}\,^{}$}(\alpha^4) = \mbox{$\mathbb{Q}\,^{}$}(\alpha)$](img137-8.gif)
.
Problem 18
Let
![$G$](img138-7.gif)
be a finite simple group of order
![$n$](img139-4.gif)
.
Determine the number of normal subgroups
in the direct product
![$G \times G$](img140-4.gif)
.
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Next: Fall99
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10