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Preliminary Exam - Fall 1982
Problem 2
Consider the polynomial ring
![% latex2html id marker 723
$R = \mbox{$\mathbb{Z}^{}$}[x]$](img12-18.gif)
and the ideal
![$\mathfrak{I}$](img13-18.gif)
of
![$R$](img14-18.gif)
generated by
![$7$](img15-18.gif)
and
![$x-3$](img16-18.gif)
.
- Show that for each
, there is an integer
satisfying
such that
.
- Find
in the special case
.
Problem 3
Let
be the Laurent expansion for
![$\cot(\pi z)$](img24-18.gif)
on the annulus
![$1<\vert z\vert<2$](img25-18.gif)
. Compute the
![$a_n$](img26-18.gif)
for
![$n<0$](img27-18.gif)
.
Problem 4
Let
![$M$](img28-18.gif)
be an
![$n\times n$](img29-18.gif)
matrix of real numbers. Prove or disprove: The
dimension of the subspace of
![% latex2html id marker 785
$\mathbb{R}^{n}$](img30-18.gif)
generated by the rows of
![$M$](img31-18.gif)
is equal to
the dimension of the subspace of
![% latex2html id marker 789
$\mathbb{R}^{n}$](img32-18.gif)
generated by the columns of
![$M$](img33-18.gif)
.
Problem 5
Let
![$\varphi_1, \varphi_2, \ldots,\varphi_n,\ldots$](img34-18.gif)
be nonnegative continuous
functions on
![$[0,1]$](img35-18.gif)
such that the limit
exists for every
![$k = 0,1,\ldots$](img37-18.gif)
. Show that the limit
exists for every continuous function f on
![$[0,1]$](img39-18.gif)
.
Problem 6
Let
![$T$](img40-18.gif)
be a linear transformation on a finite-dimensional
![% latex2html id marker 829
$\mathbb{C}\,^{}$](img41-18.gif)
-vector space V,
and let
![$f$](img42-18.gif)
be a polynomial with coefficients in
![% latex2html id marker 833
$\mathbb{C}\,^{}$](img43-18.gif)
. If
![$\lambda$](img44-18.gif)
is an
eigenvalue of
![$T$](img45-18.gif)
, show that
![$f(\lambda)$](img46-18.gif)
is an eigenvalue of
![$f(T)$](img47-18.gif)
. Is
every eigenvalue of
![$f(T)$](img48-18.gif)
necessarily obtained in this way?
Problem 8
Let
- Show that
is a normal subgroup of
and prove that
is
isomorphic to
.
- Find a normal subgroup
of
satisfying
(where the inclusions are proper), or prove that there is no such subgroup.
Problem 9
Let f be a real valued continuous nonnegative function on
![$[0,1]$](img59-18.gif)
such that
for
![$t \in [0,1]$](img61-18.gif)
. Show that
![$f(t) \leq 1 + t$](img62-18.gif)
for
![$t \in [0,1]$](img63-18.gif)
.
Problem 10
Let
![$a$](img64-18.gif)
and
![$b$](img65-18.gif)
be complex numbers whose real parts are negative or
![$0$](img66-18.gif)
.
Prove the inequality
![$\vert e^a-e^b\vert \leq \vert a-b\vert$](img67-18.gif)
.
Problem 11
- Prove that there is no continuous map from the closed
interval
onto the open interval
.
- Find a continuous surjective map from the open interval
onto the closed interval
.
- Prove that no map in Part 2 can be bijective.
Problem 12
Let
![$A$](img72-18.gif)
and
![$B$](img73-18.gif)
be complex
![$n\times n$](img74-17.gif)
matrices having the same rank. Suppose
that
![$A^2 = A$](img75-17.gif)
and
![$B^2 = B$](img76-17.gif)
. Prove that
![$A$](img77-17.gif)
and
![$B$](img78-17.gif)
are similar.
Problem 13
Let
![$f_1,f_2,\ldots$](img79-17.gif)
be continuous functions on
![$[0,1]$](img80-17.gif)
satisfying
![$f_1\geq f_2 \geq \cdots$](img81-17.gif)
and such that
![$\lim_{n \to \infty}f_n(x) = 0$](img82-17.gif)
for each x. Must the
sequence
![$\{f_n\}$](img83-17.gif)
converge to
![$0$](img84-17.gif)
uniformly on
![$[0,1]$](img85-17.gif)
?
Problem 14
Let
![$A$](img86-17.gif)
be an
![$n\times n$](img87-17.gif)
complex matrix, and let
![$B$](img88-16.gif)
be the Hermitian
transpose of
![$A$](img89-16.gif)
(i.e.,
![$b_{ij}=\overline{a}_{ji}$](img90-16.gif)
). Suppose that
![$A$](img91-16.gif)
and
![$B$](img92-16.gif)
commute with each other. Consider the linear transformations
![$\alpha$](img93-16.gif)
and
![$\beta$](img94-16.gif)
on
![% latex2html id marker 1038
$\mathbb{C}\,^{n}$](img95-16.gif)
defined by
![$A$](img96-16.gif)
and
![$B$](img97-16.gif)
. Prove that
![$\alpha$](img98-16.gif)
and
![$\beta$](img99-16.gif)
have the same image and the same kernel.
Problem 15
Let
![$A$](img100-15.gif)
be a subgroup of an abelian group
![$B$](img101-15.gif)
. Assume that
![$A$](img102-15.gif)
is a
direct summand of
![$B$](img103-13.gif)
, i.e., there exists a subgroup
![$X$](img104-13.gif)
of
![$B$](img105-12.gif)
such that
![$A\cap X =0$](img106-11.gif)
and such that
![$B = X + A$](img107-11.gif)
. Suppose that
![$C$](img108-11.gif)
is a
subgroup of
![$B$](img109-11.gif)
satisfying
![$A \subset C \subset B$](img110-11.gif)
. Is
![$A$](img111-10.gif)
necessarily a direct summand of
![$C$](img112-10.gif)
?
Problem 16
Find all pairs of
![$C^{\infty}$](img113-10.gif)
functions
![$x(t)$](img114-8.gif)
and
![$y(t)$](img115-8.gif)
on
![% latex2html id marker 1097
$\mathbb{R}^{}$](img116-8.gif)
satisfying
Problem 18
Let
![$K$](img119-7.gif)
be a continuous function on the unit square
![$0\leq x,y \leq1$](img120-7.gif)
satisfying
![$\vert K(x,y)\vert<1$](img121-7.gif)
for all
![$x$](img122-7.gif)
and
![$y$](img123-7.gif)
. Show that there is a
continuous function
![$f(x)$](img124-6.gif)
on
![$[0,1]$](img125-6.gif)
such that we have
Can there be more than one such function
![$f$](img127-6.gif)
?
Problem 19
Let
![$a$](img128-6.gif)
and
![$b$](img129-6.gif)
be nonzero complex numbers and
![$f(z)=az+bz^{-1}$](img130-6.gif)
.
Determine the image under
![$f$](img131-6.gif)
of the unit circle
![$\{z \; \vert\; \vert z\vert = 1\}$](img132-6.gif)
.
Problem 20
Let
![$G$](img133-6.gif)
be the abelian group given by generators
![$x$](img134-5.gif)
,
![$y$](img135-5.gif)
, and
![$z$](img136-5.gif)
and by
the three relations:
Write
![$G$](img138-5.gif)
as a product of cyclic groups. How many elements of
![$G$](img139-3.gif)
have
order
![$2$](img140-3.gif)
?
Previous: Summer82
Next: Spring83
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10