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Preliminary Exam - Summer 1980
Problem 1
Exhibit a real 3
![$\times $](img1-10.gif)
3 matrix having minimal polynomial
![$(t^2+1)(t-10)$](img2-10.gif)
, which, as a linear transformation of
![% latex2html id marker 663
$\mathbb{R}^{3}$](img3-10.gif)
,
leaves invariant the line
![$L$](img4-10.gif)
through
![$(0,0,0)$](img5-10.gif)
and
![$(1,1,1)$](img6-10.gif)
and the plane
through
![$(0,0,0)$](img7-10.gif)
perpendicular to
![$L$](img8-10.gif)
.
Problem 2
Which of the following matrix equations have a real matrix solution X?
(It is not necessary to exhibit solutions.)
-
-
-
-
Problem 3
Let
![$T:V \to V$](img13-10.gif)
be an invertible linear transformation of a vector
space
![$V$](img14-10.gif)
. Denote by
![$G$](img15-10.gif)
the group of all maps
![$f_{k,a}: V \to V$](img16-10.gif)
where
![% latex2html id marker 750
$k \in \mbox{$\mathbb{Z}^{}$}$](img17-10.gif)
,
![$a \in V$](img18-10.gif)
, and for
![$x \in V$](img19-10.gif)
,
Prove that the commutator subgroup
![$G'$](img21-10.gif)
of
![$G$](img22-10.gif)
is isomorphic to the
additive group of the vector space
![$(T-I)V$](img23-10.gif)
, the image of
![$T-I$](img24-10.gif)
.
(
![$G'$](img25-10.gif)
is generated by all
![$ghg^{-1}h^{-1}$](img26-10.gif)
,
![$g$](img27-10.gif)
and
![$h$](img28-10.gif)
in
![$G$](img29-10.gif)
.)
Problem 5
Consider the differential equations
Let
![$x(t)$](img42-10.gif)
and
![$y(t)$](img43-10.gif)
be a solution defined for all
![$t\geq 0$](img44-10.gif)
with
![$x(0)>0$](img45-10.gif)
and
![$y(0)>0$](img46-10.gif)
. Prove that
![$x(t)$](img47-10.gif)
and
![$y(t)$](img48-10.gif)
are bounded.
Problem 6
Let
![$C$](img49-10.gif)
denote the positively oriented circle
![$\vert z\vert = 2$](img50-10.gif)
,
![% latex2html id marker 849
$z\in \mbox{$\mathbb{C}\,^{}$}$](img51-10.gif)
. Evaluate the integral
where the branch of the square root is chosen so that
Problem 7
Exhibit a conformal map from the set
![% latex2html id marker 872
$\{z \in \mbox{$\mathbb{C}\,^{}$} \;\vert\; \vert z\vert<1,
\Re z > 0 \}$](img54-10.gif)
onto
![% latex2html id marker 876
$\mathbb{D}=\{z\in\mbox{$\mathbb{C}\,^{}$}\;\vert\;\vert z\vert<1\}$](img55-10.gif)
.
Problem 8
Give an example of a subset of
![% latex2html id marker 892
$\mathbb{R}^{}$](img56-10.gif)
having uncountably many connected
components. Can such a subset be open? Closed?
Problem 9
For each
![% latex2html id marker 914
$(a,b,c) \in \mbox{$\mathbb{R}^{3}$}$](img57-10.gif)
, consider the series
Determine the values of
![$(a,b,c)$](img59-10.gif)
for which the series
- converges absolutely;
- converges but not absolutely;
- diverges.
Problem 10
Let
![% latex2html id marker 956
$f:\mbox{$\mathbb{R}^{n}$} \to \mbox{$\mathbb{R}^{}$}$](img60-10.gif)
be a function whose partial derivatives of
order
![$\leq 2$](img61-10.gif)
are everywhere defined and continuous.
- Let
be a critical point of f (i.e.,
).
Prove that a is a local minimum provided the Hessian matrix
is positive definite at
.
- Assume the Hessian matrix is positive definite at all x.
Prove that f has, at most, one critical point.
Problem 11
Prove that every finite group is isomorphic to
- A group of permutations;
- A group of even permutations.
Problem 12
Let
![% latex2html id marker 1013
$S \subset \mbox{$\mathbb{R}^{3}$}$](img66-10.gif)
denote the ellipsoidal surface defined by
Let
![% latex2html id marker 1017
$T \subset \mbox{$\mathbb{R}^{3}$}$](img68-10.gif)
be the surface
defined by
Prove that there exist points
![$p \in S,\; q \in T$](img70-10.gif)
, such that the line
![$\overline{pq}$](img71-10.gif)
is
perpendicular to
![$S$](img72-10.gif)
at
![$p$](img73-10.gif)
and to
![$T$](img74-10.gif)
at
![$q$](img75-10.gif)
.
Problem 13
Let
![$\mathfrak{J}$](img76-10.gif)
be the ideal in the ring
![% latex2html id marker 1054
$\mbox{$\mathbb{Z}^{}$}[x]$](img77-10.gif)
(of polynomials with integer
coefficients) generated by
![$x-5$](img78-10.gif)
and
![$14$](img79-10.gif)
. Find
![% latex2html id marker 1062
$n \in \mbox{$\mathbb{Z}^{}$}$](img80-10.gif)
such
that
![$0\leq n\leq 13$](img81-10.gif)
and
![$(x^3+2x+1)^{50}-n \in \mathfrak{J}$](img82-10.gif)
.
Problem 14
Let
![$A$](img83-10.gif)
and
![$B$](img84-10.gif)
be real 2
![$\times$](img85-10.gif)
2 matrices with
![$A^2=B^2 =I,\;
AB + BA=0$](img86-10.gif)
. Prove there exists a real nonsingular matrix
![$T$](img87-10.gif)
with
Problem 15
Let
![$E$](img89-9.gif)
be a finite-dimensional vector space over a field
![% latex2html id marker 1119
$\mbox{\bf {F}}$](img90-9.gif)
. Suppose
![$B:E \times $](img91-9.gif)
E
![% latex2html id marker 1123
$ \to \mbox{\bf {F}}$](img92-9.gif)
is a bilinear map (not necessarily symmetric).
Define subspaces
Prove that
![$\dim E_1=\dim E_2$](img95-9.gif)
.
Problem 16
Let
![$(a_n)$](img96-9.gif)
be a sequence of nonzero real numbers. Prove that the
sequence of functions
has a subsequence converging to a continuous function.
Problem 17
Let
![% latex2html id marker 1175
$f:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$](img99-9.gif)
be monotonically increasing (perhaps
discontinuous). Suppose
![$0<f(0)$](img100-9.gif)
and
![$f(100)<100$](img101-9.gif)
. Prove
![$f(x)=x$](img102-9.gif)
for some
![$x$](img103-7.gif)
.
Problem 18
How many zeros does the complex polynomial
have in the annulus
![$1 < \vert z\vert < 2$](img105-6.gif)
?
Problem 19
Let
![$f$](img106-5.gif)
be a meromorphic function on
![% latex2html id marker 1211
$\mathbb{C}\,^{}$](img107-5.gif)
which is analytic in a
neighborhood of
![$0$](img108-5.gif)
. Let its Maclaurin series be
with all
![$a_k \geq 0$](img110-5.gif)
. Suppose there is a pole of modulus
![$r>0$](img111-5.gif)
and no pole has modulus
![$< r$](img112-5.gif)
. Prove there is a pole at
![$z=r$](img113-5.gif)
.
Problem 20
Prove that the initial value problem
has a solution
![$x(t)$](img115-3.gif)
defined for all
![% latex2html id marker 1240
$t \in \mbox{$\mathbb{R}^{}$}$](img116-3.gif)
.
Previous: Spring80
Next: Fall80
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10