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Preliminary Exam - Summer 1983
Problem 1
The number
![$21982145917308330487013369$](img1-19.gif)
is the thirteenth power
of a positive integer. Which positive integer?
Problem 2
Let
![% latex2html id marker 678
$f:\mbox{$\mathbb{C}\,^{}$} \to \mbox{$\mathbb{C}\,^{}$}$](img2-19.gif)
be an analytic function such that
is bounded for some k and m. Prove that
![$d^nf/dz^n$](img4-19.gif)
is
identically zero for sufficiently large n. How large must n be,
in terms of k and m?
Problem 3
Let
![$A$](img5-19.gif)
be an
![$n\times n$](img6-19.gif)
complex matrix, and let
![$\chi$](img7-19.gif)
and
![$\mu$](img8-19.gif)
be the characteristic and minimal polynomials of
![$A$](img9-19.gif)
.
Suppose that
Determine the Jordan Canonical Form of
![$A$](img12-19.gif)
.
Problem 4
Outline a proof, starting from basic properties of the real numbers, of the
following theorem: Let
![% latex2html id marker 716
$f:[a,b] \to \mbox{$\mathbb{R}^{}$}$](img13-19.gif)
be a continuous function
such that
![$f'(x)=0$](img14-19.gif)
for all
![$x\in (a,b)$](img15-19.gif)
. Then
![$f(b)=f(a)$](img16-19.gif)
.
Problem 5
Let
![$b_1,b_2,\ldots$](img17-19.gif)
be positive real numbers with
Assume also that
![$b_1<b_2<b_3<\cdots$](img19-19.gif)
. Show that
the set of quotients
![$(b_m/b_n)_{1\leq n<m}$](img20-19.gif)
is dense in
![$(1,\infty)$](img21-19.gif)
.
Problem 6
Let
![$V$](img22-19.gif)
be a real vector space of dimension
![$n$](img23-19.gif)
with a positive definite
inner product. We say that two bases
![$(a_i)$](img24-19.gif)
and
![$(b_i)$](img25-19.gif)
have the
same orientation if the matrix of the change of basis from
![$(a_i)$](img26-19.gif)
to
![$(b_i)$](img27-19.gif)
has a positive determinant. Suppose now that
![$(a_i)$](img28-19.gif)
and
![$(b_i)$](img29-19.gif)
are orthonormal bases with the same orientation. Show that
![$(a_i+2b_i)$](img30-19.gif)
is again a basis of
![$V$](img31-19.gif)
with the same orientation as
![$(a_i)$](img32-19.gif)
.
Problem 7
Compute
where
![$a>0$](img34-19.gif)
is a constant.
Problem 8
Let
![$G_1$](img35-19.gif)
,
![$G_2$](img36-19.gif)
, and
![$G_3$](img37-19.gif)
be finite groups, each of which is generated
by its commutators (elements of the form
![$xyx^{-1}y^{-1}$](img38-19.gif)
). Let
![$A$](img39-19.gif)
be
a subgroup of
![$G_1 \times G_2 \times G_3$](img40-19.gif)
, which maps surjectively,
by the natural projection map, to
the partial products
![$G_1 \times G_2$](img41-19.gif)
,
![$G_1 \times G_3$](img42-19.gif)
and
![$G_2 \times G_3$](img43-19.gif)
.
Show that
![$A$](img44-19.gif)
is equal to
![$G_1 \times G_2 \times G_3$](img45-19.gif)
.
Problem 9
Suppose
![$\Omega$](img46-19.gif)
is a bounded domain in
![% latex2html id marker 830
$\mathbb{C}\,^{}$](img47-19.gif)
with a boundary consisting
of a smooth Jordan curve
![$\gamma$](img48-19.gif)
. Let
![$f$](img49-19.gif)
be holomorphic in a
neighborhood of the closure of
![$\Omega$](img50-19.gif)
, and suppose that
![$f(z) \neq 0$](img51-19.gif)
for
![$z \in \gamma$](img52-19.gif)
. Let
![$z_1,\ldots,z_k$](img53-19.gif)
be the zeros of
![$f$](img54-19.gif)
in
![$\Omega$](img55-19.gif)
, and let
![$n_j$](img56-19.gif)
be the order of the zero of
![$f$](img57-19.gif)
at
![$z_j$](img58-19.gif)
(for
![$j=1,\ldots,k$](img59-19.gif)
).
- Use Cauchy's integral formula to show that
- Suppose that
has only one zero
in
with
multiplicity
. Find a boundary integral involving
whose
value is the point
.
Problem 10
Let
![$f$](img67-19.gif)
be a twice differentiable real valued function on
![$[0,2\pi]$](img68-19.gif)
, with
![$\int_{0}^{2\pi}f(x)dx=0=f(2\pi)-f(0)$](img69-19.gif)
. Show that
Problem 11
Find the eigenvalues, eigenvectors, and the Jordan Canonical Form of
considered as a matrix with entries in
![% latex2html id marker 919
$\mbox{\bf {F}}_3 = \mbox{$\mathbb{Z}^{}$}/3\mbox{$\mathbb{Z}^{}$}$](img72-19.gif)
.
Problem 12
Prove that a continuous function from
![% latex2html id marker 939
$\mathbb{R}^{}$](img73-19.gif)
to
![% latex2html id marker 941
$\mathbb{R}^{}$](img74-18.gif)
which maps
open sets to open sets must be monotonic.
Problem 13
Let
![$A$](img75-18.gif)
be an n
![$\times$](img76-18.gif)
n complex matrix, all of whose eigenvalues are
equal to
![$1$](img77-18.gif)
. Suppose that the set
![$\{A^n \;\vert\;n=1,2,\ldots\}$](img78-18.gif)
is
bounded. Show that
![$A$](img79-18.gif)
is the identity matrix.
Problem 14
Let
![$G$](img80-18.gif)
be a transitive subgroup of the group
![$S_n$](img81-18.gif)
of permutations of
the set
![$\{1,\ldots,n\}$](img82-18.gif)
. Suppose that
![$G$](img83-18.gif)
is a simple group and that
![$\sim$](img84-18.gif)
is an equivalence relation on
![$\{1,\ldots,n\}$](img85-18.gif)
such that
![$i\sim j$](img86-18.gif)
implies that
![$\sigma(i)\sim \sigma(j)$](img87-18.gif)
for all
![$\sigma \in G$](img88-17.gif)
. What can
you conclude about the relation
![$\sim$](img89-17.gif)
?
Problem 15
Let
![$f$](img90-17.gif)
be analytic on and inside the unit circle
![$C = \{z \;\vert\; \vert z\vert = 1\}$](img91-17.gif)
.
Let
![$L$](img92-17.gif)
be the length of the image of
![$C$](img93-17.gif)
under
![$f$](img94-17.gif)
. Show that
![$L \geq 2\pi \vert f'(0)\vert$](img95-17.gif)
.
Problem 16
Let
![$\Omega$](img96-17.gif)
be an open subset of
![% latex2html id marker 1025
$\mathbb{R}^{2}$](img97-17.gif)
, and let
![% latex2html id marker 1029
$f:\Omega \to \mbox{$\mathbb{R}^{2}$}$](img98-17.gif)
be a smooth map. Assume that
![$f$](img99-17.gif)
preserves orientation and maps any pair
of orthogonal curves to a pair of orthogonal curves. Show that
![$f$](img100-16.gif)
is
holomorphic.
Note: Here we identify
with
.
Problem 17
Let
![$A$](img103-14.gif)
be an
![$n\times n$](img104-14.gif)
Hermitian matrix satisfying the condition
Show that
![$A = I$](img106-12.gif)
.
Problem 18
Find all real valued
![$C^1$](img107-12.gif)
solutions
![$u$](img108-12.gif)
of the differential equation
Problem 19
Compute the area of the image of the unit disc
![$\{z \;\vert\; \vert z\vert < 1\}$](img110-12.gif)
under the map
![$f(z)=z+z^2/2$](img111-11.gif)
.
Problem 20
Let
![% latex2html id marker 1102
$f:\mbox{$\mathbb{R}^{}$} \to \mbox{$\mathbb{R}^{}$}$](img112-11.gif)
be continuously differentiable, periodic of
period
![$1$](img113-11.gif)
, and nonnegative. Show that
uniformly in
![$x$](img115-9.gif)
.
Previous: Spring83
Next: Fall83
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10