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Preliminary Exam - Fall 1979
Problem 1
Prove that the polynomial
has at least one root in the disc
![$\vert z\vert<1$](img2-8.gif)
.
Problem 2
Suppose that
![$f$](img3-8.gif)
is analytic on the open upper half-plane and
satisfies
![$\vert f(z)\vert\leq 1$](img4-8.gif)
for all
![$z$](img5-8.gif)
,
![$f(i)=0$](img6-8.gif)
. How large can
![$\vert f(2i)\vert$](img7-8.gif)
be under these conditions?
Problem 3
Prove that every finite group of order
![$> 2$](img8-8.gif)
has a nontrivial
automorphism.
Problem 4
Consider the polynomial ring
![% latex2html id marker 702
$\mbox{$\mathbb{Z}^{}$}[x]$](img9-8.gif)
and the ideal
![$\mathfrak{I}$](img10-8.gif)
generated by
![$7$](img11-8.gif)
and
![$x-3$](img12-8.gif)
.
- Show that for each
, there is an integer
satisfying
such that
.
- Find
in the special case
.
Problem 5
Let
![$A$](img19-8.gif)
be a real
skew-symmetric matrix
![$(A_{ij} = -A_{ji})$](img20-8.gif)
. Prove that
![$A$](img21-8.gif)
has even rank.
_matrix,>skew-symmetric
Problem 6
Let
![$N$](img22-8.gif)
be a linear operator on an
![$n$](img23-8.gif)
-dimensional vector space,
![$n>1$](img24-8.gif)
,
such that
![$N^n = 0, \; N^{n-1} \; \neq 0$](img25-8.gif)
. Prove there is no operator
X with
![$X^2=N$](img26-8.gif)
.
Problem 9
Given that
find
![$f'(t)$](img45-8.gif)
explicitly, where
Problem 10
Solve the differential equations
Problem 11
An accurate map of California is spread out flat on a table in Evans Hall,
in Berkeley. Prove that there is exactly one point on the map lying directly
over the point it represents.
Problem 12
Consider the following properties of a map
![% latex2html id marker 883
$f:\mbox{$\mathbb{R}^{n}$} \to \mbox{$\mathbb{R}^{}$}$](img48-8.gif)
:
is continuous.
- The graph of
is connected in
.
Prove or disprove the implications 1
![$\Rightarrow$](img52-8.gif)
2, 2
![$\Rightarrow$](img53-8.gif)
1.
Problem 13
Let
![$\{P_n\}$](img54-8.gif)
be a sequence of real polynomials of degree
![$\leq D$](img55-8.gif)
, a fixed
integer. Suppose that
![$P_n(x) \to 0$](img56-8.gif)
pointwise for
![$0\leq x \leq 1$](img57-8.gif)
.
Prove that
![$P_n \to 0$](img58-8.gif)
uniformly on
![$[0,1]$](img59-8.gif)
.
Problem 14
Let
![$y=y(x)$](img60-8.gif)
be a solution of the differential equation
![$y''=-\vert y\vert$](img61-8.gif)
with
![$-\infty<x<\infty$](img62-8.gif)
,
![$y(0)=1$](img63-8.gif)
and
![$y'(0)=0$](img64-8.gif)
.
- Show that
is an even function.
- Show that
has exactly one zero on the positive real axis.
Problem 15
Suppose
![$f$](img67-8.gif)
and
![$g$](img68-8.gif)
are entire functions with
![$\vert f(z)\vert\leq \vert g(z)\vert$](img69-8.gif)
for all
![$z$](img70-8.gif)
. Prove that
![$f(z) = cg(z)$](img71-8.gif)
for
some constant
![$c$](img72-8.gif)
.
Problem 16
Prove that
What restrictions must be placed on
![$\alpha$](img74-8.gif)
?
Problem 17
Let
![$A$](img75-8.gif)
be an
![$n\times n$](img76-8.gif)
complex matrix. Prove there is a unitary
matrix
![$U$](img77-8.gif)
such that
![$B =UAU^{-1}$](img78-8.gif)
is upper-triangular:
![$B_{jk}=0$](img79-8.gif)
for
![$j>k$](img80-8.gif)
.
Problem 18
Let
![$B$](img81-8.gif)
denote the matrix
where
![$a$](img83-8.gif)
,
![$b$](img84-8.gif)
, and
![$c$](img85-8.gif)
are real and
![$\vert a\vert$](img86-8.gif)
,
![$\vert b\vert$](img87-8.gif)
, and
![$\vert c\vert$](img88-7.gif)
are distinct. Show that there are exactly four symmetric
matrices of the form
![$BQ$](img89-7.gif)
, where
![$Q$](img90-7.gif)
is a real orthogonal matrix
of determinant
![$1$](img91-7.gif)
.
Problem 19
Let
![% latex2html id marker 1047
$M_{n\times n}(\mbox{\bf {F}})$](img92-7.gif)
be the ring of
![$n\times n$](img93-7.gif)
matrices over a field
![% latex2html id marker 1051
$\mbox{\bf {F}}$](img94-7.gif)
. Prove that it has no 2-sided ideals except
![% latex2html id marker 1053
$M_{n\times n}(\mbox{\bf {F}})$](img95-7.gif)
and
![$\{0\}$](img96-7.gif)
.
Problem 20
Let
![$G$](img97-7.gif)
be the abelian group defined by generators
![$x$](img98-7.gif)
,
![$y$](img99-7.gif)
, and
![$z$](img100-7.gif)
, and
relations
- Express
as a direct product of two cyclic groups.
- Express
as a direct product of cyclic groups of prime
power order.
- How many elements of
have order
?
Previous: Summer79
Next: Spring80
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10