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Preliminary Exam - Spring 1982
Problem 1
Prove the Fundamental Theorem of Algebra: Every nonconstant polynomial
with complex coefficients has a complex root.
_Fundamental Theorem>of Algebra
Problem 2
Let
![% latex2html id marker 666
$S \subset \mbox{$\mathbb{R}^{n}$}$](img1-14.gif)
be a subset which is uncountable. Prove that there
is a sequence of distinct points in
![$S$](img2-14.gif)
converging to a point of
![$S$](img3-14.gif)
.
Problem 3
Let
![$A$](img4-14.gif)
and
![$B$](img5-14.gif)
be n
![$\times$](img6-14.gif)
n complex matrices. Prove that
Problem 4
Let
![% latex2html id marker 705
$f:\mbox{$\mathbb{R}^{2}$} \to \mbox{$\mathbb{R}^{}$}$](img8-14.gif)
have directional derivatives in all
directions at the origin. Is
![$f$](img9-14.gif)
differentiable at the origin?
Prove or give a counterexample.
Problem 5
Let
![$\{g_n\}$](img10-14.gif)
be a sequence of twice differentiable functions on
![$[0,1]$](img11-14.gif)
such that
![$g_n(0)=g_n'(0)=0$](img12-14.gif)
for all n. Suppose also that
![$\vert g_n''(x)\vert \leq 1$](img13-14.gif)
for all n and all
![$x \in [0,1]$](img14-14.gif)
. Prove that there
is a subsequence of
![$\{g_n\}$](img15-14.gif)
which converges uniformly on
![$[0,1]$](img16-14.gif)
.
Problem 6
Suppose that
![$f(x)$](img17-14.gif)
is a polynomial with real coefficients and
![$a$](img18-14.gif)
is a
real number with
![$f(a)\neq 0$](img19-14.gif)
. Show that there exists a real polynomial
![$g(x)$](img20-14.gif)
such that if we define
![$p$](img21-14.gif)
by
![$p(x)=f(x)g(x)$](img22-14.gif)
, we have
![$p(a)=1$](img23-14.gif)
,
![$p'(a)=0$](img24-14.gif)
, and
![$p''(a)=0$](img25-14.gif)
.
Problem 7
Suppose that the group
![$G$](img26-14.gif)
is generated by elements
![$x$](img27-14.gif)
and
![$y$](img28-14.gif)
that satisfy
![$x^{5}y^{3}=x^{8}y^{5}=1$](img29-14.gif)
. Is
![$G$](img30-14.gif)
the trivial group?
Problem 8
Find
by contour integration.
Problem 9
Find the Jordan Canonical Form for the matrix (over
![% latex2html id marker 799
$\mathbb{R}^{}$](img32-14.gif)
)
Problem 10
Prove that any group of order
![$77$](img34-14.gif)
is cyclic.
Problem 11
Decide, without too much computation, whether a finite limit
exists, where z is a complex variable, and if yes, compute the limit.
Problem 12
Prove or give a counterexample: Every connected, locally pathwise
connected set in
![% latex2html id marker 826
$\mathbb{R}^{n}$](img36-14.gif)
is pathwise connected.
Problem 13
Let
![$T:V \to W$](img37-14.gif)
be a linear transformation between finite-dimensional
vector spaces. Prove that
Problem 14
Let
![% latex2html id marker 862
$f:I \to \mbox{$\mathbb{R}^{}$}$](img39-14.gif)
(where
![$I$](img40-14.gif)
is an interval of
![% latex2html id marker 866
$\mathbb{R}^{}$](img41-14.gif)
) be such
that
![$f(x)>0,\;x \in I$](img42-14.gif)
. Suppose that
![$e^{cx}f(x)$](img43-14.gif)
is
convex in
![$I$](img44-14.gif)
for every real number c. Show that
![$\log f(x)$](img45-14.gif)
is
convex in
![$I$](img46-14.gif)
.
Note: A function
is convex if
for all
![$x$](img49-14.gif)
and
![$y$](img50-14.gif)
in
![$I$](img51-14.gif)
and
![$0\leq t \leq 1$](img52-14.gif)
.
_function,>convex
Problem 15
How many nonsingular 2
![$\times$](img53-14.gif)
2 matrices are there over the field of p
elements?
Problem 16
Prove that if
![$G$](img54-14.gif)
is a group containing no subgroup of index
![$2$](img55-14.gif)
,
then any subgroup of index
![$3$](img56-14.gif)
is normal.
Problem 17
Let
![$\{f_n\}$](img57-14.gif)
be a sequence of continuous functions from
![$[0,1]$](img58-14.gif)
to
![% latex2html id marker 935
$\mbox{$\mathbb{R}^{}$}$](img59-14.gif)
. Suppose that
![$f_n(x) \to 0$](img60-14.gif)
as
![$n \to \infty$](img61-14.gif)
for
each
![$x \in [0,1]$](img62-14.gif)
and also that, for some constant
![$K$](img63-14.gif)
, we have
for all n. Does
Problem 18
For
![$\Re z \geq 0$](img66-14.gif)
, define
Show that F(z) is continuous for
![$\Re z\geq 0$](img68-14.gif)
and analytic
for
![$\Re z>0$](img69-14.gif)
.
Problem 19
Show that the initial value problem
has a solution defined for
![$-\infty < x < \infty$](img71-14.gif)
.
Problem 20
Prove that the polynomial
![$x^4 + x + 1$](img72-14.gif)
is irreducible over
![% latex2html id marker 979
$\mathbb{Q}\,^{}$](img73-14.gif)
.
Previous: Fall81
Next: Summer82
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10