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Preliminary Exam - Spring 1984
Problem 1
Evaluate
for
![$a>0$](img2-20.gif)
.
Problem 2
For a
![$p$](img3-20.gif)
-group of order
![$p^4$](img4-20.gif)
, assume the center of
![$G$](img5-20.gif)
has order
![$p^2$](img6-20.gif)
.
Determine the number of conjugacy classes of
![$G$](img7-20.gif)
.
Problem 3
Let
![% latex2html id marker 686
$f : [0,1] \to \mbox{$\mathbb{R}^{}$}$](img8-20.gif)
be continuous, with
![$f(0)=f(1)=0$](img9-20.gif)
. Assume
that
![$f''$](img10-20.gif)
exists on
![$0<x<1$](img11-20.gif)
, with
![$f''+2f'+f\geq 0$](img12-20.gif)
. Show that
![$f(x)\leq 0$](img13-20.gif)
for all
![$0\leq x \leq 1$](img14-20.gif)
.
Problem 4
Which number is larger,
![$\pi^3$](img15-20.gif)
or
![$3^{\pi}$](img16-20.gif)
?
Problem 5
Let
![$A$](img17-20.gif)
and
![$B$](img18-20.gif)
be complex
![$n\times n$](img19-20.gif)
matrices such that
![$AB=BA^2$](img20-20.gif)
, and assume
![$A$](img21-20.gif)
has no eigenvalues of absolute
value
![$1$](img22-20.gif)
. Prove that
![$A$](img23-20.gif)
and
![$B$](img24-20.gif)
have a common (nonzero) eigenvector.
Problem 6
Let
![$a$](img25-20.gif)
be a positive real number. Define a sequence
![$(x_n)$](img26-20.gif)
by
Find a necessary and sufficient condition on
![$a$](img28-20.gif)
in order that a
finite limit
![$\displaystyle{\lim_{n \to \infty} x_n}$](img29-20.gif)
should exist.
Problem 7
Find the number of roots of
which lie between the two circles
![$\vert z\vert=1$](img31-20.gif)
and
![$\vert z\vert=2$](img32-20.gif)
.
Problem 8
Show that the system of differential equations
has a solution which tends to
![$\infty$](img34-20.gif)
as
![$t \to -\infty$](img35-20.gif)
and tends to
the origin as
![$t \to +\infty$](img36-20.gif)
.
Problem 9
Let
![$A$](img37-20.gif)
be a real
![$m\times n$](img38-20.gif)
matrix with rational entries and let b be
an
![$m$](img39-20.gif)
-tuple of rational numbers. Assume that the system of
equations
Ax = b has a solution x in complex
![$n$](img40-20.gif)
-space
![% latex2html id marker 809
$\mathbb{C}\,^{n}$](img41-20.gif)
. Show that
the equation
has a solution vector with rational components, or give a counterexample.
Problem 10
Let
![$R$](img42-20.gif)
be a principal ideal domain and let
![$\mathfrak{I}$](img43-20.gif)
and
![$\mathfrak{J}$](img44-20.gif)
be nonzero ideals in
![$R$](img45-20.gif)
. Show that
![$\mathfrak{IJ} = \mathfrak{I} \cap
\mathfrak{J}$](img46-20.gif)
if and only if
![$\mathfrak{I} + \mathfrak{J} = R$](img47-20.gif)
.
Problem 11
Prove the following statement or supply a counterexample: If
![$A$](img48-20.gif)
and
![$B$](img49-20.gif)
are real
![$n\times n$](img50-20.gif)
matrices which are similar over
![% latex2html id marker 854
$\mathbb{C}\,^{}$](img51-20.gif)
,
then
![$A$](img52-20.gif)
and
![$B$](img53-20.gif)
are similar over
![% latex2html id marker 860
$\mathbb{R}^{}$](img54-20.gif)
.
Problem 12
Consider the equation
Show that there is an
![$\varepsilon > 0$](img56-20.gif)
such that if
![$\vert y_0\vert< \varepsilon$](img57-20.gif)
,
then the solution
![$y = f(x)$](img58-20.gif)
with
![$f(0) = y_0$](img59-20.gif)
satisfies
Problem 13
Let
![$I$](img61-20.gif)
be an open interval in
![% latex2html id marker 899
$\mathbb{R}^{}$](img62-20.gif)
containing zero. Assume that
![$f'$](img63-20.gif)
exists on a neighborhood of zero and
![$f''(0)$](img64-20.gif)
exists. Show
that
(
![$o(x^2)$](img66-20.gif)
denotes a quantity such that
![$o(x^2)/x^2 \to 0$](img67-20.gif)
as
![$x \to 0$](img68-20.gif)
).
Problem 14
Let
![% latex2html id marker 929
$\mbox{\bf {F}}$](img69-20.gif)
be a field and let
![$X$](img70-20.gif)
be a finite set. Let
![% latex2html id marker 933
$R(X,\mbox{\bf {F}})$](img71-20.gif)
be the
ring of all functions from
![$X$](img72-20.gif)
to
![% latex2html id marker 937
$\mbox{\bf {F}}$](img73-20.gif)
, endowed with the pointwise
operations. What are the maximal ideals of
![% latex2html id marker 939
$R(X,\mbox{\bf {F}})$](img74-19.gif)
?
Problem 15
Let
![$F$](img75-19.gif)
be a continuous complex valued function on the interval
![$[0,1]$](img76-19.gif)
. Let
for
![$z$](img78-19.gif)
a complex number not in
![$[0,1]$](img79-19.gif)
.
- Prove that
is an analytic function.
- Express the coefficients of the Laurent series of
about
in terms of
. Use your result to show that
is uniquely determined by
.
Problem 16
Prove, or supply a counterexample: If
![$A$](img86-19.gif)
is an invertible
![$n \times n$](img87-19.gif)
complex matrix and some power of
![$A$](img88-18.gif)
is diagonal,
then
![$A$](img89-18.gif)
can be diagonalized.
Problem 17
Prove that the Taylor coefficients at the origin of the function
are rational numbers.
Problem 18
Prove or supply a counterexample: If the function
![$f$](img91-18.gif)
from
![% latex2html id marker 1023
$\mathbb{R}^{}$](img92-18.gif)
to
![% latex2html id marker 1025
$\mathbb{R}^{}$](img93-18.gif)
has both a left limit and a right limit at each point of
![% latex2html id marker 1027
$\mathbb{R}^{}$](img94-18.gif)
, then the set of discontinuities of
![$f$](img95-18.gif)
is, at most, countable.
Problem 19
Let
![$f(x)=x\log\left( 1+x^{-1} \right)$](img96-18.gif)
,
![$0<x<\infty$](img97-18.gif)
.
- Show that
is strictly monotonically increasing.
- Compute
as
and
.
Problem 20
Determine all finitely generated abelian groups
![$G$](img102-17.gif)
which have only
finitely many automorphisms.
Previous: Fall83
Next: Summer84
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10