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Preliminary Exam - Spring 1991
Problem 1
List, to within isomorphism, all the finite groups whose orders do not
exceed
![$5$](img1-37.gif)
. Explain why your list is complete and why no two
groups on the list are isomorphic.
Problem 2
Let
![$f$](img2-37.gif)
be a continuous complex valued function on
![$[0,1]$](img3-37.gif)
, and
define the function
![$g$](img4-37.gif)
by
Prove that
![$g$](img6-37.gif)
is analytic in the entire complex plane.
Problem 3
For
![$n$](img7-37.gif)
a positive integer, let
![$d(n)$](img8-37.gif)
denote the number of
positive integers that divide
![$n$](img9-37.gif)
. Prove that
![$d(n)$](img10-37.gif)
is odd
if and only if n is a perfect square.
Problem 4
Let
![$p$](img11-37.gif)
be a prime number and
![$R$](img12-37.gif)
a ring with identity containing
![$p^2$](img13-37.gif)
elements. Prove that
![$R$](img14-37.gif)
is commutative.
Problem 6
Let the vector field
![$F$](img27-37.gif)
in
![% latex2html id marker 758
$\mathbb{R}^{3}$](img28-37.gif)
have the form
where
![$g$](img30-37.gif)
is a real valued smooth function on
![$(0,\infty)$](img31-37.gif)
and
![$\Vert\cdot\Vert$](img32-37.gif)
denotes the Euclidean norm. (
![$F$](img33-37.gif)
is undefined
at
![$(0,0,0)$](img34-37.gif)
.) Prove that
for any smooth closed path
![$C$](img36-37.gif)
in
![% latex2html id marker 772
$\mathbb{R}^{3}$](img37-37.gif)
that does not pass
through the origin.
Problem 7
Let the function
![$f$](img38-37.gif)
be analytic in the unit disc, with
![$\vert f(z)\vert\leq 1$](img39-37.gif)
and
![$f(0)=0$](img40-37.gif)
. Assume that there is a number
![$r$](img41-37.gif)
in
![$(0,1)$](img42-37.gif)
such that
![$f(r)=f(-r)=0$](img43-37.gif)
. Prove that
Problem 8
Let
![${T}$](img45-37.gif)
be a real, symmetric,
![$n \times n$](img46-37.gif)
,
tridiagonal matrix:
(All entries not on the main diagonal or the diagonals just above
and below the main one are zero.) Assume
![$b_j\neq 0$](img48-37.gif)
for all
![$j$](img49-37.gif)
.
_matrix,>tridiagonal
Prove:
-
.
has
distinct eigenvalues.
Problem 9
Let
![$f$](img53-37.gif)
be a continuous function from the ball
![% latex2html id marker 855
$B_n = \{x \in \mbox{$\mathbb{R}^{n}$} \;\vert\;
\Vert x\Vert < 1\}$](img54-37.gif)
into itself. (Here,
![$\Vert\cdot\Vert$](img55-37.gif)
denotes the Euclidean
norm.) Assume
![$\Vert f(x)\Vert<\Vert x\Vert$](img56-37.gif)
for all nonzero
![$x \in B_n$](img57-37.gif)
. Let
![$x_0$](img58-37.gif)
be a nonzero point of
![$B_n$](img59-37.gif)
, and define the sequence
![$(x_k)$](img60-37.gif)
by setting
![$x_k = f(x_{k-1})$](img61-37.gif)
. Prove
that
![$\lim x_k = 0$](img62-37.gif)
.
Problem 10
Prove that
![% latex2html id marker 887
$\mathbb{Q}\,^{}$](img63-37.gif)
, the additive group of rational numbers, cannot be
written as the direct sum of two nontrivial subgroups.
Note: See also Problems
and
.
Problem 11
For which real numbers x does the infinite series
converge?
Problem 12
Let
![$A$](img65-37.gif)
be the set of positive integers that do not contain the
digit
![$9$](img66-37.gif)
in their decimal expansions. Prove that
that is,
![$A$](img68-37.gif)
defines a convergent subseries of the harmonic series.
Problem 13
Prove that
exists and find its value.
Problem 14
Let
![$x(t)$](img70-37.gif)
be a nontrivial solution to the system
where
Prove that
![$\Vert x(t)\Vert$](img73-37.gif)
is an increasing function of t. (Here,
![$\Vert\cdot\Vert$](img74-36.gif)
denotes the Euclidean norm.)
Problem 15
Let
![$G$](img75-36.gif)
be a finite nontrivial group with the property that for
any two elements
![$a$](img76-36.gif)
and
![$b$](img77-36.gif)
in
![$G$](img78-36.gif)
different from the identity,
there is an element
![$c$](img79-36.gif)
in
![$G$](img80-36.gif)
such that
![$b= c^{-1}ac$](img81-35.gif)
. Prove
that
![$G$](img82-35.gif)
has order
![$2$](img83-35.gif)
.
Problem 16
Let
![$A$](img84-35.gif)
be a linear transformation on an n-dimensional vector
space over
![% latex2html id marker 986
$\mathbb{C}\,^{}$](img85-35.gif)
with characteristic polynomial
![$(x-1)^n$](img86-35.gif)
. Prove that
![$A$](img87-35.gif)
is similar to
![$A^{-1}$](img88-34.gif)
.
Problem 17
Let the function
![$f$](img89-34.gif)
be analytic in the punctured disc
![$0<\vert z\vert<r_0$](img90-34.gif)
, with Laurent series
Assume there is a positive number
![$M$](img92-34.gif)
such that
Prove that
![$c_n = 0$](img94-34.gif)
for
![$n< -2$](img95-33.gif)
.
Problem 18
Let the real valued function
![$f$](img96-33.gif)
be defined in an open interval
about the point
![$a$](img97-33.gif)
on the real line and be differentiable at
![$a$](img98-33.gif)
.
Prove that if
![$(x_n)$](img99-32.gif)
is an increasing sequence and
![$(y_n)$](img100-31.gif)
is a decreasing sequence in the domain of
![$f$](img101-31.gif)
, and both sequences converge to
![$a$](img102-31.gif)
, then
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Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10