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Preliminary Exam - Spring 1993
Problem 1
Let
![$(a_{n})$](img1-41.gif)
and
![$(\varepsilon_{n})$](img2-41.gif)
be sequences of positive numbers. Assume that
![$\lim_{n \to \infty}\varepsilon_{n}=0$](img3-41.gif)
and that there is a number
![$k$](img4-41.gif)
in
![$(0,1)$](img5-41.gif)
such that
![$a_{n+1}
\leq k a_{n} + \varepsilon_{n}$](img6-41.gif)
for every
![$n$](img7-41.gif)
. Prove that
![$\lim_{n \to \infty}a_{n}=0$](img8-41.gif)
.
Problem 2
Let
![$A =(a_{ij})$](img9-41.gif)
be an
![$n \times n$](img10-41.gif)
matrix such that
![$\sum_{j=1}^{n} \vert a_{ij}\vert < 1$](img11-41.gif)
for each
![$i$](img12-41.gif)
. Prove that
![$I-A$](img13-41.gif)
is invertible.
Problem 4
Suppose that the group
![$G$](img15-41.gif)
is generated by elements
![$x$](img16-41.gif)
and
![$y$](img17-41.gif)
that satisfy
![$x^{5}y^{3}=x^{8}y^{5}=1$](img18-41.gif)
. Does it follow that
![$G$](img19-41.gif)
is the trivial group?
Problem 5
Let
![$k$](img20-41.gif)
be a positive integer. For which values of the real number
![$c$](img21-41.gif)
does
the differential equation
have a solution satisfying
![$x(0)=x(2 \pi k)=0$](img23-41.gif)
?
Problem 6
Find a list of real matrices, as long as possible, such that
- the characteristic polynomial of each matrix is
,
- the minimal polynomial of each matrix is
,
- no two matrices in the list are similar to each other.
Problem 8
Classify up to isomorphism all groups of order
![$45$](img35-41.gif)
.
Problem 9
Let
![$a$](img36-41.gif)
be a complex number and
![$\varepsilon$](img37-41.gif)
a positive number.
Prove that the function
![$f(z)= \sin z + \frac{1}{z-a}$](img38-41.gif)
has
infinitely many zeros in the strip
![$\vert \Im z\vert < \varepsilon$](img39-41.gif)
.
Problem 10
Let
![$f$](img40-41.gif)
be a real valued
![$C^{1}$](img41-41.gif)
function on
![$[0, \infty )$](img42-41.gif)
such that
the improper integral
![$\int_{1}^{\infty}\vert f'(x)\vert dx$](img43-41.gif)
converges. Prove that the
infinite series
![$\sum_{n=1}^{\infty} f(n)$](img44-41.gif)
converges if and only if
the integral
![$\int_{1}^{\infty}f(x)dx$](img45-41.gif)
converges.
Problem 11
Let
![$P$](img46-41.gif)
be the vector space of polynomials over
![% latex2html id marker 845
$\mathbb{R}$](img47-41.gif)
. Let
the linear transformation
![$E : P \to P$](img48-41.gif)
be defined by
![$Ef=f+f'$](img49-41.gif)
, where
![$f'$](img50-41.gif)
is the derivative of
![$f$](img51-41.gif)
. Prove that
![$E$](img52-41.gif)
is invertible.
Problem 12
Prove that for any fixed complex number
![$\zeta$](img53-41.gif)
,
Problem 13
Prove that no commutative ring with identity has additive group
isomorphic to
![% latex2html id marker 886
$\mathbb{Q}/\mathbb{Z}$](img55-41.gif)
.
Problem 14
Prove that every solution
![$x(t)\; (t \geq 0)$](img56-41.gif)
of the
differential equation
with
![$x(0) > 0$](img58-41.gif)
satisfies
![$\displaystyle{\lim_{t \to \infty} x(t)=1}$](img59-41.gif)
.
Problem 15
Let
![$\Lambda$](img60-41.gif)
be the set of 2
![$\times $](img61-41.gif)
2 matrices of the form
where a and b are elements of a given field
![% latex2html id marker 942
$\mbox{\bf {F}}$](img63-41.gif)
. Prove
that
![$\Lambda$](img64-41.gif)
, with the usual matrix operations, is a commutative
ring with identity. For which of the following fields
![% latex2html id marker 946
$\mbox{\bf {F}}$](img65-41.gif)
is
![$\Lambda$](img66-41.gif)
a field?
![% latex2html id marker 954
$\mbox{\bf {F}}= \mathbb{Q}, \; \mathbb{C}, \; \mbox{$\mathbb{Z}^{}$}_5, \; \mbox{$\mathbb{Z}^{}$}_7$](img67-41.gif)
.
Problem 16
Prove that
![$\displaystyle{\frac{x^{2}+y^{2}}{4} \leq e^{x+y-2}}$](img68-41.gif)
for
![$x \geq 0\,,\; y \geq 0\,$](img69-41.gif)
.
Problem 17
Prove that if
![$G$](img70-41.gif)
is a group containing no subgroup of index
![$2$](img71-41.gif)
,
then any subgroup of index
![$3$](img72-41.gif)
in
![$G$](img73-41.gif)
is a normal subgroup.
Previous: Fall92
Next: Fall93
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10