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Preliminary Exam - Fall 1980
Problem 1
Define
Compute
![$F'(0)$](img2-11.gif)
.
Problem 2
Let
Is
![$A$](img4-11.gif)
similar to
Problem 3
Do there exist functions
![$f(z)$](img6-11.gif)
and
![$g(z)$](img7-11.gif)
that are analytic at
![$z=0$](img8-11.gif)
and
that satisfy
-
,
,
-
,
?
Problem 5
Evaluate
where
![$n$](img28-11.gif)
and
![$m$](img29-11.gif)
are positive integers and
![$0<m<n$](img30-11.gif)
.
Problem 6
Let
![$M$](img31-11.gif)
be the ring of real 2
![$\times$](img32-11.gif)
2 matrices and
![$S \subset M$](img33-11.gif)
the
subring of matrices of the form
- Exhibit (without proof) an isomorphism between
and
.
- Prove that
lies in a subring isomorphic to
.
- Prove that there is an
such that
.
Problem 7
Let
![$g$](img41-11.gif)
be
![$2\pi$](img42-11.gif)
-periodic, continuous on
![$[-\pi,\pi]$](img43-11.gif)
and have Fourier series
Let
![$f$](img45-11.gif)
be
![$2\pi$](img46-11.gif)
-periodic and satisfy the differential equation
where
![$k\neq n^2, n= 1, 2, 3,\ldots$](img48-11.gif)
.
Find the Fourier series of
![$f$](img49-11.gif)
and prove that it converges everywhere.
Problem 8
Let
![% latex2html id marker 881
$\mbox{\bf {F}}_2 = \{0,1\}$](img50-11.gif)
be the field with two elements. Let
![$G$](img51-11.gif)
be the group
of invertible 2
![$\times $](img52-11.gif)
2 matrices with entries in
![% latex2html id marker 887
$\mbox{\bf {F}}_2$](img53-11.gif)
. Show that
![$G$](img54-11.gif)
is isomorphic to
![$S_3$](img55-11.gif)
, the group of permutations of three objects.
Problem 9
For a real 2
![$\times$](img56-11.gif)
2 matrix
let
![$\Vert X\Vert = x^2+y^2+z^2+t^2$](img58-11.gif)
, and define a metric by
![$d(X,Y) = \Vert X-Y\Vert$](img59-11.gif)
. Let
![$\Sigma = \{X \;\vert\; \det(X) = 0\}$](img60-11.gif)
. Let
Find the minimum distance from
![$A$](img62-11.gif)
to
![$\Sigma$](img63-11.gif)
and exhibit an
![$S \in \Sigma$](img64-11.gif)
that achieves this minimum.
Problem 10
Show that there is an
![$\varepsilon>0$](img65-11.gif)
such that if
![$A$](img66-11.gif)
is any real 2
![$\times
$](img67-11.gif)
2 matrix satisfying
![$\vert a_{ij}\vert\leq \varepsilon$](img68-11.gif)
for all entries
![$a_{ij}$](img69-11.gif)
of
![$A$](img70-11.gif)
, then there is a real 2
![$\times$](img71-11.gif)
2 matrix
![$X$](img72-11.gif)
such that
![$X^2+X^t=A$](img73-11.gif)
,
where
![$X^{t}$](img74-11.gif)
is the transpose of
![$X$](img75-11.gif)
. Is
![$X$](img76-11.gif)
unique?
Problem 11
Let
![$f(z)$](img77-11.gif)
be an analytic function defined for
![$\vert z\vert\leq 1$](img78-11.gif)
and let
Prove that
where
![$C$](img81-11.gif)
is the unit circle,
![$x^2+y^2=1$](img82-11.gif)
.
Problem 12
Prove that any group of order
![$6$](img83-11.gif)
is isomorphic to either
![% latex2html id marker 995
$\mbox{$\mathbb{Z}^{}$}_6$](img84-11.gif)
or
![$S_3$](img85-11.gif)
(the group of permutations of three objects).
Problem 13
Let
![$f(x)=\frac{1}{4}+x-x^2$](img86-11.gif)
. For any real number
![$x$](img87-11.gif)
, define a sequence
![$(x_n)$](img88-10.gif)
by
![$x_0=x$](img89-10.gif)
and
![$x_{n+1}=f(x_n)$](img90-10.gif)
. If the sequence converges, let
![$x_{\infty}$](img91-10.gif)
denote the limit.
- For
, show that the sequence is bounded and nondecreasing and
find
.
- Find all
such that
.
Problem 14
Exhibit a set of 2
![$\times $](img96-10.gif)
2 real matrices with the following
property: A matrix
![$A$](img97-10.gif)
is similar to exactly one matrix in
![$S$](img98-10.gif)
provided
![$A$](img99-10.gif)
is a 2
![$\times $](img100-10.gif)
2 invertible matrix of integers with all the roots of
its characteristic polynomial on the unit circle.
Problem 15
Consider the differential equation
![$x''+x'+x^3=0$](img101-10.gif)
and the function
![$f(x,x')= (x+x')^2+(x')^2+x^4$](img102-10.gif)
.
- Show that
decreases along trajectories of the differential
equation.
- Show that if
is any solution, then
tends to
as
.
Problem 16
Suppose that
![$A$](img108-6.gif)
and
![$B$](img109-6.gif)
are real matrices such that
![$A^t=A$](img110-6.gif)
,
for all
![% latex2html id marker 1107
$v \in \mbox{$\mathbb{R}^{n}$}$](img112-6.gif)
and
Show that
![$AB=BA=0$](img114-4.gif)
and give an example where neither
![$A$](img115-4.gif)
nor
![$B$](img116-4.gif)
is zero.
Problem 17
Let
![$P_n$](img117-3.gif)
be a sequence of real polynomials of degree
![$\leq D$](img118-3.gif)
, a fixed
integer. Suppose that
![$P_n(x) \to 0$](img119-3.gif)
pointwise for
![$0\leq x \leq 1$](img120-3.gif)
.
Prove that
![$P_n \to 0$](img121-3.gif)
uniformly on
![$[0,1]$](img122-3.gif)
.
Problem 18
Suppose that
![$f$](img123-3.gif)
is analytic inside and on the unit circle
![$\vert z\vert=1$](img124-2.gif)
and
satisfies
![$\vert f(z)\vert<1$](img125-2.gif)
for
![$\vert z\vert=1$](img126-2.gif)
. Show that the equation
![$f(z)=z^3$](img127-2.gif)
has exactly three solutions (counting multiplicities) inside the unit
circle.
Problem 19
Let
![$X$](img128-2.gif)
be a compact metric space and
![$f : X \to X$](img129-2.gif)
an isometry.
Show that
![$f(X) = X$](img130-2.gif)
.
Problem 20
Let
![$R$](img131-2.gif)
be a ring with multiplicative identity
![$1$](img132-2.gif)
. Call
![$x \in R$](img133-2.gif)
a
unit if
![$xy=yx=1$](img134-2.gif)
for some
![$y \in R$](img135-2.gif)
. Let
![$G(R)$](img136-2.gif)
denote the set of units. _ring>unit
- Prove
is a multiplicative group.
- Let
be the ring of complex numbers
, where
and
are
integers. Prove
is isomorphic to
(the additive group
of integers modulo
).
Previous: Summer80
Next: Spring81
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10