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Preliminary Exam - Spring 1977
Problem 1
Suppose
![$f$](img1.gif)
is a differentiable function from the reals into the reals.
Suppose
![$f'(x)>f(x)$](img2.gif)
for all
![% latex2html id marker 665
$x\in\mbox{$\mathbb{R}^{}$}$](img3.gif)
, and
![$f(x_0)=0$](img4.gif)
. Prove
that
![$f(x)>0$](img5.gif)
for all
![$x>x_0$](img6.gif)
.
Problem 2
Suppose that
![$f$](img7.gif)
is a real valued function of one real variable such
that
exists for all
![$c \in [a,b]$](img9.gif)
. Show that
![$f$](img10.gif)
is Riemann integrable
on
![$[a,b]$](img11.gif)
.
Problem 3
- Evaluate
, where
is the polynomial
- Consider a circle of radius
, and let
be the vertices of a regular
-gon inscribed in the circle. Join
to
by segments of a straight line. You
obtain
segments of lengths
.
Show that
file=../Fig/Pr/Sp77-3,width=2.7in
Problem 4
Prove the Fundamental Theorem of Algebra: Every nonconstant polynomial
with complex coefficients has a complex root.
_Fundamental Theorem>of Algebra
Problem 5
Let
![% latex2html id marker 761
$\mbox{\bf {F}} \subset \mbox{\bf {K}}$](img23.gif)
be fields, and
![$a$](img24.gif)
and
![$b$](img25.gif)
elements of
![% latex2html id marker 767
$\mbox{\bf {K}}$](img26.gif)
which are algebraic over
![% latex2html id marker 769
$\mbox{\bf {F}}$](img27.gif)
. Show that
![$a+b$](img28.gif)
is algebraic over
![% latex2html id marker 773
$\mbox{\bf {F}}$](img29.gif)
.
Problem 7
A matrix of the form
where the
![$a_i$](img48.gif)
are complex numbers, is called a Vandermonde matrix.
- Prove that the Vandermonde matrix is invertible if
are all different.
- If
are all different, and
are complex numbers, prove that there is a
unique polynomial
of degree
with complex coefficients such
that
,
,...,
.
Problem 8
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 9
Find the solution of the differential equation
subject to the conditions
Problem 10
In
![% latex2html id marker 911
$\mathbb{R}^{2}$](img61.gif)
, consider the region
![${\cal A}$](img62.gif)
defined by
![$x^2+y^2>1$](img63.gif)
. Find
differentiable real valued functions
![$f$](img64.gif)
and
![$g$](img65.gif)
on
![${\cal A}$](img66.gif)
such that
![$ \frac{\partial f}{\partial x} = \frac{\partial g}{\partial y} $](img67.gif)
but there is no real valued function
![$h$](img68.gif)
on
![${\cal A}$](img69.gif)
such that
![$f = \frac{\partial h}{\partial y}$](img70.gif)
and
Problem 11
Let the sequence
![$a_0,a_1,\ldots$](img72.gif)
be defined by the equation
Find
Problem 12
Let
![$p$](img75.gif)
be an odd prime. Let
![$Q(p)$](img76.gif)
be the set of integers
![$a$](img77.gif)
,
![$0 \leq a \leq p-1$](img78.gif)
, for which the congruence
has a solution. Show that
![$Q(p)$](img80.gif)
has cardinality
![$(p+1)/2$](img81.gif)
.
Problem 13
Consider the family of square matrices
![$A(\theta)$](img82.gif)
defined by the
solution of the matrix differential equation _matrix,>orthogonal
with the initial condition
![$A(0) = I$](img84.gif)
, where
![$B$](img85.gif)
is a constant square matrix.
- Find a property of
which is necessary and sufficient for
to be orthogonal for all
; that is,
, where
denotes the
transpose of
.
- Find the matrices
corresponding to
and give a geometric interpretation.
Problem 15
Let
![$f(z)$](img107.gif)
be a nonconstant meromorphic function. A complex number
![$w$](img108.gif)
is called
a period of
![$f$](img109.gif)
if
![$f(z+w)=f(z)$](img110.gif)
for all
![$z$](img111.gif)
.
- Show that if
and
are periods, so are
for all integers
and
.
- Show that there are, at most, a finite number of periods of
in any bounded region of the complex plane.
Problem 17
Let
![% latex2html id marker 1124
$\mbox{$\mathbb{Q}\,^{}$}_{+}$](img119.gif)
be the multiplicative group of positive rational numbers.
- Is
torsion free?
- Is
free?
Problem 18
- In
, consider the set of polynomials
for which
. Prove that this set forms an ideal
and find its monic generator.
- Do the polynomials such that
and
form
an ideal?
Problem 19
Suppose that
![$u(x,t)$](img127.gif)
is a continuous function of the real variables
x and t with continuous second partial derivatives. Suppose that
u and its first partial derivatives are periodic in
![$x$](img128.gif)
with period
![$1$](img129.gif)
,
and that
Prove that
is a constant independent of
![$t$](img132.gif)
.
Problem 20
Let
![% latex2html id marker 1202
$h:[0,1)\to\mbox{$\mathbb{R}^{}$}$](img133.gif)
be a function defined on the half-open interval
![$[0,1)$](img134.gif)
. Prove that if
![$h$](img135.gif)
is
uniformly continuous, there exists a unique continuous function
![% latex2html id marker 1210
$g:[0,1]\to\mbox{$\mathbb{R}^{}$}$](img136.gif)
such that
![$g(x)=h(x)$](img137.gif)
for all
![$x\in [0,1)$](img138.gif)
.
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Next: Summer77
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10