Previous: Spring86
Next: Spring87
Preliminary Exam - Fall 1986
Problem 1
The Arzelà-Ascoli Theorem asserts that the sequence
![$\left\{f_n\right\}$](img1-30.gif)
of continuous real valued functions on
a metric space
![$\Omega$](img2-30.gif)
is precompact (i.e., has a uniformly convergent
subsequence) if
- (i)
is compact,
- (ii)
-
(where
),
- (iii)
- the sequence is equicontinuous.
Give examples of sequences which are not precompact such that:
(i) and (ii) hold but (iii) fails; (i) and (iii) hold but (ii) fails;
(ii) and (iii) hold but (i) fails. Take
![$\Omega$](img6-30.gif)
to be a subset
of the real line. Sketch the graph of a typical member of the
sequence in each case.
Problem 2
Let the points
![$a$](img7-30.gif)
,
![$b$](img8-30.gif)
, and
![$c$](img9-30.gif)
lie on the unit circle of the complex
plane and satisfy
![$a+b+c=0$](img10-30.gif)
. Prove that
![$a$](img11-30.gif)
,
![$b$](img12-30.gif)
, and
![$c$](img13-30.gif)
form the
vertices of an equilateral triangle.
Problem 4
Show that the polynomial
![$p(z) = z^5-6z+3$](img16-30.gif)
has five distinct complex
roots, of which exactly three (and not five) are real.
Problem 5
Let
![$M_{2\times 2}$](img17-30.gif)
denote the vector space of complex 2
![$\times $](img18-30.gif)
2 matrices.
Let
and let the linear transformation
![$T: M_{2\times 2} \to M_{2\times 2}$](img20-30.gif)
be
defined by
![$T(X) = XA-AX$](img21-30.gif)
. Find the Jordan Canonical Form for
![$T$](img22-30.gif)
.
Problem 6
Prove the following theorem, or find a counterexample: If
![$p$](img23-30.gif)
and
![$q$](img24-30.gif)
are continuous real valued functions on
![% latex2html id marker 767
$\mathbb{R}^{}$](img25-30.gif)
such that
![$\vert q(x)\vert\leq \vert p(x)\vert$](img26-30.gif)
for all
![$x$](img27-30.gif)
, and if every solution
![$f$](img28-30.gif)
of the differential
equation
satisfies
![$\displaystyle{\lim_{x\to +\infty}f(x)=0}$](img30-30.gif)
, then every solution
![$f$](img31-30.gif)
of the differential equation
satisfies
![$\displaystyle{\lim_{x\to +\infty}f(x)=0}$](img33-30.gif)
.
Problem 7
Let
![% latex2html id marker 809
$\mbox{\bf {F}}$](img34-30.gif)
be a field containing
![% latex2html id marker 811
$\mathbb{Q}\,^{}$](img35-30.gif)
such that
![% latex2html id marker 815
$[\mbox{\bf {F}}:\mbox{$\mathbb{Q}\,^{}$}] = 2$](img36-30.gif)
.
Prove that there exists a unique integer
![$m$](img37-30.gif)
such that
![$m$](img38-30.gif)
has no multiple
prime factors and
![% latex2html id marker 821
$\mbox{\bf {F}}$](img39-30.gif)
is isomorphic to
![% latex2html id marker 825
$\mbox{$\mathbb{Q}\,^{}$}(\sqrt{m}\,)$](img40-30.gif)
.
Problem 8
Let
![$f$](img41-30.gif)
be a continuous real valued function on
![$[0,1]$](img42-30.gif)
such that,
for each
![$x_0 \in [0,1)$](img43-30.gif)
,
Prove that
![$f$](img45-30.gif)
is nondecreasing.
Problem 10
For
![$f$](img47-30.gif)
a real valued function on the real line, define the function
![$\triangle f$](img48-30.gif)
by
![$\triangle f(x) = f(x+1)-f(x)$](img49-30.gif)
. For
![$n\geq2$](img50-30.gif)
,
define
![$\triangle^nf$](img51-30.gif)
recursively by
![$\triangle^nf =
\triangle(\triangle^{n-1}f)$](img52-30.gif)
. Prove that
![$\triangle^nf=0$](img53-30.gif)
if
and only if
![$f$](img54-30.gif)
has the form
![$f(x)=a_0(x)+a_1(x)x+\cdots+a_{n-1}(x)x^{n-1}$](img55-30.gif)
where
![$a_0,a_1,\ldots,a_{n-1}$](img56-30.gif)
are periodic functions of period
![$1$](img57-30.gif)
.
Problem 11
Let
![$A$](img58-30.gif)
be an
![$m\times n$](img59-30.gif)
matrix with entries in a field
![% latex2html id marker 906
$\mbox{\bf {F}}$](img60-30.gif)
. Define
the row rank and the column rank of
![$A$](img61-30.gif)
and show from first principles
that they are equal.
Problem 12
Let
![$\{U_1, U_2, \ldots \}$](img62-30.gif)
be a cover of
![% latex2html id marker 928
$\mathbb{R}^{n}$](img63-30.gif)
by open sets. Prove
that there is a cover
![$\{V_1, V_2, \ldots \}$](img64-30.gif)
such that
-
for each
;
- each compact subset of
is disjoint from all but
finitely many of the
.
Problem 13
Let
![% latex2html id marker 974
$f: \mbox{$\mathbb{R}^{2}$} \to \mbox{$\mathbb{R}^{}$}$](img69-30.gif)
be defined by:
Determine all points at which
![$f$](img71-30.gif)
is differentiable.
Problem 14
Let
![$a$](img72-30.gif)
and
![$b$](img73-30.gif)
be real numbers. Prove that there are two
orthogonal unit vectors
![$u$](img74-29.gif)
and
![$v$](img75-29.gif)
in
![% latex2html id marker 998
$\mathbb{R}^{3}$](img76-29.gif)
such that
![$u = (u_1,u_2,a)$](img77-29.gif)
and
![$v = (v_1,v_2,b)$](img78-29.gif)
if and only if
![$a^2+b^2\leq 1$](img79-29.gif)
.
Problem 15
Prove that if
![$p$](img80-29.gif)
is a prime number
![$(>0)$](img81-29.gif)
then the polynomial
is irreducible in
![% latex2html id marker 1021
$\mathbb{Q}[x]$](img83-29.gif)
.
Problem 16
Discuss the solvability of the differential equation
with the initial condition
![$y(0)=0$](img85-29.gif)
. Does a solution exist
in some interval about
![$0$](img86-29.gif)
? If so, is it unique?
Problem 17
Let
![$G$](img87-29.gif)
be a subgroup of
![$S_5$](img88-28.gif)
, the group of all permutations on the
set
![$\{1, 2, 3, 4, 5\}$](img89-28.gif)
. Prove that if
![$G$](img90-28.gif)
contains a
![$5$](img91-28.gif)
-cycle and a
![$2$](img92-28.gif)
-cycle, then
![$G = S_5$](img93-28.gif)
.
Problem 18
Evaluate
where the direction of integration is counterclockwise.
Problem 19
Prove that if six people are riding together in an Evans Hall
elevator, there is either a three-person subset of mutual friends
(each knows the other two) or a three-person subset of mutual
strangers (each knows neither of the other two).
Problem 20
Let
![$f$](img95-28.gif)
be a real valued continuous function on
![$[0,\infty)$](img96-28.gif)
such
that
exists. Prove that
Previous: Spring86
Next: Spring87
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10