Previous: Fall97
Next: Fall98
Preliminary Exam - Spring 1998
Problem 1
Prove that the polynomial
![$z^4+z^3+1$](img1-50.gif)
has exactly one root
in the quadrant
![$\{z=x+iy\,\vert\, x,y >0\}$](img2-50.gif)
.
Problem 2
Let
![$f$](img3-50.gif)
be analytic in an open set containing the closed unit disc.
Suppose that
![$\vert f(z)\vert > m$](img4-50.gif)
for
![$\vert z\vert=1$](img5-50.gif)
and
![$\vert f(0)\vert < m$](img6-50.gif)
.
Prove that
![$f(z)$](img7-50.gif)
has at least one zero in the open unit disc
![$\vert z\vert < 1$](img8-50.gif)
.
Problem 3
Let
![$M$](img9-50.gif)
be a non-empty complete metric space. Let
![$T:M\to M$](img10-50.gif)
such that
![$T\circ T=T^2$](img11-50.gif)
is a strict contraction. Prove that
![$T$](img12-50.gif)
has a unique fixed point in
![$M$](img13-50.gif)
, i.e., there is a unique
point
![$x_0$](img14-50.gif)
with
![$T(x_0)=x_0$](img15-50.gif)
.
Problem 4
Using the properties of the Riemann integral, show that if
![$f$](img16-50.gif)
is a non-negative continuous function on
![$[0,1]$](img17-50.gif)
, and
![$\int^1_0 f(x)dx=0$](img18-50.gif)
, then
![$f(x)=0$](img19-50.gif)
for all
![$x\in [0,1]$](img20-50.gif)
.
Problem 5
Let
![$A$](img21-50.gif)
be the ring of real 2
![$\times $](img22-50.gif)
2 matrices of the form
![$\bigl({a\atop 0}{b\atop c}\bigr)$](img23-50.gif)
.
What are the 2-sided ideals in
![$A$](img24-50.gif)
? Justify your answer.
Problem 6
Let
![$G$](img25-50.gif)
be the group
![% latex2html id marker 743
$\mbox{$\mathbb{Q}\,^{}$}/\mbox{$\mathbb{Z}^{}$}$](img26-50.gif)
. Show that for every positive
integer
![$t$](img27-50.gif)
,
![$G$](img28-50.gif)
has a unique cyclic subgroup of order
![$t$](img29-50.gif)
.
Problem 7
Suppose that
![$A$](img30-50.gif)
and
![$B$](img31-50.gif)
are two commuting
![$n\times n$](img32-50.gif)
complex
matrices. Show that they have a common eigenvector.
Problem 8
Let
![$m\geq 0$](img33-50.gif)
be an integer. Let
![$a_1,a_2,\dots ,a_m$](img34-50.gif)
be integers and let
Show that if
![$d\geq 0$](img36-50.gif)
is an integer then
![$f(x)^d/d!$](img37-50.gif)
can be expressed in the form
where the
![$b_i$](img39-50.gif)
are integers.
Problem 9
Let
![$M_1= \bigl({3\atop 1}{2\atop 4}\bigr)$](img40-50.gif)
,
![$M_2= \bigl({5\atop -3}{7\atop -4}\bigr)$](img41-50.gif)
,
![$M_3= \bigl({5\atop -3}{6.9\atop -4}\bigr)$](img42-50.gif)
.
For which (if any)
![$i$](img43-50.gif)
,
![$1\leq i\leq 3$](img44-50.gif)
, is the sequence
![$(M_i^n)$](img45-50.gif)
bounded away from
![$\infty$](img46-50.gif)
?
For which
![$i$](img47-50.gif)
is the sequence bounded away from
![$0$](img48-50.gif)
?
Problem 10
Let
![$P_i=(a\cos\theta_i,b\sin\theta_i)$](img49-50.gif)
,
![$i=1,2,3$](img50-50.gif)
, be a triple
of points on the ellipse
![$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$](img51-50.gif)
.
What is the maximal value of the area of the triangle
![$\Delta P_1P_2P_3$](img52-50.gif)
? Determine those triangles with maximal area.
Problem 11
Let
![$A,B,\dots ,F$](img53-50.gif)
be real coefficients. Show that the
quadratic form
is positive definite if and only if
Problem 12
Given the fact that
![$\displaystyle{\int^\infty_{-\infty}e^{-x^2}dx=\sqrt{\pi}}$](img56-50.gif)
,
evaluate the integral
Problem 13
Let
![$a$](img58-50.gif)
be a complex number with
![$\vert a\vert < 1$](img59-50.gif)
.
Evaluate the integral
Problem 14
Let
![$K$](img61-50.gif)
be a real constant. Suppose that
![$y(t)$](img62-50.gif)
is a positive
differentiable function satisfying
![$y'(t)\leq Ky(t)$](img63-50.gif)
for
![$t\geq 0$](img64-50.gif)
. Prove that
![$y(t)\leq e^{Kt}y(0)$](img65-50.gif)
for
![$t\geq 0$](img66-50.gif)
.
Problem 15
For continuous real valued functions
![$f,g$](img67-50.gif)
on the interval
![$[-1,1]$](img68-50.gif)
define the inner product
![$\langle f,g\rangle=\int^1_{-1} f(x)g(x)dx$](img69-50.gif)
.
Find that polynomial of the form
![$p(x)=a+bx^2-x^4$](img70-50.gif)
which is orthogonal on
![$[-1,1]$](img71-50.gif)
to all lower order polynomials.
Problem 16
Let
![$a > 0$](img72-50.gif)
. Show that the complex function
satisfies
![$\vert f(z)\vert <1$](img74-49.gif)
for all
![$z$](img75-49.gif)
in the open left half-plane
![$\Re z < 0$](img76-49.gif)
.
Problem 17
Let
![$A$](img77-49.gif)
be an
![$n\times n$](img78-49.gif)
complex matrix with
![$\mathrm{tr}\,A=0$](img79-49.gif)
.
Show that
![$A$](img80-49.gif)
is similar to a matrix with all
![$0$](img81-48.gif)
's along the main diagonal.
Problem 18
Let
![$N$](img82-48.gif)
be a nilpotent complex matrix. Let
![$r$](img83-47.gif)
be a positive
integer. Show that there is a
![$n\times n$](img84-46.gif)
complex matrix
![$A$](img85-45.gif)
with
Previous: Fall97
Next: Fall98
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10