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Preliminary Exam - Spring 1985
Problem 1
Let
![$f(x)$](img1-24.gif)
,
![$0\leq x<\infty$](img2-24.gif)
, be continuous, differentiable,
with
![$f(0) = 0$](img3-24.gif)
, and that
![$f'(x)$](img4-24.gif)
is an increasing function
of
![$x$](img5-24.gif)
for
![$x \geq 0$](img6-24.gif)
. Prove that
is an increasing function of
![$x$](img8-24.gif)
.
Problem 2
In a commutative group
![$G$](img9-24.gif)
, let the element
![$a$](img10-24.gif)
have order
![$r$](img11-24.gif)
, let
![$b$](img12-24.gif)
have order
![$s\; (r, s < \infty )$](img13-24.gif)
, and assume that the greatest
common divisor of
![$r$](img14-24.gif)
and
![$s$](img15-24.gif)
is
![$1$](img16-24.gif)
. Show that
![$ab$](img17-24.gif)
has order
![$rs$](img18-24.gif)
.
Problem 3
Show that a necessary and sufficient condition for three points
![$a$](img19-24.gif)
,
![$b$](img20-24.gif)
, and
![$c$](img21-24.gif)
in the complex plane to form an equilateral triangle
is that
Problem 4
Let
![$R>1$](img23-24.gif)
and let
![$f$](img24-24.gif)
be analytic on
![$\vert z\vert<R$](img25-24.gif)
except at
![$z=1$](img26-24.gif)
,
where
![$f$](img27-24.gif)
has a simple pole. If
is the Maclaurin series for
![$f$](img29-24.gif)
, show that
![$\lim_{n\to\infty}a_n$](img30-24.gif)
exists.
Problem 5
Factor
![$x^4+x^3+x+3$](img31-24.gif)
completely in
![% latex2html id marker 745
$\mbox{$\mathbb{Z}^{}$}_5[x]$](img32-24.gif)
.
Problem 6
Let
![$A$](img33-24.gif)
and
![$B$](img34-24.gif)
be two
![$n\times n$](img35-24.gif)
self-adjoint (i.e., Hermitian) matrices
over
![% latex2html id marker 763
$\mathbb{C}\,^{}$](img36-24.gif)
such that all eigenvalues of
![$A$](img37-24.gif)
lie in
![$[a,a']$](img38-24.gif)
and all
eigenvalues of
![$B$](img39-24.gif)
lie in
![$[b,b']$](img40-24.gif)
. Show that all eigenvalues of
![$A + B$](img41-24.gif)
lie in
![$[a+b,a'+b']$](img42-24.gif)
.
Problem 7
Prove that
What restrictions, if any, need be placed on
![$b$](img44-24.gif)
?
Problem 8
Let
![$h>0$](img45-24.gif)
be given. Consider the linear difference equation
(Note the analogy with the differential equation
![$y''= -y$](img47-24.gif)
.)
- Find the general solution of
by trying suitable
exponential substitutions.
- Find the solution with
and
. Denote it by
,
.
- Let
be fixed and
. Show that
Problem 9
Define the function
![$\zeta$](img56-24.gif)
by
Prove that
![$\zeta(x)$](img58-24.gif)
is defined and has continuous derivatives
of all orders in the interval
![$1<x<\infty$](img59-24.gif)
.
Problem 10
For arbitrary elements
![$a$](img60-24.gif)
,
![$b$](img61-24.gif)
, and
![$c$](img62-24.gif)
in a field
![% latex2html id marker 874
$\mbox{\bf {F}}$](img63-24.gif)
, compute the
minimal polynomial of the matrix
Problem 11
Let
![% latex2html id marker 910
$\mbox{\bf {F}} = \{a+b\sqrt[3]{2}+c\sqrt[3]{4} \;\vert\; a, b, c \in \mbox{$\mathbb{Q}\,^{}$}\}$](img65-24.gif)
.
Prove that
![% latex2html id marker 912
$\mbox{\bf {F}}$](img66-24.gif)
is a field and each element in
![% latex2html id marker 914
$\mbox{\bf {F}}$](img67-24.gif)
has a unique
representation
as
![$a+b\sqrt[3]{2}+c\sqrt[3]{4}$](img68-24.gif)
with
![% latex2html id marker 920
$a, b, c \in \mbox{$\mathbb{Q}\,^{}$}$](img69-24.gif)
. Find
![$\left( 1-\sqrt[3]{2} \, \right)^{-1}$](img70-24.gif)
in
![% latex2html id marker 924
$\mbox{\bf {F}}$](img71-24.gif)
.
Problem 12
Prove that
What restrictions must be placed on
![$\alpha$](img73-24.gif)
?
Problem 13
Prove that for any
![% latex2html id marker 963
$a\in\mbox{$\mathbb{C}\,^{}$}$](img74-23.gif)
and any integer
![$n\geq 2$](img75-23.gif)
, the
equation
![$1+z+az^n=0$](img76-23.gif)
has at least one root in the disc
![$\vert z\vert\leq~2$](img77-23.gif)
.
Problem 14
Show that
converges as an improper Riemann integral. Evaluate
![$I$](img79-23.gif)
.
Problem 15
Let
![$\displaystyle{\zeta = e^{\frac{2\pi i}{7}}}$](img80-23.gif)
be a primitive
![$7^{th}$](img81-23.gif)
root of
unity. Find a cubic polynomial with integer coefficients having
![$\alpha = \zeta + \zeta^{-1}$](img82-23.gif)
as a root.
Problem 16
Let f be continuous on
![% latex2html id marker 1014
$\mathbb{R}^{}$](img83-23.gif)
, and let
Prove that
![$f_n(x)$](img85-23.gif)
converges uniformly to a limit on every finite
interval
![$[a,b]$](img86-23.gif)
.
Problem 17
Let
![$v_1$](img87-23.gif)
and
![$v_2$](img88-22.gif)
be two real valued continuous functions on
![% latex2html id marker 1044
$\mathbb{R}^{}$](img89-22.gif)
such that
![$v_1(x)<v_2(x)$](img90-22.gif)
for all
![% latex2html id marker 1050
$x\in\mbox{$\mathbb{R}^{}$}$](img91-22.gif)
. Let
![$\varphi_1(t)$](img92-22.gif)
and
![$\varphi_2(t)$](img93-22.gif)
be, respectively, solutions of the differential equations
for
![$a<t<b$](img95-22.gif)
. If
![$\varphi_1(t_0)=\varphi_2(t_0)$](img96-22.gif)
for some
![$t_0\in (a,b)$](img97-22.gif)
,
show that
![$\varphi_1(t)\leq\varphi_2(t)$](img98-22.gif)
for all
![$t\in (t_0,b)$](img99-22.gif)
.
Problem 18
Let
![$A$](img100-21.gif)
and
![$B$](img101-21.gif)
be two
![$n\times n$](img102-21.gif)
self-adjoint (i.e., Hermitian) matrices
over
![% latex2html id marker 1084
$\mathbb{C}\,^{}$](img103-18.gif)
and assume
![$A$](img104-18.gif)
is positive definite. Prove that all eigenvalues
of AB are real.
Problem 19
Let
![% latex2html id marker 1100
$\mbox{\bf {F}}$](img105-17.gif)
be a finite field. Give a complete proof of the fact that
the number of elements of
![% latex2html id marker 1102
$\mbox{\bf {F}}$](img106-16.gif)
is of the form
![$p^r$](img107-16.gif)
, where
![$p\geq 2$](img108-16.gif)
is
a prime number and
![$r$](img109-16.gif)
is an integer
![$\geq 1$](img110-16.gif)
.
Problem 20
Let
![$f(z)$](img111-15.gif)
be an analytic function that maps the open disc
![$\vert z\vert<1$](img112-15.gif)
into itself. Show that
![$\vert f'(z)\vert\leq 1/(1-\vert z\vert^2)$](img113-15.gif)
.
Previous: Fall84
Next: Summer85
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10