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Preliminary Exam - Fall 1990
Problem 1
Find all pairs of integers
![$a$](img1-35.gif)
and
![$b$](img2-35.gif)
satisfying
![$0<a<b$](img3-35.gif)
and
![$a^b=b^a$](img4-35.gif)
.
Problem 2
Evaluate the integral
where
![$C$](img6-35.gif)
is the circle
![$\vert z\vert=1$](img7-35.gif)
with counterclockwise orientation.
Problem 3
Let
![$R$](img8-35.gif)
be a ring with
![$1$](img9-35.gif)
, and let
![$\mathfrak{I}$](img10-35.gif)
be the left ideal of
![$R$](img11-35.gif)
generated by
![$\{ab-ba \;\vert\; a, b \in R\}$](img12-35.gif)
. Prove that
![$\mathfrak{I}$](img13-35.gif)
is a two-sided ideal.
Problem 4
Suppose
![$f$](img14-35.gif)
is a continuous real valued function. Show that
for some
![$\xi\in [0,1]$](img16-35.gif)
.
Problem 5
Let
![${A}$](img17-35.gif)
be a real symmetric
![$n\times n$](img18-35.gif)
matrix that is positive definite.
Let
![% latex2html id marker 739
$y \in \mbox{$\mathbb{R}^{n}$}$](img19-35.gif)
,
![$y\neq0$](img20-35.gif)
. Prove that the limit
exists and is an eigenvalue of
![${A}$](img22-35.gif)
.
Problem 6
Let the function
![$f$](img23-35.gif)
be analytic in the entire complex plane, and
suppose that
![$f(z)/z \to 0$](img24-35.gif)
as
![$\vert z\vert \to \infty$](img25-35.gif)
. Prove that
![$f$](img26-35.gif)
is constant.
Problem 7
Let
![$G$](img27-35.gif)
be a group and
![$N$](img28-35.gif)
be a normal subgroup of
![$G$](img29-35.gif)
with
![$N \neq G$](img30-35.gif)
.
Suppose that there does not exist a subgroup
![$H$](img31-35.gif)
of
![$G$](img32-35.gif)
satisfying
![$N \subset H \subset G$](img33-35.gif)
and
![$N \neq H \neq G$](img34-35.gif)
. Prove that the index of
![$N$](img35-35.gif)
in
![$G$](img36-35.gif)
is finite and equal to a prime number.
Problem 8
Let
![$f$](img37-35.gif)
be a continuous real valued function satisfying
![$f(x) \geq 0$](img38-35.gif)
, for all x, and
Prove that
as
![$n \to \infty$](img41-35.gif)
.
Problem 9
Let
![% latex2html id marker 816
$\mathbb{R}^{3}$](img42-35.gif)
be 3-space with the usual inner product, and
![% latex2html id marker 820
$(a,b,c)
\in \mbox{$\mathbb{R}^{3}$}$](img43-35.gif)
a vector of length
![$1$](img44-35.gif)
. Let
![$W$](img45-35.gif)
be the plane defined by
![$ax+by+cz=0$](img46-35.gif)
. Find, in the standard basis, the matrix representing
the orthogonal projection of
![% latex2html id marker 828
$\mathbb{R}^{3}$](img47-35.gif)
onto
![$W$](img48-35.gif)
.
Problem 10
Determine the Jordan Canonical Form of the matrix
Problem 11
Suppose that
![$f$](img50-35.gif)
maps the compact interval
![$I$](img51-35.gif)
into itself and that
for all
![$x, y \in I$](img53-35.gif)
,
![$x\neq y$](img54-35.gif)
. Can one conclude that there
is some constant
![$M<1$](img55-35.gif)
such that, for all
![$x, y \in I$](img56-35.gif)
,
Problem 12
Let
![$A$](img58-35.gif)
be an additively written abelian group, and
![$f, g:A \to A$](img59-35.gif)
two group homomorphisms. Define the group homomorphisms
![$i,j:A \to A$](img60-35.gif)
by
Prove that the kernel of i is isomorphic to the kernel of j.
Problem 13
Suppose that
![$f$](img62-35.gif)
is analytic on the open upper half-plane and
satisfies
![$\vert f(z)\vert\leq 1$](img63-35.gif)
for all
![$z$](img64-35.gif)
,
![$f(i)=0$](img65-35.gif)
. How large can
![$\vert f(2i)\vert$](img66-35.gif)
be under these conditions?
Problem 14
Prove that
![$\sqrt{2}+\sqrt[3]{3}$](img67-35.gif)
is irrational.
Problem 15
Let
![$n$](img68-35.gif)
be a positive integer and let
![$P_{2n+1}$](img69-35.gif)
be the vector space
of real polynomials whose degrees are, at most,
![$2n+1$](img70-35.gif)
. Prove that
there exist unique real numbers
![$c_1,\ldots,c_n$](img71-35.gif)
such that
for all
![$p \in P_{2n+1}$](img73-35.gif)
.
Problem 16
Evaluate the limit
Problem 17
Does the set
![% latex2html id marker 952
$G=\{a \in \mbox{$\mathbb{R}^{}$} \;\vert\; a>0, a\neq 1\}$](img75-34.gif)
form a group
with the operation
![$a*b=a^{\log b}$](img76-34.gif)
?
Problem 18
Let the function
![$f$](img77-34.gif)
be analytic in the entire complex plane and
satisfy
for all
![$r>0$](img79-34.gif)
. Prove that
![$f$](img80-34.gif)
is the zero function.
Previous: Spring90
Next: Spring91
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10