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Preliminary Exam - Fall 1988
Problem 1
Let
![$R$](img1-34.gif)
be a finite ring. Prove that there are positive integers m and
n with
![$m>n$](img2-34.gif)
such that
![$x^m=x^n$](img3-34.gif)
for every
![$x$](img4-34.gif)
in
![$R$](img5-34.gif)
.
Problem 2
Determine the group
![% latex2html id marker 684
${\rm Aut}(\mbox{$\mathbb{C}\,^{}$})$](img6-34.gif)
of all one-to-one analytic maps of
![% latex2html id marker 686
$\mathbb{C}\,^{}$](img7-34.gif)
onto
![% latex2html id marker 688
$\mathbb{C}\,^{}$](img8-34.gif)
.
Problem 3
Let the real valued functions
![$f_1,\ldots,f_{n+1}$](img9-34.gif)
on
![% latex2html id marker 719
$\mathbb{R}^{}$](img10-34.gif)
satisfy
the system of differential equations
Prove that for each
![$k$](img12-34.gif)
,
Problem 4
Find the Jordan Canonical Form of the matrix
Problem 5
Let
![$f$](img15-34.gif)
be a continuous, strictly increasing function from
![$[0,\infty)$](img16-34.gif)
onto
![$[0,\infty)$](img17-34.gif)
and let
![$g=f^{-1}$](img18-34.gif)
.
Prove that
for all positive numbers
![$a$](img20-34.gif)
and
![$b$](img21-34.gif)
, and determine the condition
for equality.
Problem 6
Let
![$f$](img22-34.gif)
be a function from
![$[0,1]$](img23-34.gif)
into itself whose graph
is a closed subset of the unit square. Prove that
![$f$](img25-34.gif)
is continuous.
Note: See also Problem
.
Problem 7
Find all abelian groups of order
![$8$](img26-34.gif)
, up to isomorphism. Then identify
which type occurs in each of
-
,
-
,
- the roots of
in
,
-
,
-
.
![% latex2html id marker 825
$\mbox{\bf {F}}_8$](img33-34.gif)
is the field of eight elements, and
![% latex2html id marker 827
$\mbox{\bf {F}}_8^+$](img34-34.gif)
is
its underlying additive group;
![$R^*$](img35-34.gif)
is the group of invertible
elements in the ring
![$R$](img36-34.gif)
, under multiplication.
Problem 8
Do the functions
![$f(z)=e^z+z$](img37-34.gif)
and
![$g(z)=ze^z+1$](img38-34.gif)
have the same
number of zeros in the strip
![$-\frac{\pi}{2} < \Im z < \frac{\pi}{2}$](img39-34.gif)
?
Problem 9
Let
![$A$](img40-34.gif)
and
![$B$](img41-34.gif)
be real symmetric n
![$\times$](img42-34.gif)
n matrices. Assume that
the eigenvalues of
![$A$](img43-34.gif)
all lie in the interval
![$[a_1,a_2]$](img44-34.gif)
and those
of
![$B$](img45-34.gif)
all lie in the interval
![$[b_1,b_2]$](img46-34.gif)
. Prove that the eigenvalues
of
![$A + B$](img47-34.gif)
all lie in the interval
![$[a_1+b_1, a_2+b_2]$](img48-34.gif)
.
Problem 10
Find (up to isomorphism) all groups of order
![$2p$](img49-34.gif)
, where
![$p$](img50-34.gif)
is a
prime (
![$p\geq 2$](img51-34.gif)
).
Problem 11
Let
![$f$](img52-34.gif)
be an analytic function on a disc
![$D$](img53-34.gif)
whose center is the
point
![$z_0$](img54-34.gif)
. Assume that
![$\vert f'(z)-f'(z_0)\vert<\vert f'(z_0)\vert$](img55-34.gif)
on
D. Prove that
![$f$](img56-34.gif)
is one-to-one on
![$D$](img57-34.gif)
.
Problem 12
Let n be a positive integer and let
![$f$](img58-34.gif)
be a polynomial in
![% latex2html id marker 938
$\mbox{$\mathbb{R}^{}$}[x]$](img59-34.gif)
of degree
![$n$](img60-34.gif)
. Prove that there are real numbers
![$a_0,a_1,\ldots,a_n$](img61-34.gif)
,
not all equal to zero, such that the polynomial
is divisible by f.
Problem 13
Let
![$A$](img63-34.gif)
be a complex
![$n\times n$](img64-34.gif)
matrix, and let
![$C(A)$](img65-34.gif)
be the
commutant
of
![$A$](img66-34.gif)
; that is, the set of complex
![$n\times n$](img67-34.gif)
matrices
![$B$](img68-34.gif)
such that
![$AB = BA$](img69-34.gif)
. (It is obviously a subspace of
![$M_{n\times n}$](img70-34.gif)
, the vector space of
all complex
![$n\times n$](img71-34.gif)
matrices.) Prove that
![$\dim C(A) \geq n$](img72-34.gif)
.
_matrix,>commutant of a
Problem 14
Let the group
![$G$](img73-34.gif)
be generated by two elements,
![$a$](img74-33.gif)
and
![$b$](img75-33.gif)
, both of
order
![$2$](img76-33.gif)
. Prove that
![$G$](img77-33.gif)
has a subgroup of index
![$2$](img78-33.gif)
.
Problem 15
Prove that a real valued
![$C^3$](img79-33.gif)
function
![$f$](img80-33.gif)
on
![% latex2html id marker 1011
$\mathbb{R}^{2}$](img81-33.gif)
whose Laplacian,
is everywhere positive cannot have a local maximum.
Problem 16
Let
![$n$](img83-33.gif)
be a positive integer. Prove that the polynomial
in
![% latex2html id marker 1040
$\mbox{$\mathbb{R}^{}$}[x]$](img85-33.gif)
has
![$n$](img86-33.gif)
distinct complex zeros,
![$z_1,z_2,\ldots,z_n$](img87-33.gif)
, and that
they satisfy
Problem 18
Let
![$g$](img90-32.gif)
be a continuous real valued function on
![$[0,1]$](img91-32.gif)
. Prove that
there exists a continuous real valued function
![$f$](img92-32.gif)
on
![$[0,1]$](img93-32.gif)
satisfying the equation
Previous: Spring88
Next: Spring89
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10