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Preliminary Exam - Spring 1988
Problem 1
Suppose that
![$f(x)$](img1-29.gif)
,
![$-\infty < x < \infty$](img2-29.gif)
, is a continuous
real valued function, that
![$f'(x)$](img3-29.gif)
exists for
![$x \neq 0$](img4-29.gif)
, and that
![$\lim_{x \to 0}f'(x)$](img5-29.gif)
exists. Prove that
![$f'(0)$](img6-29.gif)
exists.
Problem 2
Determine the last digit of
in the decimal system.
Problem 3
If a finite homogeneous system of linear equations with rational
coefficients has a nontrivial complex solution, need it have a
nontrivial rational solution? Give a proof or a counterexample.
Problem 4
True or false: A function
![$f(z)$](img8-29.gif)
analytic on
![$\vert z-a\vert<r$](img9-29.gif)
and
continuous on
![$\vert z-a\vert\leq r$](img10-29.gif)
extends, for some
![$\delta > 0$](img11-29.gif)
, to
a function analytic on
![$\vert z-a\vert<r+\delta$](img12-29.gif)
? Give a proof
or a counterexample.
Problem 5
Let
![$D$](img13-29.gif)
be a group of order
![$2n$](img14-29.gif)
, where
![$n$](img15-29.gif)
is odd, with a subgroup
![$H$](img16-29.gif)
of
order
![$n$](img17-29.gif)
satisfying
![$xhx^{-1} =h^{-1}$](img18-29.gif)
for all h in
![$H$](img19-29.gif)
and all
x in
![$D\setminus H$](img20-29.gif)
. Prove that
![$H$](img21-29.gif)
is commutative and that every
element of
![$D\setminus H$](img22-29.gif)
is of order
![$2$](img23-29.gif)
.
Problem 6
Prove or disprove: There is a real
![$n\times n$](img24-29.gif)
matrix
![${A}$](img25-29.gif)
such that
if and only if
![$n$](img27-29.gif)
is even.
Problem 7
Let
![$R$](img28-29.gif)
be a commutative ring with unit element and
![$a\in R$](img29-29.gif)
. Let
![$n$](img30-29.gif)
and
![$m$](img31-29.gif)
be positive integers, and write
![$d=\gcd\{n,m\}$](img32-29.gif)
. Prove that
the ideal of
![$R$](img33-29.gif)
generated by
![$a^n-1$](img34-29.gif)
and
![$a^m-1$](img35-29.gif)
is the same as the
ideal generated by
![$a^d-1$](img36-29.gif)
.
Problem 8
For
![$a>1$](img37-29.gif)
and
![$n= 0, 1, 2, \ldots$](img38-29.gif)
, evaluate the integrals
Problem 9
Prove that the integrals
converge.
Problem 11
Let
![$S_9$](img52-29.gif)
denote the group of permutations of
![$\{1,2,\ldots,9\}$](img53-29.gif)
and let
![$A_9$](img54-29.gif)
be the subgroup consisting of all even permutations. Denote by
![$1\in S_9$](img55-29.gif)
the identity permutation. Determine the minimum of all
positive integers
![$m$](img56-29.gif)
such that every
![$\sigma\in S_9$](img57-29.gif)
satisfies
![$\sigma^m=1$](img58-29.gif)
. Determine also the minimum of all positive integers
![$m$](img59-29.gif)
such that every
![$\sigma\in A_9$](img60-29.gif)
satisfies
![$\sigma^m=1$](img61-29.gif)
.
Problem 12
For each real value of the parameter
![$t$](img62-29.gif)
,
determine the number of real roots, counting multiplicities, of
the cubic polynomial
![$p_t(x) = (1+t^2)x^3 - 3t^3x + t^4$](img63-29.gif)
.
Problem 13
Find all groups
![$G$](img64-29.gif)
such that every automorphism of
![$G$](img65-29.gif)
is trivial.
Problem 14
Let the function f be analytic in the open unit disc of the
complex plane and real valued on the radii
![$[0,1)$](img66-29.gif)
and
![$[0,e^{i\pi
\sqrt{2}}\,)$](img67-29.gif)
. Prove that f is constant.
Problem 15
Compute
![${A}^{10}$](img68-29.gif)
for the matrix
Problem 16
Let
![$X$](img70-29.gif)
be a set and
![$V$](img71-29.gif)
a real vector space of real valued functions
on
![$X$](img72-29.gif)
of dimension
![$n$](img73-29.gif)
,
![$0 < n < \infty$](img74-28.gif)
. Prove that there are
![$n$](img75-28.gif)
points
![$x_1, x_2, \ldots, x_n$](img76-28.gif)
in
![$X$](img77-28.gif)
such that the map
![$f \to (f(x_1), \ldots, f(x_n))$](img78-28.gif)
of
![$V$](img79-28.gif)
to
![% latex2html id marker 923
$\mathbb{R}^{n}$](img80-28.gif)
is an
isomorphism (i.e., one-to-one and onto). (The operations of
addition and scalar multiplication in
![$V$](img81-28.gif)
are assumed to be the
natural ones.)
Problem 18
For which positive integers
![$n$](img89-27.gif)
is there a 2
![$\times$](img90-27.gif)
2
matrix
with integer entries and order
![$n$](img92-27.gif)
; that is,
![$A^n=I$](img93-27.gif)
but
![$A^k\neq I$](img94-27.gif)
for
![$0 < k < n$](img95-27.gif)
? _matrix,>order n
Note: See also Problem
.
Problem 19
Show that you can represent the set of nonnegative integers,
![% latex2html id marker 1003
$\mbox{$\mathbb{Z}^{}$}_+$](img96-27.gif)
,
as the union of two disjoint subsets
![$N_1$](img97-27.gif)
and
![% latex2html id marker 1011
$(N_1 \cap
N_2 = \emptyset,\; N_1 \cup N_2 = \mbox{$\mathbb{Z}^{}$}_+)$](img99-27.gif)
such that
neither
![$N_1$](img100-26.gif)
nor
![$N_2$](img101-26.gif)
contains an infinite arithmetic progression.
Problem 20
Does there exist a continuous real valued function
![$f(x)$](img102-26.gif)
,
![$0\leq x \leq 1$](img103-23.gif)
, such that
for
![$n = 0, 2, 3, 4, \ldots$](img105-22.gif)
? Give an example or a proof that no
such
![$f$](img106-21.gif)
exists.
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Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10