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Preliminary Exam - Spring 1990
Problem 1
Let
![% latex2html id marker 670
$y : \mbox{$\mathbb{R}^{}$} \to \mbox{$\mathbb{R}^{}$}$](img1-36.gif)
be a
![$C^{\infty}$](img2-36.gif)
function that satisfies
the differential equation
for
![$x \in [0,L]$](img4-36.gif)
, where
![$L$](img5-36.gif)
is a positive real number. Suppose
that
![$y(0)=y(L)=0$](img6-36.gif)
. Prove that
![$y \equiv 0$](img7-36.gif)
on
![$[0,L]$](img8-36.gif)
.
Problem 2
Let
![$A$](img9-36.gif)
be a complex
![$n \times n$](img10-36.gif)
matrix that has
finite order; that is,
![$A^k=I$](img11-36.gif)
for some positive integer
![$k$](img12-36.gif)
. Prove that
![$A$](img13-36.gif)
is diagonalizable.
_matrix,>finite order
Problem 3
Let
![$c_0,c_1,\ldots,c_{n-1}$](img14-36.gif)
be complex numbers. Prove that all the
zeros of the polynomial
lie in the open disc with center
![$0$](img16-36.gif)
and radius
Problem 4
Let
![$R$](img18-36.gif)
be a commutative ring with
![$1$](img19-36.gif)
, and
![$R^*$](img20-36.gif)
be its group
of units. Suppose that the additive group of
![$R$](img21-36.gif)
is generated by
![$\{u^2 \;\vert\; u \in R^*\}$](img22-36.gif)
. Prove that
![$R$](img23-36.gif)
has, at most, one ideal
![$\mathfrak{I}$](img24-36.gif)
for which
![$R/\mathfrak{I}$](img25-36.gif)
has cardinality
![$3$](img26-36.gif)
.
Problem 5
Suppose
![$x_1,x_2,x_3,\ldots$](img27-36.gif)
is a sequence of nonnegative real numbers
satisfying
for all
![$n\geq 1$](img29-36.gif)
. Prove that
![$\lim_{n\to\infty}x_n$](img30-36.gif)
exists.
Problem 6
Give an example of a continuous function
![% latex2html id marker 787
$v : \mbox{$\mathbb{R}^{}$} \to \mbox{$\mathbb{R}^{3}$}$](img31-36.gif)
with
the property that
![$v(t_1)$](img32-36.gif)
,
![$v(t_2)$](img33-36.gif)
, and
![$v(t_3)$](img34-36.gif)
form a basis for
![% latex2html id marker 795
$\mathbb{R}^{3}$](img35-36.gif)
whenever
![$t_1$](img36-36.gif)
,
![$t_2$](img37-36.gif)
, and
![$t_3$](img38-36.gif)
are distinct points of
![% latex2html id marker 803
$\mathbb{R}^{}$](img39-36.gif)
.
Problem 7
Let
![$a$](img40-36.gif)
be a positive real number. Evaluate the improper integral
Problem 8
Let
![% latex2html id marker 840
$\mathbb{C}\,^{*}$](img42-36.gif)
be the multiplicative group of nonzero complex numbers.
Suppose that
![$H$](img43-36.gif)
is a subgroup of finite index of
![% latex2html id marker 844
$\mathbb{C}\,^{*}$](img44-36.gif)
. Prove that
![% latex2html id marker 848
$H=\mbox{$\mathbb{C}\,^{*}$}$](img45-36.gif)
.
Problem 9
Let the real valued function
![$f$](img46-36.gif)
on
![$[0,1]$](img47-36.gif)
have the following two
properties:
Prove that
![$f$](img54-36.gif)
is continuous.
Problem 10
Show that there are at least two nonisomorphic nonabelian groups
of order
![$24$](img55-36.gif)
, of order
![$30$](img56-36.gif)
and order
![$40$](img57-36.gif)
.
Problem 11
Let the function
![$f$](img58-36.gif)
be analytic and bounded in the complex half-plane
![$\Re z>0$](img59-36.gif)
. Prove that for any positive real number
![$c$](img60-36.gif)
, the
function
![$f$](img61-36.gif)
is uniformly continuous in the half-plane
![$\Re z>c$](img62-36.gif)
.
Problem 12
Let
![$n$](img63-36.gif)
be a positive integer, and let
![$A= (a_{ij})_{i,j=1}^n$](img64-36.gif)
be
the n
![$\times$](img65-36.gif)
n matrix with
![$a_{ii} = 2,\, a_{i\,i\pm 1} = -1$](img66-36.gif)
,
and
![$a_{ij} = 0$](img67-36.gif)
otherwise; that is,
Prove that every eigenvalue of
![$A$](img69-36.gif)
is a positive real number.
Problem 13
Let
![$f$](img70-36.gif)
be an infinitely differentiable function from
![% latex2html id marker 966
$\mathbb{R}^{}$](img71-36.gif)
to
![% latex2html id marker 968
$\mathbb{R}^{}$](img72-36.gif)
. Suppose that, for some positive integer
![$n$](img73-36.gif)
,
Prove that
![$f^{(n+1)}(x)=0$](img75-35.gif)
for some
![$x$](img76-35.gif)
in
![$(0,1)$](img77-35.gif)
.
Problem 14
Let
![$A$](img78-35.gif)
and
![$B$](img79-35.gif)
be subspaces of a finite-dimensional vector space
![$V$](img80-35.gif)
such
that
![$A+B=V$](img81-34.gif)
. Write
![$n =\dim V$](img82-34.gif)
,
![$a=\dim A$](img83-34.gif)
, and
![$b=\dim B$](img84-34.gif)
.
Let
![$S$](img85-34.gif)
be the set of those
endomorphisms
![$f$](img86-34.gif)
of
![$V$](img87-34.gif)
for which
![$f(A)\subset~A$](img88-33.gif)
and
![$f(B)\subset B$](img89-33.gif)
. Prove that
![$S$](img90-33.gif)
is a subspace of the set of
all endomorphisms of
![$V$](img91-33.gif)
, and express the dimension of
![$S$](img92-33.gif)
in terms of
![$n$](img93-33.gif)
,
![$a$](img94-33.gif)
, and
![$b$](img95-32.gif)
.
Problem 15
Find a one-to-one conformal map of the semidisc
onto the upper half-plane.
Problem 16
Determine the greatest common divisor of the elements of the set
![% latex2html id marker 1047
$\{n^{13}-n \;\vert\; n \in \mbox{$\mathbb{Z}^{}$}\}$](img97-32.gif)
.
Problem 17
Let
![$f$](img98-32.gif)
be a differentiable function on
![$[0,1]$](img99-31.gif)
and let
Let
![$n$](img101-30.gif)
be a positive integer. Prove that
Problem 18
Let
![$z_1,z_2,\ldots,z_n$](img103-27.gif)
be complex numbers. Prove that there exists
a subset
![$J \subset \{1,2,\ldots,n$](img104-27.gif)
}
such that
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Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10