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Preliminary Exam - Spring 1992
Problem 2
Find a square root of the matrix
How many square roots does this matrix have?
Problem 3
Let
![$f$](img6-39.gif)
be an analytic function in the connected open subset
![$G$](img7-39.gif)
of
the complex plane. Assume that for each point
![$z$](img8-39.gif)
in
![$G$](img9-39.gif)
, there is a
positive integer
![$n$](img10-39.gif)
such that the
![$n^{th}$](img11-39.gif)
derivative
of
![$f$](img12-39.gif)
vanishes at
![$z$](img13-39.gif)
. Prove that
![$f$](img14-39.gif)
is a polynomial.
Problem 4
Show that every infinite closed subset of
![% latex2html id marker 740
$\mbox{$\mathbb{R}^{n}$}$](img15-39.gif)
is the closure
of a countable set.
Problem 5
Let
![$f$](img16-39.gif)
be a differentiable function from
![% latex2html id marker 766
$\mathbb{R}^{n}$](img17-39.gif)
to
![% latex2html id marker 768
$\mathbb{R}^{n}$](img18-39.gif)
. Assume that
there is a differentiable function
![$g$](img19-39.gif)
from
![% latex2html id marker 772
$\mathbb{R}^{n}$](img20-39.gif)
to
![% latex2html id marker 774
$\mathbb{R}^{}$](img21-39.gif)
having
no critical points such that
![$g\circ f$](img22-39.gif)
vanishes identically.
Prove that the Jacobian determinant of
![$f$](img23-39.gif)
vanishes identically.
Problem 6
Let
![$p$](img24-39.gif)
be a prime integer,
![$p\equiv 3 \pmod{4}$](img25-39.gif)
, and let
![% latex2html id marker 818
$\mbox{\bf {F}}_p=\mbox{$\mathbb{Z}^{}$}/p\mbox{$\mathbb{Z}^{}$}$](img26-39.gif)
. If
![$x^4+1$](img27-39.gif)
factors into a product
![$g(x)h(x)$](img28-39.gif)
of two quadratic polynomials in
![% latex2html id marker 824
$\mbox{\bf {F}}_p[x]$](img29-39.gif)
, prove that
![$g(x)$](img30-39.gif)
and
![$h(x)$](img31-39.gif)
are both irreducible over
![% latex2html id marker 830
$\mbox{\bf {F}}_p$](img32-39.gif)
.
Problem 7
Let
![$a_1,a_2,\ldots,a_{10}$](img33-39.gif)
be integers with
![$1\leq a_i \leq 25$](img34-39.gif)
,
for
![$1\leq i \leq 10$](img35-39.gif)
. Prove that there exist integers
![$n_1,n_2,
\ldots,n_{10}$](img36-39.gif)
, not all zero, such that
Problem 8
Evaluate the integral
Problem 9
Let
![$p$](img39-39.gif)
be a nonconstant polynomial with real coefficients and only
real roots. Prove that for each real number
![$r$](img40-39.gif)
, the polynomial
![$p-rp'$](img41-39.gif)
has only real roots.
Problem 10
Let
![$A$](img42-39.gif)
denote the matrix
For which positive integers
![$n$](img44-39.gif)
is there a complex 4
![$\times$](img45-39.gif)
4
matrix
![$X$](img46-39.gif)
such that
![$X^n=A$](img47-39.gif)
?
Problem 11
Find a Laurent series that converges in the annulus
![$1<\vert z\vert<2$](img48-39.gif)
to a
branch of the function
![$\displaystyle{\log\left(\frac{z(2-z)}{1-z}\right)}$](img49-39.gif)
.
Problem 12
Let
![$A$](img50-39.gif)
be a real symmetric
![$n\times n$](img51-39.gif)
matrix with nonnegative entries.
Prove that
![$A$](img52-39.gif)
has an eigenvector with nonnegative entries.
Problem 13
Let
![$f$](img53-39.gif)
be a one-to-one
![$C^1$](img54-39.gif)
map of
![% latex2html id marker 960
$\mathbb{R}^{3}$](img55-39.gif)
into
![% latex2html id marker 962
$\mathbb{R}^{3}$](img56-39.gif)
, and let
![$J$](img57-39.gif)
denote its Jacobian determinant. Prove that if
![$x_0$](img58-39.gif)
is any
point of
![% latex2html id marker 968
$\mathbb{R}^{3}$](img59-39.gif)
and
![$Q_r(x_0)$](img60-39.gif)
denotes the cube with center
![$x_0$](img61-39.gif)
, side length
![$r$](img62-39.gif)
, and edges parallel to the coordinate
axes, then
Here,
is the Euclidean norm in
.
Problem 15
Let
![$S_{999}$](img74-38.gif)
denote the group of all permutations of
![$\{1,\ldots,999\}$](img75-38.gif)
, and let
![$G\subset S_{999}$](img76-38.gif)
be an abelian
subgroup of order
![$1111$](img77-38.gif)
. Prove that there exists
![$i\in\{1,\ldots,999\}$](img78-38.gif)
such that for all
![$\sigma\in G$](img79-38.gif)
, one has
![$\sigma(i)=i$](img80-38.gif)
.
Problem 16
Let
![$x_0=1$](img81-37.gif)
and
Prove that
![$\displaystyle{x_{\infty} = \lim_{n\to\infty}x_n}$](img83-37.gif)
exists, and find its value.
Problem 17
For which positive numbers
![$a$](img84-37.gif)
and
![$b$](img85-37.gif)
, with
![$a>1$](img86-37.gif)
, does the
equation
![$\log_a x = x^b$](img87-37.gif)
have a positive solution for
![$x$](img88-36.gif)
?
Problem 18
Let the function
![$f$](img89-36.gif)
be analytic in the entire complex plane,
real valued on the real axis, and of positive imaginary part in
the upper half-plane. Prove
![$f'(x)>0$](img90-36.gif)
for
![$x$](img91-36.gif)
real.
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Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10