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Preliminary Exam - Summer 1982
Problem 1
Determine the Jordan Canonical Form of the matrix
Problem 2
Compute the integral
Problem 3
Let
be a nonempty compact set in a metric space with distance
function
. Suppose that
satisfies
for all
![$x\ne y$](img7-15.gif)
in
![$K$](img8-15.gif)
. Show there exists precisely one point
![$x\in K$](img9-15.gif)
such
that
![$x=\varphi(x)$](img10-15.gif)
.
Problem 4
Let
![$G$](img11-15.gif)
be a group with generators
![$a$](img12-15.gif)
and
![$b$](img13-15.gif)
satisfying
Is
![$G$](img15-15.gif)
trivial?
Problem 5
Let
![$0<a_0\leq a_1\leq \cdots \leq a_n$](img16-15.gif)
. Prove that the equation
has no roots in the disc
![$\vert z\vert < 1$](img18-15.gif)
.
Problem 6
Suppose
![$f$](img19-15.gif)
is a differentiable real valued function such that
![$f'(x)>f(x)$](img20-15.gif)
for all
![% latex2html id marker 740
$x \in \mbox{$\mathbb{R}^{}$}$](img21-15.gif)
and
![$f(0)=0$](img22-15.gif)
. Prove that
![$f(x)>0$](img23-15.gif)
for all positive
![$x$](img24-15.gif)
.
Problem 7
Let
![$V$](img25-15.gif)
be the vector space of all real 3
![$\times $](img26-15.gif)
3 matrices and let
![${A}$](img27-15.gif)
be
the diagonal matrix
Calculate the determinant of the linear transformation
![${T}$](img29-15.gif)
on
![$V$](img30-15.gif)
defined by
![$T(X) = \frac{1}{2}(AX + XA)$](img31-15.gif)
.
Problem 8
Let n be a positive integer.
- Show that the binomial coefficient
is even.
- Prove that
is divisible by
if and only if
n is not a power of
.
Problem 9
Determine the complex numbers
![$z$](img36-15.gif)
for which the power series
and its term by term derivatives of all orders converge absolutely.
Problem 10
For complex numbers
![$\alpha_1,\alpha_2,\ldots,\alpha_k$](img38-15.gif)
, prove
Note: See also Problem
![[*]](file:/local/sol/2.6/depot/tetex-20000212/lib/latex2html/icons/crossref.gif)
.
Problem 11
Let
![$s(y)$](img40-15.gif)
and
![$t(y)$](img41-15.gif)
be real differentiable functions of
![$y$](img42-15.gif)
,
![$-\infty < y < \infty$](img43-15.gif)
, such that the complex function
is complex analytic with
![$s(0)=1$](img45-15.gif)
and
![$t(0)=0$](img46-15.gif)
. Determine
![$s(y)$](img47-15.gif)
and
![$t(y)$](img48-15.gif)
.
Problem 12
Determine (with proofs) which of the following polynomials are irreducible
over the field
![% latex2html id marker 861
$\mathbb{Q}\,^{}$](img49-15.gif)
of rationals.
.
Problem 13
Let
![% latex2html id marker 896
$f:[0,\pi] \to \mbox{$\mathbb{R}^{}$}$](img54-15.gif)
be continuous and such that
for all integers
![$n \geq 1$](img56-15.gif)
. Is
![$f$](img57-15.gif)
identically
![$0$](img58-15.gif)
?
Problem 14
Let
![$A$](img59-15.gif)
be a real
![$n \times n$](img60-15.gif)
matrix such that
![$\langle Ax, x \rangle \geq 0$](img61-15.gif)
for
every real
![$n$](img62-15.gif)
-vector
![$x$](img63-15.gif)
. Show that
![$Au=0$](img64-15.gif)
if and only if
![$A^tu=0$](img65-15.gif)
.
Problem 15
Let
![$f(z)$](img66-15.gif)
be analytic on the open unit disc
![% latex2html id marker 934
$\mathbb{D}= \{z \;\vert\; \vert z\vert < 1\}$](img67-15.gif)
.
Prove that there is a sequence
![$(z_n)$](img68-15.gif)
in
![% latex2html id marker 938
$\mathbb{D}$](img69-15.gif)
such that
![$\vert z_n\vert \to 1$](img70-15.gif)
and
![$(f(z_n))$](img71-15.gif)
is bounded.
Problem 17
Let
![% latex2html id marker 992
$f:\mbox{$\mathbb{R}^{3}$} \to \mbox{$\mathbb{R}^{2}$}$](img80-14.gif)
and assume that
![$0$](img81-14.gif)
is a regular value of f
(i.e., the differential of
![$f$](img82-14.gif)
has rank
![$2$](img83-14.gif)
at each point of
![$f^{-1}(0)$](img84-14.gif)
).
Prove that
![% latex2html id marker 1004
$\mbox{$\mathbb{R}^{3}$}\setminus f^{-1}(0)$](img85-14.gif)
is arcwise connected.
Problem 18
Let
![$E$](img86-14.gif)
be the set of all continuous real valued functions
![% latex2html id marker 1026
\(u:[0,1]\to\mbox{$\mathbb{R}^{}$}\)](img87-14.gif)
satisfying
Let
![% latex2html id marker 1038
$\varphi : E \to \mbox{$\mathbb{R}^{}$}$](img89-13.gif)
be defined by
Show that
![$\varphi$](img91-13.gif)
achieves its maximum value at some element
of
![$E$](img92-13.gif)
.
Problem 19
Let
![$V$](img93-13.gif)
be a finite-dimensional vector space over the rationals
![% latex2html id marker 1064
$\mathbb{Q}\,^{}$](img94-13.gif)
and
let
![${M}$](img95-13.gif)
be an automorphism of
![$V$](img96-13.gif)
such that
![${M}$](img97-13.gif)
fixes no
nonzero vector in
![$V$](img98-13.gif)
. Suppose that
![${M}^p$](img99-13.gif)
is the identity map on V,
where
![$p$](img100-13.gif)
is a prime number. Show that the dimension of
![$V$](img101-13.gif)
is
divisible by
![$p-1$](img102-13.gif)
.
Problem 20
Let
![$M_{2\times 2}$](img103-11.gif)
be the four-dimensional vector space of all 2
![$\times $](img104-11.gif)
2 real matrices
and define
![$f:M_{2\times 2} \to M_{2\times 2}$](img105-10.gif)
by
![$f(X)=X^2$](img106-9.gif)
.
- Show that
has a local inverse near the point
- Show that
does not have a local inverse near the point
Previous: Spring82
Next: Fall82
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10