Previous: Fall86
Next: Fall87
Preliminary Exam - Spring 1987
Problem 1
A standard theorem states that a continuous real valued function
on a compact set is bounded. Prove the converse: If
![$K$](img1-31.gif)
is a subset
of
![% latex2html id marker 661
$\mathbb{R}^{n}$](img2-31.gif)
and if every continuous real valued function on
![$K$](img3-31.gif)
is bounded, then
![$K$](img4-31.gif)
is compact.
Problem 2
Let the transformation
![$T$](img5-31.gif)
from the subset
![$U=\{(u,v)\;\vert\; u > v \}$](img6-31.gif)
of
![% latex2html id marker 687
$\mathbb{R}^{2}$](img7-31.gif)
into
![% latex2html id marker 689
$\mathbb{R}^{2}$](img8-31.gif)
be defined by
![$T(u,v)=(u+v, u^2+v^2)$](img9-31.gif)
.
- Prove that
is locally one-to-one.
- Determine the range of
, and show that
is globally one-to-one.
Problem 3
Let
![$f$](img13-31.gif)
be a complex valued function in the open unit disc,
![% latex2html id marker 715
$\mathbb{D}$](img14-31.gif)
,
of the complex plane such that the functions
![$g = f^2$](img15-31.gif)
and
![$h = f^3$](img16-31.gif)
are both analytic. Prove that
![\(f\)](img17-31.gif)
is analytic in
![% latex2html id marker 721
$\mathbb{D}$](img18-31.gif)
.
Problem 4
Let
![% latex2html id marker 755
$\mbox{\bf {F}}$](img19-31.gif)
be a finite field with
![$q$](img20-31.gif)
elements and let
![$x$](img21-31.gif)
be an
indeterminate. For
![$f$](img22-31.gif)
a polynomial in
![% latex2html id marker 763
$\mbox{\bf {F}}[x]$](img23-31.gif)
, let
![$\varphi_f$](img24-31.gif)
denote
the corresponding function of
![% latex2html id marker 767
$\mbox{\bf {F}}$](img25-31.gif)
into
![% latex2html id marker 769
$\mbox{\bf {F}}$](img26-31.gif)
, defined by
![% latex2html id marker 771
$\varphi_f(a)=f(a),\, (a\in \mbox{\bf {F}})$](img27-31.gif)
. Prove that if
![$\varphi$](img28-31.gif)
is
any function of
![% latex2html id marker 775
$\mbox{\bf {F}}$](img29-31.gif)
into
![% latex2html id marker 777
$\mbox{\bf {F}}$](img30-31.gif)
, then there is an
![$f$](img31-31.gif)
in
![% latex2html id marker 781
$\mbox{\bf {F}}[x]$](img32-31.gif)
such that
![$\varphi=\varphi_f$](img33-31.gif)
. Prove that
![$f$](img34-31.gif)
is uniquely determined by
![$\varphi$](img35-31.gif)
to within addition of a multiple of
![$x^q-x$](img36-31.gif)
.
Problem 5
Let
![$f$](img37-31.gif)
be a continuous real valued function on
![% latex2html id marker 835
$\mathbb{R}^{}$](img38-31.gif)
satisfying
where
![$C$](img40-31.gif)
is a positive constant.
Define the function
![$F$](img41-31.gif)
on
![% latex2html id marker 841
$\mathbb{R}^{}$](img42-31.gif)
by
- Prove that
is continuous and periodic with period
.
- Prove that if
is continuous and periodic with period
, then
Problem 6
Let
![$f$](img49-31.gif)
be an analytic function in the open unit disc of the
complex plane such that
![$\vert f(z)\vert\leq C/(1-\vert z\vert)$](img50-31.gif)
for all
![$z$](img51-31.gif)
in the disc, where
![$C$](img52-31.gif)
is a positive constant. Prove
that
![$\vert f'(z)\vert\leq 4C/(1-\vert z\vert)^2$](img53-31.gif)
.
Problem 7
Let p, q and r be continuous real valued functions on
![% latex2html id marker 883
$\mathbb{R}^{}$](img54-31.gif)
, with
![$p>0$](img55-31.gif)
. Prove that the differential equation
is equivalent to (i.e., has exactly the same solutions as) a
differential equation of the
form
where a is continuously differentiable and b is continuous.
Problem 8
Prove that if the nonconstant polynomial
![$p(z)$](img58-31.gif)
, with complex
coefficients, has all of its roots in the half-plane
![$\Re z > 0$](img59-31.gif)
,
then all of the roots of its derivative are in the same half-plane.
Problem 9
Let
![$A$](img60-31.gif)
be an
![$m\times n$](img61-31.gif)
matrix with rational entries and
![$b$](img62-31.gif)
an
![$m$](img63-31.gif)
-dimensional column vector with rational entries. Prove or
disprove: If the equation
![$Ax = b$](img64-31.gif)
has a solution
![$x$](img65-31.gif)
in
![% latex2html id marker 921
$\mathbb{C}\,^{n}$](img66-31.gif)
,
then it has a solution with
![$x$](img67-31.gif)
in
![% latex2html id marker 925
$\mathbb{Q}\,^{n}$](img68-31.gif)
.
Problem 10
Prove that any finite group of order
![$n$](img69-31.gif)
is isomorphic to a
subgroup of
![% latex2html id marker 938
$\mathbb{O}(n)$](img70-31.gif)
, the group of
![$n\times n$](img71-31.gif)
orthogonal real
matrices.
Problem 11
Show that the equation
![$ae^x=1+x+x^2/2$](img72-31.gif)
, where
![$a$](img73-31.gif)
is
a positive constant, has exactly one real root.
Problem 12
Evaluate the integral
to an accuracy of two decimal places; that is, find a number
![$I^*$](img75-30.gif)
such that
![$\vert I-I^*\vert<0.005$](img76-30.gif)
.
Problem 13
Let
![$f$](img77-30.gif)
be a real valued
![$C^1$](img78-30.gif)
function defined in the punctured
plane
![% latex2html id marker 985
$\mbox{$\mathbb{R}^{2}$}\setminus\{(0,0)\}$](img79-30.gif)
. Assume that the partial derivatives
![$\partial f/\partial x$](img80-30.gif)
and
![$\partial f/\partial y$](img81-30.gif)
are
uniformly bounded; that is, there exists a positive constant
![$M$](img82-30.gif)
such that
![$\vert\partial f/\partial x\vert\leq M$](img83-30.gif)
and
![$\vert\partial f/\partial y\vert\leq M$](img84-30.gif)
for all points
![$(x,y) \neq (0,0)$](img85-30.gif)
. Prove that
exists. _function,>uniformly bounded
Problem 15
Prove or disprove: If the function
![$f$](img91-29.gif)
is analytic in the
entire complex plane, and if
![$f$](img92-29.gif)
maps every unbounded sequence to
an unbounded sequence, then
![$f$](img93-29.gif)
is a polynomial.
Problem 16
Let
![$\cal F$](img94-29.gif)
be a uniformly bounded, equicontinuous family of real valued
functions on the metric space
![$(X,d)$](img95-29.gif)
. Prove that the function
is continuous.
Problem 17
Let
![$V$](img97-29.gif)
be a finite-dimensional linear subspace of
![% latex2html id marker 1059
$C^{\infty}(\mbox{$\mathbb{R}^{}$})$](img98-29.gif)
(the space of complex valued, infinitely differentiable functions).
Assume that
![$V$](img99-28.gif)
is closed under
![$D$](img100-27.gif)
, the operator of differentiation
(i.e.,
![$f\in V\Rightarrow Df\in V$](img101-27.gif)
). Prove that there is
a constant coefficient differential operator
such that
![$V$](img103-24.gif)
consists of all solutions of the differential equation
![$Lf=0$](img104-24.gif)
.
Problem 18
Let
![${A}$](img105-23.gif)
and
![${B}$](img106-22.gif)
be two diagonalizable
![$n\times n$](img107-21.gif)
complex matrices
such that
![% latex2html id marker 1095
$\rm AB = BA$](img108-21.gif)
. Prove that there is a basis for
![% latex2html id marker 1097
$\mathbb{C}\,^{n}$](img109-21.gif)
that
simultaneously diagonalizes
![${A}$](img110-21.gif)
and
![${B}$](img111-20.gif)
.
Problem 19
Let
![% latex2html id marker 1117
$\mbox{\bf {F}}$](img112-19.gif)
be a field. Prove that every finite subgroup of the
multiplicative group of nonzero elements of
![% latex2html id marker 1119
$\mbox{\bf {F}}$](img113-19.gif)
is cyclic.
Previous: Fall86
Next: Fall87
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10