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Preliminary Exam - Spring 1995
Problem 1
For each positive integer
![$n$](img1-46.gif)
, define
![% latex2html id marker 659
$f_n : \mathbb{R}\to \mathbb{R}$](img2-46.gif)
by
![$f_n(x)=\cos nx$](img3-46.gif)
. Prove that the sequence of functions
![$\{f_n\}$](img4-46.gif)
has no uniformly convergent subsequence.
Problem 2
Let
![$A$](img5-46.gif)
be the 3
![$\times$](img6-46.gif)
3 matrix
Determine all real numbers a for which the limit
![$\displaystyle{\lim_{n\to\infty}a^nA^n}$](img8-46.gif)
exists and is nonzero (as a matrix).
Problem 3
Let n be a positive integer and
![$0<\theta<\pi$](img9-46.gif)
. Prove that
where the circle
![$\vert z\vert=2$](img11-46.gif)
is oriented counterclockwise.
Problem 5
Let
be a bounded continuously differentiable function.
Show that every solution of
is monotone.
Problem 6
Suppose that
![$R$](img24-46.gif)
is a subring of a commutative ring
![$S$](img25-46.gif)
and that
![$R$](img26-46.gif)
is of
finite index
![$n$](img27-46.gif)
in
![$S$](img28-46.gif)
. Let
![$m$](img29-46.gif)
be an integer that is relatively prime
to
![$n$](img30-46.gif)
. Prove that the natural map
![$R/mR\to S/mS$](img31-46.gif)
is a ring isomorphism.
Problem 7
Let
,
be continuous functions satisfying
Prove that there exists
![$t\in[0,1]$](img35-46.gif)
with
![$f(t)^2+3f(t)=g(t)^2+3g(t)$](img36-46.gif)
.
Problem 8
Suppose that
![$W\subset V$](img37-46.gif)
are finite-dimensional vector spaces over a field,
and let
![$L\colon V\to V$](img38-46.gif)
be a linear transformation with
![$L(V)\subset W$](img39-46.gif)
.
Denote the restriction of
![$L$](img40-46.gif)
to
![$W$](img41-46.gif)
by
![$L_W$](img42-46.gif)
. Prove that
![$\det(1-tL)=\det(1-tL_W)$](img43-46.gif)
.
Problem 10
Let
![$f_n\colon[0,1]\to[0,\infty)$](img49-46.gif)
be a continuous function, for
![$n=1, 2, \ldots $](img50-46.gif)
. Suppose that one has
Let
![$f(x)=\lim_{n\to\infty}f_n(x)$](img52-46.gif)
and
![$M=\sup_{0\le x\le1}f(x)$](img53-46.gif)
.
- Prove that there exists
with
.
- Show by example that the conclusion of Part 1 need not hold if
instead of
we merely know that for each
there exists
such that for all
one has
.
Problem 11
Let
![$n$](img61-46.gif)
be a positive integer, and let
![% latex2html id marker 886
$S\subset \mathbb{R}^n$](img62-46.gif)
a finite subset
with
![$0\in S$](img63-46.gif)
. Suppose that
![$\varphi : S \to S$](img64-46.gif)
is a map satisfying
where
![$d(\;,\;)$](img66-46.gif)
denotes Euclidean distance. Prove that
there is a linear map
![% latex2html id marker 894
$f : \mathbb{R}^{n} \to \mathbb{R}^{n}$](img67-46.gif)
whose restriction to
![$S$](img68-46.gif)
is
![$\varphi$](img69-46.gif)
.
Problem 12
Let
![$n$](img70-46.gif)
be a positive integer. Compute
Problem 15
Let
![% latex2html id marker 990
$\mbox{\bf {F}}$](img89-41.gif)
be a finite field, and suppose that the subfield of
![% latex2html id marker 992
$\mbox{\bf {F}}$](img90-41.gif)
generated by
![% latex2html id marker 994
$\{x^3 \;\vert\;x\in \mbox{\bf {F}}\}$](img91-41.gif)
is different from
![% latex2html id marker 996
$\mbox{\bf {F}}$](img92-40.gif)
. Show that
![% latex2html id marker 998
$\mbox{\bf {F}}$](img93-40.gif)
has cardinality
![$4$](img94-40.gif)
.
Problem 16
Let
be a nonempty compact set in a metric space with distance
function
. Suppose that
satisfies
for all
![$x\ne y$](img99-36.gif)
in
![$K$](img100-35.gif)
. Show there exists precisely one point
![$x\in K$](img101-35.gif)
such
that
![$x=\varphi(x)$](img102-35.gif)
.
Problem 17
Let
![$V$](img103-32.gif)
be a finite-dimensional vector space over a field
![% latex2html id marker 1046
$\mbox{\bf {F}}$](img104-31.gif)
, and let
![$L : V \to V$](img105-30.gif)
be a linear transformation. Suppose that the characteristic
polynomial
![$\chi$](img106-28.gif)
of
![$L$](img107-27.gif)
is written as
![$\chi=\chi_1\chi_2$](img108-27.gif)
, where
![$\chi_1$](img109-27.gif)
and
![$\chi_2$](img110-27.gif)
are two relatively prime polynomials with coefficients in
![% latex2html id marker 1060
$\mbox{\bf {F}}$](img111-26.gif)
.
Show that
![$V$](img112-25.gif)
can be written as the direct sum of two subspaces
![$V_1$](img113-25.gif)
and
![$V_2$](img114-22.gif)
with the property that
![$\chi_i(L)V_i=0$](img115-18.gif)
for
![$i=1,2$](img116-17.gif)
.
Problem 18
Prove that there is no one-to-one conformal map of the punctured disc
![% latex2html id marker 1080
$G=\{z\in \mathbb{C}\;\vert\; 0<\vert z\vert<1\}$](img117-16.gif)
onto the
annulus
![% latex2html id marker 1082
$A=\{z\in \mathbb{C}\;\vert\;1<\vert z\vert<2\}$](img118-14.gif)
.
Previous: Fall94
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Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10