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Preliminary Exam - Fall 1978
Problem 1
Let
![% latex2html id marker 667
$f:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$](img1-4.gif)
satisfy
![$f(x)\leq f(y)$](img2-4.gif)
for
![$x\leq y$](img3-4.gif)
. Prove
that the set where
![$f$](img4-4.gif)
is not continuous is finite or countably infinite.
Problem 2
Let
![$\{g_n\}$](img5-4.gif)
be a sequence of Riemann integrable functions from
![$[0,1]$](img6-4.gif)
into
![% latex2html id marker 693
$\mathbb{R}^{}$](img7-4.gif)
such that
![$\vert g_n(x)\vert\leq 1$](img8-4.gif)
for all
![$n,x$](img9-4.gif)
.
Define
Prove that a subsequence of
![$\{G_n\}$](img11-4.gif)
converges uniformly.
Problem 3
Let
![$M_{n\times n}$](img12-4.gif)
denote the vector space of
![$n\times n$](img13-4.gif)
real matrices
(identified with
![% latex2html id marker 717
$\mathbb{R}^{n^2}$](img14-4.gif)
). Prove that there are neighborhoods
![$U$](img15-4.gif)
and
![$V$](img16-4.gif)
in
![$M_{n\times n}$](img17-4.gif)
of the identity matrix such that for every
![$A$](img18-4.gif)
in
![$U$](img19-4.gif)
,
there is a unique
![$X$](img20-4.gif)
in
![$V$](img21-4.gif)
such that
![$X^4 = A$](img22-4.gif)
.
Problem 4
Evaluate
where
![$r^2 \neq 1$](img24-4.gif)
.
Problem 5
Let
![$f(z) = a_0 + a_1z + \cdots + a_nz^n$](img25-4.gif)
be a complex polynomial
of degree
![$n > 0$](img26-4.gif)
. Prove
Problem 6
Solve the differential equation
Problem 7
Let
![$H$](img29-4.gif)
be a subgroup of a finite group
![$G$](img30-4.gif)
.
- Show that
has the same number of left cosets as
right cosets.
- Let
be the group of symmetries of the square. Find
a subgroup
such that xH
Hx for some x.
Problem 9
For
![% latex2html id marker 851
$x, y \in \mbox{$\mathbb{C}\,^{n}$}$](img43-4.gif)
, let
![$\langle x, y \rangle $](img44-4.gif)
be the Hermitian inner product
![$\sum_j x_j \overline{y}_j$](img45-4.gif)
. Let
![$T$](img46-4.gif)
be a linear operator
on
![% latex2html id marker 859
$\mathbb{C}\,^{n}$](img47-4.gif)
such that
![$\langle Tx, Ty \rangle = 0$](img48-4.gif)
if
![$\langle x, y \rangle = 0$](img49-4.gif)
. Prove
that
![$T = kS$](img50-4.gif)
for some scalar
![$k$](img51-4.gif)
and some operator
![$S$](img52-4.gif)
which is
unitary:
![$\langle Sx, Sy \rangle = \langle x, y \rangle $](img53-4.gif)
for all
![$x$](img54-4.gif)
and
![$y$](img55-4.gif)
.
_matrix,>unitary
Problem 10
How many homomorphisms are there from the group
![% latex2html id marker 897
$\mbox{$\mathbb{Z}^{}$}_2 \times \mbox{$\mathbb{Z}^{}$}_2$](img56-4.gif)
to the symmetric group on three letters?
Problem 11
Let
![% latex2html id marker 915
$W\subset \mbox{$\mathbb{R}^{n}$}$](img57-4.gif)
be an open connected set and
![$f$](img58-4.gif)
a real valued
function on
![$W$](img59-4.gif)
such that all partial derivatives of
![$f$](img60-4.gif)
are
![$0$](img61-4.gif)
. Prove
that
![$f$](img62-4.gif)
is constant.
Problem 12
Let
![% latex2html id marker 954
$f:\mbox{$\mathbb{R}^{n}$} \to \mbox{$\mathbb{R}^{}$}$](img63-4.gif)
have the following properties:
![$f$](img64-4.gif)
is
differentiable on
![% latex2html id marker 960
$\mbox{$\mathbb{R}^{n}$} \setminus \{0\}$](img65-4.gif)
,
![$f$](img66-4.gif)
is continuous at
![$0$](img67-4.gif)
,
and
for
![$i = 1, \ldots, n$](img69-4.gif)
. Prove that
![$f$](img70-4.gif)
is differentiable at
![$0$](img71-4.gif)
.
Problem 15
Suppose
![$h(z)$](img86-4.gif)
is analytic in the whole plane,
![$h(0)=3+4i$](img87-4.gif)
, and
![$\vert h(z)\vert\leq 5$](img88-3.gif)
if
![$\vert z\vert< 1$](img89-3.gif)
. What is
![$h'(0)$](img90-3.gif)
?
Problem 16
Which pairs of the following matrices are similar?
Problem 17
Find all automorphisms of the additive group of rational numbers.
Problem 18
Prove that every finite multiplicative group of complex numbers is cyclic.
Problem 19
Let
Express
![$A^{-1}$](img94-3.gif)
as a polynomial in
![$A$](img95-3.gif)
with real coefficients.
Problem 20
Let
![$M_{n \times n}$](img96-3.gif)
be the vector space of real
![$n\times n$](img97-3.gif)
matrices, identified with
![% latex2html id marker 1203
$\mathbb{R}^{n^2}$](img98-3.gif)
. Let
![% latex2html id marker 1207
$X \subset \mbox{$M_{n \times n}$}$](img99-3.gif)
be a compact set. Let
![% latex2html id marker 1211
$S \subset \mbox{$\mathbb{C}\,^{}$}$](img100-3.gif)
be
the set of all numbers that are eigenvalues of at least one element of
![$X$](img101-3.gif)
.
Prove that
![$S$](img102-3.gif)
is compact.
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Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10