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Preliminary Exam - Summer 1985
Problem 2
- For
, show that
- By using Part 1, or by any other method, show that if
, then
Problem 3
Let
![${A}$](img16-25.gif)
be a nonsingular real
![$n\times n$](img17-25.gif)
matrix. Prove that there
exists a unique orthogonal matrix
![$Q$](img18-25.gif)
and a unique positive definite
symmetric matrix
![$B$](img19-25.gif)
such that
![$A = QB$](img20-25.gif)
.
Problem 4
Let
![$G$](img21-25.gif)
be a group of order
![$120$](img22-25.gif)
, let
![$H$](img23-25.gif)
be a subgroup of order
![$24$](img24-25.gif)
, and
assume that there is at least one left coset of
![$H$](img25-25.gif)
(other than
![$H$](img26-25.gif)
itself)
which is equal to some right coset of
![$H$](img27-25.gif)
. Prove that
![$H$](img28-25.gif)
is a normal
subgroup of
![$G$](img29-25.gif)
.
Problem 5
By the Fundamental Theorem of Algebra, the polynomial
![$x^3+2x^2+7x+1$](img30-25.gif)
has
three complex roots,
![$\alpha_1$](img31-25.gif)
,
![$\alpha_2$](img32-25.gif)
, and
![$\alpha_3$](img33-25.gif)
. Compute
![$\alpha_1^3+\alpha_2^3+\alpha_3^3$](img34-25.gif)
.
_Fundamental Theorem>of Algebra
Problem 6
Evaluate the integral
where
![$a$](img36-25.gif)
is a complex number. What restrictions must be put on
![$a$](img37-25.gif)
?
Problem 7
Let
for
![$x>0$](img39-25.gif)
.
- Show that
.
- Show that
is strictly decreasing for
.
Problem 8
Let f be a real valued continuous function on a compact interval
![$[a,b]$](img43-25.gif)
. Given
![$\varepsilon > 0$](img44-25.gif)
, show that there is a polynomial p
such that
![$p(a)= f(a)$](img45-25.gif)
,
![$p'(a) = 0$](img46-25.gif)
, and
![$\vert p(x)-f(x)\vert < \varepsilon$](img47-25.gif)
for
![$x\in [a,b]$](img48-25.gif)
.
Problem 10
Prove that for each
![$\lambda>1$](img60-25.gif)
the equation
![$z= \lambda - e^{-z}$](img61-25.gif)
in
the half-plane
![$\Re z \geq 0$](img62-25.gif)
has exactly one root, and that this root is real.
Problem 11
- Let
be a cyclic group, and let
be elements which
are not squares. Prove that
is a square.
- Give an example to show that this result is false if the group
is not cyclic.
Problem 12
Let
![${A}$](img66-25.gif)
be an
![$n\times n$](img67-25.gif)
real matrix and
![${A}^t$](img68-25.gif)
its
transpose. Show that
![${A}^t{A}$](img69-25.gif)
and
![${A}^t$](img70-25.gif)
have the same
range.
Problem 13
Let
![$P(z)$](img71-25.gif)
be a polynomial of degree
![$< k$](img72-25.gif)
with complex coefficients.
Let
![$\omega_1,\ldots,\omega_k$](img73-25.gif)
be the
![$k^{th}$](img74-24.gif)
roots of unity in
![% latex2html id marker 940
$\mathbb{C}\,^{}$](img75-24.gif)
.
Prove that
Problem 14
Let
![% latex2html id marker 961
$M_n(\mbox{\bf {F}})$](img77-24.gif)
denote the ring of
![$n\times n$](img78-24.gif)
matrices over a field
![% latex2html id marker 965
$\mbox{\bf {F}}$](img79-24.gif)
. For
![$n\geq 1$](img80-24.gif)
, does there exist a ring homomorphism from
![% latex2html id marker 969
$M_{n+1}(\mbox{\bf {F}})$](img81-24.gif)
onto
![% latex2html id marker 971
$M_n(\mbox{\bf {F}})$](img82-24.gif)
?
Problem 15
For each
![$k>0$](img83-24.gif)
, let
![$X_k$](img84-24.gif)
be the set of analytic functions
![$f(z)$](img85-24.gif)
on the open unit disc
![% latex2html id marker 1000
$\mathbb{D}$](img86-24.gif)
such that
is finite. Show that
![$f \in X_k$](img88-23.gif)
if and only if
![$f' \in X_{k+1}$](img89-23.gif)
.
Problem 16
A function
![% latex2html id marker 1023
$f:[0,1]\to\mbox{$\mathbb{R}^{}$}$](img90-23.gif)
is said to be upper semicontinuous
if given
![$x \in [0,1]$](img91-23.gif)
and
![$\varepsilon > 0$](img92-23.gif)
, there exists a
![$\delta > 0$](img93-23.gif)
such that if
![$\vert y-x\vert< \delta$](img94-23.gif)
, then
![$f(y)< f(x) + \varepsilon$](img95-23.gif)
. Prove that
an upper semicontinuous function f on
![$[0,1]$](img96-23.gif)
is bounded above
and attains its maximum value at some point
![$p \in [0,1]$](img97-23.gif)
.
_function,>semicontinuous
Problem 17
Let
where all the
![$a_n$](img99-23.gif)
are nonnegative reals, and the series
has radius of convergence
![$1$](img100-22.gif)
. Prove that
![$f(z)$](img101-22.gif)
cannot be analytically
continued to a function analytic in a neighborhood of
![$z=1$](img102-22.gif)
.
Problem 18
Solve the differential equations
Problem 19
Let
![$A_1 \geq A_2 \geq \cdots \geq A_k \geq 0$](img105-18.gif)
. Evaluate
Note: See also Problem
![[*]](file:/local/sol/2.6/depot/tetex-20000212/lib/latex2html/icons/crossref.gif)
.
Problem 20
Let
F be a field of characteristic
![$p>0$](img107-17.gif)
,
![$p\neq 3$](img108-17.gif)
. If
![$\alpha$](img109-17.gif)
is a zero of the polynomial
![$f(x) = x^p-x+3$](img110-17.gif)
in an extension field
of
![% latex2html id marker 1099
$\mbox{\bf {F}}$](img111-16.gif)
, show that
![$f(x)$](img112-16.gif)
has
![$p$](img113-16.gif)
distinct zeros in the field
![% latex2html id marker 1105
$\mbox{\bf {F}}(\alpha)$](img114-13.gif)
.
Previous: Spring85
Next: Fall85
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10