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Preliminary Exam - Fall 1985
Problem 1
Evaluate the integral
for
![% latex2html id marker 661
$a \in \mathbb{R}$](img2-26.gif)
.
Problem 2
Prove that for every
![$\lambda >1$](img3-26.gif)
, the equation
![$ze^{\lambda-z}=1$](img4-26.gif)
has exactly one root in the disc
![$\vert z\vert<1$](img5-26.gif)
and that this root is real.
Problem 4
Let
![$G$](img10-26.gif)
be a finite subgroup of the group
![% latex2html id marker 722
$\mathbb{C}\,^{*}$](img11-26.gif)
of nonzero complex
numbers under multiplication. Prove that
![$G$](img12-26.gif)
is cyclic.
Problem 5
How many roots does the polynomial
![$z^4+3z^2+z+1$](img13-26.gif)
have in the right half
![$z$](img14-26.gif)
-plane?
Problem 6
Let
![$k$](img15-26.gif)
be real,
![$n$](img16-26.gif)
an integer
![$\geq 2$](img17-26.gif)
, and let
![${A}=(a_{ij})$](img18-26.gif)
be the
![$n\times n$](img19-26.gif)
matrix such that all diagonal entries
![$a_{ii}=k$](img20-26.gif)
,
all entries
![$a_{i\,i\pm1}$](img21-26.gif)
immediately above or below the diagonal
equal
![$1$](img22-26.gif)
, and all other entries equal
![$0$](img23-26.gif)
. For example, if
![$n=5$](img24-26.gif)
,
Let
![$\lambda_{min}$](img26-26.gif)
and
![$\lambda_{max}$](img27-26.gif)
denote the smallest and
largest eigenvalues of
![${A}$](img28-26.gif)
, respectively. Show that
![$\lambda_{min}\leq k-1$](img29-26.gif)
and
![$\lambda_{max}\geq k+1$](img30-26.gif)
.
Problem 7
Let
![$y(t)$](img31-26.gif)
be a real valued solution, defined for
![$0<t<\infty$](img32-26.gif)
, of
the differential equation
Show that
![$y(t) \to +\infty$](img34-26.gif)
as
![$t \to +\infty$](img35-26.gif)
.
Problem 8
Let
![$f(x)$](img36-26.gif)
,
![$0\leq x \leq 1$](img37-26.gif)
, be a real valued continuous function.
Show that
Problem 9
Let
![$A$](img39-26.gif)
be the symmetric matrix
We denote by
![$x$](img41-26.gif)
the column vector
![% latex2html id marker 855
$x_i\in\mbox{$\mathbb{R}^{}$}$](img43-26.gif)
, and by
![$x^t$](img44-26.gif)
its transpose
![$(x_1,x_2,x_3)$](img45-26.gif)
.
Let
![$\vert x\vert$](img46-26.gif)
denote the length of the vector
![$x$](img47-26.gif)
. As
![$x$](img48-26.gif)
ranges
over the set of vectors for which
![$x^t A x = 1$](img49-26.gif)
, show that
![$\vert x\vert$](img50-26.gif)
is bounded, and determine its least upper bound.
Problem 10
Let
![$f$](img51-26.gif)
and
![$f_n$](img52-26.gif)
,
![$n=1,2,\ldots$](img53-26.gif)
, be functions from
![% latex2html id marker 891
$\mathbb{R}^{}$](img54-26.gif)
to
![% latex2html id marker 893
$\mathbb{R}^{}$](img55-26.gif)
.
Assume that
![$f_n(x_n)\to f(x)$](img56-26.gif)
as
![$n\to \infty$](img57-26.gif)
whenever
![$x_n \to x$](img58-26.gif)
. Show that
![$f$](img59-26.gif)
is continuous. Note: The functions
![$f_n$](img60-26.gif)
are not assumed to be continuous.
Problem 11
Let
![$G$](img61-26.gif)
be a subgroup of the symmetric group on six letters,
![$\mathrm{S}_{6}$](img62-26.gif)
. Assume that
![$G$](img63-26.gif)
has an element of order
![$6$](img64-26.gif)
. Prove that
![$G$](img65-26.gif)
has a normal subgroup
![$H$](img66-26.gif)
of index
![$2$](img67-26.gif)
.
Problem 13
Let
![$f(z)$](img69-26.gif)
be analytic on the right half-plane
![$H = \{ z\;\vert\; \Re z > 0 \}$](img70-26.gif)
and suppose
![$\vert f(z)\vert \leq 1$](img71-26.gif)
for
![$z \in H$](img72-26.gif)
. Suppose also that
![$f(1) = 0$](img73-26.gif)
.
What is the largest possible value of
![$\vert f'(1)\vert$](img74-25.gif)
?
Problem 15
Let
![$0 \leq a \leq 1$](img83-25.gif)
be given. Determine all nonnegative
continuous functions
![$f$](img84-25.gif)
on
![$[0,1]$](img85-25.gif)
which satisfy the following
three conditions:
Problem 16
Let
![$f(x)=x^5-8x^3+9x-3$](img89-24.gif)
and
![$g(x)=x^4-5x^2-6x+3$](img90-24.gif)
.
Prove that there is
an integer
![$d$](img91-24.gif)
such that the polynomials
![$f(x)$](img92-24.gif)
and
![$g(x)$](img93-24.gif)
have
a common root in the field
![% latex2html id marker 1026
$\mbox{$\mathbb{Q}\,^{}$}(\sqrt{d}\,)$](img94-24.gif)
. What is
![$d$](img95-24.gif)
?
Problem 17
Let
![$(M,d)$](img96-24.gif)
be a nonempty complete metric space. Let
![$S$](img97-24.gif)
map
![$M$](img98-24.gif)
into
![$M$](img99-24.gif)
,
and write
![$S^2$](img100-23.gif)
for
![$S\circ S$](img101-23.gif)
; that is,
![$S^2(x) = S \left( S(x) \right)$](img102-23.gif)
.
Suppose that
![$S^2$](img103-20.gif)
is a
strict contraction; that is, there is a constant
![$\lambda < 1$](img104-20.gif)
such that for all points
![$x, y \in M,
d \left( S^2(x),S^2(y) \right) \leq \lambda d(x,y)$](img105-19.gif)
. Show that
![$S$](img106-18.gif)
has
a unique fixed point in
![$M$](img107-18.gif)
. _map>strict contraction
Problem 18
Let
![$G$](img108-18.gif)
be a group. For any subset
![$X$](img109-18.gif)
of
![$G$](img110-18.gif)
, define its centralizer
![$C(X)$](img111-17.gif)
to be
![$\{y \in G \;\vert\; xy = yx, \forall x \in X\}$](img112-17.gif)
. Prove the
following:
- If
, then
.
-
.
-
.
Problem 19
An n
![$\times$](img117-11.gif)
n real matrix
![$T$](img118-10.gif)
is
positive definite if
_matrix,>positive definite
![$T$](img119-10.gif)
is symmetric and
![$\langle Tx,x \rangle >0$](img120-10.gif)
for all nonzero vectors
![% latex2html id marker 1132
$x \in \mbox{$\mathbb{R}^{n}$}$](img121-10.gif)
, where
![$\langle u, v \rangle$](img122-10.gif)
is the standard inner product.
Suppose that
![$A$](img123-10.gif)
and
![$B$](img124-9.gif)
are two positive definite real matrices.
- Show that there is a basis
of
and real numbers
such that,
for
:
and
- Deduce from Part 1 that there is an invertible real matrix
such that
is the identity matrix and
is
diagonal.
Problem 20
Let
![$f$](img134-7.gif)
be a differentiable function on
![$[0,1]$](img135-7.gif)
and let
Let
![$n$](img137-6.gif)
be a positive integer. Prove that
Previous: Summer85
Next: Spring86
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10