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Preliminary Exam - Spring 1986
Problem 1
Let
![$e = (a, b, c)$](img1-27.gif)
be a unit vector in
![% latex2html id marker 670
$\mathbb{R}^{3}$](img2-27.gif)
and let
![${T}$](img3-27.gif)
be the
linear transformation on
![% latex2html id marker 674
$\mathbb{R}^{3}$](img4-27.gif)
of rotation by
![$180^{\circ}$](img5-27.gif)
about
e. Find the matrix for
![${T}$](img6-27.gif)
with respect to the standard basis.
Problem 2
Let f be a continuous real valued function on
![% latex2html id marker 695
$\mathbb{R}^{}$](img7-27.gif)
such that
for all x. Prove that f is constant.
Problem 3
Let
![$C$](img9-27.gif)
be a simple closed contour enclosing the points
![$0, 1, 2, \ldots, k$](img10-27.gif)
in the complex plane, with positive
orientation. Evaluate the integrals
Problem 4
Let
![$f$](img13-27.gif)
be a positive differentiable function on
![$(0,\infty)$](img14-27.gif)
.
Prove that
exists (finitely) and is nonzero for each x.
Problem 5
Prove that there exists only one automorphism of the field of
real numbers; namely the identity automorphism.
Problem 7
For
![$\lambda$](img36-27.gif)
a real number, find all solutions of the integral
equations
Problem 8
Let the 3
![$\times$](img39-27.gif)
3 matrix function
![$A$](img40-27.gif)
be defined on the complex
plane by
How many distinct values of
![$z$](img42-27.gif)
are there such that
![$\vert z\vert<1$](img43-27.gif)
and
![$A(z)$](img44-27.gif)
is not invertible?
Problem 9
Let
![% latex2html id marker 847
$\mathbb{Z}^{2}$](img45-27.gif)
be the group of lattice points in the plane (ordered pairs
of integers, with coordinatewise addition as the group operation).
Let
![$H_1$](img46-27.gif)
be the subgroup generated by the two elements
![$(1,2)$](img47-27.gif)
and
![$(4,1)$](img48-27.gif)
, and
![$H_2$](img49-27.gif)
the subgroup generated by the two elements
![$(3,2)$](img50-27.gif)
and
![$(1,3)$](img51-27.gif)
. Are the quotient groups
![% latex2html id marker 863
$G_1 = \mbox{$\mathbb{Z}^{2}$}/H_1$](img52-27.gif)
and
![% latex2html id marker 867
$G_2 = \mbox{$\mathbb{Z}^{2}$}/H_2$](img53-27.gif)
isomorphic?
Problem 10
Suppose addition and multiplication are defined on
![% latex2html id marker 896
$\mathbb{C}\,^{n}$](img54-27.gif)
, complex
![$n$](img55-27.gif)
-space, coordinatewise, making
![% latex2html id marker 900
$\mathbb{C}\,^{n}$](img56-27.gif)
into a ring. Find all
ring homomorphisms of
![% latex2html id marker 902
$\mathbb{C}\,^{n}$](img57-27.gif)
onto
![% latex2html id marker 904
$\mathbb{C}\,^{}$](img58-27.gif)
.
Problem 11
Let the complex valued functions
![$f_n$](img59-27.gif)
,
![% latex2html id marker 946
$n \in \mbox{$\mathbb{Z}^{}$}$](img60-27.gif)
, be defined
on
![% latex2html id marker 948
$\mathbb{R}^{}$](img61-27.gif)
by _function,>orthonormal
Prove that these functions are
orthonormal; that is,
Problem 12
Let
![$f$](img64-27.gif)
be a real valued continuous function on
![% latex2html id marker 982
$\mathbb{R}^{}$](img65-27.gif)
satisfying
the
mean value inequality below:
Prove:
- The maximum of
on any closed interval is assumed at one
of the endpoints.
is convex.
_function,>convex
Problem 13
Let
![$S$](img69-27.gif)
be a nonempty commuting set of
![$n\times n$](img70-27.gif)
complex matrices
![$(n\geq 1)$](img71-27.gif)
.
Prove that the members of
![$S$](img72-27.gif)
have a common eigenvector.
Problem 14
Let
![$K$](img73-27.gif)
be a compact subset of
![% latex2html id marker 1020
$\mathbb{R}^{n}$](img74-26.gif)
and
![$\{B_j\}$](img75-26.gif)
a
sequence of open balls that covers
![$K$](img76-26.gif)
. Prove that there is a positive
number
![$\varepsilon$](img77-26.gif)
such that each
![$\varepsilon$](img78-26.gif)
-ball centered at a point
of
![$K$](img79-26.gif)
is contained in one of the balls
![$B_j$](img80-26.gif)
.
Problem 15
Consider
![% latex2html id marker 1052
$\mathbb{R}^{2}$](img81-26.gif)
be equipped with the Euclidean metric
![$d(x,y) = \Vert x-y\Vert$](img82-26.gif)
. Let
![$T$](img83-26.gif)
be an isometry of
![% latex2html id marker 1058
$\mathbb{R}^{2}$](img84-26.gif)
into itself.
Prove that
![$T$](img85-26.gif)
can be represented as
![$T(x) = a + U(x)$](img86-26.gif)
, where a is
a vector in
![% latex2html id marker 1064
$\mathbb{R}^{2}$](img87-26.gif)
and
![$U$](img88-25.gif)
is an orthogonal linear transformation.
Problem 16
Let
![% latex2html id marker 1101
$\mathbb{Z}^{}$](img89-25.gif)
be the ring of integers,
![$p$](img90-25.gif)
a prime, and
![% latex2html id marker 1109
$\mbox{\bf {F}}_p = \mbox{$\mathbb{Z}^{}$}/p\mbox{$\mathbb{Z}^{}$}$](img91-25.gif)
the field of
![$p$](img92-25.gif)
elements.
Let
![$x$](img93-25.gif)
be an indeterminate, and set
![% latex2html id marker 1115
$R_1=\mbox{\bf {F}}_p[x]/\langle x^2-2 \rangle$](img94-25.gif)
,
![% latex2html id marker 1117
$R_2 = \mbox{\bf {F}}_p[x]/\langle x^2-3\rangle $](img95-25.gif)
.
Determine whether the rings
![$R_1$](img96-25.gif)
and
![$R_2$](img97-25.gif)
are isomorphic in each of the cases
![$p = 2, 5, 11$](img98-25.gif)
.
Problem 17
Let
![$V$](img99-25.gif)
be a finite-dimensional vector space (over
![% latex2html id marker 1156
$\mathbb{C}\,^{}$](img100-24.gif)
) of
![$C^{\infty}$](img101-24.gif)
complex valued functions on
![% latex2html id marker 1160
$\mathbb{R}^{}$](img102-24.gif)
(the linear operations being defined
pointwise). Prove that if
![$V$](img103-21.gif)
is closed under differentiation (i.e.,
![$f'(x)$](img104-21.gif)
belongs to
![$V$](img105-20.gif)
whenever
![$f(x)$](img106-19.gif)
does), then
![$V$](img107-19.gif)
is
closed under translations (i.e.,
![$f(x+a)$](img108-19.gif)
belongs to
![$V$](img109-19.gif)
whenever
![$f(x)$](img110-19.gif)
does, for all real numbers
![$a$](img111-18.gif)
).
Problem 18
Let
![$f$](img112-18.gif)
,
![$g_1$](img113-18.gif)
,
![$g_2,\ldots$](img114-15.gif)
be entire functions. Assume that
-
for all
and
;
-
exists for all
.
Prove that the sequence
![$\{g_n\}$](img120-11.gif)
converges uniformly
on compact sets and that its limit is an entire function.
Problem 19
Prove that the additive group of
![% latex2html id marker 1229
$\mathbb{Q}\,^{}$](img121-11.gif)
, the rational number field,
is not finitely generated.
Note: See also Problems
and
.
Problem 20
Evaluate
where the integral is taken in counterclockwise direction.
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Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10