Let X be a banach space such that X= C([a,b]) where - a
b
+
with the sup
norm. Let x
and f
X. Show
that the non linear integral equation
u(x) = (
sin
u(y) dy + f(x) ) has a solution u
X. (the integral is
from a to b).
Let X be a banach space such that X= C([a,b]) where - ab+ with the sup...
Let f(x)=
if
,
if
if
a) What is the fomain of f(x)? Write in interval notation.
b) Determine the y-intercept of the function, if any. Make sure
to justify your answer.
c) Determine the x-intercepts of the function, if any. Justify
your answer.
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Partial Differential Equations:
Calculate the eigenvalues and eigenfunctions for the eigenvalue
problem associated with the vibrating string problem with
homogeneous boundary conditions. i.e.,
,
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A Pareto distribution is often used in economics to explain a
distribution of wealth. Let a random variable X have a Pareto
distribution with parameter θ so that its probability distribution
function is
for
and 0 otherwise. The parameters and
are
known and fixed; is a constant to
be determined.
a) Assuming that
find the expected value and variance of ?
b) Show that for 3 ≥ θ > 2 the Pareto distribution has a
finite mean but infinite variance,...
Evaluate the flux F across the positively oriented surface
S
where
and S is the boundary of
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sin 0, cos 0
Name the quadrant in which the angle lies
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A probability density function f of a continuous random variable
x satisfies all of the following conditions except
a)
b)
c) For any a,b with
, P()
=
d) The mean and variance of a probability density function f are
both finite
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Prove the following
Let
with
Then:
i)
if and only if
where the double inequality
means
and
ii) If
,
if and only if
.
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Suppose that
is a bounded function with following Lower and Upper
Integrals:
and
a) Prove that for every
, there exists a partition
of
such that the difference between the upper and lower sums
satisfies
.
b) Furthermore, does there have to be a subdivision such that
. Either prove it or find a counterexample and show to the
contrary.
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Let
and consider the domain
(an open rectangle). Find the maximum of
on
as well as the
-value(s) at which
attains this maximum value.
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Solve the harmonic oscillator motion for initial conditions x(0)
= 0, V(0) = V0 in the case of (a) underdamped
(b) overdamped
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