For 1 < p < oo it is true that:
If we have a sequence ( x_n )_(n >= 1) in l^p that converges weakly to zero then that implies the x_n are uniformly bounded and that ((x^{m})_n) -> 0 in C (as n -> oo) for each fixed m.
(where l^p is the space of p-summable sequences of complex numbers, and I...
Suppose I have a sample X_1, ..., X_n of independently, identically distributed exponential random variables.
One result I deducted was that the ratio of any two of them (eg. X_1 / X_2) is independent of the sample average 1/n * \sum_{i=1}^{n} X_i.
(Aside: that ratio, as a random variable...
Homework Statement
Suppose u is harmonic on B\{0}, and that u is continuous in the closure of B.
How would we show that u is in fact harmonic on B?
(B = unit ball in R^n, n>=2)
Homework Equations
u is harmonic in Ω if Δu=0 in Ω.
u is weakly harmonic in Ω if integral over Ω of...
Thanks mate.
Are there any good online ebooks with more examples like these available that you know of? Most of the ones I've searched/seen have either no examples or only very straightforward ones like 'polynomial denominators'...
I still don't quite fully understand about the order of poles and calculating residues.
Take f(z) = 1/sin z at z=0 for example.
When I try putting that into Laurent expansion about z=0,
1/ sin z = 2i/ (e^z - e^(-z)) = 2ie^(-z) / ( 1 - e^(-2z))
= 2ie^(-z) [ 1 - e^(-2z) + e^(-4z) - e^(-6z) +...
If E_1, E_2, ... is a sequence (of subsets of R^n) that decreases to E
(i.e. E_m+1 is a subset of E_m for all m, and E = intersection of all the E_m's)
and some E_k has finite (lebesgue) measure, i.e. lambda(E_k) is finite
it is a known result that the measure of E is equal to the limit of...
Homework Statement
Pathwise connectedness implies connectedness.
Show that the converse is false by considering the following subspace of R^2:
X = { (t, sin(1/t)) : t is nonzero } union { (0, t) : -1 <= t <= 1 }
Homework Equations
Definitions:
A topological space X is called...
Pr( ∪_(n from m to k) ((A_n) ∩ ((A_(n+1))^c)) )
= Pr( ∪_(n from m to k) (A_n) ) - Pr( (A_k) ∩ (A_k+1) )
or
the probability of [ the union (where n goes from m to k) of [ A_n intersect (A_(n+1) compliment) ] ]
is equal to
the probability of [ the union (where n goes from m to...
That (induction) is exactly what I've been attempting to use to convince myself that it is true. I've got the base step (k=m+1) which was (for me) expand-able to see that both sides are equal.
However I couldn't get through the inductive step. Perhaps it is false then? Or it could also just...
Is it true that
Pr( ∪_(n from m to k) ((A_n) ∩ ((A_(n+1))^c)) )
= Pr( ∪_(n from m to k) (A_n) ) - Pr( (A_k) ∩ (A_k+1) )
where A_1, A_2, ... is any sequence of sets.
Well, for the (k=m+1) case I am convinced since I can see they are equal after expanding both sides out, so for...
Let (A_n)_n>=1 be any event in some probability space { Omega, F, P }, then
(i) SUM_n (P(A_n)) < oo => P( limsup_n->oo (A_n) ) = 0
(ii) If in addition the A_n are independent then
P( limsup_n->oo (A_n) ) <1 => SUM_n (P(A_n)) < oo
Does that mean if the A_n are independent...
Then the factor is a field! Yay, fantastic! Thanks!
This shows that I need to do a lot of extra reading on stuff that is not taught in order to fully understand the material.. I've never heard of a pid before, unless it's lying on some fresh page that I've never looked at in our notes, or it's...