Sure - just staying in your case of an increasing sequence, under that assumption ##x_{k+1}-x_k\leq Ca^k## and ##x_n=x_0+\sum (x_{k+1}-x_k )\leq x_0+C\sum a^k## is bounded by the (finite since a<1) sum of a geometric series - so it must converge.
Presumably the same thing that causes the universe to collapse from now towards the big bang/bounce in reverse time - gravity. Unless I'm mistaken, models such as the LCDM bounce mentionned by Marcus are time-symmetric.
If the sequence is monotonic, then there is one necessary and sufficient condition for convergence: the sequence must be bounded.
Of course this can be difficult to prove, and writing it as a series ##x_n=\sum (x_{k+1}-x_k)## can help. Then you have various convergence criteria, typically...
Close - check again your statement
In other words you're saying ##\sigma_g^X## is one-to-one hence it must be onto. You need to explain why this inference is true.
There are many big bounce models in quantum gravity, and there are models where the universe recollapses, but the two features are unrelated - a bounce is an alternative to a bang, it describes what may have happened ~14bn years ago and says nothing about what may happen some time in the distant...
Right, I should have been more specific, the map to consider is ## \sigma_g^X : X\rightarrow X, x\rightarrow gxg^{-1} ## which exists because of the assumption ##gXg^{-1}\subseteq X##. Finiteness of X is key of course.
Almost. You need to prove that if ##gXg^{-1}\subseteq X## then ##gXg^{-1}=X##. Consider the map ##\sigma_g:X\rightarrow X##. Is it injective? Surjective?
Of multiplication rather. ##x^{-1}## is the multiplicative inverse of ##x##, defined by the equation ##x^{-1}×x=1##, while ##f^{-1}## is the composition inverse of ##f##, defined by the equation ##f^{-1}\circ f=Id## (##1## and ##Id## being the identity element of the corresponding operation)...
This convention is not an oddity though, it is related to the fact that the natural, generally defined operation between functions is composition rather than multplication. ##f^{-1}## is the inverse of ##f## under the composition operation, not under multiplication, and in the same way ##f^n##...
There's a distinction between unaccelerated expansion, which as I understand it has no effect whatsoever, and a cosmological constant (leading to accelerated expansion) which is similar to a tiny repulsive force proportional to distance - the latter doesn't prevent the formation of...
OK never mind the references, could you elaborate on what you said ?
I don't really understand NCG and while I vaguely get the connection with discreteness and with QM I wasn't aware that NCG involved a non-commutative distance, nor with the connection with classical SR and GR you also...
Hah, not so easy as I thought : ) I'll have a look around. It's easy to give a simple Newtonian argument for the Friedman equation with curvature but without cosmological constant using just ##\ddot a=-C/a^2## but I'm not sure what is the natural way of introducing the CC... Where does ##\ddot...
This seems completely different from the meaning of "noncommutative geometry" I am (very vaguely) familiar with (as in Connes' NCG). Can you point to some reference ?
Regarding the shadowing, wouldn't it be best to use a separate screen to protect the system from solar radiation and keep it both cool (for cameras etc) and at a constant temperature ? This seems to be the design for the JWST if my understanding is correct.
The only thing missing I think (perhaps not on Jorrie's blog since that is covered elsewhere - though that blog entry doesn't link to it) is that short intro/motivation/pointer to other source explaining where that Friedman equation comes from : )
I realize the text says its a solution of GR...
Looks like it yes, if you compute the cross derivatives
## \frac{\partial^2x}{\partial u\partial v} =-\sin\theta\frac{\partial\theta}{\partial v}=-\cos\theta\frac{\partial\theta}{\partial u}\\\frac{\partial^2y}{\partial u\partial v} =\cos\theta\frac{\partial\theta}{\partial...
What you need to find is an example of ## x,y ## such that ## x/y=-1 ## and ## xy\neq -1 ## .
Once you've done that you've proved that your initial statement ## (xy\neq -1\Rightarrow x/y\neq -1 )## is false
Given that it's neither budgeted nor planned as far as I know, this might be wise : )
It's an interesting idea though, with a science case and one for which the moon seems rather uniquely suited. Plus liquid mirrors cost far less than solid ones so the whole thing might even be realistic in...
My favourite is still this one : http://science.nasa.gov/science-news/science-at-nasa/2008/09oct_liquidmirror/
A cryogenic liquid-mirror telescope on the moon to study the early universe