# Search results

1. ### Taylor expansion for matrix logarithm

This doesn't seem quite right, unless ## A ## and ## B ## commute.
2. ### Hope for ET?

They might be dead too. Who's to say we're going to last for 3 bn years?
3. ### How to test if a sequence converges?

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.
4. ### In big bounce models, what drives the contracting phase?

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.
5. ### How to test if a sequence converges?

True but since we it is convergent in the present case.
6. ### Normalizer N_G(X) equal to another set when o(X)<infty

Yes. In fact a set S is finite if and only if any injective map from S to itself is bijective.
7. ### How to test if a sequence converges?

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...
8. ### Normalizer N_G(X) equal to another set when o(X)<infty

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.
9. ### In big bounce models, what drives the contracting phase?

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...
10. ### Normalizer N_G(X) equal to another set when o(X)<infty

Not sure what part 2 is, but the fact that X is finite is necessary to prove statement 1, so make sure you use it explicitly : )
11. ### Normalizer N_G(X) equal to another set when o(X)<infty

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.
12. ### Normalizer N_G(X) equal to another set when o(X)<infty

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?
13. ### Negative power of function

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)...
14. ### Negative power of function

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##...
15. ### What force is needed to overcome expansion?

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...
16. ### Background independent framework

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...
17. ### From Aeon to Zeon to Zeit, simplifying the standard cosmic model

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...
18. ### Background independent framework

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 ?
19. ### Stargazing Why no telescopes on the moon?

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.
20. ### Background independent framework

A starting point is the wikipedia article on noncommutative geommetry. Not that easy a read though, maybe there's something better out there.
21. ### From Aeon to Zeon to Zeit, simplifying the standard cosmic model

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...
22. ### Math Proof

You don't leave any stone unturned :)
23. ### Question about Coordinate Change

Ah yes, signs are the spawn of the devil :)
24. ### Question about Coordinate Change

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...
25. ### Math Proof

No, this does not follow at all.
26. ### Math Proof

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
27. ### Math Proof

You don't prove it, because it isn't true. Can you find a couterexample ?
28. ### Marginal evidence for cosmic acceleration from Type Ia SNe

You cannot say it fits the data as well, that is not what any of the papers quoted here show.
29. ### Stargazing Why no telescopes on the moon?

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...
30. ### Stargazing Why no telescopes on the moon?

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