Finding Möbius transformation from fixed points

Question: Given a non-parabolic transformation which is also an orientation preserving isometry in the hyperbolic upper half plane union the boundary, if I know the two fixed points and they are two different irreducible fractions on the boundary, how can I find the corresponding Möbius transformation?
Comment: I haven't thought about this stuff in a while, but if I recall correctly this only nails down that you have an elliptic Mobius transformation (of which the standard example takes the form $z\mapsto az$ for $a>0$).  Check out Alan Beardon's "Geometry of Discrete Groups".
Comment: Small correction - they are hyperbolic, not elliptic, transformations.  Beardon's book is indeed a great reference.
Answer: It is an important and beautiful accident that the orientation preserving isometry group of the upper half plane $\bf{H}$ is $PSL(2,\bf{R})$.  The action of the matrix $A = [[a,b],[c,d]]$ is given by $A\cdot z = (az + b)/(cz + d)$.  In addition, we allow the point $\infty = 1/0$: this has image $a/c$.  Likewise the real number $-d/c$ is sent to infinity.
Now to your question: Suppose that $p, q$ are distinct real numbers, thus two points in $\partial \bf{H}$. Then there is a hyperbolic geodesic connecting them.  A calculation using the above reveals that there is a one-parameter family of hyperbolic isometries fixing $p, q$.   To get you started here is the first line of the computation supposing that you have explicit, simple, values of $p, q \in \bf{R}$ in mind.
Fix $A$.  Then $A \cdot p = p$ implies that $ap + b = cp^2 + dp$.  Solve this, do the same for $q$ and remember that $kA = A$ for non-zero $k \in \bf{R}$. (That is what the $P$ stands for in $PSL$.)
As a final note - as the comment suggests it is also possible to find a matrix $B$ sending $p, q$ to $0, \infty$, solve the problem for those very special values, and conjugate back.  But perhaps it is important to understand the problem both ways.
Comment: @Theo - It is not an accident that $PSL(2,R)$ preserves the conformal structure of the upper half-plane.  Why conformal transformations (preserving angle) should be hyperbolic transformations (preserving distance) always shocks me a little.  For example, it is false in the Euclidean and in the spherical geometries...
Comment: Why do you call that an accident?
Comment: Ah, I see. Thanks for this explanation.
Answer: Let $T:z\to z'$ be such a Moebius transform and let $p, q\in{\mathbb R}$, $\  p<q$, be its two fixed points. As $z'=p$ iff $z=p$ and $z'=q$ iff $z=q$, the two variables $z$ and $z'$ have to be related by a formula of the form
$${z'-q \over z'-p}=\lambda {z-q \over z- p}$$
for some complex constant $\lambda$. This constant has to fulfill conditions to guarantee that (a) $T$ maps the real axis to itself and (b) $T$ maps, e.g., the point $i$ to a point in the upper half plane.


Kickboxing as a fitness plan?

Question: Would kickboxing be an effective strategy for gaining muscle and losing weight? How many calories per hour would it burn?
I am considering three times per week.
Answer: If you are completely detrained (in other words, you currently do no exercise at all), then any exercise will help you build muscle.  At the very least, a good kick boxing class will have you do some body weight exercises to build up strength.  Kick boxing also has a very intense conditioning requirement.  These can only help.
The number of Calories you will burn in an hour are highly dependent on several factors such as how much lean mass you currently have, your height/limb length, and how much movement you do.  Not all activities in a typical class will burn at the same rate, and some will be more energy intensive than others.  In general:

The more mass you have, the more energy it takes to move
The more lean mass you have, the more energy it takes to maintain
The longer your limbs, taller you are, the more energy it takes to move

You will be burning several hundred Calories per hour, and depending on the factors listed above, potentially over 1,000 for an intense class.  However, the only way to know for sure is to measure it.  Working out with a heart rate monitor can give you the ballpark figures--although they are not practical when it comes to sparring work.
As long as your diet makes sense for the new activity you want to do, you will satisfy your goals.  However, if you exercise regularly, you won't necessarily add more muscle unless you find ways of increasing the load on your body.  This can be done with changing the leverages on body-weight exercises or supplementing your training with weightlifting.
Answer: I can tell you my own experience with kickboxing:
I started 4 Months ago. I can really feel my body changing and I feel really uncomfortable when I don't workout. Even if the weight loss/fat burning - process on my Body isn't going as fast as I want to, I know that it's the right way to start. The person who started kickboxing with me was a chubby person but in this 4 Months he lost weight incredibly fast. (He didn't wanted to weigh himself so I can't say you any numbers.)
Kickbox training extends very different kind of exercises: Strength- Endurance and Technical Training. It's a very nice mix of everything you need.
And like Berin said: in a session of 1,5 hours of training you lose over 1000 calories. To compare: if you want to lose 1000 Calories with jogging you have to run like 2 hours (additionally you'll have aching muscles on the next few days).
Comment: voted up although normally you burn above 100 calories per 10 minutes jog, so you need 100 minutes to burn 1000 calories, not 2 hours
Comment: well it depends on how intense you jog and on how much someone weighs :-)
Answer: Berin's answer is right, but I'd like to add something. As someone who trains martial arts to fight, rather than for fitness, I'd hope that you choose a specific cardio kickboxing class, rather than a competetive one with the side benefit of cardio.
It really sucks finding out that the person you just went hard on just wanted to get in better shape, and wasn't prepared for some tough sparring. It also sucks for the person who just got worked over when they just wanted to get in better shape, and serious bruising, fractured or broken bones and concussions could be some unexpected consequences to what you thought would be a decent fitness plan.

Kickboxing as a fitness plan?


Do solar systems have to evolve in a galaxy?

Question: Is it possible for a solar system like ours to exist and evolve apart from all other star systems and galaxies. Sort of like a stand alone solar system or rogue solar system for lack of a better name. Or is there some requirement that a solar system capable of managing and growing planets especially ones that will occur in the habitable zone originate in a galaxy?
Answer: Probably not. You'd need a could of gas dense enough to contract under its own gravity. In the space between galaxies, matter is spread too thin for a solar-system sized cloud to exist. A much larger cloud could be more diffuse and still collapse, but being much larger, it would collapse into a galaxy.
Even within a galaxy, stars seem to form when shock waves from nova or some other event initiates the gravitational collapse.
Comment: Marc, thank you for your answer, but please see our [help center](http://astronomy.stackexchange.com/help/on-topic) for guidance on how to write a good answer for this site. Answers need better backing as explained in the link.
Answer: It may be possible for a solar system like ours to exist outside of a galaxy. This topic was discussed briefly in this episode of CBC's Quirks and Quarks where they were talking about extremely fast moving stars.
These stars are believed to have originally come from the centre of the galaxy but have achieved such huge speeds that they were flung out, and some that have been observed are going in such bizarre angles that they may actually have come from other galaxies.
The conclusion was basically, if such a solar system does exist, the view from the planet is probably pretty damn good.
Comment: It would be interesting to know if a planet in a star's habitable zone could survive in the zone through the gravitational chaos of such a galactic ejection. My guess is no.
Comment: I think it could because it is really the sun of the planetary system(sun just means star with planets orbiting it) that makes the planet survive, not the galaxy.
Answer: Stars form in star-formation regions, which itself develop from molecular clouds. The paper "The Minimum Mass of Molecular Cloud Cores" [1] claims that there is a minimum mass and density for a molecular cloud to turn into a star-formation region and they claim it's $10 \times M_{sol}$ with the cloud having a core diameter of $0.1\; pc$.
So if there were regions satisfying these properties elsewhere you could say that star formations is possible, though I can't quote a source which has found such a region yet.
Edit: Apparently such regions were found [2] between M81 and M82, although their mass is quite higher that the theoretical low limit; the mass is supposed to be $10^{4.3} - 10^{6.5}\;M_{sol}$.

Kamaya, Hideyuki. "The Minimum Mass of Molecular Cloud Cores." The Astrophysical Journal Letters 466.2 (1996): L99.; http://iopscience.iop.org/1538-4357/466/2/L99/fulltext/5258.text.html
de Mello, D. F. "Blue Blobs: Star-Forming Regions Outside Galaxies." Revista Mexicana de Astronomía y Astrofísica 35 (2009): 211-212.; http://www.astroscu.unam.mx/rmaa/RMxAC..35/PDF/RMxAC..35_ddemello.pdf
Answer: Evidence for stars evolving alone outside galaxies is very hard to get. And for a good reason: those stars are very hard to detect.
Galaxies are quite easy to observe: one of the reasons for that, is that the surface brightness of a galaxy does not change with the distance (a good explanation here: http://mysite.verizon.net/vze55p46/id18.html, 5th paragraph) (Note: this is not true anymore for cosmological distances because the expansion plays a role, but this demonstration is usefull for the nearby universe in which we focus now).
However, observing stars is very hard as they do not have a surface brightness: the light we receive evolves in 1/radius2. Do stars outside galaxies exist? Yes: we have evidence for intergalactic stars. See here for example: http://hubblesite.org/newscenter/archive/releases/1997/02/text/ which I think was the first discovery (anyone can back me up?). Or see this more recent paper about "rogue stars" evaded from our Galaxy: http://lanl.arxiv.org/abs/1202.2152. Those findings are not surprising as we were expecting that some stars are sometimes ejected from galaxies by gravitational forces. Can these stars carry on their planetary system with them? On this question I am not an expert, so I won't give an answer.
That was for stars created inside galaxies. What about star formation outside galaxies? This is a recent subject of discussion but we have growing evidence that this could be possible. Here is a paper about the discovery of intergalactic HII regions, which are regions of star formation outside galaxies: http://arxiv.org/abs/astro-ph/0310674.
Therefore we showed that:
1/ there are stars outside galaxies
2/ there is evidence for star formation outside galaxies.
In the first case, it may be possible that these stars carry their planetary systems with them. In the second case, it sounds plausible that the forming stars will build their torque of material which might give birth to planetary systems. There is however no evidence yet for this (to my mind).
I hope my answer clarifies your mind!