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July 5

How to categorize these images?

I recently uploaded these two images to supplement the four referenced in the Fundamental polygon article. What categories should I put them in? In particular, I don't know if they're actually fundamental polygons or not: it's been a while since I've proven that something is a closed surface. Jbeyerl (talk) 17:37, 5 July 2014 (UTC)

It's not definite, but I lean against those ending up being manifolds. The group representation for the first would be A² = 1, and the second would be A²B⁻² = 1. — Arthur Rubin (talk) 01:17, 8 July 2014 (UTC)

A² = 1 is the projective plane. For the second, by cutting diagonally from the top left to the bottom right, then gluing along A, you get the standard presentation of the Klein bottle.--Antendren (talk) 04:42, 8 July 2014 (UTC)

July 8

How does this graph work? How is it possible?

This is the weirdest real-life graph I've ever seen: [[1]]
Source:
IMF Staff Discussions Note: Emerging Markets in Transition: Growth Prospects and challenges.
[2] (bottom of page 4)
— Preceding unsigned comment added by 14.200.130.252 (talk) 11:32, 8 July 2014 (UTC)

Graphical representations of parametric equations often look like that. Dolphin (t) 12:22, 8 July 2014 (UTC)

Thank you for that. It did help shed a bit of light and introduced me to paramtric vs non-parametric equations :) I guess there wasn't a proper function in this graph, it was just simply data points plotted and therefore the line we see in the graph is a bit misleading because we could join the 'dots' in any way we like. 14.200.130.252 (talk) 14:22, 8 July 2014 (UTC)

The dots are joined in chronological order. You can see the years marked next to each location in the graph. -- Meni Rosenfeld (talk) 15:45, 8 July 2014 (UTC)

That graph doesn't seem to show a case where one axis variable is strongly dependent on the other, which is what is most familiar (for example, population is strongly dependent on time). If you plot two largely independent values, the graph will take on a rather random look (for example, sound volume versus frequency in a piece of music). Such graphs seem less useful, to me. I'm not sure how you are supposed to use it to predict anything. StuRat (talk) 14:30, 8 July 2014 (UTC)

Such a graph can help you see whether the variables are independent. —Tamfang (talk) 04:52, 9 July 2014 (UTC)

Agreed, that's the only point I see in it. StuRat (talk) 15:11, 13 July 2014 (UTC)

It isn't intended to predict anything; it simply provides an easily read display of how the two variables related to each other at different times. It's no different in principle than plotting the path followed by a vehicle or a person over time, like this map or this map... except that with a map, latitude and longitude are obviously related, whereas with a graph showing two functions over time, their relationship may not be obvious. — Preceding unsigned comment added by 50.100.189.160 (talk) 06:22, 9 July 2014 (UTC)

Right, but just what is the point in showing how two apparently unrelated variables change relative to one another over time ? I could plot the rabbit population against the stock market indices, too, but why would I ? StuRat (talk) 15:11, 13 July 2014 (UTC)

July 10

Lie algebras of U(p, q)

Hi!

Which, if any, is correct?

${\mathfrak {u}}(p,q)=\left\{\left.\left({\begin{matrix}X_{p\times p}&Z_{p\times q}\\-{\overline {Z}}^{\rm {T}}&Y_{q\times q}\end{matrix}}\right)\right|{\overline {X}}^{\rm {T}}=-X,\quad {\overline {Y}}^{\rm {T}}=-Y\right\}.$

${\mathfrak {u}}(p,q)=\left\{\left.\left({\begin{matrix}X_{p\times p}&Z_{p\times q}\\+{\overline {Z}}^{\rm {T}}&Y_{q\times q}\end{matrix}}\right)\right|{\overline {X}}^{\rm {T}}=-X,\quad {\overline {Y}}^{\rm {T}}=-Y\right\}.$

I have one opinion and my reference another about the +/- sign in the lower left. YohanN7 (talk) 21:29, 10 July 2014 (UTC)

We only have a section on indefinite unitary groups and less on the Lie algebras, but if you know the Lie group it's generally easy enough to work out corresponding tangents at I. Take

$\left({\begin{matrix}I+tX&tZ\\tW&I+tY\end{matrix}}\right)$

to be in the unitary group U(p, q). It must preserve the form giving

$(I+tX){\overline {(I+tX)}}^{\intercal }-(tZ){\overline {(tZ)}}^{\intercal }=I\,$

$(I+tX){\overline {(tW)}}^{\intercal }-(tZ){\overline {(I+tY)}}^{\intercal }=0\,$

$(tW){\overline {(I+tX)}}^{\intercal }-(I+tY){\overline {(tZ)}}^{\intercal }=0\,$

$(tW){\overline {(tW)}}^{\intercal }-(I+tY){\overline {(I+tY)}}^{\intercal }=-I.$

Now take derivatives and evaluate at t=0 to get

$X+{\overline {X}}^{\intercal }=0\,$

${\overline {W}}^{\intercal }-Z=0\,$

$W-{\overline {Z}}^{\intercal }=0\,$

$-Y-{\overline {Y}}^{\intercal }=0\,$

in other words, according to my calculations, the plus sign is correct.

{Btw, I thought U(3, 1) was a big deal in physics, so why don't we have more material on it?)--RDBury (talk) 15:16, 11 July 2014 (UTC)

Thank you, your calculation confirms mine, done in a different way. We will in a day or two have at least something on the indefinite unitary groups and other classical groups. (This is why I asked in the first place.) I am making a complete rewrite of Classical group. Today's article manages to miss the majority of the classical groups (while it does contain stuff about fields not of characteristic 0, hardly in the spirit of classical groups). You can find my version from my user page. (My draft has a talk page if anyone wants to comment.) I expect to be done in a day or two. YohanN7 (talk) 16:26, 11 July 2014 (UTC)

Actually, the new version is in place now. YohanN7 (talk) 18:37, 11 July 2014 (UTC)

July 13

Efficient numerical integration

Runge kutta methods provide an excellent solution to systems of the form ${\dot {x}}_{i}=f_{i}\left(\{x\}\right)$ but does anyone know of a good numerical recipe for solving $\sum _{j}g_{ij}\left(\{x\}\right){\dot {x}}_{j}=f_{i}\left(\{x\}\right)$ without having to invert the matrix $g_{ij}$ at every time step? — Preceding unsigned comment added by 84.92.32.38 (talk) 14:24, 13 July 2014 (UTC)

This isn't a huge help but I think really you only need to solve the equations for the ${\dot {x}}_{j}$, which is somewhat easier that finding the inverse of the matrix. Suppose there were a more efficient method for generating a solution. Then you could apply it the case where the g's are constant, and from the solution you could determine the ${\dot {x}}_{j}$, which would then give you a more efficient method for sets of linear equations. So, and it would nice if someone checked my reasoning here, finding the solution of the differential equation must be at least as hard as solving systems of linear equations. Perhaps there is some good news though in that there are iterative methods for solving linear equations. So you could use the solution at point k as the starting point for the iteration at point k+1 and since presumably the solutions are close you may only need one or two iterations to get an accurate result. That assumes that the g is smooth and well conditioned. Of course everything here assumes g is nonsingular.--RDBury (talk) 16:11, 13 July 2014 (UTC)

I need a little help

How do you do this problem? 5`7-6 — Preceding unsigned comment added by 66.87.65.238 (talk) 23:49, 13 July 2014 (UTC)

What does that operator between 5 and 7 mean?--Jasper Deng (talk) 03:28, 14 July 2014 (UTC)

Now that is an interesting question! I can't find any real use of it as a mathematical operator. `#Use_in_programming mentions use in LaTeX for quotations, and also mentions use as command substitution in e.g. Bash shell. None of those make sense here though... SemanticMantis (talk) 15:18, 14 July 2014 (UTC)

Could it be a mistake for an uparrow, meaning 5 to the power 7 (minus 6, which for all I know is meant to be combined with the 7 rather than deducted last).→86.146.61.61 (talk) 18:56, 14 July 2014 (UTC)