Wikipedia:Reference desk/Archives/Mathematics/2011 June 22

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June 22

Commutative Diagrams in LaTeX

I need to draw a commutivity diagram with two lines of three spaces. Something like this:

A → B → C

↓        ↓        ↓

E → F → G

But I need the first two horizontal arrows( A to B and E to F) to be $\hookrightarrow$ with letter above them. I need the second horizontal arrows (B to C and F to G) to be $\twoheadrightarrow$ with letters above them. The vertices arrows (A to E, B to F and C to G) can just be normal arrows but with letter to the side of them. I can do it on one line using \stackrel $A{\stackrel {i}{\hookrightarrow }}B{\stackrel {\pi }{\twoheadrightarrow }}C$, but I've no idea how to do it a diagram. I have used the package CD, but that only allows normal arrows with, for example, @>>i>, to get ${\stackrel {i}{\longrightarrow }}$, but that's not good enought.

Any ideas boys and girls? — Fly by Night (talk) 00:08, 22 June 2011 (UTC)

I believe there are several available packages. You could try the "diagrams" package, documented here. I have never used it and couldn't tell you whether it's best for your needs. I think there's also support in TikZ, which is a package you should learn anyway, so there might be an efficiency in going that way. I don't do commutative diagrams much myself so I have no experience with this functionality. --Trovatore (talk) 00:38, 22 June 2011 (UTC)

It's been a while since I've had to make commutative diagrams, but I remember the package "pb-diagram" being good for them.--Antendren (talk) 00:40, 22 June 2011 (UTC)

You can do it using the "array" environment:

${\begin{array}{ccccc}A&{\stackrel {i}{\hookrightarrow }}&B&{\stackrel {j}{\hookrightarrow }}&C\\\downarrow ^{k}&&\downarrow ^{l}&&\downarrow ^{m}\\D&{\stackrel {n}{\twoheadrightarrow }}&E&{\stackrel {p}{\twoheadrightarrow }}&F\end{array}}$

Looie496 (talk) 04:01, 22 June 2011 (UTC)

Cool, I tried that, but couldn't get it to work. I've edited your example, and this is exactly what I wanted.

${\begin{array}{lllll}A&{\stackrel {i_{1}}{\hookrightarrow }}&B&{\stackrel {\pi _{1}}{\twoheadrightarrow }}&C\\\downarrow ^{\sigma }&&\downarrow ^{\tau }&&\downarrow ^{\nu }\\D&{\stackrel {i_{2}}{\hookrightarrow }}&E&{\stackrel {\pi _{2}}{\twoheadrightarrow }}&F\end{array}}$

Click to Enlarge

Is there a way to lengthen all of the arrows, and to get the vertical labels in the middle? — Fly by Night (talk) 20:44, 22 June 2011 (UTC)

• Thanks to you all for your help. I spend a little while learning how to use the tikzpicture package that Trovatore suggested. It wasn't easy at first. If your codes not quite right it crashes LaTeX while keeping the file in use in the CPU. Which meant I had to restart my laptop about five times until I got the hang of it. But it's well worth it once you get the hang of it, and it's quite intuitive too. I've included a screen shot of the final product. Thanks again everyone. — Fly by Night (talk) 23:47, 22 June 2011 (UTC)

true that hyper regular complex numbers are all trans regular?

Hi, Are all hyper regular complex numbers are also trans regular? It seems to me intuitively true, but maybe I'm missing something. --188.28.167.165 (talk) 12:06, 22 June 2011 (UTC)

I'm not familiar with your terminology, and a cursory search turns up nothing. Maybe you could supply a little context? --Trovatore (talk) 21:14, 22 June 2011 (UTC)

Vertical bar notation

Hi. Please refer to section 2.2 in the following paper: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.46.2009&rep=rep1&type=pdf

I am unfamiliar with the usage of the bar in this section, and can't see which of the possibilities in the vertical bar article it corresponds to. I also have no idea what the d above the equals sign represents. Any guidance would be appreciated! --Iae (talk) 12:50, 22 June 2011 (UTC)

It seems likely that the vertical bar indicates conditional probability and the d above the equals sign means "has the same probability distribution as". Gandalf61 (talk) 13:48, 22 June 2011 (UTC)

That would make sense, although even with that knowledge I can't find any reference on the internet to using an equal sign in that manner, so I can't add the info to an article. At least I'll know if there's a next time. Thanks! --Iae (talk) 14:26, 22 June 2011 (UTC)

Mathematical games

From the article linked above: "A mathematical game is a multiplayer game whose rules, strategies, and outcomes can be studied and explained by mathematics. " What does it mean "can be"? Does something exist that cannot be explained by maths? (some day, by someone). 88.8.78.155 (talk) 20:42, 22 June 2011 (UTC)

I guess it means "has been". Bo Jacoby (talk) 21:11, 22 June 2011 (UTC).

So, if I explain playing with dolls in mathematical terms, it becomes a mathematical game? Wikiweek (talk) 21:21, 22 June 2011 (UTC)

So first of all, you have to keep in mind that, just because there's a WP article on something, it doesn't follow that it's a good article or that it reflects standard usage. I'm a little undecided on whether I think that article should be deleted. Maybe it should be turned into a list, instead.

But all that aside, I don't think there was any intent to give a precise definition of mathematical game. I think the author(s) just wanted to collect together coverage of some games in which they found a commonality, and they had to give some sort of a definition at the top because it's expected that an article will start with a definition. --Trovatore (talk) 21:30, 22 June 2011 (UTC)

Honey, the social worker wants you to show her on the doll how the man touched your asymptote. :)Naraht (talk) 21:34, 22 June 2011 (UTC)

Indeed, the article Game theory has a better working definition of 'game'. I don't know if social games, like playing with dolls, could be fitted in the definition of game in the sense of the Game theory. Game theory is mostly concerned with finite, discrete games. Wikiweek (talk) 19:59, 23 June 2011 (UTC)

When I think of "mathematical games", I tend to think of Martin Gardner's long-running Scientific American column entitled Mathematical Games, which covered a very broad range of topics. Also it may be worth mentioning Ludwig Wittgenstein's famous discussion of the word "game", the thrust of which is that it is really impossible to give any concise definition that encompasses all of the ways in which the word is commonly used. Looie496 (talk) 00:34, 24 June 2011 (UTC)

Bernard J Oldfield defines a mathematical game as

1. It is an activity involving

EITHER a challenge against a task or one or more opponents

OR a common task to be tackled either individually or (more normally) in conjunction with others.

2. The activity is governed by a set of rules, and has a clear underlying structure to it.

3. The activity normally has a distinct finishing point.

4. The activity has specific mathematical cognitive objectives.

("Games in the Learning of Mathematics: 1: A Classification", Bernard J. Oldfield, Mathematics in School, Vol. 20, No. 1 (Jan., 1991), pp. 41-43) However this would exclude chess, nim, and other games not usually or not always played for mathematical reasons. (I played nim as a small child, certainly without any mathematical analysis, but according to Wikipedia it's a mathematical game.)

Nonetheless, that's the clearest definition I have found. St Andrews University's computer science department has a page on mathematical games which fails to define them at all.[1] There are patents for mathematical games, e.g. US 3204345. There doesn't seem to be a good definition of the difference between e.g. mathematical games and mathematical puzzles or diversions either. Gardner's column Mathematical Games was first collected in a book called Mathematical Puzzles and Diversions - so is a mathematical game the same as a mathematical puzzle or diversion?

At the same time, as this shows, "mathematical game" is a frequently-used phrase, referring to a game that is interesting to mathematicians, that involves mathematical concepts, that mathematics can be used to get a winning strategy, or something similar. The division is fuzzy: e.g. darts isn't normally considered a mathematical game, but it might be possible to analyse probabilities to work out the best strategy, so might it become a mathematical game, or is it too dangerous for classroom use? --Colapeninsula (talk) 12:54, 24 June 2011 (UTC)

Basically none of that justifies having an article called mathematical game. The article should be deleted or listified. I'm too lazy at the moment. --Trovatore (talk) 01:59, 25 June 2011 (UTC)