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February 12

Definition of Fundamental Domain from MathWorld--correct?

"Let be a group and be a topological G-set. Then a closed subset of is called a fundamental domain of in if is the union of conjugates of , i.e., and the intersection of any two conjugates has no interior. For example, a fundamental domain of the group of rotations by multiples of in is the upper half-plane and a fundamental domain of rotations by multiples of is the first quadrant . The concept of a fundamental domain is a generalization of a minimal group block, since while the intersection of fundamental domains has empty interior, the intersection of minimal blocks is the empty set."

I hope this is right, since the last sentence is due to me. I would say if it is right , that would be nice, because it is concise and seems easy to understand and use. The current wikipedia article, like many wikipedia math articles (such as how affine space USED to be)go on and on without giving an immediate, precise definition. Maybe that's necessary in the nature of these things and the clear, concise definition on MathWorld is wrong? I am genuinely concerned that I have contributed a plausible, userfriendly but wrong explanation to MathWorld.Rich (talk) 03:05, 10 February 2010 (UTC) Pasted to Question Desk from Fundamental Domain(Talk) just now.Rich (talk) 00:14, 12 February 2010 (UTC)[reply]

Obviously you pasted the above and some symbols got omitted. Please try again. Michael Hardy (talk) 01:40, 12 February 2010 (UTC)[reply]

sorry!98.207.84.24 (talk) 08:16, 12 February 2010 (UTC)[reply]

OK, I wondered which term was to be defined, and I was annoyed that your posting didn't tell us. But after getting a bit into it, I guessed that it's "fundamental domain". So I found that on MathWorld. I'm wondering about the word "conjugate". The way that word is used when speaking of conjugates of members of a group would make me expect to see gFg⁻¹ or the like instead of gF. But I'm not an expert in this area. Michael Hardy (talk) 01:47, 12 February 2010 (UTC)[reply]

- Here's an improved transcription from Mathworld:

"Fundamental Domain

Let G be a group and S be a topological G-set. Then a closed subset F of S is called a fundamental domain of G in S if S is the union of conjugates of F , i.e., S=Union over all g in G of all gF, and the intersection of any two conjugates has no interior. For example, a fundamental domain of the group of rotations by multiples of 180 degrees in R^2 is the upper half-plane and a fundamental domain of rotations by multiples of 90 degrees is the first quadrant. The concept of a fundamental domain is a generalization of a minimal group block, since while the intersection of fundamental domains has empty interior, the intersection of minimal blocks is the empty set." (Transcribed from Eric Weisstein's MathWorld)Rich (talk) 23:24, 14 February 2010 (UTC)[reply]

The standard definition of a group action requires that the group act on a set from either the right or the left, so it is meaningful to interpret gFg⁻¹ as gg⁻¹F which is F for all g. Thus, it would not be very interesting to study the conjugates of F in that sense, and hence gF is referred to as the conjugate of F instead. (This seems the best explanation of the terminology "conjugation" in the given context; perhaps I am missing something, but I cannot think of another motivation for the terminology) Of course, there are other meaningful ways to merge conjugation with the theory of group actions; for instance, we can allow G to act on itself via conjugation.

With regards to the question, I am fairly certain that the last sentence is correct. In the most basic sense, for "A" to be a "generalization" of "B", every instance of B should be an instance of A. If we equip the set on which a group acts with the discrete topology (in fact, any topology finer than the smallest topology under which all minimal group blocks are closed will do), the instersection of the minimal group blocks is the empty set which of course has empty interior. The concept of a fundamental domain generalizes that of a minimal group block because the requirement on the intersection is less stringent; if the intersection were empty the requirement would be satisfied, but it would also be satisfied even if the intersection is more general (namely, has empty interior). Hope this helps. PS T 03:45, 12 February 2010 (UTC)[reply]

A group block does not need to be closed in a topological sense so there are blocks that aren't fundamental domains. On the other hand the examples given in the MathWorld article are fundamental domains without being group blocks. Even if you assume a finite set with the discrete topology the concepts are still different. With a fundamental domain F=Fg implies g=1 but that's not the case with a block.--RDBury (talk) 23:37, 12 February 2010 (UTC)[reply]

What Minimal group blocks fail to be closed? Thanks, Rich (talk) 01:20, 13 February 2010 (UTC)[reply]

Let T be the smallest topology relative to which all the minimal group blocks in the G-set are closed. If we equip the G-set with certain topologies finer than T (such as T itself, or even the discrete topology), it can be ensured that all minimal group blocks are closed and that the G-set is in fact a topological G-set. Of course, fundamental domains are not necessarily minimal group blocks as RDBury points out. But fundamental domains are generalizations of minimal group blocks for transitive G-actions. Otherwise, they are not. For if we let G act trivially on a set S, the singleton sets in S are minimal group blocks but the orbit of any single point in S is trivial and thus no minimal group block can be a fundamental domain (no matter what topology we equip on the G-set).

But to clarify once more, it is indeed the case that, depending on the topology, all minimal group blocks are closed. Although the set on which the group acts initially does not have a topology, you can equip the set with a topology such that all minimal group blocks are closed (and of course such that the G-set is a topological G-set). (I made a silly error in my last post; unless we restrict our attention to transitive group actions, it is not necessarily the case that fundamental domains are generalizations of minimal group blocks). PS T 05:10, 13 February 2010 (UTC)[reply]

Thanks Michael, RDBury, and PST for your helpful thoughts.98.207.84.24 (talk) 04:44, 14 February 2010 (UTC)[reply]

Integration of square root in the complex plane

Hi,

Just a quick one: I want to find $\int _{\gamma }z^{\frac {1}{2}}dz$ on the principal branch of the square root, where gamma is the circle |z|=1 and then the circle|z-1|=1, of radii 1. I've tried to parametrize in the obvious way, $z=e^{i\theta },\,1+e^{i\theta }$ , respectively, and then I accidentally integrated for theta between 0 and 2pi - I think, retrospectively, this is wrong, since I got solutions out of -4/3 and 0: I have a feeling I should have gone from -pi to pi, but my work has been marked with my first answer correct and my second one wrong: if going from -pi to pi however, I wouldn't get -4/3 for the first integral, would I? So has my work just been marked correct wrongly, if that makes any sense?

In general, when integrating something with a branch cut, do we just choose our parameter range so that it 'goes around' the branch cut, e.g. -pi to pi doesn't cross the negative real axis, rather than 0 to 2pi which does? Thanks in advance, I don't need much detail in your answers, just yes/nos would be fine unless I've misunderstood something! 82.6.96.22 (talk) 05:47, 12 February 2010 (UTC)[reply]

You can choose the parameter range however you want, but be prepared for the possibility that the formula for the integrand might not be simple. For

0\leq \theta \leq 2\pi

you have

{\sqrt {e^{i\theta }}}={\begin{cases}e^{i\theta /2}&0\leq \theta \leq \pi \\e^{i\theta /2-\pi }&\pi <\theta \leq 2\pi \end{cases}}

, while for

-\pi \leq \theta \leq \pi

, which goes around the branch cut, you can simply use

{\sqrt {e^{i\theta }}}=e^{i\theta /2}

.

I think the answer to the second question is indeed 0, and for the first it is actually

{\frac {-4i}{3}}

. -- Meni Rosenfeld (talk) 08:35, 12 February 2010 (UTC)[reply]

Stats resources/texts

Please recommend a stat resource/text that teaches you all the different manipulations involving expectation and variance. The stat text by Devore doesn't have comprehensive intro to these things. Eg. V(constant*random variable) = constant^2 * V(random variable) and many other manipulations involving summation sign and more than one random variable and constants. —Preceding unsigned comment added by 142.58.129.94 (talk) 19:58, 12 February 2010 (UTC)[reply]

This is mostly about probability rather than statistics, but it's online and covers the topics you mention. 66.127.55.192 (talk) 08:50, 13 February 2010 (UTC)[reply]

Integer closure under addition

I am currently somewhat confused about this. One article says that the integers are closed under addition and subtraction, whereas another one seems to give an example where they are not. What am I missing? Thanks in advance. It Is Me Here ^{t / c} 21:22, 12 February 2010 (UTC)[reply]

The integers are closed under finite sums; the second link refers to an infinite sum (not a convergent one either; its sum is 1/2 by a nonstandard definition). Here's another example of an infinite sum of integers whose value is not an integer:

1+1+1+1+1+1+1+1+1+1+...

The sum of that sequence is either 'undefined' or infinity, depending on context (it's not even a real number!). My point in bringing that up is that it's easy to see that integers aren't closed under infinite sums.

HTH, --COVIZAPIBETEFOKY (talk) 21:31, 12 February 2010 (UTC)[reply]

Closure under addition merely means that the sum of two integers is an integer. Addition is thought of in that context as a binary operation.

A consequence is that the sum of any finite number of integers is an integer. Michael Hardy (talk) 23:31, 12 February 2010 (UTC)[reply]

An example where subtraction isn't closed is for the non-negative integers 0, 1, 2, 3 etc. With those 3−2 gives 1 but 2−3 doesn't have a value so subtraction isn't closed. In topology you'll find an interesting combination, an infinite union of open sets gives an open set but only a finite number of intersections is guaranteed to give an open set. Dmcq (talk) 10:24, 13 February 2010 (UTC)[reply]

Thanks, all. It Is Me Here ^{t / c} 11:33, 13 February 2010 (UTC)[reply]

"subtraction isn't closed is for the non-negative integers" is a clumsy phrasing. "The set of all non-negative integers isn't closed under subtraction", or just "The non-negative integers are not closed under subtraction" is standard language. Michael Hardy (talk) 14:11, 14 February 2010 (UTC)[reply]

0.18, 9.45, and 0.38 estimation

Is Pluto's diameter 0.18 clser to 1/5 the size of earth of 1/6 the sizeof earth. Is this conventional to say Saturn is 9 times larger than earth or 10 times larger than earth. for Mercury is it conventional to say 1/3 the size of earth or 2.5 times smaller than earth?--69.233.255.251 (talk) 21:15, 12 February 2010 (UTC)[reply]

If you want to know whether 0.18 is closer to 1/5 or a 1/6, try a calculator. --Tango (talk) 23:49, 12 February 2010 (UTC)[reply]

Also keep in mind that "one time larger" means "twice as large," so "9 times larger" means "10 times as large," and "2.5 times smaller" doesn't make sense. —Bkell (talk) 00:49, 13 February 2010 (UTC)[reply]

That doesn't seem correct to me. I'm pretty sure in normal English "9 times larger" means the same thing as "9 times as large". 75.142.246.117 (talk) 04:19, 13 February 2010 (UTC)[reply]

After some googling it seems like the usage is mixed. I think I'm technically wrong about the meaning, but "times larger" is going to cause confusion. 75.142.246.117 (talk) 04:29, 13 February 2010 (UTC)[reply]

Yes, it's ambiguous. "900% larger" definitely means "10 times as large", but "9 times larger" is unclear and should be avoided. --Tango (talk) 13:46, 13 February 2010 (UTC)[reply]

Those who care about precision of language could say "larger by a factor of 9" to avoid the confusion if they insisted on using "larger". Colloquially, many imprecise people use "9 times larger" when they mean "9 times as large", and some of them can't even see the difference! The phrase "2.5 times smaller" doesn't make any logical sense, but it is used surprisingly often to mean "smaller by a factor of 2.5". Dbfirs 22:37, 13 February 2010 (UTC)[reply]

Remember that "size" in this case usually refers to the planetary volume, not its diameter. ~A H 1^(TCU) 00:15, 14 February 2010 (UTC)[reply]

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