Knowledge (XXG):Reference desk/Archives/Mathematics/2009 August 23

5161:

an existing one (other than the kind of strange games mathematicians invent for various purposes) seems unlikely. Most games have a finite number of possible game states since there is usually a finite number of possible moves at any given point and a finite amount of time to play the game (although I suppose it is possible that time could be unbounded, leading to countably infinitely many game states, but that isn't much help anyway). If we want to have interesting geometry over the real numbers we obviously need uncountably many game states, that means uncountably many decisions at a given point. The only such games I can think of are sports where you have things moving around in space, but they aren't usefully described in the way you want. For football (soccer) you can have the position of the ball represented by a 2D vector, the possible game states are a rectangle and the winning game positions are a line segment at each end of the pitch. Not very interesting, but it fits your requirements technically. You could try snooker with 2 dimensions per ball, the possible game states would be those where the balls are in a rectangle and there is a minimum distance between them (the diameter of the balls). The end game states would be a discrete set where all the balls are in the pockets (the minimum distance rule doesn't apply there), and you need 2 extra discrete dimensions for the scores. So, you can do the kind of things you are talking about, but they aren't useful. I think it is only going to work nicely if you invent a game for that purpose. --

5408:), but within the particular Coxeter system they are quite different. For instance, take the group to be SL(4,p^f) of type A3. B is a Borel subgroup, the normalizer of a Sylow p-subgroup. B is a semidirect product of T and the Sylow p-subgroup, and N is the normalizer of T in G. N/(BnN) is isomorphic to the symmetric group on 4 points, a Coxeter group of type A3 with Coxeter generating set w1=(1,2), w2=(2,3), and w3=(3,4). Letting n1, n2, n3 be preimages of w1, w2, and w3, one has that the double cosets Bn1B and Bn2B have different sizes. For instance, when p^f=3, Bn1B has size 157464 and Bn2B has size 17496. Now, if w1 and w2 are conjugate under a graph automorphism of the Weyl group, then I think the group itself should have a graph automorphism so that Bn1B and Bn2B are conjugate. The graph automorphism is unlikely to be an inner automorphism, so probably they are not conjugate within the group itself, but they would be in the holomorph. 2101:

Durham but by the reseach council EPSRC, I already had a member of staff applying on my behalf for the grant so that I could work with him). I am currently awaiting the outcome of a research proposal submitted (to the research council EPSRC) to work at Leeds. These proposals take up to six months each to be resolved, and my situation is not uncommon; especially given that the government have cut research funding due to the economic climate. Tango, if and when you complete a PhD you will find that this situation is very common. I know a Cambridge graduate that spent three years looking for a postdoc position. So I'm just keeping my fingers crossed. As for recent mentions, my last paper wat published seven months ago; that's not too long ago is it? Tell me Tango: when was your last paper publish?

1932:. I am truly insulted by your comments Tango. As an undergraduate I took an interest in differential geometry, singularity theory, and Riemann surfaces. I never found number theory or, for that matter, real analysis very entertaining. Just because I have gaps in my knowledge in one area doesn't mean I have gaps in ever area, or more importantly that I'm lying! Just google me Tango, you'll find all the evidence you need. If you want castiron proof then why not email one of the other mathematicians on those webpages with a secret code, and ask them to email me, and I'll post it on here. I'm truly insulted! I'm sure you know next to nothing about affine differential geometry, but does that mean you're not a maths student?! 6072:

automorphism of the Coxeter diagram, an automorphism of the Weyl group described by permuting the generators according to the diagram, an automorphism of the algebraic group (over the algebraically closed field) inducing that automorphism on the Weyl group, an automorphism of the finite untwisted group that is just the restriction of the one on the algebraic group, and, when talking of a twisted type group, the automorphism of the algebraic group that gets combined with the frobenius endomorphism to define the twisted group. I think the fourth and the fifth don't conflict, since I think a twisted group doesn't have any graph automorphisms in the fourth sense, though 2E6 as F4 worries me a little.

5926:, but perhaps they happen to be). Unfortunately the Weyl group is type B_n in the SU case, and so there does seem to be a good chance the q_i will respect conjugacy. The size of the q_i though is supposed to depend on the size of the orbit of the root under the graph automorphism that defined the twisted group, and I don't see why that should be particularly related to conjugacy in the Weyl group. If it is true (in some generality), then Carter's Finite Groups of Lie Type (the thick brown one, not the thin black/gray one) might have the appropriate setup in its early chapters. I prefer the thin "Simple" book for most things, but it treats twisted and untwisted separately. 5623:

there is a subgroup U_w of w which is generated by all X_i such that w(i) is a negative root. So U_1 = 1, and U=U_{w0} where w0 is the longest element of the Weyl group. The double coset BwB is also equal to BwU_w and has cardinality |B|*|U_w|. In particular, for w=w1, U_w1 = X_1 has order q, and for w=w2, U_w2 = X_2 has order 3, so the quotient |BwB|/|B| = |U_w| is equal to q whenever w has length 1 (that is, whenever it is one of the generators from the Coxeter generating set). Indeed, if w has length l (it is a product of l generators, but not a product of fewer than l generators), then |U_w|=q^l.

775:. 10 is the number of slashes in "//////////", and is an integer with any conventional way of defining integers. "10" is a string consisting of the symbol "1" followed by the symbol "0", which can mean anything at all. Usually this string is interpreted using the decimal system, in which case it means the number 10. But the string itself is not a number and certainly not an integer (unless we choose to embed all finite ASCII strings as integers, but that is beside the point). 10 is of course also a rational number, while "10" is not. 1405:", for instance, we assume the subscript to be in decimal, and trying to explicitly state this by subscripting another "10" leads nowhere. You can go around this, of course, by writing the subscript as "ten", "dec" or even "5+5", all of which always mean the number ten, which "10" does not necessarily do. (The convention that numbers are in decimal unless otherwise stated is pretty useful.) — 1701:

you user page section on numbers and it does help a little. I am begining to think that my problem is not (thankfully) with the idea of integers and natural numbers, but with the notation used to give representations of these numbers. I have learned something over the last few days; but it has been quite painful. I'm sure people could have been more gentle. Anyway...

185:, nothing)... which can express the exact value of an irrational number; it can not be exactly bound by two other numbers greater and less than it (because for any arbitrarily sized distance, there are smaller numbers which make a better bound). There's a lot of subtle advanced mathematics here - the best place to start would be to really read and understand our 3894:

writing a reply that assumes they have the required knowledge to solve the problem. If they did then they would have solved it, and if they'd have solved it then they wouldn't be asking the question. And whatever Jao was trying to say, he didn't say it in a very nice way. Scroll up and read more of his posts, he's not the most civil or polite.

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boundary between these points. (You could do that anyway by just defining a metric on the positions, but that would introduce the possibility that the choice of metric is more important to the dimension than the game). Of course any game with a finite number of positions has dimension zero, but there are ways around that.

5887:

It is exactly those q_i that I am after. They should also be the cardinalities |B w_i B|, divided by |B| itself. And they should be equal if the corresponding generators are conjugate. (so that is why there is only one q involved in the projective case, the A_n case). A lot of people have told me

5160:

Are you restricting yourself to vector spaces or is any manifold acceptable? There is an additional requirement that the subsets you are interested in actually be subspaces/submanifolds, otherwise the dimensions you talk about aren't well defined. I'm sure we can invent a game that works, but finding

4526:

The problem with this is that the transformation removes too much information and the geometric interpretation of this set is fairly meaningless. I was trying to think of a game where there is a more natural transformation from positions to euclidean points, so that things like the euclidean distance

4057:

I'd like to thank you all for you wonderful cooperation and interaction. To tell you the truth, I feel such proofs are totally new to me because I didn't study about "modular arithmetic"; although I think the one who asked this in Arabic Knowledge (XXG) must have studied it unless there's another way

2144:

I was going to work with Farid Tari. I have even given a talk at the Durham geometry seminar. We applied to the EPSRC but one of the referees decided that he didn't like the project even though, believe it or not, we were going to try and establish some affine results that had been publish, in their

2051:

Meni's right... Just find Declan Davis' email address at Liverpool and send him and email. He'll be glad to confirm that he's the person writing this very message. Although he won't be on Liverpool's webpage since he's already left. You'll have to try one of this publications. Do you have MathSciNet?

1700:

But that oversight on my part exemplifies my point: I took π as a given, i.e. 3.14159265... = π. Most people always assume that the numbers are given to base 10. I agree that numbers exist independently of their representations. My problem seems to lie with the representations themselves. I have read

1424:

You make a good point about π! As for the point about bases, it shows that base 10 does indeed play a special role. We always assume that numbers are base 10, and then within this framework we can talk about other bases (base 10). To define integers and the like without any mention of a base requires

1248:

If you have the complete set of real numbers without having taken any of the intermediate steps in defining natural numbers, integers and rationals, you can still produce the set of all integers, as I said above, as the smallest possible closure of the set {1} (where 1 is the multiplicative identity)

265:

gave a very nice reply to the earlier post. I have only ever come across number systems to integer bases. Talking about base π seems a little wishy-washy to me. What's the expansion of 7 in base π? It would be irrational! Infact every number except multiples of π would be irrational. Would it even be

4522:

I was thinking along similar lines for Chess or Go, but I figure that even if the space of all games is near infinite, the space between 0 and 1 in even one dimension is actually infinite, so it could easily accommodate all those positions. Consider, for instance a representation of a chess position

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Hi, 129 here, I don't think my solution is convoluted. All I did was to observe 1) the LHS is 1 mod 6 so k must be 1 mod 6 (I left this as a hint, 93 spelled it out) 2) knowing this you are lead to the equation n(n+1)+2 = 0 mod 6 which is easily seen to have no solutions (in my 15:38 post). Jao is

2129:

Ok, that makes sense. I chose not to do a PhD in the end, my final year project wasn't very enjoyable so I decided to go down a different route. I'm currently job hunting myself. I'm curious, who were you going to work with in Durham? I did my MMath there and a Differential Geometer was my tutor, so

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What base is the base in? And what base is the base of the base in? We have to take base 10 as a starting point and move on. Otherwise I find myself going round and around in circles. I guess that's the problem with expressing the integers in any base. It reminds me of relativity, and there being no

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pretty, but, really, it's a rather clunky representation to go by for the purposes of mathematics (partly because it contains multiple representations for the same number, and partly because the definitions of addition and multiplication aren't exactly trivial). The integers are not defined by their

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Hello, I certainly enjoy "debating" with you (and I think we have also crossed paths elsewhere). If it suits you better, this discussion can also be continued on my talk page. I somewhat recognise the stuff you wrote about U_w. I read that in general, there is a q_i associated with each generator

5626:

For general (B,N) pairs I don't know a formula for the size, but for so called "split (B,N) pairs of characteristic p satisfying the commutator relations" this same thing occurs, BwB = BwU_w and |BwB|/|B| = |U_w|. However, even for algebraic groups, the size of U_{w1} can depend on which root you

5142:

I should perhaps elaborate a little on what I'm trying to do. One of the things I'm toying with is the idea that in such a representation you can talk about the (fractal) dimension of the total set of board positions, or the dimension of the sets of winning/losing positions, or the dimension of the

4213:

Another interpretation would be to relate this to ergodicity. (My understanding of ergodic theory isn't great, so please correct me if this makes no sense). For instance, if you were interested in which parks in a city were well-liked by people, you could can do two things. You can look at a lot of

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Well, 129's proof is clever and clear. It reduces to an elementary fact (i.e., n(n+1)+2 is never divisible by 6) that one can easily check. Most likely you are able to understand it perfectly, but of course, if any point is still not clear to you, I guess you are welcome to ask for further details.

2607:

Thank you, that excludes some of the possibility for misinterpretation. So f is constant on the equivalence classes of X induced by G. Now with the Lagrangian itself operating on functions I'm still not clear if in this case G acts on these functions (as the domain of L) or on the domain of each of

1997:

I have googled you, certainly there did used to be a PhD student with your name (I have no way to know if that is actually you), although oddly I can find no recent mentions of you. Were you unable to find a post-doc position? That whether or not a number is an integer is not dependant on what base

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does not specify a base, and says that 10 is an integer. When you talk about integers, you seem to mean those numbers with integer values when expressed to base 10. So the integers base 10 seem to hold a special place in our arithmetic. I asked this question earlier, but didn't get a reply. If not,

5426:

Hello, thanks for your reply. Indeed, I mean that they are both part of a single coxeter generating set (sorry if that was unclear). I must admit that I do not know what a "graph automorphism of a group" is. I do know "automorphisms of a graph" (and there is plenty on that on the internet) but

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I'm looking for a reasonably complex game of perfect information, where each position (ie. the state of the game after a player has made her move) can be fully expressed as an n-dimensional vector in a straightforward manner. I'm sure that the board-positions of Chess and Go can be encoded in this

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is small. An average gives only a first description of a distribution; then you can ask how close the individuals are to the average value. For instance, in a society of clones, everybody is very close to the average person (and I'm trying to recall a sentence in a novel by Oscar Wilde, asserting

1400:

As for that specific example, remember that "π" is not the base-10 representation of the number π; the base-10 representation of π is "3.14159...". π is just a symbol whose value does not depend on base—just like the symbols 1, 2 and 5 denote the same numbers in octal, decimal or hexadecimal. Base

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Cool, I checked all the twisted groups and it is always true that if w1 and w2 are conjugate in the Weyl group then q1=q2 for the root subgroups. I still don't quite understand why this is true other than as coincidence, but it seems to work. By graph automorphism I mean five related things: an

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For SL(n+1,q) of type An, B has order (q-1)^n*q^(n(n+1)/2) and it has a very important subgroup U of order q^(n(n+1)/2). U is made up of subgroups X_i, each of order q, called root subgroups, that are in one to one correspondence with the positive roots. For a given element w of the Weyl group,

3893:

I did read the thread carefully, it wasn't very long. You gave very little explanation and just wrote a lot of maths. I found that convoluted. Clearly, you wouldn't think that your solution is convoluted: you wrote it! People ask questions because they don't understand the topic. There's no point

1673:

I have no doubt that there are areas of mathematics where our levels of knowledge are reversed - but here, in this discussion, you have made several statements which reflect very fundamental misunderstandings about the subject matter. These misunderstandings don't resolve themselves, and the more

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These method will likely give very different results, whatever person you choose. This means that even if you choose the "most average" person in the city (by whatever definition) his path (the parks he visits) will still diverge from the average. In that sense human beings have a tendency to be

2100:

Tango: FYI: I graduated in July 2008. I spent a month working in Valencia, spent a month back in the UK, then spent two months working in Brazil. In September last year I returned to the UK and started to look for a postdoc. Unfortunatly a research proposal to work in Durham was rejected (not by

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exactly. If you switch to a number system based on an irrational number, you will be mapping a different subset of real numbers into rationals and irrationals; but there will still not be an exact representation for all real numbers. These are moot points though - I shouldn't have made such a

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would indeed be the vector space to use. An empty board would be prepresented by the zero vector. A board filled with naughts (although not a valid game state) would be (1,…,1) and a board filled with crosses (although not a valid game state) would be (2,…,2). Tango, thanks for the correction.

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Really? If he did then it's very convoluted. I didn't see that he said that. If I didn't see that then I'm sure the person making the original post didn't see it either. Jao, please do me a favour and calm down. From this post, and earlier ones, you seem to be quite confrontational. So what if

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I'm puzzled too, but it's not that much of a stretch, definitely not enough to openly question his credentials. I don't think any required course in my education discussed the set-theoretic (or any other foundational) construction of numbers. And of course, one can conduct fruitful research in

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is a more subtle mathematical concept than just the number of decimal-places. The fact that the digits never repeat is an artifact of the way we represent numbers with decimal place-values, but in reality, the concept of irrational numbers is much more precisely defined than the non-repeating

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be a more obvious vector space for describing games states in noughts and crosses? Each position is represented by a dimension and what is filling that position is given by the coordinate in that dimension. I'm not sure how your suggestion would work - you would need 9 points in that space to

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that Carter's book is a good reference, but I couldn't find an answer in there. That is in generaly my problem with theory like this, too much stuff I picked up "from the street". I was very curious if there is a really quick way to see that those q_i must be equal for conjugate generators

768:" doesn't make much sense. The subscript "10" means that we use the decimal system to represent a number with a sequence of decimal digits. "π" is not a sequence of decimal digits, it is a Greek letter. This letter denotes the number π in a way which does not involve any radix system. 4134:

In a philosophically mathematical manner, is it ridiculous to ask if "the average person is average"? I suppose there could be some term switching in that both averages would not refer to the same method of obtaining an average (e.g. mean vs. median), but that would sort of be a

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On another note, there is, of course, an exact decimal representation of pi. It happens to contain infinitely many digits. But that's nothing special. Even the decimal representation of the real number zero has infinitely many digits. It just so happens that they're all 0.

5139:, isn't very interesting (essentially a 3x3x3x... grid), and in that sense it does create more 'empty space'. Of course, a carefully designed game could get around this by making all the real or rational vectors possible/legal moves, which is more or less what I'm looking for. 5208:

If it helps, most variations of Nim, and most take-away games in general, can be viewed as motion in a discrete subset of a vector space in an obvious way. I've also seen an analysis of Peg Solitaire that viewed playing the game as vector addition in a high power of

1279:

Jao, thanks for your reply. The construction of the integers starts of by saying that they are equivalence classes of pairs of natural numbers. And this is okay because I understand what COVIZAPIBETEFOKY is saying about the integers being generated by {1}. Since

945:

We can't measure a diameter and a circumference with infinite precision, true; we also can't know π with infinite precision. To put it another way, we can (in principle) measure a circle to as many digits as we want, and compute π to as many digits as we want.

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Dr. Dec, I'm sure you know this but "irrational" means not a ratio of integers, which is a property of the number rather than its representation. 7 is rational whether you express it in base 10, 2 or π - "having no repeating expansion in base π" is the correct

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is an integer, gives the set of integers; but there's the problem - I'm going around in circles. We need to know what an integer is, before we can understand what an integer is. I know I sound insane, and I'm getting quite worried. But, for example, what does

409:

But the article also says that golden base ratio has this problem too, i.e. equalities along the lines of 0.999... = 1. It seems that every choice of base gives this same problem with infinite decimal expansions. But what base 10 doesn't say is that 11 = 100.

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can be equal- as far as I understand it- even if the generators are not conjugate, but conjugacy does imply that they are equal. I must have a look at that other book by Carter then. But in the mean time, could you tell me what exactly you mean by "graph

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Hi, indeed I mostly reply questions at Arabic Knowledge (XXG), and one question stopped me where I couldn't think where to classify (regardless whether or no being a homework). I'll try to translate it in English symbols as follows: For any natural numbers

5607:(which is 17496 if q=3). So I don't understand what is going on. Note that the cardinality of each double coset must be a multiple of the cardinality of a coset of B. (And to be honest : it is in fact the quotient that I am interested in). Many thanks.ıı 5403:

So by w_1 and w_2 being generators, I'm assuming you mean they are both part of a single (Coxeter) generating set. The Weyl group of type An is a little weird because individually all of its generators are conjugate (they are just two cycles in the

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you represent it in is pretty fundamental, I really don't get how you can spend 7 or 8 years surrounded by pure mathematics without coming across that fact. Your arrogance is also typical of someone claiming to be "better" than they really are. --

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as a number that can be written without a period. As you have discovered, this definition only works in base 10, 2, 16, etc—or in other words, only in bases which are, themselves, integers. A much better definition is "an integer is an element of

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Dr Dec, I am confused. I try to always assume good faith, but I am struggling to do so in this case. How can someone with a PhD in mathematics not know what integers are? The possibility that you may be lying about having a PhD concerns me since

2018:

Tango: Perhaps Declan has preferred the company of that subset of mathematicians which, like him, have little interest in these matters? If the "PhD student with his name" has contact information, perhaps you should try contacting him? --

1614:. There is no need for it. I'm sure that there are areas of mathematics where our levels of knowledge are reversed, and I would take great pleasure in fostering your interest in those areas by being patient, warm, helpful, and respectful. 5175:

I guess you're right. Specifically, the difference between a countable and an uncountable number of positions may not be so easy to gloss over as I had thought. I gues I have some more thinking to do. Thank you both for the insights.

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posibilities, and so the space of all game states would be huge – if not infinite. So you would only ever have a single vector for a given game state sat in a massive space of game states. I've thought about something similar with

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to approximate its value to any desired level of precision (if we are willing to work out the math to that accuracy). In any case, 22/7 is a terribly inaccurate approximation - they differ by more than 0.001 (which is a lot!)

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strong statement about pi. In any case, the OP is clearly unaware of the concept of rational and irrational numbers - so let's try to phrase our responses to help him/her understand that, before diving into such subtleties.

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Injecting myself into this conversation, I'd like to say that Jao has always been civil in the lengthy period of time I've seen him contributing to the RD. I can sort of see how Jao's comment in the unit normal discussion

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every point is a legal board position. In that sense, you're absolutely right, the space is fully packed with positions (though not all legal). But the structure that those points make when you view the set as a subset of

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like I asked you. If you missed the post in which I asked this, please go over this thread to locate it. I'm sure your confusion will be dispelled if you do, or at least we will be able to intelligently discuss this. --

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extends indefinitely if it is the answer to a division problem, named 22/7, in which the LCD of significant figures is 1? Is it because we are not measuring anything with the 22 and the 7, and we really therefor mean

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Can anybody think of a game that fits this description? I'm sure such a game would be very useful in Game Theory (relating game theory to geometry), so I figured someone might have invented one just for that purpose.

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What does that mean? Let's for simplicity take a system where the Lagrangian L is invariant under U(1) (which I understand contains the complex numbers with absolute value 1). But invariance can not mean -L = L.

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DrDec: "Rational" does not mean rational relative to some base. "Rational" means a ratio of two integers. Thus 22/7 is a rational number. What base you express numbers is has absolutely nothing to do with it.

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The number 7 can be expressed as the quotient 7/1, where 7 and 1 are integers. Therefore, 7 is a rational. It doesn't matter at all how you choose to represent the number. You can represent it as "7", "3+4" or

3360: 1610:" and quite strong, and could be written, IMHO, in a much more friendly way. (Although I can be quite sensative.) Many mathematicians – and I have sometimes been guilty of this myself – seem to suffer from a 804:, it gives a string of symbols (an infinite string). In theory we can take this string and interpret it in base 10, resulting in a number which is indeed irrational. But doing this doesn't make any sense. -- 5618:

Oops, sorry, my explicit calculation was wrong (it took a B and an N, but not form the same (B,N) pair, so my n1 was actually of length 3 in the true Coxeter generators). The general calculation made it

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Of course not all of these vectors and maps would represent a valid game: for example, you couldn't have a cross in all nine positions. As to your problem about big empty spaces. The layout of a game has

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The "I have already addressed all of your concerns" comment was inevitable after I explained that numbers are distinct from their representations, that you do not need a base to represent numbers, that

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as a binary string. You can interpret each binary string as a number in (0, 1), which gives you the required densely packed representation (informally; it wouldn't actually be a dense set, I guess).

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take, though this only happens in the twisted groups. Most of this can be found in Carter's Simple Groups of Lie Type. Hopefully I'll have time later today to write out the SU(n,q) case.

51: 1361: 4363: 3118: 5882: 4958: 3198: 3158: 670:, I've already confessed that number theory isn't my thing, so I would ask you to kindly tone down your rhetoric a couple of notches, and to show patience and good humour. We have that π 1465:, and is (e.g.) the cardinality of the set S={{{{{}}}}, {{{}}}, {{}}, {}}. We can specify or construct (many) numbers and all integers without resorting to a certain base X notation. -- 2429: 4995: 4851: 4799: 4758: 3714: 3489: 2727: 5899:

It's nice to have the question asked this way. I'll try to check the SU case today (I was thinking that this would be an example where the wi could be conjugate but have different

5107: 5029: 4677: 4573: 3261: 2795: 4902: 4139:. My sister-in-law asked me this question, and at first I thought it was insightful, but shortly thereafter figured that it's probably not that insightful at all. Any comments? 2557: 5137: 160:

Pi also can be derived with a variety of other techniques. The most common ones that come to mind are some easy integrals, which are part of elementary calculus. You can read

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However, I'm afraid it's not like we can prove that by proving that both double cosets are conjugate as well (because I think they aren't). Do you have any ideas? Thanks!

2439:. So in any case there's no "-L=L". Maybe you have in mind a situation where there is a G action on Y too, (usually, but not necessarily, the same action, with X=Y), and 2247: 628:

As it happens, there is a theorem that says that a rational number has a repeating expansion in any fixed integer base numeral system. However, the word "rational" does

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Similarly π has a base-π expansion "10", which is repeating (terminating, even). But that does not make it any more rational, since it is still not a ratio of integers.

623: 2353: 522:, number theory isn't really my thing so you're going to have to help me out. I'm getting a bit confussed. What is π in base-π? Well one representation would be that π 6037: 6010: 5983: 5924: 5770: 5743: 5315: 5288: 4214:

people in a single instant and see which parks are busy and which are empty. You can also follow a single person for a year, and see which parks the person frequents.

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A small nitpick: there are number systems in which pi can be expressed exactly. Base pi is the obvious one. Not that I would ever want to use base pi for anything.

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determine a game state and the first coordinate is completely redundant as long as those points are ordered (and the coordinates in a vector are always ordered). --

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are both generators of the Weyl group, and suppose that they are conjugate (within the Weyl group). Is it correct, and trivial to see(?), that the double cosets

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Now, whether Rckrone's comment is central to the discussion is irrelevant - Nimur has made a factually incorrect statement, and correcting it is appropriate. --

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So how do we know what pi is other than dividing a circumference by a diameter, neither of which we are able to measure to an infinite number of decimal places?

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have the same cardinality? (I'm rather sure it should be, because in geometric language, this is the number of chambers through one panel in a thick building).

1598:

Meni, your rhetoric was quite strong. I felt like you were speaking down to me and trying to belittle me. Just stop and read what you've written. Phrases like "

193:

articles. That being said, we can both prove that pi is irrational (meaning we can not find an exact value for it); and at the same time, we have lots of

2866: 718:

I guess you need to say that a number is rational when it can be writen as the quotient of two integer, base-10. Like we have seen: the rational number 7

85:

The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the

45: 1670:

Frankly, I think you really are too sensitive. I completely agree that we should be friendly and respectful, but I still don't see where I failed that.

782:

doesn't say anything about the rationality of the number it denotes in base π. And indeed, we know that the latter is π, which is an irrational number.

117:

22.000000000000000000000000000000000000000000000000000/7.0000000000000000000000000000000000000000000, but of course with the zeros continuing forever?

5427:

that's something else (it seems). I was looking at the numbers you gave me so I could try it myself. Do you also happen to know the size of B? If

4723:, I see what you're saying about the interval. Is there really a difference? Is a point any more or less isolated in the interval as a vector is in 1401:

dependence only comes into play when two or more such symbols are juxtaposed. But with another example, your remark is quite apt. When we write "30A6

1733:, where I try to define what a decimal expansion really is. If you're interested in more general radix representations, you can read pages such as 4108:

I started reading about modular arithmetic in order to learn more, and I hope that you give me some more recommendations to study this subject. --

1786: 1738: 37: 778:

As you have noted, the representation of the number π in base π is the string "10". But the fact that the string "10" denotes a rational number

635:

So the number 7 has an expansion in base π which starts with "20.202112..." and happens to be non-repeating. But the number 7 is still rational.

3849:

only saying (I think!) that you should read the thread carefully before you reply, it gets confusing otherwise. Anyway, the problem is done.

5463: 5146:

Another option would be to treat a move as a map. This would open up the possibility of treating the game like a discrete dynamical system.

4026: 3492: 3033: 21: 3850: 3051: 2609: 2203: 4236:

way, but any method I can think of is rather convoluted, 'hiding' the relevant information, or leads to a massive, very empty space.

3266: 1191:

why it's not an integer. (By the way, I apologize if my earlier answers have offended you, and I hope you'll find this one better.) —

754:

I'm not sure what you mean by "tone down my rhetoric". I'm using whatever rhetoric I think would be effective, while remaining civil.

395:

That is no worse than base 10 where, for example, 0.999...=1. This kind of notation is usually non-unique, regardless of the base. --

3991: 3960: 3644:. This has reduced the problem from a quadratic to a linear problem. Why not try more modular arithmetic on this linear expression. 3050:

Taking the equation mod 6 and probing for the possible values of x^3 mod 6 shows k = 1 mod 6. But I can't see any use of this fact.

2635: 1816: 1796: 1791: 1425:

more sophisticated machinery. It's just like coordinate geometry and coordinate-free geometry. Jao, thanks for your patience :o)

1237: 194: 1088:

People keep saying the same thing without ever actually addressing my question. I think we do need a base. Why do we say that 10

984:

you're not being very clear. There's no point being curt and just barking a definition at me without giving an explanation. The

6084: 5935: 5745:, turned negative by w. (so we use weighted lengths instead. So for SU(6,q) there would be 6 positive roots associated with 5636: 5417: 5397: 5236: 5222: 5213:. And of course, there's the classic probabilistic games in economic Game Theory, which are explicitly viewed as points in R. 5185: 5170: 5155: 5054: 4703: 4585: 4536: 4517: 4249: 4202: 4188: 4167: 4147: 4117: 4082: 4067: 4034: 4016: 3999: 3983: 3968: 3917: 3858: 3843: 3790: 3739: 3667: 3500: 3209: 3059: 3041: 2754: 2643: 2617: 2598: 2211: 2172: 2139: 2124: 2075: 2028: 2007: 1992: 1955: 1915: 1900: 1877: 1860: 1750: 1724: 1695: 1637: 1569: 1542: 1524: 1474: 1448: 1419: 1395: 1258: 1205: 1125: 1075: 1057: 975: 955: 922: 866: 813: 749: 713: 662: 557: 514: 463: 433: 404: 390: 376: 312: 289: 237: 211: 155: 142: 125: 5260:, but I still have a question about the correspondence described there between elements of the Weyl group and double cosets. 4601: 4494:

and power and other such factors you can think of your pockets being multi-dimensional pockets on a multi-dimensional table.

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No, base 10 is not "special", its just customary. You confuse numbers with their representation in a certain writing system.

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Me neither. Maybe there is an elementary reason, maybe not so elementary; for instance the classical diophantine equation

1826: 1811: 165: 6012:

they all are except that one on the end. In F_4 the two on the left are conjugate, and so are the two on the left. The

1821: 4483: 2462: 131: 3521:

I'm not very good with number theory: it's not my thing. I've given this a once over and here are my thoughts. The LHS

562:

There is no such thing as "rational in base 10" or "rational in base π". A real number is either rational or not. From

5657: 1556: 1240:? Very few of these constructions make any mention of base representations. A base-ten representation of a number may 758: 4262: 86: 17: 1254: 1071: 722:

written in base-π gives a number that would be irrational if it were to base-10. Similarily the irrational number π

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The integers are the closure under addition and negation of {1}. Just as with rationals, no mention of bases. --

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non-average (which, I think is more subtle than just having a high variance, although that may be an effect).

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Declan: Tango has the right not to believe something said to him on the internet by a person he's never met.

1981: 1944: 1856: 1713: 1626: 1437: 1384: 1114: 1046: 971: 738: 702: 546: 422: 365: 278: 2384: 5218: 4963: 4804: 4767: 4726: 4113: 4063: 3675: 3451: 2750: 2738: 2674: 1801: 632:

mean "has a repeating expansion", and the distinction becomes important when we discuss non-integer bases.

381:

A standard convention for eliminating these duplicates is to insist that there are no consecutive 1's. --

5078: 5000: 4648: 4544: 2634:", for those days when you get tired of writing "equivalence class induced by the group action". Eric. 1187:, or just as the symbol π (with no need for a base), we cannot create a set with that many elements, and 6080: 5931: 5632: 5413: 4136: 3995: 3964: 3229: 2763: 2639: 2149:! (By a Japanese collegue that I worked with in Sapporo, Saji-san) Feel free to ask him if he knows me. 2146: 1250: 1067: 5214: 5036: 4856: 4685: 4499: 3899: 3825: 3721: 3649: 2540: 2154: 2106: 2057: 1974: 1937: 1921: 1706: 1619: 1430: 1377: 1107: 1039: 731: 695: 539: 415: 358: 271: 1841: 1179:

is a natural number/a finite cardinal/a finite ordinal. Regardless of whether we write π as 3.14159...

5113: 2631: 443: 3820:

someone said something earlier? Does it really matter? Is there any need to make a sly comment? No!

4256: 3013: 2844: 2822: 2800: 2252: 2020: 1907: 1869: 1868:

Off-topic for this desk and unlikely to be productive. Take it to your user pages if you need to.--

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I'd say not; I guess it can be translated in a more mathematical language as asking whether the

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If "the average person" refers to someone chosen at random, there is a 1 in 2 chance that he is

3569: 3524: 1962: 4413: 4109: 4059: 2746: 951: 862: 342: 295: 233: 186: 169: 2220: 168:. I think you might be stuck on the "repeating decimals" thing - you have to recognize that 6076: 6045: 5927: 5893: 5628: 5430: 5409: 5393: 5353: 5320: 4163: 4078: 4012: 3979: 3205: 2594: 601: 447: 182: 2332: 6015: 5988: 5961: 5902: 5748: 5721: 5654:

w_i of the coxeter generating set. The correct formula for the size of U_w would then be

5460:, I find that it should be 2^3 3^6=5832. In general, if I let q denote p^f, I find that: 5405: 5293: 5266: 5166: 4581: 4198: 4182: 4141: 2581: 2135: 2003: 1965: 1896: 1233: 918: 563: 534:

has a repeating decimal expansion. Now π is irrational base 10, but π is rational base π.

400: 207: 149: 119: 4960:

What about irrational numbers like π/4 ≈ 0.7854? They don't correspond to any element of

3614: 2760:

These problems are usually best done with modular arithmetic. Take the equation mod 6 (

1925: 645:

means "the number whose expansion in the given base consists of the given digits". So 10

1836: 1781: 1734: 1369: 1225: 904:

I apologize; I should not have said "there is no number system..." which can represent

455: 438:

Yes, this is topological. Infinite strings from a finite alphabet are homeomorphic to

198: 2562: 2442: 2304: 2284: 353:. So in this base 11 = 100. So there's no uniqueness without talking a standard form. 5228: 5177: 5147: 4720: 4528: 4241: 4219: 3782: 3003:{\displaystyle \scriptstyle 3n(n+1)+6+1=(m+6r)^{3}=m^{3}+18m^{2}r+108mr^{2}+216r^{3}} 2630:(By the way, the equivalence classes induced by a group action have a special term, " 1411: 1197: 304: 134: 2278: 1929: 1164: 1034:. Once again, I would appreciate patience and good humour in any following posts. 947: 858: 678:. Now 10 is an integer and so a rational number. Now 10 is a rational number, but π 439: 258: 229: 4180:

would be considered average. So perhaps it's not such a bad statement after all.

6041: 5889: 5389: 5257: 4159: 4074: 4008: 3975: 3201: 2590: 1888: 586: 190: 726:

written in base-π gives a number that would be rational if it were to base-10.

5162: 4577: 4194: 2131: 1999: 1924:, along with some of my publications. I am listed on the math geneology site, 1892: 914: 396: 262: 203: 4312:), and each place can have a naught, a cross, or be empty (represent this by 3776:+ 1 a cube) is the second. (This is what 129.67.186.200 did two hours ago.) — 74: 5249:

BN-pairs: conjugate generators weyl group give double cosets of same size

4491: 4479: 4193:

I've never heard of "average" meaning "random" or "in the middle 50%". --

4154: 3778: 1407: 1193: 908:. What I meant to say is, there is no number system which can represent 790: 345:

article, it seems to support what I was saying. The arcticle says that 11

300: 5600:{\displaystyle B=(q-1)^{3}q^{6},Bn_{1}B=Bn_{2}B=Bn_{3}B=q^{7}(q-1)^{3}} 4487: 4478:. The pocket is on a two-dimensional table and the trajectories of the 4475: 1229: 1160: 1143: 985: 582: 1142:

I think you have a very good point here: even the introduction of our

4470: 1928:. If you really feel the need then please contact my PhD supervisor: 174: 1555:

Dr Dec - I have already addressed all of your concerns. Please read

789:. I can represent these integers in decimal, in Roman numerals, in 6075:

Is there a simple reason why conjugate generators have equal q_i?

4007:

PS: I just meant to agree with your remark, it was no objection --

178: 79:

Welcome to the Knowledge (XXG) Mathematics Reference Desk Archives

1686:" means -- and yet you continued to repeat the same mistakes. -- 3355:{\displaystyle \scriptstyle 3n(n+1)+6+1=1+18r+108r^{2}+216r^{3}} 793:, in base π or whatever - they are the same numbers in any case. 4997:

In fact the numbers in which don't correspond to elements of

3974:

I found Jao's remark a gentleman post. Very opportune, too. --

3959:

sharp, but that can't possibly be a big deal, can it? Eric.

1682:" doesn't make sense, and what the "10" in the subscript of "7 298:

comes to mind. But π is a quite different matter, of course. —

1674:

deeply they are rooted, the stronger the rhetoric required.

161: 5772:, and 3 positive roots associated with q. So the size of 1017:

can be expressed as an integer, since 10 is a (baseless) "

5718:

where a_i is the number of positive roots, conjugate to

4638:{\displaystyle \mathbb {Z} _{9}\times \mathbb {Z} _{3}.} 4405:{\displaystyle \mathbb {Z} _{9}\times \mathbb {Z} _{3}.} 173:

decimal representation. There is no number system (not

5985:

all generators (and hence all roots) are conjugate, in

4461:{\displaystyle f:\mathbb {Z} _{9}\to \mathbb {Z} _{3}.} 1146:

article seems to say that the concept "integer" can be

450:), whereas the reals are connected and one-dimensional. 4176:

average because only 50 of the population placed on a

3455: 3441:{\displaystyle \scriptstyle n(n+1)+2=6r(1+6r+12r^{2})} 3369: 3270: 3233: 3167: 3127: 3079: 3017: 2870: 2848: 2826: 2804: 2767: 2544: 2522: 6018: 5991: 5964: 5905: 5818: 5778: 5751: 5724: 5660: 5466: 5433: 5356: 5323: 5296: 5269: 5116: 5081: 5003: 4966: 4910: 4859: 4807: 4770: 4729: 4651: 4604: 4547: 4421: 4371: 4318: 4265: 3678: 3617: 3572: 3527: 3454: 3368: 3269: 3232: 3166: 3126: 3078: 3016: 2869: 2847: 2825: 2803: 2766: 2677: 2565: 2543: 2521: 2465: 2445: 2387: 2361: 2335: 2307: 2287: 2255: 2223: 1891:

did that here once before and it did not end well. --

1323: 857:

I didn't see anything offensive in Meni's language. —

641:

Anyway, you have also mixed up your notation - digits

604: 573:

is any number that can be expressed as the quotient

3566:is always congruent to 1 modulo 6. That means that 2508:{\displaystyle f(\gamma \cdot x)=\gamma \cdot f(x)} 6031: 6004: 5977: 5918: 5876: 5804: 5764: 5737: 5710: 5599: 5452: 5375: 5342: 5309: 5282: 5131: 5101: 5023: 4989: 4952: 4896: 4845: 4793: 4752: 4671: 4637: 4567: 4482:which lead to a pot are limited. But when you add 4460: 4404: 4357: 4304: 3708: 3632: 3603: 3558: 3483: 3440: 3354: 3255: 3192: 3152: 3112: 3024: 3002: 2855: 2833: 2811: 2789: 2721: 2571: 2551: 2529: 2507: 2451: 2423: 2373: 2347: 2313: 2293: 2269: 2241: 1355: 617: 5711:{\displaystyle q_{1}^{a_{1}}\ldots q_{n}^{a_{n}}} 1906:differential geometry without this knowledge. -- 761:(work in progress), and then the rest of my post. 4305:{\displaystyle \{0,\ldots ,8\}=\mathbb {Z} _{9}} 4158:that the typical Englishman is not typical...)-- 1729:Perhaps you will be interested in the last post 530:has a non-repeating decimal expansion where as π 261:, the nitpick wasn't very constructive. I think 5227:Those are very helpful suggestions, thank you. 4645:I was thinking one thing and writing another. 4259:. There are nine positions (represent this by 1368:idea of absolute space or absolute time; only 1171:is an integer iff at least one of the numbers 625:" - it is the same number, and it is rational. 1608:I have already addressed all of your concerns 8: 4891: 4873: 4337: 4319: 4284: 4266: 4231:A game with positions expressable as vectors 3768:+ 1 is certainly the first step. Seeing why 1604:Anyway, you have also mixed up your notation 1356:{\displaystyle 10_{\pi _{10_{\ddots }}}\ ,} 1224:Are you familiar with the constructions of 1102:an integer? Base 10 holds a special place. 4764:isolated in the interval than a vector in 4412:Alternatively, a game could be given by a 4358:{\displaystyle \{0,1,2\}=\mathbb {Z} _{3}} 3113:{\displaystyle \scriptstyle x^{2}+2=y^{3}} 2741:but failed. This is the original question 1864: 1001:would be a rational number. But hang on: π 294:Some irrational bases are perfectly sound— 6023: 6017: 5996: 5990: 5969: 5963: 5910: 5904: 5877:{\displaystyle (q^{2})^{6}(q)^{3}=q^{1}5} 5865: 5852: 5836: 5826: 5817: 5794: 5789: 5777: 5756: 5750: 5729: 5723: 5700: 5695: 5690: 5675: 5670: 5665: 5659: 5591: 5569: 5553: 5534: 5515: 5499: 5489: 5465: 5438: 5432: 5364: 5355: 5331: 5322: 5301: 5295: 5274: 5268: 5123: 5119: 5118: 5115: 5093: 5088: 5084: 5083: 5080: 5015: 5010: 5006: 5005: 5002: 4978: 4973: 4969: 4968: 4965: 4953:{\displaystyle 0.v_{1}v_{2}\ldots v_{9}.} 4941: 4928: 4918: 4909: 4904:could be set to correspond to the number 4864: 4858: 4834: 4815: 4806: 4782: 4777: 4773: 4772: 4769: 4741: 4736: 4732: 4731: 4728: 4663: 4658: 4654: 4653: 4650: 4626: 4622: 4621: 4611: 4607: 4606: 4603: 4559: 4554: 4550: 4549: 4546: 4449: 4445: 4444: 4434: 4430: 4429: 4420: 4393: 4389: 4388: 4378: 4374: 4373: 4370: 4349: 4345: 4344: 4317: 4296: 4292: 4291: 4264: 3772:can't be 57 (or any other number making 6 3677: 3616: 3580: 3571: 3535: 3526: 3453: 3428: 3367: 3345: 3329: 3268: 3231: 3193:{\displaystyle \scriptstyle \mathbb {Z} } 3176: 3169: 3168: 3165: 3153:{\displaystyle \scriptstyle \mathbb {Z} } 3136: 3129: 3128: 3125: 3103: 3084: 3077: 3015: 2993: 2977: 2955: 2939: 2926: 2868: 2846: 2824: 2802: 2765: 2713: 2685: 2676: 2564: 2542: 2520: 2464: 2444: 2386: 2360: 2334: 2306: 2286: 2263: 2262: 2254: 2222: 1338: 1333: 1328: 1322: 1167:). Or more loosely but more intuitively, 609: 603: 108:In the same vein as my question above on 2797:is even), what does this tell you about 1920:Tango, my PhD thesis can be found on my 690:writen to the base π a rational number? 49: 36: 1739:Non-standard positional numeral systems 1163:(with the natural numbers given by the 65: 2424:{\displaystyle f(\gamma \cdot x)=f(x)} 785:A rational number is the ratio of two 43: 4990:{\displaystyle \mathbb {Z} _{3}^{9}.} 4846:{\displaystyle (v_{1},\ldots ,v_{9})} 4794:{\displaystyle \mathbb {Z} _{3}^{9}.} 4753:{\displaystyle \mathbb {Z} _{3}^{9}?} 3709:{\displaystyle 0\leq n\leq 1,000,000} 3484:{\displaystyle \scriptstyle n(n+1)+2} 2722:{\displaystyle 3n^{2}+3n+7\neq k^{3}} 2431:. One also says that G is a group of 1306:} under addition and negation, where 7: 5102:{\displaystyle \mathbb {Z} _{3}^{9}} 5024:{\displaystyle \mathbb {Z} _{3}^{9}} 4672:{\displaystyle \mathbb {Z} _{3}^{9}} 4568:{\displaystyle \mathbb {Z} _{3}^{9}} 4365:). So any game would be a vector in 3990:Added emphasis to "appears". Eric. 1968:. We shared the same PhD supervisor! 1600:As it happens, there is a theorem... 5031:form an open and dense subset of . 3256:{\displaystyle \scriptstyle k=1+6r} 2790:{\displaystyle \scriptstyle n(n+1)} 800:written in base-π" does not give a 5958:Perhaps you know all this, but in 4897:{\displaystyle v_{i}\in \{0,1,2\}} 3120:is studied factorizing the LHS in 2552:{\displaystyle \textstyle \gamma } 2012:Whoa, can we all please calm down? 32: 2130:I know that group pretty well. -- 5132:{\displaystyle \mathbb {R} ^{9}} 2217:Just to fix the vocaboulary: if 1961:Better still Tango, contact the 4598:Yeah, I'm not sure why I wrote 4255:A very simple example would be 1249:under addition and negation. -- 5849: 5842: 5833: 5819: 5588: 5575: 5486: 5473: 4840: 4808: 4440: 3471: 3459: 3434: 3403: 3385: 3373: 3289: 3277: 3186: 3173: 3146: 3133: 3025:{\displaystyle \scriptstyle r} 2923: 2907: 2889: 2877: 2856:{\displaystyle \scriptstyle m} 2834:{\displaystyle \scriptstyle k} 2812:{\displaystyle \scriptstyle k} 2783: 2771: 2502: 2496: 2481: 2469: 2418: 2412: 2403: 2391: 2277:) and if G is a group with an 2270:{\displaystyle Y=\mathbb {R} } 2249:is any function (usually with 2233: 1817:Proof that π is transcendental 1797:Chronology of computation of π 1787:Bailey–Borwein–Plouffe formula 1300:. And indeed the closure of {1 18:Knowledge (XXG):Reference desk 1: 3672:p.s. I've just tested it for 2559:; in this case one says that 1792:Numerical approximations of π 1741:and those linked by them. -- 1372:. I'm going for a lie down... 33: 4760:I would say that a point is 2737:I guessed it can be done by 2530:{\displaystyle \textstyle x} 2374:{\displaystyle \gamma \in G} 6085:21:33, 24 August 2009 (UTC) 5936:15:52, 24 August 2009 (UTC) 5805:{\displaystyle U=U_{w_{0}}} 5637:13:06, 24 August 2009 (UTC) 5418:00:34, 24 August 2009 (UTC) 5398:21:25, 23 August 2009 (UTC) 5237:15:08, 27 August 2009 (UTC) 5223:06:31, 26 August 2009 (UTC) 5186:11:11, 24 August 2009 (UTC) 5171:22:29, 23 August 2009 (UTC) 5156:21:40, 23 August 2009 (UTC) 5055:20:42, 23 August 2009 (UTC) 4704:20:26, 23 August 2009 (UTC) 4586:20:03, 23 August 2009 (UTC) 4537:19:37, 23 August 2009 (UTC) 4518:18:34, 23 August 2009 (UTC) 4250:16:02, 23 August 2009 (UTC) 4203:19:43, 23 August 2009 (UTC) 4189:14:36, 23 August 2009 (UTC) 4168:14:16, 23 August 2009 (UTC) 4148:13:38, 23 August 2009 (UTC) 4118:10:14, 24 August 2009 (UTC) 4083:06:57, 24 August 2009 (UTC) 4068:18:27, 23 August 2009 (UTC) 4035:19:36, 26 August 2009 (UTC) 4017:15:37, 26 August 2009 (UTC) 4000:07:53, 25 August 2009 (UTC) 3984:06:57, 24 August 2009 (UTC) 3969:21:14, 23 August 2009 (UTC) 3918:20:54, 23 August 2009 (UTC) 3859:19:33, 23 August 2009 (UTC) 3844:18:09, 23 August 2009 (UTC) 3791:17:49, 23 August 2009 (UTC) 3740:17:30, 23 August 2009 (UTC) 3668:17:19, 23 August 2009 (UTC) 3604:{\displaystyle 3n^{2}+3n+7} 3559:{\displaystyle 3n^{2}+3n+7} 3501:15:38, 23 August 2009 (UTC) 3210:13:51, 23 August 2009 (UTC) 3160:(so here one would work in 3060:13:31, 23 August 2009 (UTC) 3042:11:40, 23 August 2009 (UTC) 2755:11:11, 23 August 2009 (UTC) 2644:21:01, 23 August 2009 (UTC) 2618:13:26, 23 August 2009 (UTC) 2599:10:19, 23 August 2009 (UTC) 2585:wrto the two G actions, or 2212:07:58, 23 August 2009 (UTC) 2173:19:47, 25 August 2009 (UTC) 2140:19:41, 25 August 2009 (UTC) 2125:19:31, 25 August 2009 (UTC) 2076:19:39, 25 August 2009 (UTC) 2029:19:27, 25 August 2009 (UTC) 2008:19:12, 25 August 2009 (UTC) 1993:18:54, 25 August 2009 (UTC) 1956:18:52, 25 August 2009 (UTC) 1916:18:33, 25 August 2009 (UTC) 1901:18:16, 25 August 2009 (UTC) 1878:19:44, 25 August 2009 (UTC) 1861:23:39, 24 August 2009 (UTC) 1847:List of topics related to π 1751:19:46, 25 August 2009 (UTC) 1725:19:20, 25 August 2009 (UTC) 1696:19:09, 25 August 2009 (UTC) 1638:18:24, 25 August 2009 (UTC) 1570:18:02, 25 August 2009 (UTC) 1557:User:Meni Rosenfeld/Numbers 1543:19:47, 25 August 2009 (UTC) 1529:Well, of course I meant "10 1525:19:11, 25 August 2009 (UTC) 1475:18:36, 25 August 2009 (UTC) 1449:18:24, 25 August 2009 (UTC) 1420:17:07, 25 August 2009 (UTC) 1396:13:29, 25 August 2009 (UTC) 1259:13:11, 25 August 2009 (UTC) 1206:13:07, 25 August 2009 (UTC) 1159:has been constructed as in 1126:12:48, 25 August 2009 (UTC) 1076:12:34, 25 August 2009 (UTC) 1058:11:09, 25 August 2009 (UTC) 976:23:26, 24 August 2009 (UTC) 956:17:05, 27 August 2009 (UTC) 923:21:55, 23 August 2009 (UTC) 867:17:05, 27 August 2009 (UTC) 814:20:50, 24 August 2009 (UTC) 759:User:Meni Rosenfeld/Numbers 750:13:53, 24 August 2009 (UTC) 714:13:46, 24 August 2009 (UTC) 663:13:22, 24 August 2009 (UTC) 558:12:39, 24 August 2009 (UTC) 515:19:29, 23 August 2009 (UTC) 464:20:57, 24 August 2009 (UTC) 434:12:30, 24 August 2009 (UTC) 405:19:39, 23 August 2009 (UTC) 391:19:29, 23 August 2009 (UTC) 377:18:47, 23 August 2009 (UTC) 313:17:28, 23 August 2009 (UTC) 290:17:06, 23 August 2009 (UTC) 238:05:53, 23 August 2009 (UTC) 212:03:02, 23 August 2009 (UTC) 156:02:56, 23 August 2009 (UTC) 143:02:21, 23 August 2009 (UTC) 126:02:18, 23 August 2009 (UTC) 6111: 4058:to treat such a problem.-- 1827:Software for calculating π 1812:Proof that π is irrational 773:10 is an integer, not "10" 166:Proof that π is irrational 1511:Of course, you meant "100 993:an integer? If so then 10 5075:That's a good point. In 2325:for the action of G, or 2242:{\displaystyle f:X\to Y} 1822:Gauss–Legendre algorithm 5453:{\displaystyle p^{f}=3} 5376:{\displaystyle Bw_{2}B} 5343:{\displaystyle Bw_{1}B} 3764:Yes, seeing that it's 6 3611:can always of the form 2745:. Thanks in advance, -- 2197:Lagrangian is invariant 1461:is the same number as 4 1245:decimal representation. 618:{\displaystyle 111_{2}} 6033: 6006: 5979: 5920: 5878: 5806: 5766: 5739: 5712: 5601: 5454: 5377: 5344: 5311: 5284: 5133: 5103: 5025: 4991: 4954: 4898: 4847: 4795: 4754: 4673: 4639: 4569: 4462: 4406: 4359: 4306: 3710: 3634: 3605: 3560: 3485: 3442: 3356: 3257: 3194: 3154: 3114: 3026: 3004: 2857: 2835: 2813: 2791: 2739:mathematical induction 2723: 2573: 2553: 2531: 2509: 2453: 2425: 2375: 2349: 2348:{\displaystyle x\in X} 2315: 2295: 2271: 2243: 1802:Leibniz formula for pi 1357: 910:all irrational numbers 796:"the rational number 7 619: 112:, how can we say that 87:current reference desk 6034: 6032:{\displaystyle q_{i}} 6007: 6005:{\displaystyle B_{n}} 5980: 5978:{\displaystyle A_{n}} 5921: 5919:{\displaystyle q_{i}} 5879: 5807: 5767: 5765:{\displaystyle q^{2}} 5740: 5738:{\displaystyle w_{i}} 5713: 5602: 5455: 5378: 5345: 5312: 5310:{\displaystyle w_{2}} 5285: 5283:{\displaystyle w_{1}} 5134: 5104: 5026: 4992: 4955: 4899: 4848: 4796: 4755: 4674: 4640: 4570: 4527:are more meaningful. 4463: 4407: 4360: 4307: 4137:fallacy of four terms 3711: 3635: 3606: 3561: 3486: 3443: 3357: 3258: 3195: 3155: 3115: 3027: 3005: 2858: 2836: 2814: 2792: 2724: 2574: 2554: 2532: 2510: 2454: 2426: 2376: 2350: 2316: 2296: 2272: 2244: 1807:Liu Hui's π algorithm 1358: 1292:for all real numbers 620: 6016: 5989: 5962: 5903: 5816: 5776: 5749: 5722: 5658: 5464: 5431: 5354: 5321: 5294: 5267: 5114: 5079: 5001: 4964: 4908: 4857: 4805: 4768: 4727: 4649: 4602: 4545: 4419: 4369: 4316: 4263: 3676: 3633:{\displaystyle 6m+1} 3615: 3570: 3525: 3491:has no roots mod 6. 3452: 3366: 3267: 3230: 3164: 3124: 3076: 3014: 2867: 2863:mod 6 then you have 2845: 2823: 2801: 2764: 2675: 2563: 2541: 2519: 2463: 2443: 2385: 2359: 2333: 2305: 2285: 2253: 2221: 1321: 1161:Integer#Construction 780:in some other system 602: 444:totally disconnected 5707: 5682: 5098: 5020: 4983: 4787: 4746: 4668: 4564: 4257:naughts and crosses 3226:Once you know that 2145:Euclidean form, in 1842:Borwein's algorithm 1612:superiority complex 1013:. Any real number 592:not equal to zero". 110:significant figures 6029: 6002: 5975: 5916: 5874: 5802: 5762: 5735: 5708: 5686: 5661: 5597: 5450: 5373: 5340: 5307: 5280: 5129: 5099: 5082: 5021: 5004: 4987: 4967: 4950: 4894: 4843: 4791: 4771: 4750: 4730: 4669: 4652: 4635: 4565: 4548: 4458: 4402: 4355: 4302: 4025:Ah, cool. Eric. 3706: 3630: 3601: 3556: 3481: 3480: 3438: 3437: 3352: 3351: 3253: 3252: 3190: 3189: 3150: 3149: 3110: 3109: 3022: 3021: 3000: 2999: 2853: 2852: 2831: 2830: 2809: 2808: 2787: 2786: 2719: 2569: 2549: 2548: 2527: 2526: 2505: 2449: 2421: 2371: 2345: 2311: 2291: 2267: 2239: 1353: 1095:an integer, but 10 989:well then isn't 10 615: 4038: 4037: 4020: 4019: 4002: 3986: 3789: 3640:for some integer 3184: 3144: 2608:these functions. 2572:{\displaystyle f} 2452:{\displaystyle f} 2314:{\displaystyle f} 2294:{\displaystyle X} 2194: 2193: 1418: 1349: 1204: 869: 649:= π rather than π 343:Golden ratio base 311: 296:Golden ratio base 187:irrational number 170:irrational number 93: 92: 73: 72: 6102: 6038: 6036: 6035: 6030: 6028: 6027: 6011: 6009: 6008: 6003: 6001: 6000: 5984: 5982: 5981: 5976: 5974: 5973: 5925: 5923: 5922: 5917: 5915: 5914: 5883: 5881: 5880: 5875: 5870: 5869: 5857: 5856: 5841: 5840: 5831: 5830: 5811: 5809: 5808: 5803: 5801: 5800: 5799: 5798: 5771: 5769: 5768: 5763: 5761: 5760: 5744: 5742: 5741: 5736: 5734: 5733: 5717: 5715: 5714: 5709: 5706: 5705: 5704: 5694: 5681: 5680: 5679: 5669: 5606: 5604: 5603: 5598: 5596: 5595: 5574: 5573: 5558: 5557: 5539: 5538: 5520: 5519: 5504: 5503: 5494: 5493: 5459: 5457: 5456: 5451: 5443: 5442: 5382: 5380: 5379: 5374: 5369: 5368: 5349: 5347: 5346: 5341: 5336: 5335: 5316: 5314: 5313: 5308: 5306: 5305: 5289: 5287: 5286: 5281: 5279: 5278: 5138: 5136: 5135: 5130: 5128: 5127: 5122: 5108: 5106: 5105: 5100: 5097: 5092: 5087: 5053: 5052: 5049: 5046: 5039: 5035: 5030: 5028: 5027: 5022: 5019: 5014: 5009: 4996: 4994: 4993: 4988: 4982: 4977: 4972: 4959: 4957: 4956: 4951: 4946: 4945: 4933: 4932: 4923: 4922: 4903: 4901: 4900: 4895: 4869: 4868: 4852: 4850: 4849: 4844: 4839: 4838: 4820: 4819: 4800: 4798: 4797: 4792: 4786: 4781: 4776: 4759: 4757: 4756: 4751: 4745: 4740: 4735: 4702: 4701: 4698: 4695: 4688: 4684: 4678: 4676: 4675: 4670: 4667: 4662: 4657: 4644: 4642: 4641: 4636: 4631: 4630: 4625: 4616: 4615: 4610: 4574: 4572: 4571: 4566: 4563: 4558: 4553: 4516: 4515: 4512: 4509: 4502: 4498: 4467: 4465: 4464: 4459: 4454: 4453: 4448: 4439: 4438: 4433: 4411: 4409: 4408: 4403: 4398: 4397: 4392: 4383: 4382: 4377: 4364: 4362: 4361: 4356: 4354: 4353: 4348: 4311: 4309: 4308: 4303: 4301: 4300: 4295: 4185: 4144: 4024: 4023: 4006: 4005: 3989: 3973: 3916: 3915: 3912: 3909: 3902: 3898: 3842: 3841: 3838: 3835: 3828: 3824: 3777: 3738: 3737: 3734: 3731: 3724: 3720: 3715: 3713: 3712: 3707: 3666: 3665: 3662: 3659: 3652: 3648: 3639: 3637: 3636: 3631: 3610: 3608: 3607: 3602: 3585: 3584: 3565: 3563: 3562: 3557: 3540: 3539: 3490: 3488: 3487: 3482: 3447: 3445: 3444: 3439: 3433: 3432: 3361: 3359: 3358: 3353: 3350: 3349: 3334: 3333: 3262: 3260: 3259: 3254: 3199: 3197: 3196: 3191: 3185: 3177: 3172: 3159: 3157: 3156: 3151: 3145: 3137: 3132: 3119: 3117: 3116: 3111: 3108: 3107: 3089: 3088: 3031: 3029: 3028: 3023: 3009: 3007: 3006: 3001: 2998: 2997: 2982: 2981: 2960: 2959: 2944: 2943: 2931: 2930: 2862: 2860: 2859: 2854: 2840: 2838: 2837: 2832: 2818: 2816: 2815: 2810: 2796: 2794: 2793: 2788: 2728: 2726: 2725: 2720: 2718: 2717: 2690: 2689: 2578: 2576: 2575: 2570: 2558: 2556: 2555: 2550: 2536: 2534: 2533: 2528: 2514: 2512: 2511: 2506: 2458: 2456: 2455: 2450: 2430: 2428: 2427: 2422: 2380: 2378: 2377: 2372: 2354: 2352: 2351: 2346: 2320: 2318: 2317: 2312: 2301:, one says that 2300: 2298: 2297: 2292: 2276: 2274: 2273: 2268: 2266: 2248: 2246: 2245: 2240: 2171: 2170: 2167: 2164: 2157: 2153: 2123: 2122: 2119: 2116: 2109: 2105: 2074: 2073: 2070: 2067: 2060: 2056: 1991: 1990: 1987: 1984: 1977: 1973: 1954: 1953: 1950: 1947: 1940: 1936: 1865: 1723: 1722: 1719: 1716: 1709: 1705: 1636: 1635: 1632: 1629: 1622: 1618: 1447: 1446: 1443: 1440: 1433: 1429: 1406: 1394: 1393: 1390: 1387: 1380: 1376: 1362: 1360: 1359: 1354: 1348: 1347: 1346: 1345: 1344: 1343: 1342: 1251:COVIZAPIBETEFOKY 1234:rational numbers 1192: 1124: 1123: 1120: 1117: 1110: 1106: 1068:COVIZAPIBETEFOKY 1056: 1055: 1052: 1049: 1042: 1038: 856: 748: 747: 744: 741: 734: 730: 712: 711: 708: 705: 698: 694: 686:, so why isn't π 624: 622: 621: 616: 614: 613: 556: 555: 552: 549: 542: 538: 448:zero-dimensional 432: 431: 428: 425: 418: 414: 375: 374: 371: 368: 361: 357: 299: 288: 287: 284: 281: 274: 270: 183:rational numbers 152: 122: 75: 38:Mathematics desk 34: 6110: 6109: 6105: 6104: 6103: 6101: 6100: 6099: 6019: 6014: 6013: 5992: 5987: 5986: 5965: 5960: 5959: 5906: 5901: 5900: 5861: 5848: 5832: 5822: 5814: 5813: 5790: 5785: 5774: 5773: 5752: 5747: 5746: 5725: 5720: 5719: 5696: 5671: 5656: 5655: 5587: 5565: 5549: 5530: 5511: 5495: 5485: 5462: 5461: 5434: 5429: 5428: 5406:symmetric group 5360: 5352: 5351: 5327: 5319: 5318: 5297: 5292: 5291: 5270: 5265: 5264: 5251: 5212: 5117: 5112: 5111: 5077: 5076: 5050: 5044: 5041: 5037: 5033: 5032: 4999: 4998: 4962: 4961: 4937: 4924: 4914: 4906: 4905: 4860: 4855: 4854: 4830: 4811: 4803: 4802: 4766: 4765: 4725: 4724: 4699: 4693: 4690: 4686: 4682: 4681: 4647: 4646: 4620: 4605: 4600: 4599: 4543: 4542: 4513: 4507: 4504: 4500: 4496: 4495: 4443: 4428: 4417: 4416: 4387: 4372: 4367: 4366: 4343: 4314: 4313: 4290: 4261: 4260: 4233: 4183: 4142: 4132: 3913: 3907: 3904: 3900: 3896: 3895: 3839: 3833: 3830: 3826: 3822: 3821: 3735: 3729: 3726: 3722: 3718: 3717: 3674: 3673: 3663: 3657: 3654: 3650: 3646: 3645: 3613: 3612: 3576: 3568: 3567: 3531: 3523: 3522: 3450: 3449: 3424: 3364: 3363: 3341: 3325: 3265: 3264: 3228: 3227: 3162: 3161: 3122: 3121: 3099: 3080: 3074: 3073: 3012: 3011: 2989: 2973: 2951: 2935: 2922: 2865: 2864: 2843: 2842: 2821: 2820: 2799: 2798: 2762: 2761: 2709: 2681: 2673: 2672: 2657: 2561: 2560: 2539: 2538: 2517: 2516: 2461: 2460: 2441: 2440: 2383: 2382: 2357: 2356: 2331: 2330: 2303: 2302: 2283: 2282: 2251: 2250: 2219: 2218: 2199: 2168: 2162: 2159: 2155: 2151: 2150: 2120: 2114: 2111: 2107: 2103: 2102: 2071: 2065: 2062: 2058: 2054: 2053: 1988: 1982: 1979: 1975: 1971: 1970: 1951: 1945: 1942: 1938: 1934: 1933: 1832:Viète's formula 1720: 1714: 1711: 1707: 1703: 1702: 1685: 1681: 1633: 1627: 1624: 1620: 1616: 1615: 1532: 1514: 1464: 1460: 1444: 1438: 1435: 1431: 1427: 1426: 1404: 1391: 1385: 1382: 1378: 1374: 1373: 1370:inertial frames 1334: 1329: 1324: 1319: 1318: 1315:actually mean? 1314: 1305: 1291: 1285: 1226:natural numbers 1186: 1182: 1121: 1115: 1112: 1108: 1104: 1103: 1098: 1091: 1053: 1047: 1044: 1040: 1036: 1035: 1033: 1027: 1012: 1008: 1004: 1000: 996: 992: 986:integer article 799: 767: 745: 739: 736: 732: 728: 727: 725: 721: 709: 703: 700: 696: 692: 691: 689: 685: 681: 677: 673: 652: 648: 644: 605: 600: 599: 571:rational number 564:Rational number 553: 547: 544: 540: 536: 535: 533: 529: 525: 429: 423: 420: 416: 412: 411: 372: 366: 363: 359: 355: 354: 352: 348: 341:Looking at the 285: 279: 276: 272: 268: 267: 150: 120: 106: 101: 30: 29: 28: 12: 11: 5: 6108: 6106: 6098: 6097: 6096: 6095: 6094: 6093: 6092: 6091: 6090: 6089: 6088: 6087: 6073: 6058: 6057: 6056: 6055: 6054: 6053: 6052: 6051: 6050: 6049: 6040:automorphism"? 6026: 6022: 5999: 5995: 5972: 5968: 5947: 5946: 5945: 5944: 5943: 5942: 5941: 5940: 5939: 5938: 5913: 5909: 5885: 5873: 5868: 5864: 5860: 5855: 5851: 5847: 5844: 5839: 5835: 5829: 5825: 5821: 5797: 5793: 5788: 5784: 5781: 5759: 5755: 5732: 5728: 5703: 5699: 5693: 5689: 5685: 5678: 5674: 5668: 5664: 5644: 5643: 5642: 5641: 5640: 5639: 5624: 5620: 5611: 5610: 5609: 5608: 5594: 5590: 5586: 5583: 5580: 5577: 5572: 5568: 5564: 5561: 5556: 5552: 5548: 5545: 5542: 5537: 5533: 5529: 5526: 5523: 5518: 5514: 5510: 5507: 5502: 5498: 5492: 5488: 5484: 5481: 5478: 5475: 5472: 5469: 5449: 5446: 5441: 5437: 5421: 5420: 5372: 5367: 5363: 5359: 5339: 5334: 5330: 5326: 5304: 5300: 5277: 5273: 5250: 5247: 5246: 5245: 5244: 5243: 5242: 5241: 5240: 5239: 5210: 5201: 5200: 5199: 5198: 5197: 5196: 5195: 5194: 5193: 5192: 5191: 5190: 5189: 5188: 5144: 5140: 5126: 5121: 5096: 5091: 5086: 5064: 5063: 5062: 5061: 5060: 5059: 5058: 5057: 5018: 5013: 5008: 4986: 4981: 4976: 4971: 4949: 4944: 4940: 4936: 4931: 4927: 4921: 4917: 4913: 4893: 4890: 4887: 4884: 4881: 4878: 4875: 4872: 4867: 4863: 4842: 4837: 4833: 4829: 4826: 4823: 4818: 4814: 4810: 4790: 4785: 4780: 4775: 4749: 4744: 4739: 4734: 4711: 4710: 4709: 4708: 4707: 4706: 4666: 4661: 4656: 4634: 4629: 4624: 4619: 4614: 4609: 4591: 4590: 4589: 4588: 4562: 4557: 4552: 4539: 4524: 4457: 4452: 4447: 4442: 4437: 4432: 4427: 4424: 4401: 4396: 4391: 4386: 4381: 4376: 4352: 4347: 4342: 4339: 4336: 4333: 4330: 4327: 4324: 4321: 4299: 4294: 4289: 4286: 4283: 4280: 4277: 4274: 4271: 4268: 4232: 4229: 4228: 4227: 4215: 4210: 4209: 4208: 4207: 4206: 4205: 4131: 4128: 4127: 4126: 4125: 4124: 4123: 4122: 4121: 4120: 4098: 4096: 4095: 4094: 4093: 4092: 4091: 4090: 4089: 4088: 4087: 4086: 4085: 4055: 4054: 4053: 4052: 4051: 4050: 4049: 4048: 4047: 4046: 4045: 4044: 4043: 4042: 4041: 4040: 4039: 4027:216.27.191.178 3935: 3934: 3933: 3932: 3931: 3930: 3929: 3928: 3927: 3926: 3925: 3924: 3923: 3922: 3921: 3920: 3876: 3875: 3874: 3873: 3872: 3871: 3870: 3869: 3868: 3867: 3866: 3865: 3864: 3863: 3862: 3861: 3804: 3803: 3802: 3801: 3800: 3799: 3798: 3797: 3796: 3795: 3794: 3793: 3751: 3750: 3749: 3748: 3747: 3746: 3745: 3744: 3743: 3742: 3716:and it holds. 3705: 3702: 3699: 3696: 3693: 3690: 3687: 3684: 3681: 3670: 3629: 3626: 3623: 3620: 3600: 3597: 3594: 3591: 3588: 3583: 3579: 3575: 3555: 3552: 3549: 3546: 3543: 3538: 3534: 3530: 3510: 3509: 3508: 3507: 3506: 3505: 3504: 3503: 3493:129.67.186.200 3479: 3476: 3473: 3470: 3467: 3464: 3461: 3458: 3436: 3431: 3427: 3423: 3420: 3417: 3414: 3411: 3408: 3405: 3402: 3399: 3396: 3393: 3390: 3387: 3384: 3381: 3378: 3375: 3372: 3362:. This gives 3348: 3344: 3340: 3337: 3332: 3328: 3324: 3321: 3318: 3315: 3312: 3309: 3306: 3303: 3300: 3297: 3294: 3291: 3288: 3285: 3282: 3279: 3276: 3273: 3251: 3248: 3245: 3242: 3239: 3236: 3217: 3216: 3215: 3214: 3213: 3212: 3188: 3183: 3180: 3175: 3171: 3148: 3143: 3140: 3135: 3131: 3106: 3102: 3098: 3095: 3092: 3087: 3083: 3065: 3064: 3063: 3062: 3045: 3044: 3034:129.67.186.200 3020: 2996: 2992: 2988: 2985: 2980: 2976: 2972: 2969: 2966: 2963: 2958: 2954: 2950: 2947: 2942: 2938: 2934: 2929: 2925: 2921: 2918: 2915: 2912: 2909: 2906: 2903: 2900: 2897: 2894: 2891: 2888: 2885: 2882: 2879: 2876: 2873: 2851: 2829: 2807: 2785: 2782: 2779: 2776: 2773: 2770: 2735: 2734: 2730: 2729: 2716: 2712: 2708: 2705: 2702: 2699: 2696: 2693: 2688: 2684: 2680: 2656: 2653: 2652: 2651: 2650: 2649: 2648: 2647: 2623: 2622: 2621: 2620: 2602: 2601: 2568: 2547: 2525: 2504: 2501: 2498: 2495: 2492: 2489: 2486: 2483: 2480: 2477: 2474: 2471: 2468: 2448: 2420: 2417: 2414: 2411: 2408: 2405: 2402: 2399: 2396: 2393: 2390: 2370: 2367: 2364: 2344: 2341: 2338: 2329:, iff for all 2310: 2290: 2265: 2261: 2258: 2238: 2235: 2232: 2229: 2226: 2198: 2195: 2192: 2191: 2190: 2189: 2188: 2187: 2186: 2185: 2184: 2183: 2182: 2181: 2180: 2179: 2178: 2177: 2176: 2175: 2087: 2086: 2085: 2084: 2083: 2082: 2081: 2080: 2079: 2078: 2040: 2039: 2038: 2037: 2036: 2035: 2034: 2033: 2032: 2031: 2021:Meni Rosenfeld 2016: 2013: 1995: 1908:Meni Rosenfeld 1881: 1880: 1870:Stephan Schulz 1850: 1849: 1844: 1839: 1837:Wallis product 1834: 1829: 1824: 1819: 1814: 1809: 1804: 1799: 1794: 1789: 1784: 1772: 1771: 1770: 1769: 1768: 1767: 1766: 1765: 1764: 1763: 1762: 1761: 1760: 1759: 1758: 1757: 1756: 1755: 1754: 1753: 1743:Meni Rosenfeld 1735:Numeral system 1688:Meni Rosenfeld 1683: 1679: 1675: 1671: 1653: 1652: 1651: 1650: 1649: 1648: 1647: 1646: 1645: 1644: 1643: 1642: 1641: 1640: 1583: 1582: 1581: 1580: 1579: 1578: 1577: 1576: 1575: 1574: 1573: 1572: 1562:Meni Rosenfeld 1553: 1552: 1551: 1550: 1549: 1548: 1547: 1546: 1545: 1535:Stephan Schulz 1530: 1517:Meni Rosenfeld 1512: 1492: 1491: 1490: 1489: 1488: 1487: 1486: 1485: 1484: 1483: 1482: 1481: 1480: 1479: 1478: 1477: 1467:Stephan Schulz 1462: 1458: 1402: 1365: 1364: 1363: 1352: 1341: 1337: 1332: 1327: 1312: 1301: 1287: 1281: 1268: 1267: 1266: 1265: 1264: 1263: 1262: 1261: 1246: 1215: 1214: 1213: 1212: 1211: 1210: 1209: 1208: 1184: 1180: 1133: 1132: 1131: 1130: 1129: 1128: 1096: 1089: 1081: 1080: 1079: 1078: 1061: 1060: 1029: 1025: 1010: 1006: 1002: 998: 994: 990: 963: 962: 961: 960: 959: 958: 938: 937: 936: 935: 934: 933: 932: 931: 930: 929: 928: 927: 926: 925: 889: 888: 887: 886: 885: 884: 883: 882: 881: 880: 879: 878: 877: 876: 875: 874: 873: 872: 871: 870: 835: 834: 833: 832: 831: 830: 829: 828: 827: 826: 825: 824: 823: 822: 821: 820: 819: 818: 817: 816: 806:Meni Rosenfeld 797: 794: 783: 776: 769: 765: 762: 755: 723: 719: 716: 687: 683: 679: 675: 671: 668:Meni Rosenfeld 655:Meni Rosenfeld 650: 646: 642: 639: 636: 633: 626: 612: 608: 595: 594: 593: 531: 527: 523: 520:Meni Rosenfeld 507:Meni Rosenfeld 503: 487: 485: 484: 483: 482: 481: 480: 479: 478: 477: 476: 475: 474: 473: 472: 471: 470: 469: 468: 467: 466: 451: 442:and therefore 393: 383:Meni Rosenfeld 350: 346: 326: 325: 324: 323: 322: 321: 320: 319: 318: 317: 316: 315: 266:well defined? 247: 246: 245: 244: 243: 242: 241: 240: 219: 218: 217: 216: 215: 214: 105: 102: 100: 97: 95: 91: 90: 82: 81: 71: 70: 64: 48: 41: 40: 31: 15: 14: 13: 10: 9: 6: 4: 3: 2: 6107: 6086: 6082: 6078: 6074: 6070: 6069: 6068: 6067: 6066: 6065: 6064: 6063: 6062: 6061: 6060: 6059: 6047: 6043: 6024: 6020: 5997: 5993: 5970: 5966: 5957: 5956: 5955: 5954: 5953: 5952: 5951: 5950: 5949: 5948: 5937: 5933: 5929: 5911: 5907: 5898: 5897: 5895: 5891: 5886: 5871: 5866: 5862: 5858: 5853: 5845: 5837: 5827: 5823: 5795: 5791: 5786: 5782: 5779: 5757: 5753: 5730: 5726: 5701: 5697: 5691: 5687: 5683: 5676: 5672: 5666: 5662: 5652: 5651: 5650: 5649: 5648: 5647: 5646: 5645: 5638: 5634: 5630: 5625: 5621: 5617: 5616: 5615: 5614: 5613: 5612: 5592: 5584: 5581: 5578: 5570: 5566: 5562: 5559: 5554: 5550: 5546: 5543: 5540: 5535: 5531: 5527: 5524: 5521: 5516: 5512: 5508: 5505: 5500: 5496: 5490: 5482: 5479: 5476: 5470: 5467: 5447: 5444: 5439: 5435: 5425: 5424: 5423: 5422: 5419: 5415: 5411: 5407: 5402: 5401: 5400: 5399: 5395: 5391: 5387: 5384: 5370: 5365: 5361: 5357: 5337: 5332: 5328: 5324: 5302: 5298: 5275: 5271: 5261: 5259: 5254: 5248: 5238: 5234: 5230: 5226: 5225: 5224: 5220: 5216: 5207: 5206: 5205: 5204: 5203: 5202: 5187: 5183: 5179: 5174: 5173: 5172: 5168: 5164: 5159: 5158: 5157: 5153: 5149: 5145: 5141: 5124: 5094: 5089: 5074: 5073: 5072: 5071: 5070: 5069: 5068: 5067: 5066: 5065: 5056: 5047: 5040: 5016: 5011: 4984: 4979: 4974: 4947: 4942: 4938: 4934: 4929: 4925: 4919: 4915: 4911: 4888: 4885: 4882: 4879: 4876: 4870: 4865: 4861: 4835: 4831: 4827: 4824: 4821: 4816: 4812: 4788: 4783: 4778: 4763: 4747: 4742: 4737: 4722: 4719: 4718: 4717: 4716: 4715: 4714: 4713: 4712: 4705: 4696: 4689: 4664: 4659: 4632: 4627: 4617: 4612: 4597: 4596: 4595: 4594: 4593: 4592: 4587: 4583: 4579: 4560: 4555: 4540: 4538: 4534: 4530: 4525: 4521: 4520: 4519: 4510: 4503: 4493: 4489: 4485: 4481: 4477: 4472: 4455: 4450: 4435: 4425: 4422: 4415: 4399: 4394: 4384: 4379: 4350: 4340: 4334: 4331: 4328: 4325: 4322: 4297: 4287: 4281: 4278: 4275: 4272: 4269: 4258: 4254: 4253: 4252: 4251: 4247: 4243: 4237: 4230: 4225: 4221: 4216: 4212: 4211: 4204: 4200: 4196: 4192: 4191: 4190: 4187: 4186: 4179: 4175: 4171: 4170: 4169: 4165: 4161: 4156: 4152: 4151: 4150: 4149: 4146: 4145: 4138: 4129: 4119: 4115: 4111: 4107: 4106: 4105: 4104: 4103: 4102: 4101: 4100: 4099: 4084: 4080: 4076: 4071: 4070: 4069: 4065: 4061: 4056: 4036: 4032: 4028: 4022: 4021: 4018: 4014: 4010: 4004: 4003: 4001: 3997: 3993: 3988: 3987: 3985: 3981: 3977: 3972: 3971: 3970: 3966: 3962: 3958: 3953: 3952: 3951: 3950: 3949: 3948: 3947: 3946: 3945: 3944: 3943: 3942: 3941: 3940: 3939: 3938: 3937: 3936: 3919: 3910: 3903: 3892: 3891: 3890: 3889: 3888: 3887: 3886: 3885: 3884: 3883: 3882: 3881: 3880: 3879: 3878: 3877: 3860: 3856: 3852: 3851:87.194.213.98 3847: 3846: 3845: 3836: 3829: 3818: 3817: 3816: 3815: 3814: 3813: 3812: 3811: 3810: 3809: 3808: 3807: 3806: 3805: 3792: 3788: 3784: 3780: 3775: 3771: 3767: 3763: 3762: 3761: 3760: 3759: 3758: 3757: 3756: 3755: 3754: 3753: 3752: 3741: 3732: 3725: 3703: 3700: 3697: 3694: 3691: 3688: 3685: 3682: 3679: 3671: 3669: 3660: 3653: 3643: 3627: 3624: 3621: 3618: 3598: 3595: 3592: 3589: 3586: 3581: 3577: 3573: 3553: 3550: 3547: 3544: 3541: 3536: 3532: 3528: 3520: 3519: 3518: 3517: 3516: 3515: 3514: 3513: 3512: 3511: 3502: 3498: 3494: 3477: 3474: 3468: 3465: 3462: 3456: 3429: 3425: 3421: 3418: 3415: 3412: 3409: 3406: 3400: 3397: 3394: 3391: 3388: 3382: 3379: 3376: 3370: 3346: 3342: 3338: 3335: 3330: 3326: 3322: 3319: 3316: 3313: 3310: 3307: 3304: 3301: 3298: 3295: 3292: 3286: 3283: 3280: 3274: 3271: 3249: 3246: 3243: 3240: 3237: 3234: 3225: 3224: 3223: 3222: 3221: 3220: 3219: 3218: 3211: 3207: 3203: 3181: 3178: 3141: 3138: 3104: 3100: 3096: 3093: 3090: 3085: 3081: 3071: 3070: 3069: 3068: 3067: 3066: 3061: 3057: 3053: 3052:93.132.179.71 3049: 3048: 3047: 3046: 3043: 3039: 3035: 3018: 2994: 2990: 2986: 2983: 2978: 2974: 2970: 2967: 2964: 2961: 2956: 2952: 2948: 2945: 2940: 2936: 2932: 2927: 2919: 2916: 2913: 2910: 2904: 2901: 2898: 2895: 2892: 2886: 2883: 2880: 2874: 2871: 2849: 2827: 2805: 2780: 2777: 2774: 2768: 2759: 2758: 2757: 2756: 2752: 2748: 2744: 2740: 2733:always holds. 2732: 2731: 2714: 2710: 2706: 2703: 2700: 2697: 2694: 2691: 2686: 2682: 2678: 2671: 2670: 2669: 2668:; Prove that 2667: 2663: 2654: 2645: 2641: 2637: 2633: 2629: 2628: 2627: 2626: 2625: 2624: 2619: 2615: 2611: 2610:93.132.179.71 2606: 2605: 2604: 2603: 2600: 2596: 2592: 2588: 2587:G-equivariant 2584: 2583: 2566: 2545: 2523: 2499: 2493: 2490: 2487: 2484: 2478: 2475: 2472: 2466: 2446: 2438: 2434: 2415: 2409: 2406: 2400: 2397: 2394: 2388: 2368: 2365: 2362: 2342: 2339: 2336: 2328: 2324: 2308: 2288: 2280: 2259: 2256: 2236: 2230: 2227: 2224: 2216: 2215: 2214: 2213: 2209: 2205: 2204:93.132.179.71 2196: 2174: 2165: 2158: 2148: 2143: 2142: 2141: 2137: 2133: 2128: 2127: 2126: 2117: 2110: 2099: 2098: 2097: 2096: 2095: 2094: 2093: 2092: 2091: 2090: 2089: 2088: 2077: 2068: 2061: 2050: 2049: 2048: 2047: 2046: 2045: 2044: 2043: 2042: 2041: 2030: 2026: 2022: 2017: 2014: 2011: 2010: 2009: 2005: 2001: 1996: 1994: 1985: 1978: 1969: 1967: 1964: 1959: 1958: 1957: 1948: 1941: 1931: 1927: 1923: 1919: 1918: 1917: 1913: 1909: 1904: 1903: 1902: 1898: 1894: 1890: 1885: 1884: 1883: 1882: 1879: 1875: 1871: 1867: 1866: 1863: 1862: 1858: 1854: 1853:Michael Hardy 1848: 1845: 1843: 1840: 1838: 1835: 1833: 1830: 1828: 1825: 1823: 1820: 1818: 1815: 1813: 1810: 1808: 1805: 1803: 1800: 1798: 1795: 1793: 1790: 1788: 1785: 1783: 1780: 1779: 1778: 1776: 1752: 1748: 1744: 1740: 1736: 1732: 1728: 1727: 1726: 1717: 1710: 1699: 1698: 1697: 1693: 1689: 1676: 1672: 1669: 1668: 1667: 1666: 1665: 1664: 1663: 1662: 1661: 1660: 1659: 1658: 1657: 1656: 1655: 1654: 1639: 1630: 1623: 1613: 1609: 1605: 1601: 1597: 1596: 1595: 1594: 1593: 1592: 1591: 1590: 1589: 1588: 1587: 1586: 1585: 1584: 1571: 1567: 1563: 1558: 1554: 1544: 1540: 1536: 1528: 1527: 1526: 1522: 1518: 1510: 1509: 1508: 1507: 1506: 1505: 1504: 1503: 1502: 1501: 1500: 1499: 1498: 1497: 1496: 1495: 1494: 1493: 1476: 1472: 1468: 1456: 1452: 1451: 1450: 1441: 1434: 1423: 1422: 1421: 1417: 1413: 1409: 1399: 1398: 1397: 1388: 1381: 1371: 1366: 1350: 1339: 1335: 1330: 1325: 1317: 1316: 1309: 1304: 1299: 1295: 1290: 1284: 1278: 1277: 1276: 1275: 1274: 1273: 1272: 1271: 1270: 1269: 1260: 1256: 1252: 1247: 1243: 1239: 1235: 1231: 1227: 1223: 1222: 1221: 1220: 1219: 1218: 1217: 1216: 1207: 1203: 1199: 1195: 1190: 1178: 1174: 1170: 1166: 1162: 1158: 1154: 1149: 1145: 1141: 1140: 1139: 1138: 1137: 1136: 1135: 1134: 1127: 1118: 1111: 1101: 1094: 1087: 1086: 1085: 1084: 1083: 1082: 1077: 1073: 1069: 1065: 1064: 1063: 1062: 1059: 1050: 1043: 1032: 1024: 1020: 1016: 987: 983: 980: 979: 978: 977: 973: 969: 968:Michael Hardy 957: 953: 949: 944: 943: 942: 941: 940: 939: 924: 920: 916: 911: 907: 903: 902: 901: 900: 899: 898: 897: 896: 895: 894: 893: 892: 891: 890: 868: 864: 860: 855: 854: 853: 852: 851: 850: 849: 848: 847: 846: 845: 844: 843: 842: 841: 840: 839: 838: 837: 836: 815: 811: 807: 803: 795: 792: 788: 784: 781: 777: 774: 770: 763: 760: 756: 753: 752: 751: 742: 735: 717: 715: 706: 699: 669: 666: 665: 664: 660: 656: 640: 637: 634: 631: 627: 610: 606: 596: 591: 588: 584: 580: 576: 572: 568: 567: 565: 561: 560: 559: 550: 543: 521: 518: 517: 516: 512: 508: 504: 500: 499: 498: 497: 496: 495: 494: 493: 492: 491: 490: 489: 488: 465: 461: 457: 452: 449: 445: 441: 437: 436: 435: 426: 419: 408: 407: 406: 402: 398: 394: 392: 388: 384: 380: 379: 378: 369: 362: 344: 340: 339: 338: 337: 336: 335: 334: 333: 332: 331: 330: 329: 328: 327: 314: 310: 306: 302: 297: 293: 292: 291: 282: 275: 264: 260: 257: 256: 255: 254: 253: 252: 251: 250: 249: 248: 239: 235: 231: 227: 226: 225: 224: 223: 222: 221: 220: 213: 209: 205: 200: 196: 192: 188: 184: 180: 176: 171: 167: 163: 159: 158: 157: 154: 153: 146: 145: 144: 140: 136: 133: 130: 129: 128: 127: 124: 123: 115: 111: 103: 98: 96: 88: 84: 83: 80: 77: 76: 68: 61: 57: 53: 47: 42: 39: 35: 27: 23: 19: 5388: 5385: 5262: 5255: 5252: 5215:Black Carrot 4761: 4238: 4234: 4181: 4173: 4140: 4133: 4110:Email4mobile 4097: 4060:Email4mobile 3956: 3773: 3769: 3765: 3641: 3200:instead). -- 2747:Email4mobile 2736: 2665: 2661: 2658: 2586: 2580: 2436: 2432: 2381:there holds 2326: 2322: 2200: 1960: 1930:Peter Giblin 1851: 1774: 1773: 1607: 1603: 1599: 1454: 1307: 1302: 1297: 1293: 1288: 1282: 1241: 1238:real numbers 1188: 1176: 1172: 1168: 1165:Peano axioms 1156: 1152: 1147: 1099: 1092: 1030: 1022: 1018: 1014: 964: 909: 905: 801: 786: 779: 772: 757:Please read 629: 589: 578: 574: 570: 486: 440:Cantor space 148: 118: 113: 107: 94: 78: 6077:JackSchmidt 5928:JackSchmidt 5629:JackSchmidt 5410:JackSchmidt 5258:(B, N) pair 4801:The vector 3992:98.207.86.2 3961:98.207.86.2 2819:mod 6? If 2636:98.207.86.2 2582:equivariant 2327:G-invariant 1782:Computing π 587:denominator 585:, with the 191:real number 26:Mathematics 4184:DRosenbach 4178:bell curve 4143:DRosenbach 3263:, you get 2459:satisfies 2433:symmetries 1966:Salix alba 1775:DRosenbach 526:= 1. Now π 199:algorithms 195:techniques 151:DRosenbach 121:DRosenbach 5812:would be 4541:Wouldn't 3010:for some 2743:in Arabic 2323:invariant 1922:user page 1533:" ;-). -- 1155:", after 653:= 10. -- 456:Trovatore 446:(indeed, 99:August 23 67:August 24 50:<< 46:August 22 5263:Suppose 4480:cue ball 4155:variance 2515:for all 1926:see here 1606:" and " 1230:integers 791:Gematria 787:integers 583:integers 135:Staecker 132:π ≠ 22/7 24:‎ | 22:Archives 20:‎ | 5256:I read 5253:Hello, 4476:snooker 4414:mapping 4130:Average 3957:appears 1889:someone 1777:: See: 1183:, as 10 1148:defined 1144:Integer 1019:integer 982:Michael 948:Tamfang 859:Tamfang 581:of two 259:Rckrone 230:Rckrone 89:pages. 6042:Evilbu 5890:Evilbu 5619:clear: 5390:Evilbu 5038:Dr Dec 4853:where 4687:Dr Dec 4501:Dr Dec 4492:bottom 4471:myriad 3901:Dr Dec 3827:Dr Dec 3723:Dr Dec 3651:Dr Dec 3448:, but 2655:Proof? 2632:orbits 2279:action 2156:Dr Dec 2147:Annals 2108:Dr Dec 2059:Dr Dec 1976:Dr Dec 1939:Dr Dec 1708:Dr Dec 1621:Dr Dec 1515:". -- 1432:Dr Dec 1379:Dr Dec 1189:that's 1109:Dr Dec 1100:is not 1041:Dr Dec 1021:" and 802:number 733:Dr Dec 697:Dr Dec 541:Dr Dec 417:Dr Dec 360:Dr Dec 273:Dr Dec 181:, not 177:, not 175:binary 164:, and 56:August 5163:Tango 4578:Tango 4195:Tango 2589:. -- 2132:Tango 2000:Tango 1963:admin 1893:Tango 1175:and − 915:Nimur 771:Now, 502:term. 397:Tango 349:= 100 263:Nimur 204:Nimur 179:octal 69:: --> 63:: --> 62:: --> 44:< 16:< 6081:talk 6046:talk 5932:talk 5894:talk 5633:talk 5414:talk 5394:talk 5350:and 5290:and 5233:talk 5229:risk 5219:talk 5182:talk 5178:risk 5167:talk 5152:talk 5148:risk 5045:Talk 4762:more 4721:Risk 4694:Talk 4582:talk 4533:talk 4529:risk 4508:Talk 4484:side 4246:talk 4242:risk 4224:talk 4220:risk 4199:talk 4164:talk 4114:talk 4079:talk 4064:talk 4031:talk 4013:talk 3996:talk 3980:talk 3965:talk 3908:Talk 3855:talk 3834:Talk 3730:Talk 3658:Talk 3497:talk 3206:talk 3056:talk 3038:talk 3032:... 2751:talk 2640:talk 2614:talk 2595:talk 2537:and 2355:and 2208:talk 2163:Talk 2136:talk 2115:Talk 2066:Talk 2025:talk 2004:talk 1983:Talk 1946:Talk 1912:talk 1897:talk 1874:talk 1857:talk 1747:talk 1731:here 1715:Talk 1692:talk 1628:Talk 1602:", " 1566:talk 1539:talk 1521:talk 1471:talk 1439:Talk 1386:Talk 1296:and 1255:talk 1242:look 1236:and 1116:Talk 1072:talk 1048:Talk 1028:= 10 1005:= 10 972:talk 952:talk 919:talk 863:talk 810:talk 740:Talk 704:Talk 682:= 10 674:= 10 659:talk 643:base 548:Talk 511:talk 460:talk 424:Talk 401:talk 387:talk 367:Talk 280:Talk 234:talk 208:talk 197:and 189:and 139:talk 4488:top 4174:not 4160:pma 4075:pma 4009:pma 3976:pma 3779:JAO 3704:000 3698:000 3339:216 3323:108 3202:pma 2987:216 2968:108 2841:is 2591:pma 2579:is 2435:of 2321:is 2281:on 1457:100 1408:JAO 1286:= 1 1194:JAO 630:not 607:111 569:"a 301:JAO 60:Sep 52:Jul 6083:) 5934:) 5896:) 5684:… 5635:) 5582:− 5480:− 5416:) 5396:) 5235:) 5221:) 5184:) 5169:) 5154:) 5051:~~ 5034:~~ 4935:… 4912:0. 4871:∈ 4825:… 4700:~~ 4683:~~ 4618:× 4584:) 4535:) 4514:~~ 4497:~~ 4490:, 4486:, 4441:→ 4385:× 4276:… 4248:) 4201:) 4166:) 4116:) 4081:) 4073:-- 4066:) 4033:) 4015:) 3998:) 3982:) 3967:) 3914:~~ 3897:~~ 3857:) 3840:~~ 3823:~~ 3785:• 3781:• 3736:~~ 3719:~~ 3689:≤ 3683:≤ 3664:~~ 3647:~~ 3499:) 3422:12 3314:18 3208:) 3179:− 3139:− 3058:) 3040:) 2949:18 2753:) 2707:≠ 2664:, 2642:) 2616:) 2597:) 2546:γ 2491:⋅ 2488:γ 2476:⋅ 2473:γ 2398:⋅ 2395:γ 2366:∈ 2363:γ 2340:∈ 2234:→ 2210:) 2169:~~ 2152:~~ 2138:) 2121:~~ 2104:~~ 2072:~~ 2055:~~ 2027:) 2006:) 1989:~~ 1972:~~ 1952:~~ 1935:~~ 1914:) 1899:) 1876:) 1859:) 1749:) 1737:, 1721:~~ 1704:~~ 1694:) 1684:10 1680:10 1678:"π 1634:~~ 1617:~~ 1568:) 1541:) 1523:) 1473:) 1463:10 1455:10 1445:~~ 1428:~~ 1414:• 1410:• 1403:16 1392:~~ 1375:~~ 1340:⋱ 1336:10 1331:π 1326:10 1311:10 1257:) 1232:, 1228:, 1200:• 1196:• 1181:10 1122:~~ 1105:~~ 1093:is 1090:10 1074:) 1054:~~ 1037:~~ 1026:10 1009:/1 1003:10 997:/1 974:) 954:) 921:) 906:pi 865:) 812:) 798:10 766:10 764:"π 746:~~ 729:~~ 724:10 720:10 710:~~ 693:~~ 688:10 680:10 672:10 661:) 566:: 554:~~ 537:~~ 528:10 513:) 462:) 454:-- 430:~~ 413:~~ 403:) 389:) 373:~~ 356:~~ 307:• 303:• 286:~~ 269:~~ 236:) 210:) 162:pi 141:) 114:pi 104:Pi 58:| 54:| 6079:( 6048:) 6044:( 6025:i 6021:q 5998:n 5994:B 5971:n 5967:A 5930:( 5912:i 5908:q 5892:( 5884:. 5872:5 5867:1 5863:q 5859:= 5854:3 5850:) 5846:q 5843:( 5838:6 5834:) 5828:2 5824:q 5820:( 5796:0 5792:w 5787:U 5783:= 5780:U 5758:2 5754:q 5731:i 5727:w 5702:n 5698:a 5692:n 5688:q 5677:1 5673:a 5667:1 5663:q 5631:( 5593:3 5589:) 5585:1 5579:q 5576:( 5571:7 5567:q 5563:= 5560:B 5555:3 5551:n 5547:B 5544:= 5541:B 5536:2 5532:n 5528:B 5525:= 5522:B 5517:1 5513:n 5509:B 5506:, 5501:6 5497:q 5491:3 5487:) 5483:1 5477:q 5474:( 5471:= 5468:B 5448:3 5445:= 5440:f 5436:p 5412:( 5392:( 5371:B 5366:2 5362:w 5358:B 5338:B 5333:1 5329:w 5325:B 5303:2 5299:w 5276:1 5272:w 5231:( 5217:( 5211:2 5209:Z 5180:( 5165:( 5150:( 5125:9 5120:R 5095:9 5090:3 5085:Z 5048:) 5042:( 5017:9 5012:3 5007:Z 4985:. 4980:9 4975:3 4970:Z 4948:. 4943:9 4939:v 4930:2 4926:v 4920:1 4916:v 4892:} 4889:2 4886:, 4883:1 4880:, 4877:0 4874:{ 4866:i 4862:v 4841:) 4836:9 4832:v 4828:, 4822:, 4817:1 4813:v 4809:( 4789:. 4784:9 4779:3 4774:Z 4748:? 4743:9 4738:3 4733:Z 4697:) 4691:( 4665:9 4660:3 4655:Z 4633:. 4628:3 4623:Z 4613:9 4608:Z 4580:( 4561:9 4556:3 4551:Z 4531:( 4511:) 4505:( 4456:. 4451:3 4446:Z 4436:9 4431:Z 4426:: 4423:f 4400:. 4395:3 4390:Z 4380:9 4375:Z 4351:3 4346:Z 4341:= 4338:} 4335:2 4332:, 4329:1 4326:, 4323:0 4320:{ 4298:9 4293:Z 4288:= 4285:} 4282:8 4279:, 4273:, 4270:0 4267:{ 4244:( 4226:) 4222:( 4197:( 4162:( 4112:( 4077:( 4062:( 4029:( 4011:( 3994:( 3978:( 3963:( 3911:) 3905:( 3853:( 3837:) 3831:( 3787:C 3783:T 3774:m 3770:m 3766:m 3733:) 3727:( 3701:, 3695:, 3692:1 3686:n 3680:0 3661:) 3655:( 3642:m 3628:1 3625:+ 3622:m 3619:6 3599:7 3596:+ 3593:n 3590:3 3587:+ 3582:2 3578:n 3574:3 3554:7 3551:+ 3548:n 3545:3 3542:+ 3537:2 3533:n 3529:3 3495:( 3478:2 3475:+ 3472:) 3469:1 3466:+ 3463:n 3460:( 3457:n 3435:) 3430:2 3426:r 3419:+ 3416:r 3413:6 3410:+ 3407:1 3404:( 3401:r 3398:6 3395:= 3392:2 3389:+ 3386:) 3383:1 3380:+ 3377:n 3374:( 3371:n 3347:3 3343:r 3336:+ 3331:2 3327:r 3320:+ 3317:r 3311:+ 3308:1 3305:= 3302:1 3299:+ 3296:6 3293:+ 3290:) 3287:1 3284:+ 3281:n 3278:( 3275:n 3272:3 3250:r 3247:6 3244:+ 3241:1 3238:= 3235:k 3204:( 3187:] 3182:3 3174:[ 3170:Z 3147:] 3142:2 3134:[ 3130:Z 3105:3 3101:y 3097:= 3094:2 3091:+ 3086:2 3082:x 3054:( 3036:( 3019:r 2995:3 2991:r 2984:+ 2979:2 2975:r 2971:m 2965:+ 2962:r 2957:2 2953:m 2946:+ 2941:3 2937:m 2933:= 2928:3 2924:) 2920:r 2917:6 2914:+ 2911:m 2908:( 2905:= 2902:1 2899:+ 2896:6 2893:+ 2890:) 2887:1 2884:+ 2881:n 2878:( 2875:n 2872:3 2850:m 2828:k 2806:k 2784:) 2781:1 2778:+ 2775:n 2772:( 2769:n 2749:( 2715:3 2711:k 2704:7 2701:+ 2698:n 2695:3 2692:+ 2687:2 2683:n 2679:3 2666:k 2662:n 2646:) 2638:( 2612:( 2593:( 2567:f 2524:x 2503:) 2500:x 2497:( 2494:f 2485:= 2482:) 2479:x 2470:( 2467:f 2447:f 2437:f 2419:) 2416:x 2413:( 2410:f 2407:= 2404:) 2401:x 2392:( 2389:f 2369:G 2343:X 2337:x 2309:f 2289:X 2264:R 2260:= 2257:Y 2237:Y 2231:X 2228:: 2225:f 2206:( 2166:) 2160:( 2134:( 2118:) 2112:( 2069:) 2063:( 2023:( 2002:( 1986:) 1980:( 1949:) 1943:( 1910:( 1895:( 1872:( 1855:( 1745:( 1718:) 1712:( 1690:( 1631:) 1625:( 1564:( 1537:( 1531:4 1519:( 1513:2 1469:( 1459:2 1442:) 1436:( 1416:C 1412:T 1389:) 1383:( 1351:, 1313:π 1308:n 1303:n 1298:y 1294:x 1289:y 1283:x 1280:1 1253:( 1202:C 1198:T 1185:π 1177:n 1173:n 1169:n 1157:Z 1153:Z 1119:) 1113:( 1097:π 1070:( 1051:) 1045:( 1031:x 1023:x 1015:x 1011:π 1007:π 999:π 995:π 991:π 970:( 950:( 946:— 917:( 861:( 808:( 743:) 737:( 707:) 701:( 684:π 676:π 657:( 651:π 647:π 611:2 598:" 590:b 579:b 577:/ 575:a 551:) 545:( 532:π 524:π 509:( 458:( 427:) 421:( 399:( 385:( 370:) 364:( 351:φ 347:φ 309:C 305:T 283:) 277:( 232:( 206:( 137:(

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Knowledge (XXG)

:Reference desk/Archives/Mathematics/2009 August 23 - Knowledge (XXG)

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