Talk:Lagrange's identity

84: 74: 53: 7033:, and that within these 252 terms we have three groups of 84 terms, two of which are mutually cancelling. That is not straightforward mathematics. If we look at the case of 5D, the 252 terms reduce to 80 terms. 60 of those terms reduce to the 15 terms of the Lagrange identity, but we are then left with an extra 20 which means that the 5D cross product does not match with the Pythagorean identity. Strangely, it only works for 3 and 7. 22: 6655: 3519: 2321: 461: 7487: 4769: 2676:

On your question to me, I made a mistake in segregating the equation into seven parts. I'm not going to cover up the mistake and it's important that these issues are corrected, but it didn't have any bearing on the wider issue as regards the fact that the equation holds in seven dimensions. The seven

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Why does it? You have provided no source for this, just some very flawed reasoning. What you've written above does not even come close to a mathematical proof, and even in it's incomplete state contains some basic mathematical errors that you'd expect a high school student to make. So, until you come

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The maths is all in 7D surely, so there should be no 3D Levi-Civita symbols, but it's not important as it's a result we already know. My point was, with the trig example, that two things being equal does not make them the same identity. Lounesto does not connect them in seven dimensions (he does in

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article gave the misleading information that it only holds in 3D. But it does hold in 7D even though proofs of that fact are anything but straightforward. It's far from immediately obvious that the 21 terms on the right hand side in the 7D case expand into 84 terms, and that these 84 terms are in

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John: First, no confusion has been made between the 3D epsilon tensor and the 7D one. Second, if the contraction of the epsilon tensors is used to form products of Kronecker deltas, exactly the same summation formula appears with the components just as they appear in the Lagrange form. Third, the

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The question which we were tackling was how to relate the 21 terms to the magnitude of a cross product. I had my doubts, having been mislead by two wikipedia articles into thinking that this could only be done in 3D. You finally convinced me with your numerical examples. It then became a case of

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Yes - although as I note above it's not so much that it holds, it's more that it's proposed as a condition the cross product must satisfy. From it and the other properties it's shown that a product only exists in 3 and 7 dimensions. For other reasons there are multiple products in 7D. Meanwhile

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Lagrange's theorem. Nowhere in Lounesto (I have the book in front of me) does it refer to Lagrange's identity. The two are only the same in 3D, so it has no place here. David tried to prove that they are the same but his maths is simply wrong: again here is David's calculation

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terms in the equation can be paired off with their family members. Your point above, in that respect was correct. The equation only holds over the entire set of relationships. But we still don't know why you are resisting the correction at this article and at the

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are dual, so the above can be written in terms of the cross product, as the article says. But this only works in 3D. In 7D there is no such dual relation so the above formula cannot be written in terms of the cross product in

2628: 6307: 6650:{\displaystyle {\biggl (}\sum _{k=1}^{n}x_{k}^{2}{\biggr )}{\biggl (}\sum _{k=1}^{n}y_{k}^{2}{\biggr )}-{\biggl (}\sum _{k=1}^{n}x_{k}y_{k}{\biggr )}^{2}=\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}(x_{i}y_{j}-x_{j}y_{i})^{2},} 3514:{\displaystyle {\biggl (}\sum _{k=1}^{n}a_{k}^{2}{\biggr )}{\biggl (}\sum _{k=1}^{n}b_{k}^{2}{\biggr )}-{\biggl (}\sum _{k=1}^{n}a_{k}b_{k}{\biggr )}^{2}=\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2},} 2316:{\displaystyle {\biggl (}\sum _{k=1}^{n}a_{k}^{2}{\biggr )}{\biggl (}\sum _{k=1}^{n}b_{k}^{2}{\biggr )}-{\biggl (}\sum _{k=1}^{n}a_{k}b_{k}{\biggr )}^{2}=\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2},} 456:{\displaystyle {\biggl (}\sum _{k=1}^{n}x_{k}^{2}{\biggr )}{\biggl (}\sum _{k=1}^{n}y_{k}^{2}{\biggr )}-{\biggl (}\sum _{k=1}^{n}x_{k}y_{k}{\biggr )}^{2}=\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}(x_{i}y_{j}-x_{j}y_{i})^{2},} 7645: 5748: 4149: 3534: 5640: 5449:

is not the usual Levi-Civita symbol: it is a shorthand for the operation and restricted to a limited range of indices (which depend on your choice of 7D cross product), so you can't do algebra with it as in

1699:. It is clear that if the first equation is squared you do not get the second equation, so it is completely wrong, or at least it is something completely different. So they are not the same, as you claim.-- 2338: 5585:

it would be surprising if you got anything different - you could have copied it from the 7D cross product article. That these two things are equal does not make the identity the same. E.g. consider

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Your spreadsheet calculates the cross product in 7D. This is Lagrange's identity. The reason it works in 3D is the dual relation between vectors and bivectors, as Lagrange's identity is the same as

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are equal does not mean the second is Lagrange's identity in 7D. If there were a source that they are the same then we could use it to support the changes, otherwise it is synthesis.--

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mapping the 7-space to the scalars both produce the same scalar for every permissible argument, they are the same function for all intents and purposes. Moreover, in this case, both

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arguing that it holds in 7D. In fact you even provided sources in the form of Silagadze and Lounesto which were adamant that the Pythagorean identity holds in both 3D and 7D.

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And so it's correct that this article now mentions the 7D case, and I support the changes that introduce this fact into the article. As regards terminologies, the equation,

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in the special cases of both 3 and 7 dimensions. Establishing this relationship in 3D is a relatively straightforward business, and that is perhaps why this article and the

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cross product. That means the Pythagorean theorem is holds in 3D and in 7D, as Lounesto says point blank. Hopefully, the wording of the new subsection makes all this clear.

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in the special case of 3 dimensions. This is also true for 7 dimensions. In the seven dimensional case, we get 21 terms on the right hand side. These terms take the form,

7482:{\displaystyle |\mathbf {a\times b} |^{2}=\Sigma _{k}\left(\Sigma _{i}\Sigma _{j}a_{i}b_{j}\varepsilon _{ijk}\Sigma _{p}\Sigma _{q}a_{p}b_{q}\varepsilon _{pqk}\right)\ ,} 4764:{\displaystyle |\mathbf {a\times b} |^{2}=\Sigma _{k}\left(\Sigma _{i}\Sigma _{j}a_{i}b_{j}\varepsilon _{ijk}\Sigma _{p}\Sigma _{q}a_{p}b_{q}\varepsilon _{pqk}\right)\ } 7031: 7004: 5651: 5126: 1716:

I've contributed a subsection on 7D that is rigorously sourced to Lounesto. Yes, the cross product must satisfy this relation, and it is a property proposed for

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I didn't like that usage of the word, but I didn't quibble about it. Now you are trying to wriggle out of this by jumping to the more accurate usage of the term

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John, you believe that the Lagrange's identity holds true for any dimensionality of space, and the Pythagorean theorem only when dimensionality of the space is

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Nonetheless, you are proposing (using an algebraic instance based on incompatible multiplication tables) that Lagrange's identity does not reduce in 7D to:

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Do you care if you are right or not? Or is this really about strict enforcement of the WP challenge of unsourced material, regardless of common sense?

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holds in 3D and 7D. Or more precisely it is a condition of the cross product in 7D. But that is a different article: that and Lagrange's identity are

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can be manipulated using the contraction of the epsilon tensor as sums of Kronecker deltas to produce the result. This epsilon tensor is not the

7841:{\displaystyle |\mathbf {a} |^{2}|\mathbf {b} |^{2}-(\mathbf {a\cdot b} )^{2}=\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2}\ ,} 5944:{\displaystyle |\mathbf {a} |^{2}|\mathbf {b} |^{2}-(\mathbf {a\cdot b} )^{2}=\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2}\ ,} 4345:{\displaystyle |\mathbf {a} |^{2}|\mathbf {b} |^{2}-(\mathbf {a\cdot b} )^{2}=\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2}\ ,} 3730:{\displaystyle |\mathbf {a} |^{2}|\mathbf {b} |^{2}-(\mathbf {a\cdot b} )^{2}=\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2}\ ,} 106: 8154: 1529:

John, I'll gladly discuss your question, but only after you have first answered the question below on both this page and on the talk page of

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These are obviously equal, but that does not make them the same identity: the second is a half angle formula, but the first one is not.

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Right? My spreadsheet calculates the Lagrange terms and shows the Pythagorean identity and the Lagrange identity are identical for all

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Given these two beliefs, the issue, then, as to whether the Lagrange identity is the same as the Pythagorean identity, is whether:

2460:{\displaystyle |\mathbf {a} |^{2}|\mathbf {b} |^{2}-|\mathbf {a} \cdot \mathbf {b} |^{2}=|\mathbf {a} \times \mathbf {b} |^{2}\ .} 4110: 2007: 7501: 7159:{\displaystyle |\mathbf {x} |^{2}|\mathbf {y} |^{2}-|\mathbf {x} \cdot \mathbf {y} |^{2}=|\mathbf {x} \times \mathbf {y} |^{2}} 6783:{\displaystyle |\mathbf {x} |^{2}|\mathbf {y} |^{2}-|\mathbf {x} \cdot \mathbf {y} |^{2}=|\mathbf {x} \times \mathbf {y} |^{2}} 3185:{\displaystyle |\mathbf {x} |^{2}|\mathbf {y} |^{2}-|\mathbf {x} \cdot \mathbf {y} |^{2}=|\mathbf {x} \times \mathbf {y} |^{2}} 2003: 1659:{\displaystyle |\mathbf {x} |^{2}|\mathbf {y} |^{2}-|\mathbf {x} \cdot \mathbf {y} |^{2}=|\mathbf {x} \times \mathbf {y} |^{2}} 1530: 1232:{\displaystyle |\mathbf {x} |^{2}|\mathbf {y} |^{2}-|\mathbf {x} \cdot \mathbf {y} |^{2}=|\mathbf {x} \times \mathbf {y} |^{2}} 1098: 1087:{\displaystyle |\mathbf {x} |^{2}|\mathbf {y} |^{2}-|\mathbf {x} \cdot \mathbf {y} |^{2}=|\mathbf {x} \times \mathbf {y} |^{2}} 918: 672: 589:{\displaystyle |\mathbf {x} |^{2}|\mathbf {y} |^{2}-|\mathbf {x} \cdot \mathbf {y} |^{2}=|\mathbf {x} \times \mathbf {y} |^{2}} 1097:

whatever name you like to call it, holds in both 3 and 7 dimensions and that is exactly what you were arguing over at the

33: 7629:{\displaystyle |\mathbf {a} |^{2}|\mathbf {b} |^{2}-\left(\mathbf {a\cdot b} \right)^{2}=|\mathbf {a\times b} |^{2}\ ,} 6076:{\displaystyle |\mathbf {a\times b} |^{2}=|\mathbf {a} |^{2}|\mathbf {b} |^{2}-\left(\mathbf {a\cdot b} \right)^{2}\ .} 5574:{\displaystyle |\mathbf {a\times b} |^{2}=|\mathbf {a} |^{2}|\mathbf {b} |^{2}-\left(\mathbf {a\cdot b} \right)^{2}\ .} 2680:

terms do indeed equate to the twenty-one xy terms on the right hand side. The correct equation should look like this,

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However, algebra to one side, Blackburne's suggestion is that the two sourced results that no-one disputes, namely:

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I think that does it. In 7D the Lagrange's identity and the generalized Pythagorean theorem are the same condition.

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The article has been amended to reflect this fact. It is wrong to say that the above reduction only applies in 3D.

6317:)}. Can you really suggest that they are not the same polynomial when equality applies for arbitrary choices of {a 8105:

It is then also much easier to prove the identity by expanding the squares (in the rhs) and separating the sums.

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Using identities from the 7D cross product article for the triple cross product with only two different vectors:

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John: There are a multitude of possible multiplication tables for the 7D cross-product. You have proposed one at

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So, although the algebra is complicated, I don't think it even is necessary. Blackburne's concern is unfounded.

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3). No source says they are the same identity in 7D, as they are not, and to deduce the connection yourself is

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terms. But we agreed in the end that it works in 7D. And yes, I shouldn't have assumed that the individual z

8136: 8010:{\displaystyle |\mathbf {a\times b} |^{2}=\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2}\ .} 7182: 5385:{\displaystyle |\mathbf {a\times b} |^{2}=\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2}\ .} 4779: 3200: 1674: 1251: 926: 886: 7291:{\displaystyle \mathbf {a\times b} =\Sigma _{i}\Sigma _{j}a_{i}b_{j}\varepsilon _{ijk}\mathbf {e_{k}} \ ,} 4576:{\displaystyle \mathbf {a\times b} =\Sigma _{i}\Sigma _{j}a_{i}b_{j}\varepsilon _{ijk}\mathbf {e_{k}} \ .} 8087: 6339: 6113: 5430: 5400: 4126: 4100: 2508: 2486: 1725: 39: 4442: 83: 8132: 7178: 3196: 1670: 1247: 922: 882: 8111: 7174: 1243: 1106: 1102: 936: 6108:

Your final comment is that if A = B and B = C, then A = C is synthesis. In other words, why bother?

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The second form of the right-hand looks much nicer if you drop the superfluous condition i<: -->

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It is clear the above squared does not equal what David has, so the two things are not the same.--

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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John, Now that you have admitted that it holds in seven dimensions, why did you make this revert

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holds in both 3 and 7 dimensions, and you have been deliberately obstructing this correction.

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Using this cross product, the generalized Pythagorean theorem is established as follows:

7009: 6982: 6967:{\displaystyle z_{1}=(x_{2}y_{4}-x_{4}y_{2}+x_{3}y_{7}-x_{7}y_{3}+x_{5}y_{6}-x_{6}y_{5})} 1973:{\displaystyle z_{1}=(x_{2}y_{4}-x_{4}y_{2}+x_{3}y_{7}-x_{7}y_{3}+x_{5}y_{6}-x_{6}y_{5})} 1434:{\displaystyle z_{1}=(x_{2}y_{4}-x_{4}y_{2}+x_{3}y_{7}-x_{7}y_{3}+x_{5}y_{6}-x_{6}y_{5})} 8039:. Inasmuch as we are dealing with only the space of 7-vectors, I'd say that if function 8148: 6794: 6125: 3059: 1101:

page. You cannot wriggle out of this by supplying an alternative definition for the

4429:{\displaystyle \mathbf {e_{i}\times e_{j}} =\varepsilon _{ijk}\mathbf {e_{k}} \ ,} 2623:{\displaystyle (a\cdot a)(b\cdot b)-(a\cdot b)^{2}=(a\wedge b)\cdot (a\wedge b).} 8140: 8091: 7186: 6343: 6302:{\displaystyle \sum _{i=1}^{n-1}\sum _{j=i+1}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2}\ .} 6139: 6117: 6098: 5434: 5404: 4130: 4104: 3204: 2662: 2512: 2490: 1995: 1729: 1710: 1678: 1523: 1255: 954: 930: 908: 890: 102: 6356:

Brews, I agree that it has been established to everybody's satisfaction that,

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As you very well know from the intro to this article, Lagrange's identity is:

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That's very strange John. You just spent about five days at the talk page of

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in 7D, and finds them both to be the same for every case that I have tried.

1533:. The question is very simple, and it only requires a 'yes' or 'no' answer. 7500:. An alternative is to use the representation of the 7D cross product as a 5743:

In the same way just because this, one way of writing Lagrange's identity:

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Hi David: Well, you are right that the algebra could be a mess. The form:

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which in fact is more accurate. The point is that you were using the name

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then perhaps you will see for yourself how they are not the same in 7D.--

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Of course, if the equality held only for a particular pair of vectors

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up with a proper source for it, it does not belong in the article.--

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is a completely antisymmetric tensor with a positive value +1 when

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Lev Vasilʹevitch Sabinin, Larissa Sbitneva & I. P. Shestakov

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and this, the Pythagorean condition on the cross product in 7D:

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article. These two articles should now read that the equation,

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from the 7D cross product page, the other is what you claim is

3882:= 7. I believe that is consensus, and supported by Lounesto. 1685:

please look at the two lines above. One is the expression for

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I can see a couple of problems with your maths, mostly that

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trying to rationalize how the 21 xy terms related to the 7 z

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is based upon an equally valid, but different one found at

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spreadsheet shows the two are identical component wise.

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etc. and avoid the use of the triple product identity.

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Please show how you get from that to what you claim is

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Or am I missing something? 11:08, 29 July 2016 (UTC)

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calculates both sides of this equation for arbitrary

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If as you believe, John, the two are different when

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that I have tried - a rather large and varied set.

101:, a collaborative effort to improve the coverage of 6152:is carefully defined for 7D in the exposition, and 4476:= 123, 145, 176, 246, 257, 347, 365. Consequently, 8009: 7840: 7628: 7481: 7290: 7158: 7025: 6998: 6966: 6782: 6649: 6301: 6075: 5943: 5728: 5634: 5573: 5384: 5213: 5114: 5005: 4928: 4763: 4575: 4464: 4428: 4344: 4050: 3863: 3729: 3513: 3184: 2622: 2459: 2315: 1972: 1658: 1433: 1231: 1086: 964:John, You are now playing on words. The equation, 588: 455: 6520: 6471: 6461: 6418: 6411: 6368: 4355:One possible multiplication table in 7-space is: 3384: 3335: 3325: 3282: 3275: 3232: 2186: 2137: 2127: 2084: 2077: 2034: 326: 277: 267: 224: 217: 174: 4143:has shown Lagrange's identity is equivalent to: 162:It stated in the article that the general form, 6798:fact the 252 terms that we get when we square, 8102:j (the terms corresponding to i=j are zero). 8: 5409:An alternative approach is to contract the ε 3740:The generalized Pythagorean theorem states: 5115:{\displaystyle =\mathbf {a\cdot } \left\ ,} 5006:{\displaystyle =\mathbf {a\cdot } \left\ .} 8109: 671:, and based on the relationships from the 47: 7996: 7986: 7976: 7963: 7953: 7940: 7923: 7907: 7896: 7883: 7878: 7866: 7861: 7859: 7827: 7817: 7807: 7794: 7784: 7771: 7754: 7738: 7727: 7714: 7699: 7687: 7682: 7676: 7671: 7665: 7660: 7654: 7649: 7647: 7615: 7610: 7598: 7593: 7584: 7569: 7555: 7550: 7544: 7539: 7533: 7528: 7522: 7517: 7515: 7457: 7447: 7437: 7427: 7417: 7401: 7391: 7381: 7371: 7361: 7346: 7333: 7328: 7316: 7311: 7309: 7276: 7271: 7259: 7249: 7239: 7229: 7219: 7201: 7199: 7150: 7145: 7139: 7131: 7126: 7117: 7112: 7106: 7098: 7093: 7084: 7079: 7073: 7068: 7062: 7057: 7051: 7046: 7044: 7017: 7011: 6990: 6984: 6955: 6945: 6932: 6922: 6909: 6899: 6886: 6876: 6863: 6853: 6840: 6830: 6814: 6808: 6774: 6769: 6763: 6755: 6750: 6741: 6736: 6730: 6722: 6717: 6708: 6703: 6697: 6692: 6686: 6681: 6675: 6670: 6668: 6638: 6628: 6618: 6605: 6595: 6582: 6565: 6549: 6538: 6525: 6519: 6518: 6511: 6501: 6491: 6480: 6470: 6469: 6460: 6459: 6453: 6448: 6438: 6427: 6417: 6416: 6410: 6409: 6403: 6398: 6388: 6377: 6367: 6366: 6364: 6288: 6278: 6268: 6255: 6245: 6232: 6215: 6199: 6188: 6182: 6062: 6047: 6033: 6028: 6022: 6017: 6011: 6006: 6000: 5995: 5986: 5981: 5969: 5964: 5962: 5930: 5920: 5910: 5897: 5887: 5874: 5857: 5841: 5830: 5817: 5802: 5790: 5785: 5779: 5774: 5768: 5763: 5757: 5752: 5750: 5713: 5707: 5679: 5667: 5653: 5606: 5592: 5560: 5545: 5531: 5526: 5520: 5515: 5509: 5504: 5498: 5493: 5484: 5479: 5467: 5462: 5460: 5371: 5361: 5351: 5338: 5328: 5315: 5298: 5282: 5271: 5258: 5253: 5241: 5236: 5234: 5200: 5185: 5171: 5166: 5160: 5155: 5149: 5144: 5138: 5133: 5128: 5097: 5082: 5070: 5062: 5057: 5051: 5046: 5033: 5028: 4974: 4960: 4947: 4942: 4911: 4901: 4885: 4875: 4865: 4855: 4839: 4829: 4819: 4800: 4790: 4781: 4742: 4732: 4722: 4712: 4702: 4686: 4676: 4666: 4656: 4646: 4631: 4618: 4613: 4601: 4596: 4594: 4561: 4556: 4544: 4534: 4524: 4514: 4504: 4486: 4484: 4450: 4444: 4414: 4409: 4397: 4383: 4370: 4365: 4363: 4331: 4321: 4311: 4298: 4288: 4275: 4258: 4242: 4231: 4218: 4203: 4191: 4186: 4180: 4175: 4169: 4164: 4158: 4153: 4151: 4065:, the two are the same. If the answer is 4037: 4027: 4017: 4004: 3994: 3981: 3964: 3948: 3937: 3924: 3919: 3903: 3898: 3896: 3850: 3845: 3829: 3824: 3815: 3800: 3788: 3783: 3777: 3772: 3766: 3761: 3755: 3750: 3748: 3716: 3706: 3696: 3683: 3673: 3660: 3643: 3627: 3616: 3603: 3588: 3576: 3571: 3565: 3560: 3554: 3549: 3543: 3538: 3536: 3502: 3492: 3482: 3469: 3459: 3446: 3429: 3413: 3402: 3389: 3383: 3382: 3375: 3365: 3355: 3344: 3334: 3333: 3324: 3323: 3317: 3312: 3302: 3291: 3281: 3280: 3274: 3273: 3267: 3262: 3252: 3241: 3231: 3230: 3228: 3176: 3171: 3165: 3157: 3152: 3143: 3138: 3132: 3124: 3119: 3110: 3105: 3099: 3094: 3088: 3083: 3077: 3072: 3070: 2575: 2524: 2446: 2441: 2435: 2427: 2422: 2413: 2408: 2402: 2394: 2389: 2380: 2375: 2369: 2364: 2358: 2353: 2347: 2342: 2340: 2304: 2294: 2284: 2271: 2261: 2248: 2231: 2215: 2204: 2191: 2185: 2184: 2177: 2167: 2157: 2146: 2136: 2135: 2126: 2125: 2119: 2114: 2104: 2093: 2083: 2082: 2076: 2075: 2069: 2064: 2054: 2043: 2033: 2032: 2030: 1961: 1951: 1938: 1928: 1915: 1905: 1892: 1882: 1869: 1859: 1846: 1836: 1820: 1814: 1650: 1645: 1639: 1631: 1626: 1617: 1612: 1606: 1598: 1593: 1584: 1579: 1573: 1568: 1562: 1557: 1551: 1546: 1544: 1422: 1412: 1399: 1389: 1376: 1366: 1353: 1343: 1330: 1320: 1307: 1297: 1281: 1275: 1223: 1218: 1212: 1204: 1199: 1190: 1185: 1179: 1171: 1166: 1157: 1152: 1146: 1141: 1135: 1130: 1124: 1119: 1117: 1078: 1073: 1067: 1059: 1054: 1045: 1040: 1034: 1026: 1021: 1012: 1007: 1001: 996: 990: 985: 979: 974: 972: 580: 575: 569: 561: 556: 547: 542: 536: 528: 523: 514: 509: 503: 498: 492: 487: 481: 476: 474: 444: 434: 424: 411: 401: 388: 371: 355: 344: 331: 325: 324: 317: 307: 297: 286: 276: 275: 266: 265: 259: 254: 244: 233: 223: 222: 216: 215: 209: 204: 194: 183: 173: 172: 170: 8063:of exactly the same variables, namely {a 7504:using the standard Levi-Civita symbol. 49: 19: 7169:could rightfully be called either the 4929:{\displaystyle =\Sigma _{i}a_{i}\left} 1807:, from the 7D cross product article: 1268:, from the 7D cross product article: 7: 8030:for every possible pair of 7 vectors 5226:Combining this with Jones' result: 3885:Please let me know if you disagree. 1735:But that's the Pythagorean theorem, 158:The special case of seven dimensions 95:This article is within the scope of 8131:Whats the use of j not equal to i. 7502:sum of 3-dimensional cross products 4983: 4978: 4964: 3912: 3907: 3838: 3833: 38:It is of interest to the following 7424: 7414: 7368: 7358: 7343: 7226: 7216: 6352:Complicated algebra; is it needed? 4872: 4862: 4826: 4816: 4787: 4709: 4699: 4653: 4643: 4628: 4511: 4501: 4465:{\displaystyle \varepsilon _{ijk}} 14: 8160:Low-priority mathematics articles 115:Knowledge:WikiProject Mathematics 8059:general functions, but both are 7873: 7867: 7706: 7700: 7677: 7655: 7605: 7599: 7576: 7570: 7545: 7523: 7323: 7317: 7277: 7273: 7208: 7202: 7140: 7132: 7107: 7099: 7074: 7052: 6764: 6756: 6731: 6723: 6698: 6676: 6054: 6048: 6023: 6001: 5976: 5970: 5809: 5803: 5780: 5758: 5552: 5546: 5521: 5499: 5474: 5468: 5248: 5242: 5192: 5186: 5161: 5139: 5098: 5089: 5083: 5071: 5052: 5034: 4985: 4975: 4961: 4948: 4608: 4602: 4562: 4558: 4493: 4487: 4415: 4411: 4384: 4380: 4371: 4367: 4210: 4204: 4181: 4159: 3914: 3904: 3840: 3830: 3807: 3801: 3778: 3756: 3595: 3589: 3566: 3544: 3166: 3158: 3133: 3125: 3100: 3078: 2436: 2428: 2403: 2395: 2370: 2348: 2006:. David another. My spreadsheet 1640: 1632: 1607: 1599: 1574: 1552: 1213: 1205: 1180: 1172: 1147: 1125: 1068: 1060: 1035: 1027: 1002: 980: 570: 562: 537: 529: 504: 482: 118:Template:WikiProject Mathematics 82: 72: 51: 20: 6169:takes it is a polynomial in {(a 5453:But you've only asserted that: 4080:satisfied by the components of 2495:Or at least, identical for all 2004:Seven-dimensional cross product 1531:seven dimensional cross product 1099:seven dimensional cross product 919:seven dimensional cross product 673:seven dimensional cross product 135:This article has been rated as 7993: 7946: 7879: 7862: 7824: 7777: 7711: 7696: 7683: 7672: 7661: 7650: 7611: 7594: 7551: 7540: 7529: 7518: 7329: 7312: 7146: 7127: 7113: 7094: 7080: 7069: 7058: 7047: 6961: 6823: 6770: 6751: 6737: 6718: 6704: 6693: 6682: 6671: 6635: 6588: 6285: 6238: 6148:Again, the use of the symbol ε 6029: 6018: 6007: 5996: 5982: 5965: 5927: 5880: 5814: 5799: 5786: 5775: 5764: 5753: 5527: 5516: 5505: 5494: 5480: 5463: 5368: 5321: 5254: 5237: 5167: 5156: 5145: 5134: 5058: 5047: 4614: 4597: 4328: 4281: 4215: 4200: 4187: 4176: 4165: 4154: 4034: 3987: 3920: 3899: 3846: 3825: 3812: 3797: 3784: 3773: 3762: 3751: 3713: 3666: 3600: 3585: 3572: 3561: 3550: 3539: 3499: 3452: 3172: 3153: 3139: 3120: 3106: 3095: 3084: 3073: 2614: 2602: 2596: 2584: 2572: 2559: 2553: 2541: 2538: 2526: 2442: 2423: 2409: 2390: 2376: 2365: 2354: 2343: 2301: 2254: 1967: 1829: 1646: 1627: 1613: 1594: 1580: 1569: 1558: 1547: 1428: 1290: 1219: 1200: 1186: 1167: 1153: 1142: 1131: 1120: 1074: 1055: 1041: 1022: 1008: 997: 986: 975: 576: 557: 543: 524: 510: 499: 488: 477: 441: 394: 1: 6156:to the 3D Levi-Cevita symbol. 109:and see a list of open tasks. 8155:C-Class mathematics articles 8141:08:31, 25 January 2021 (UTC) 7496:in 3-D, but that defined by 8176: 8092:15:05, 23 April 2010 (UTC) 7301:or, for the norm squared: 7187:03:48, 23 April 2010 (UTC) 6344:23:59, 22 April 2010 (UTC) 6140:23:01, 22 April 2010 (UTC) 6118:22:39, 22 April 2010 (UTC) 6099:22:09, 22 April 2010 (UTC) 5435:21:52, 22 April 2010 (UTC) 5405:21:33, 22 April 2010 (UTC) 4131:18:36, 22 April 2010 (UTC) 4105:17:56, 22 April 2010 (UTC) 4069:, the two are different. 3205:16:15, 22 April 2010 (UTC) 2663:16:26, 22 April 2010 (UTC) 2513:16:01, 22 April 2010 (UTC) 2491:15:57, 22 April 2010 (UTC) 1996:15:26, 22 April 2010 (UTC) 1730:14:50, 22 April 2010 (UTC) 1711:11:18, 22 April 2010 (UTC) 1679:11:04, 22 April 2010 (UTC) 1669:hold in seven dimensions? 1524:07:45, 22 April 2010 (UTC) 1256:00:08, 22 April 2010 (UTC) 955:17:38, 21 April 2010 (UTC) 931:14:57, 21 April 2010 (UTC) 909:13:09, 21 April 2010 (UTC) 891:07:42, 21 April 2010 (UTC) 134: 67: 46: 3220:Lagrange's identity is: 141:project's priority scale 6177:)}. Likewise, the sum : 4095:Is that your position? 4078:two distinct identities 98:WikiProject Mathematics 8011: 7945: 7918: 7842: 7776: 7749: 7630: 7483: 7292: 7160: 7027: 7000: 6968: 6784: 6651: 6587: 6560: 6496: 6443: 6393: 6309:is a polynomial in {(a 6303: 6237: 6210: 6077: 5945: 5879: 5852: 5730: 5636: 5575: 5386: 5320: 5293: 5215: 5116: 5007: 4930: 4765: 4577: 4466: 4430: 4346: 4280: 4253: 4052: 3986: 3959: 3865: 3731: 3665: 3638: 3515: 3451: 3424: 3360: 3307: 3257: 3186: 2624: 2461: 2317: 2253: 2226: 2162: 2109: 2059: 1974: 1660: 1435: 1233: 1088: 590: 457: 393: 366: 302: 249: 199: 28:This article is rated 8012: 7919: 7892: 7843: 7750: 7723: 7631: 7484: 7293: 7161: 7028: 7026:{\displaystyle z_{7}} 7001: 6999:{\displaystyle z_{1}} 6969: 6785: 6652: 6561: 6534: 6476: 6423: 6373: 6304: 6211: 6184: 6078: 5946: 5853: 5826: 5731: 5637: 5576: 5387: 5294: 5267: 5216: 5117: 5008: 4931: 4766: 4578: 4467: 4431: 4347: 4254: 4227: 4053: 3960: 3933: 3866: 3732: 3639: 3612: 3516: 3425: 3398: 3340: 3287: 3237: 3187: 2625: 2462: 2318: 2227: 2200: 2142: 2089: 2039: 1975: 1661: 1452:(copied from above): 1436: 1234: 1089: 591: 458: 367: 340: 282: 229: 179: 7858: 7646: 7514: 7308: 7198: 7175:Pythagorean identity 7043: 7010: 6983: 6807: 6667: 6363: 6181: 5961: 5749: 5652: 5591: 5459: 5233: 5127: 5027: 4941: 4780: 4593: 4483: 4443: 4362: 4150: 4109:BTW, my spreadsheet 4076:= 7, then there are 3895: 3747: 3535: 3227: 3069: 2523: 2339: 2029: 1813: 1543: 1274: 1244:Pythagorean identity 1116: 1107:Pythagorean identity 1103:Pythagorean identity 971: 937:Pythagorean identity 473: 169: 121:mathematics articles 8061:linear combinations 6458: 6408: 3322: 3272: 2124: 2074: 1536:Does the equation, 264: 214: 8007: 7838: 7626: 7494:Levi-Civita symbol 7479: 7288: 7156: 7023: 6996: 6964: 6780: 6647: 6444: 6394: 6299: 6073: 5941: 5726: 5632: 5571: 5382: 5211: 5112: 5003: 4926: 4761: 4573: 4462: 4426: 4342: 4048: 3861: 3727: 3524:or, equivalently, 3511: 3308: 3258: 3210:Summary of 7D case 3182: 2620: 2457: 2313: 2110: 2060: 1970: 1656: 1431: 1229: 1109:for the equation, 1084: 586: 453: 250: 200: 90:Mathematics portal 34:content assessment 8123: 8114:comment added by 8003: 7834: 7622: 7475: 7284: 7171:Lagrange identity 6295: 6133: 6092: 6069: 5937: 5724: 5721: 5687: 5630: 5567: 5378: 5207: 5108: 5069: 4999: 4984: 4979: 4965: 4760: 4569: 4422: 4338: 4061:If the answer is 4044: 3913: 3908: 3857: 3839: 3834: 3723: 2656: 2453: 1989: 1704: 1517: 948: 943:the same thing.-- 902: 155: 154: 151: 150: 147: 146: 8167: 8127:j not equal to i 8016: 8014: 8013: 8008: 8002: 8001: 8000: 7991: 7990: 7981: 7980: 7968: 7967: 7958: 7957: 7944: 7939: 7917: 7906: 7888: 7887: 7882: 7876: 7865: 7847: 7845: 7844: 7839: 7833: 7832: 7831: 7822: 7821: 7812: 7811: 7799: 7798: 7789: 7788: 7775: 7770: 7748: 7737: 7719: 7718: 7709: 7692: 7691: 7686: 7680: 7675: 7670: 7669: 7664: 7658: 7653: 7635: 7633: 7632: 7627: 7621: 7620: 7619: 7614: 7608: 7597: 7589: 7588: 7583: 7579: 7560: 7559: 7554: 7548: 7543: 7538: 7537: 7532: 7526: 7521: 7488: 7486: 7485: 7480: 7474: 7473: 7469: 7468: 7467: 7452: 7451: 7442: 7441: 7432: 7431: 7422: 7421: 7412: 7411: 7396: 7395: 7386: 7385: 7376: 7375: 7366: 7365: 7351: 7350: 7338: 7337: 7332: 7326: 7315: 7297: 7295: 7294: 7289: 7283: 7282: 7281: 7280: 7270: 7269: 7254: 7253: 7244: 7243: 7234: 7233: 7224: 7223: 7211: 7165: 7163: 7162: 7157: 7155: 7154: 7149: 7143: 7135: 7130: 7122: 7121: 7116: 7110: 7102: 7097: 7089: 7088: 7083: 7077: 7072: 7067: 7066: 7061: 7055: 7050: 7032: 7030: 7029: 7024: 7022: 7021: 7005: 7003: 7002: 6997: 6995: 6994: 6973: 6971: 6970: 6965: 6960: 6959: 6950: 6949: 6937: 6936: 6927: 6926: 6914: 6913: 6904: 6903: 6891: 6890: 6881: 6880: 6868: 6867: 6858: 6857: 6845: 6844: 6835: 6834: 6819: 6818: 6789: 6787: 6786: 6781: 6779: 6778: 6773: 6767: 6759: 6754: 6746: 6745: 6740: 6734: 6726: 6721: 6713: 6712: 6707: 6701: 6696: 6691: 6690: 6685: 6679: 6674: 6656: 6654: 6653: 6648: 6643: 6642: 6633: 6632: 6623: 6622: 6610: 6609: 6600: 6599: 6586: 6581: 6559: 6548: 6530: 6529: 6524: 6523: 6516: 6515: 6506: 6505: 6495: 6490: 6475: 6474: 6465: 6464: 6457: 6452: 6442: 6437: 6422: 6421: 6415: 6414: 6407: 6402: 6392: 6387: 6372: 6371: 6308: 6306: 6305: 6300: 6294: 6293: 6292: 6283: 6282: 6273: 6272: 6260: 6259: 6250: 6249: 6236: 6231: 6209: 6198: 6129: 6088: 6082: 6080: 6079: 6074: 6068: 6067: 6066: 6061: 6057: 6038: 6037: 6032: 6026: 6021: 6016: 6015: 6010: 6004: 5999: 5991: 5990: 5985: 5979: 5968: 5950: 5948: 5947: 5942: 5936: 5935: 5934: 5925: 5924: 5915: 5914: 5902: 5901: 5892: 5891: 5878: 5873: 5851: 5840: 5822: 5821: 5812: 5795: 5794: 5789: 5783: 5778: 5773: 5772: 5767: 5761: 5756: 5735: 5733: 5732: 5727: 5725: 5723: 5722: 5714: 5712: 5711: 5689: 5688: 5680: 5668: 5641: 5639: 5638: 5633: 5631: 5629: 5618: 5607: 5580: 5578: 5577: 5572: 5566: 5565: 5564: 5559: 5555: 5536: 5535: 5530: 5524: 5519: 5514: 5513: 5508: 5502: 5497: 5489: 5488: 5483: 5477: 5466: 5391: 5389: 5388: 5383: 5377: 5376: 5375: 5366: 5365: 5356: 5355: 5343: 5342: 5333: 5332: 5319: 5314: 5292: 5281: 5263: 5262: 5257: 5251: 5240: 5220: 5218: 5217: 5212: 5206: 5205: 5204: 5199: 5195: 5176: 5175: 5170: 5164: 5159: 5154: 5153: 5148: 5142: 5137: 5121: 5119: 5118: 5113: 5107: 5106: 5102: 5101: 5096: 5092: 5074: 5068: 5067: 5066: 5061: 5055: 5050: 5040: 5012: 5010: 5009: 5004: 4998: 4997: 4993: 4992: 4988: 4969: 4954: 4935: 4933: 4932: 4927: 4925: 4921: 4917: 4916: 4915: 4906: 4905: 4896: 4895: 4880: 4879: 4870: 4869: 4860: 4859: 4850: 4849: 4834: 4833: 4824: 4823: 4805: 4804: 4795: 4794: 4770: 4768: 4767: 4762: 4759: 4758: 4754: 4753: 4752: 4737: 4736: 4727: 4726: 4717: 4716: 4707: 4706: 4697: 4696: 4681: 4680: 4671: 4670: 4661: 4660: 4651: 4650: 4636: 4635: 4623: 4622: 4617: 4611: 4600: 4582: 4580: 4579: 4574: 4568: 4567: 4566: 4565: 4555: 4554: 4539: 4538: 4529: 4528: 4519: 4518: 4509: 4508: 4496: 4471: 4469: 4468: 4463: 4461: 4460: 4435: 4433: 4432: 4427: 4421: 4420: 4419: 4418: 4408: 4407: 4389: 4388: 4387: 4375: 4374: 4351: 4349: 4348: 4343: 4337: 4336: 4335: 4326: 4325: 4316: 4315: 4303: 4302: 4293: 4292: 4279: 4274: 4252: 4241: 4223: 4222: 4213: 4196: 4195: 4190: 4184: 4179: 4174: 4173: 4168: 4162: 4157: 4057: 4055: 4054: 4049: 4043: 4042: 4041: 4032: 4031: 4022: 4021: 4009: 4008: 3999: 3998: 3985: 3980: 3958: 3947: 3929: 3928: 3923: 3917: 3902: 3870: 3868: 3867: 3862: 3856: 3855: 3854: 3849: 3843: 3828: 3820: 3819: 3810: 3793: 3792: 3787: 3781: 3776: 3771: 3770: 3765: 3759: 3754: 3736: 3734: 3733: 3728: 3722: 3721: 3720: 3711: 3710: 3701: 3700: 3688: 3687: 3678: 3677: 3664: 3659: 3637: 3626: 3608: 3607: 3598: 3581: 3580: 3575: 3569: 3564: 3559: 3558: 3553: 3547: 3542: 3520: 3518: 3517: 3512: 3507: 3506: 3497: 3496: 3487: 3486: 3474: 3473: 3464: 3463: 3450: 3445: 3423: 3412: 3394: 3393: 3388: 3387: 3380: 3379: 3370: 3369: 3359: 3354: 3339: 3338: 3329: 3328: 3321: 3316: 3306: 3301: 3286: 3285: 3279: 3278: 3271: 3266: 3256: 3251: 3236: 3235: 3191: 3189: 3188: 3183: 3181: 3180: 3175: 3169: 3161: 3156: 3148: 3147: 3142: 3136: 3128: 3123: 3115: 3114: 3109: 3103: 3098: 3093: 3092: 3087: 3081: 3076: 2652: 2629: 2627: 2626: 2621: 2580: 2579: 2466: 2464: 2463: 2458: 2452: 2451: 2450: 2445: 2439: 2431: 2426: 2418: 2417: 2412: 2406: 2398: 2393: 2385: 2384: 2379: 2373: 2368: 2363: 2362: 2357: 2351: 2346: 2322: 2320: 2319: 2314: 2309: 2308: 2299: 2298: 2289: 2288: 2276: 2275: 2266: 2265: 2252: 2247: 2225: 2214: 2196: 2195: 2190: 2189: 2182: 2181: 2172: 2171: 2161: 2156: 2141: 2140: 2131: 2130: 2123: 2118: 2108: 2103: 2088: 2087: 2081: 2080: 2073: 2068: 2058: 2053: 2038: 2037: 1985: 1979: 1977: 1976: 1971: 1966: 1965: 1956: 1955: 1943: 1942: 1933: 1932: 1920: 1919: 1910: 1909: 1897: 1896: 1887: 1886: 1874: 1873: 1864: 1863: 1851: 1850: 1841: 1840: 1825: 1824: 1700: 1665: 1663: 1662: 1657: 1655: 1654: 1649: 1643: 1635: 1630: 1622: 1621: 1616: 1610: 1602: 1597: 1589: 1588: 1583: 1577: 1572: 1567: 1566: 1561: 1555: 1550: 1513: 1440: 1438: 1437: 1432: 1427: 1426: 1417: 1416: 1404: 1403: 1394: 1393: 1381: 1380: 1371: 1370: 1358: 1357: 1348: 1347: 1335: 1334: 1325: 1324: 1312: 1311: 1302: 1301: 1286: 1285: 1238: 1236: 1235: 1230: 1228: 1227: 1222: 1216: 1208: 1203: 1195: 1194: 1189: 1183: 1175: 1170: 1162: 1161: 1156: 1150: 1145: 1140: 1139: 1134: 1128: 1123: 1093: 1091: 1090: 1085: 1083: 1082: 1077: 1071: 1063: 1058: 1050: 1049: 1044: 1038: 1030: 1025: 1017: 1016: 1011: 1005: 1000: 995: 994: 989: 983: 978: 944: 898: 595: 593: 592: 587: 585: 584: 579: 573: 565: 560: 552: 551: 546: 540: 532: 527: 519: 518: 513: 507: 502: 497: 496: 491: 485: 480: 462: 460: 459: 454: 449: 448: 439: 438: 429: 428: 416: 415: 406: 405: 392: 387: 365: 354: 336: 335: 330: 329: 322: 321: 312: 311: 301: 296: 281: 280: 271: 270: 263: 258: 248: 243: 228: 227: 221: 220: 213: 208: 198: 193: 178: 177: 123: 122: 119: 116: 113: 92: 87: 86: 76: 69: 68: 63: 55: 48: 31: 25: 24: 16: 8175: 8174: 8170: 8169: 8168: 8166: 8165: 8164: 8145: 8144: 8129: 8099: 8078: 8074: 8070: 8066: 7992: 7982: 7972: 7959: 7949: 7877: 7856: 7855: 7823: 7813: 7803: 7790: 7780: 7710: 7681: 7659: 7644: 7643: 7609: 7565: 7564: 7549: 7527: 7512: 7511: 7453: 7443: 7433: 7423: 7413: 7397: 7387: 7377: 7367: 7357: 7356: 7352: 7342: 7327: 7306: 7305: 7272: 7255: 7245: 7235: 7225: 7215: 7196: 7195: 7144: 7111: 7078: 7056: 7041: 7040: 7013: 7008: 7007: 6986: 6981: 6980: 6951: 6941: 6928: 6918: 6905: 6895: 6882: 6872: 6859: 6849: 6836: 6826: 6810: 6805: 6804: 6768: 6735: 6702: 6680: 6665: 6664: 6634: 6624: 6614: 6601: 6591: 6517: 6507: 6497: 6361: 6360: 6354: 6324: 6320: 6316: 6312: 6284: 6274: 6264: 6251: 6241: 6179: 6178: 6176: 6172: 6151: 6138: 6097: 6043: 6042: 6027: 6005: 5980: 5959: 5958: 5926: 5916: 5906: 5893: 5883: 5813: 5784: 5762: 5747: 5746: 5703: 5690: 5669: 5650: 5649: 5619: 5608: 5589: 5588: 5541: 5540: 5525: 5503: 5478: 5457: 5456: 5448: 5424: 5420: 5416: 5412: 5367: 5357: 5347: 5334: 5324: 5252: 5231: 5230: 5181: 5180: 5165: 5143: 5125: 5124: 5078: 5056: 5045: 5041: 5025: 5024: 4970: 4959: 4955: 4939: 4938: 4907: 4897: 4881: 4871: 4861: 4851: 4835: 4825: 4815: 4814: 4810: 4806: 4796: 4786: 4778: 4777: 4738: 4728: 4718: 4708: 4698: 4682: 4672: 4662: 4652: 4642: 4641: 4637: 4627: 4612: 4591: 4590: 4557: 4540: 4530: 4520: 4510: 4500: 4481: 4480: 4446: 4441: 4440: 4410: 4393: 4379: 4366: 4360: 4359: 4327: 4317: 4307: 4294: 4284: 4214: 4185: 4163: 4148: 4147: 4138: 4033: 4023: 4013: 4000: 3990: 3918: 3893: 3892: 3844: 3811: 3782: 3760: 3745: 3744: 3712: 3702: 3692: 3679: 3669: 3599: 3570: 3548: 3533: 3532: 3498: 3488: 3478: 3465: 3455: 3381: 3371: 3361: 3225: 3224: 3212: 3170: 3137: 3104: 3082: 3067: 3066: 3056: 3053: 3046: 3042: 3038: 3034: 3030: 3026: 3022: 3018: 3014: 3010: 3006: 3002: 2998: 2994: 2990: 2986: 2982: 2978: 2974: 2970: 2966: 2962: 2958: 2954: 2950: 2946: 2942: 2938: 2934: 2930: 2926: 2922: 2918: 2914: 2910: 2906: 2902: 2898: 2894: 2890: 2886: 2882: 2878: 2874: 2870: 2866: 2862: 2858: 2854: 2850: 2846: 2842: 2838: 2834: 2830: 2826: 2822: 2818: 2814: 2810: 2806: 2802: 2798: 2794: 2790: 2786: 2782: 2778: 2774: 2770: 2766: 2762: 2758: 2754: 2750: 2746: 2742: 2738: 2734: 2730: 2726: 2722: 2718: 2714: 2710: 2706: 2702: 2698: 2694: 2690: 2686: 2679: 2661: 2571: 2521: 2520: 2440: 2407: 2374: 2352: 2337: 2336: 2300: 2290: 2280: 2267: 2257: 2183: 2173: 2163: 2027: 2026: 1994: 1957: 1947: 1934: 1924: 1911: 1901: 1888: 1878: 1865: 1855: 1842: 1832: 1816: 1811: 1810: 1806: 1794: 1790: 1786: 1782: 1778: 1774: 1770: 1766: 1762: 1758: 1754: 1750: 1746: 1709: 1698: 1691: 1644: 1611: 1578: 1556: 1541: 1540: 1522: 1506: 1502: 1498: 1494: 1490: 1486: 1482: 1478: 1474: 1470: 1466: 1462: 1458: 1451: 1418: 1408: 1395: 1385: 1372: 1362: 1349: 1339: 1326: 1316: 1303: 1293: 1277: 1272: 1271: 1267: 1217: 1184: 1151: 1129: 1114: 1113: 1072: 1039: 1006: 984: 969: 968: 953: 907: 670: 666: 656: 652: 648: 644: 640: 636: 632: 628: 624: 620: 616: 612: 608: 574: 541: 508: 486: 471: 470: 440: 430: 420: 407: 397: 323: 313: 303: 167: 166: 160: 120: 117: 114: 111: 110: 88: 81: 61: 32:on Knowledge's 29: 12: 11: 5: 8173: 8171: 8163: 8162: 8157: 8147: 8146: 8128: 8125: 8098: 8095: 8076: 8072: 8068: 8064: 8018: 8017: 8006: 7999: 7995: 7989: 7985: 7979: 7975: 7971: 7966: 7962: 7956: 7952: 7948: 7943: 7938: 7935: 7932: 7929: 7926: 7922: 7916: 7913: 7910: 7905: 7902: 7899: 7895: 7891: 7886: 7881: 7875: 7872: 7869: 7864: 7851:do not imply: 7849: 7848: 7837: 7830: 7826: 7820: 7816: 7810: 7806: 7802: 7797: 7793: 7787: 7783: 7779: 7774: 7769: 7766: 7763: 7760: 7757: 7753: 7747: 7744: 7741: 7736: 7733: 7730: 7726: 7722: 7717: 7713: 7708: 7705: 7702: 7698: 7695: 7690: 7685: 7679: 7674: 7668: 7663: 7657: 7652: 7637: 7636: 7625: 7618: 7613: 7607: 7604: 7601: 7596: 7592: 7587: 7582: 7578: 7575: 7572: 7568: 7563: 7558: 7553: 7547: 7542: 7536: 7531: 7525: 7520: 7490: 7489: 7478: 7472: 7466: 7463: 7460: 7456: 7450: 7446: 7440: 7436: 7430: 7426: 7420: 7416: 7410: 7407: 7404: 7400: 7394: 7390: 7384: 7380: 7374: 7370: 7364: 7360: 7355: 7349: 7345: 7341: 7336: 7331: 7325: 7322: 7319: 7314: 7299: 7298: 7287: 7279: 7275: 7268: 7265: 7262: 7258: 7252: 7248: 7242: 7238: 7232: 7228: 7222: 7218: 7214: 7210: 7207: 7204: 7167: 7166: 7153: 7148: 7142: 7138: 7134: 7129: 7125: 7120: 7115: 7109: 7105: 7101: 7096: 7092: 7087: 7082: 7076: 7071: 7065: 7060: 7054: 7049: 7020: 7016: 6993: 6989: 6977: 6976: 6975: 6974: 6963: 6958: 6954: 6948: 6944: 6940: 6935: 6931: 6925: 6921: 6917: 6912: 6908: 6902: 6898: 6894: 6889: 6885: 6879: 6875: 6871: 6866: 6862: 6856: 6852: 6848: 6843: 6839: 6833: 6829: 6825: 6822: 6817: 6813: 6791: 6790: 6777: 6772: 6766: 6762: 6758: 6753: 6749: 6744: 6739: 6733: 6729: 6725: 6720: 6716: 6711: 6706: 6700: 6695: 6689: 6684: 6678: 6673: 6658: 6657: 6646: 6641: 6637: 6631: 6627: 6621: 6617: 6613: 6608: 6604: 6598: 6594: 6590: 6585: 6580: 6577: 6574: 6571: 6568: 6564: 6558: 6555: 6552: 6547: 6544: 6541: 6537: 6533: 6528: 6522: 6514: 6510: 6504: 6500: 6494: 6489: 6486: 6483: 6479: 6473: 6468: 6463: 6456: 6451: 6447: 6441: 6436: 6433: 6430: 6426: 6420: 6413: 6406: 6401: 6397: 6391: 6386: 6383: 6380: 6376: 6370: 6353: 6350: 6349: 6348: 6347: 6346: 6329: 6328: 6327: 6326: 6322: 6318: 6314: 6310: 6298: 6291: 6287: 6281: 6277: 6271: 6267: 6263: 6258: 6254: 6248: 6244: 6240: 6235: 6230: 6227: 6224: 6221: 6218: 6214: 6208: 6205: 6202: 6197: 6194: 6191: 6187: 6174: 6170: 6165:Whatever form 6160: 6159: 6158: 6157: 6154:does not refer 6149: 6143: 6142: 6134: 6131:JohnBlackburne 6102: 6101: 6093: 6090:JohnBlackburne 6085: 6084: 6083: 6072: 6065: 6060: 6056: 6053: 6050: 6046: 6041: 6036: 6031: 6025: 6020: 6014: 6009: 6003: 5998: 5994: 5989: 5984: 5978: 5975: 5972: 5967: 5953: 5952: 5951: 5940: 5933: 5929: 5923: 5919: 5913: 5909: 5905: 5900: 5896: 5890: 5886: 5882: 5877: 5872: 5869: 5866: 5863: 5860: 5856: 5850: 5847: 5844: 5839: 5836: 5833: 5829: 5825: 5820: 5816: 5811: 5808: 5805: 5801: 5798: 5793: 5788: 5782: 5777: 5771: 5766: 5760: 5755: 5741: 5738: 5737: 5736: 5720: 5717: 5710: 5706: 5702: 5699: 5696: 5693: 5686: 5683: 5678: 5675: 5672: 5666: 5663: 5660: 5657: 5644: 5643: 5642: 5628: 5625: 5622: 5617: 5614: 5611: 5605: 5602: 5599: 5596: 5583: 5582: 5581: 5570: 5563: 5558: 5554: 5551: 5548: 5544: 5539: 5534: 5529: 5523: 5518: 5512: 5507: 5501: 5496: 5492: 5487: 5482: 5476: 5473: 5470: 5465: 5451: 5444: 5422: 5418: 5414: 5410: 5393: 5392: 5381: 5374: 5370: 5364: 5360: 5354: 5350: 5346: 5341: 5337: 5331: 5327: 5323: 5318: 5313: 5310: 5307: 5304: 5301: 5297: 5291: 5288: 5285: 5280: 5277: 5274: 5270: 5266: 5261: 5256: 5250: 5247: 5244: 5239: 5224: 5223: 5222: 5221: 5210: 5203: 5198: 5194: 5191: 5188: 5184: 5179: 5174: 5169: 5163: 5158: 5152: 5147: 5141: 5136: 5132: 5122: 5111: 5105: 5100: 5095: 5091: 5088: 5085: 5081: 5077: 5073: 5065: 5060: 5054: 5049: 5044: 5039: 5036: 5032: 5016: 5015: 5014: 5013: 5002: 4996: 4991: 4987: 4982: 4977: 4973: 4968: 4963: 4958: 4953: 4950: 4946: 4936: 4924: 4920: 4914: 4910: 4904: 4900: 4894: 4891: 4888: 4884: 4878: 4874: 4868: 4864: 4858: 4854: 4848: 4845: 4842: 4838: 4832: 4828: 4822: 4818: 4813: 4809: 4803: 4799: 4793: 4789: 4785: 4772: 4771: 4757: 4751: 4748: 4745: 4741: 4735: 4731: 4725: 4721: 4715: 4711: 4705: 4701: 4695: 4692: 4689: 4685: 4679: 4675: 4669: 4665: 4659: 4655: 4649: 4645: 4640: 4634: 4630: 4626: 4621: 4616: 4610: 4607: 4604: 4599: 4584: 4583: 4572: 4564: 4560: 4553: 4550: 4547: 4543: 4537: 4533: 4527: 4523: 4517: 4513: 4507: 4503: 4499: 4495: 4492: 4489: 4459: 4456: 4453: 4449: 4437: 4436: 4425: 4417: 4413: 4406: 4403: 4400: 4396: 4392: 4386: 4382: 4378: 4373: 4369: 4353: 4352: 4341: 4334: 4330: 4324: 4320: 4314: 4310: 4306: 4301: 4297: 4291: 4287: 4283: 4278: 4273: 4270: 4267: 4264: 4261: 4257: 4251: 4248: 4245: 4240: 4237: 4234: 4230: 4226: 4221: 4217: 4212: 4209: 4206: 4202: 4199: 4194: 4189: 4183: 4178: 4172: 4167: 4161: 4156: 4137: 4134: 4059: 4058: 4047: 4040: 4036: 4030: 4026: 4020: 4016: 4012: 4007: 4003: 3997: 3993: 3989: 3984: 3979: 3976: 3973: 3970: 3967: 3963: 3957: 3954: 3951: 3946: 3943: 3940: 3936: 3932: 3927: 3922: 3916: 3911: 3906: 3901: 3872: 3871: 3860: 3853: 3848: 3842: 3837: 3832: 3827: 3823: 3818: 3814: 3809: 3806: 3803: 3799: 3796: 3791: 3786: 3780: 3775: 3769: 3764: 3758: 3753: 3738: 3737: 3726: 3719: 3715: 3709: 3705: 3699: 3695: 3691: 3686: 3682: 3676: 3672: 3668: 3663: 3658: 3655: 3652: 3649: 3646: 3642: 3636: 3633: 3630: 3625: 3622: 3619: 3615: 3611: 3606: 3602: 3597: 3594: 3591: 3587: 3584: 3579: 3574: 3568: 3563: 3557: 3552: 3546: 3541: 3522: 3521: 3510: 3505: 3501: 3495: 3491: 3485: 3481: 3477: 3472: 3468: 3462: 3458: 3454: 3449: 3444: 3441: 3438: 3435: 3432: 3428: 3422: 3419: 3416: 3411: 3408: 3405: 3401: 3397: 3392: 3386: 3378: 3374: 3368: 3364: 3358: 3353: 3350: 3347: 3343: 3337: 3332: 3327: 3320: 3315: 3311: 3305: 3300: 3297: 3294: 3290: 3284: 3277: 3270: 3265: 3261: 3255: 3250: 3247: 3244: 3240: 3234: 3215:JohnBlackburne 3211: 3208: 3193: 3192: 3179: 3174: 3168: 3164: 3160: 3155: 3151: 3146: 3141: 3135: 3131: 3127: 3122: 3118: 3113: 3108: 3102: 3097: 3091: 3086: 3080: 3075: 3055: 3052: 3044: 3040: 3036: 3032: 3028: 3024: 3020: 3016: 3012: 3008: 3004: 3000: 2996: 2992: 2988: 2984: 2980: 2976: 2972: 2968: 2964: 2960: 2956: 2952: 2948: 2944: 2940: 2936: 2932: 2928: 2924: 2920: 2916: 2912: 2908: 2904: 2900: 2896: 2892: 2888: 2884: 2880: 2876: 2872: 2868: 2864: 2860: 2856: 2852: 2848: 2844: 2840: 2836: 2832: 2828: 2824: 2820: 2816: 2812: 2808: 2804: 2800: 2796: 2792: 2788: 2784: 2780: 2776: 2772: 2768: 2764: 2760: 2756: 2752: 2748: 2744: 2740: 2736: 2732: 2728: 2724: 2720: 2716: 2712: 2708: 2704: 2700: 2696: 2692: 2688: 2684: 2678: 2668: 2667: 2666: 2665: 2657: 2654:JohnBlackburne 2632: 2631: 2630: 2619: 2616: 2613: 2610: 2607: 2604: 2601: 2598: 2595: 2592: 2589: 2586: 2583: 2578: 2574: 2570: 2567: 2564: 2561: 2558: 2555: 2552: 2549: 2546: 2543: 2540: 2537: 2534: 2531: 2528: 2493: 2470: 2469: 2468: 2467: 2456: 2449: 2444: 2438: 2434: 2430: 2425: 2421: 2416: 2411: 2405: 2401: 2397: 2392: 2388: 2383: 2378: 2372: 2367: 2361: 2356: 2350: 2345: 2331: 2330: 2326: 2325: 2324: 2323: 2312: 2307: 2303: 2297: 2293: 2287: 2283: 2279: 2274: 2270: 2264: 2260: 2256: 2251: 2246: 2243: 2240: 2237: 2234: 2230: 2224: 2221: 2218: 2213: 2210: 2207: 2203: 2199: 2194: 2188: 2180: 2176: 2170: 2166: 2160: 2155: 2152: 2149: 2145: 2139: 2134: 2129: 2122: 2117: 2113: 2107: 2102: 2099: 2096: 2092: 2086: 2079: 2072: 2067: 2063: 2057: 2052: 2049: 2046: 2042: 2036: 2021: 2020: 2016: 2015: 1999: 1998: 1990: 1987:JohnBlackburne 1982: 1981: 1980: 1969: 1964: 1960: 1954: 1950: 1946: 1941: 1937: 1931: 1927: 1923: 1918: 1914: 1908: 1904: 1900: 1895: 1891: 1885: 1881: 1877: 1872: 1868: 1862: 1858: 1854: 1849: 1845: 1839: 1835: 1831: 1828: 1823: 1819: 1804: 1798: 1797: 1796: 1792: 1788: 1784: 1780: 1776: 1772: 1768: 1764: 1760: 1756: 1752: 1748: 1744: 1714: 1713: 1705: 1702:JohnBlackburne 1696: 1689: 1667: 1666: 1653: 1648: 1642: 1638: 1634: 1629: 1625: 1620: 1615: 1609: 1605: 1601: 1596: 1592: 1587: 1582: 1576: 1571: 1565: 1560: 1554: 1549: 1527: 1526: 1518: 1515:JohnBlackburne 1510: 1509: 1508: 1504: 1500: 1496: 1492: 1488: 1484: 1480: 1476: 1472: 1468: 1464: 1460: 1456: 1449: 1443: 1442: 1441: 1430: 1425: 1421: 1415: 1411: 1407: 1402: 1398: 1392: 1388: 1384: 1379: 1375: 1369: 1365: 1361: 1356: 1352: 1346: 1342: 1338: 1333: 1329: 1323: 1319: 1315: 1310: 1306: 1300: 1296: 1292: 1289: 1284: 1280: 1265: 1240: 1239: 1226: 1221: 1215: 1211: 1207: 1202: 1198: 1193: 1188: 1182: 1178: 1174: 1169: 1165: 1160: 1155: 1149: 1144: 1138: 1133: 1127: 1122: 1095: 1094: 1081: 1076: 1070: 1066: 1062: 1057: 1053: 1048: 1043: 1037: 1033: 1029: 1024: 1020: 1015: 1010: 1004: 999: 993: 988: 982: 977: 962: 961: 960: 959: 958: 957: 949: 946:JohnBlackburne 912: 911: 903: 900:JohnBlackburne 668: 664: 661: 660: 659: 658: 654: 650: 646: 642: 638: 634: 630: 626: 622: 618: 614: 610: 606: 597: 596: 583: 578: 572: 568: 564: 559: 555: 550: 545: 539: 535: 531: 526: 522: 517: 512: 506: 501: 495: 490: 484: 479: 464: 463: 452: 447: 443: 437: 433: 427: 423: 419: 414: 410: 404: 400: 396: 391: 386: 383: 380: 377: 374: 370: 364: 361: 358: 353: 350: 347: 343: 339: 334: 328: 320: 316: 310: 306: 300: 295: 292: 289: 285: 279: 274: 269: 262: 257: 253: 247: 242: 239: 236: 232: 226: 219: 212: 207: 203: 197: 192: 189: 186: 182: 176: 159: 156: 153: 152: 149: 148: 145: 144: 133: 127: 126: 124: 107:the discussion 94: 93: 77: 65: 64: 56: 44: 43: 37: 26: 13: 10: 9: 6: 4: 3: 2: 8172: 8161: 8158: 8156: 8153: 8152: 8150: 8143: 8142: 8138: 8134: 8126: 8124: 8121: 8117: 8116:110.23.118.21 8113: 8106: 8103: 8096: 8094: 8093: 8089: 8085: 8080: 8062: 8058: 8054: 8050: 8046: 8043:and function 8042: 8038: 8034: 8031: 8027: 8023: 8004: 7997: 7987: 7983: 7977: 7973: 7969: 7964: 7960: 7954: 7950: 7941: 7936: 7933: 7930: 7927: 7924: 7920: 7914: 7911: 7908: 7903: 7900: 7897: 7893: 7889: 7884: 7870: 7854: 7853: 7852: 7835: 7828: 7818: 7814: 7808: 7804: 7800: 7795: 7791: 7785: 7781: 7772: 7767: 7764: 7761: 7758: 7755: 7751: 7745: 7742: 7739: 7734: 7731: 7728: 7724: 7720: 7715: 7703: 7693: 7688: 7666: 7642: 7641: 7640: 7623: 7616: 7602: 7590: 7585: 7580: 7573: 7566: 7561: 7556: 7534: 7510: 7509: 7508: 7505: 7503: 7499: 7495: 7476: 7470: 7464: 7461: 7458: 7454: 7448: 7444: 7438: 7434: 7428: 7418: 7408: 7405: 7402: 7398: 7392: 7388: 7382: 7378: 7372: 7362: 7353: 7347: 7339: 7334: 7320: 7304: 7303: 7302: 7285: 7266: 7263: 7260: 7256: 7250: 7246: 7240: 7236: 7230: 7220: 7212: 7205: 7194: 7193: 7192: 7189: 7188: 7184: 7180: 7176: 7172: 7151: 7136: 7123: 7118: 7103: 7090: 7085: 7063: 7039: 7038: 7037: 7034: 7018: 7014: 6991: 6987: 6956: 6952: 6946: 6942: 6938: 6933: 6929: 6923: 6919: 6915: 6910: 6906: 6900: 6896: 6892: 6887: 6883: 6877: 6873: 6869: 6864: 6860: 6854: 6850: 6846: 6841: 6837: 6831: 6827: 6820: 6815: 6811: 6803: 6802: 6801: 6800: 6799: 6796: 6795:cross product 6775: 6760: 6747: 6742: 6727: 6714: 6709: 6687: 6663: 6662: 6661: 6644: 6639: 6629: 6625: 6619: 6615: 6611: 6606: 6602: 6596: 6592: 6583: 6578: 6575: 6572: 6569: 6566: 6562: 6556: 6553: 6550: 6545: 6542: 6539: 6535: 6531: 6526: 6512: 6508: 6502: 6498: 6492: 6487: 6484: 6481: 6477: 6466: 6454: 6449: 6445: 6439: 6434: 6431: 6428: 6424: 6404: 6399: 6395: 6389: 6384: 6381: 6378: 6374: 6359: 6358: 6357: 6351: 6345: 6341: 6337: 6333: 6332: 6331: 6330: 6296: 6289: 6279: 6275: 6269: 6265: 6261: 6256: 6252: 6246: 6242: 6233: 6228: 6225: 6222: 6219: 6216: 6212: 6206: 6203: 6200: 6195: 6192: 6189: 6185: 6168: 6164: 6163: 6162: 6161: 6155: 6147: 6146: 6145: 6144: 6141: 6137: 6132: 6127: 6122: 6121: 6120: 6119: 6115: 6111: 6106: 6100: 6096: 6091: 6086: 6070: 6063: 6058: 6051: 6044: 6039: 6034: 6012: 5992: 5987: 5973: 5957: 5956: 5954: 5938: 5931: 5921: 5917: 5911: 5907: 5903: 5898: 5894: 5888: 5884: 5875: 5870: 5867: 5864: 5861: 5858: 5854: 5848: 5845: 5842: 5837: 5834: 5831: 5827: 5823: 5818: 5806: 5796: 5791: 5769: 5745: 5744: 5742: 5739: 5718: 5715: 5708: 5704: 5700: 5697: 5694: 5691: 5684: 5681: 5676: 5673: 5670: 5664: 5661: 5658: 5655: 5648: 5647: 5645: 5626: 5623: 5620: 5615: 5612: 5609: 5603: 5600: 5597: 5594: 5587: 5586: 5584: 5568: 5561: 5556: 5549: 5542: 5537: 5532: 5510: 5490: 5485: 5471: 5455: 5454: 5452: 5447: 5443: 5439: 5438: 5437: 5436: 5432: 5428: 5407: 5406: 5402: 5398: 5379: 5372: 5362: 5358: 5352: 5348: 5344: 5339: 5335: 5329: 5325: 5316: 5311: 5308: 5305: 5302: 5299: 5295: 5289: 5286: 5283: 5278: 5275: 5272: 5268: 5264: 5259: 5245: 5229: 5228: 5227: 5208: 5201: 5196: 5189: 5182: 5177: 5172: 5150: 5130: 5123: 5109: 5103: 5093: 5086: 5079: 5075: 5063: 5042: 5037: 5030: 5023: 5022: 5021: 5020: 5019: 5000: 4994: 4989: 4980: 4971: 4966: 4956: 4951: 4944: 4937: 4922: 4918: 4912: 4908: 4902: 4898: 4892: 4889: 4886: 4882: 4876: 4866: 4856: 4852: 4846: 4843: 4840: 4836: 4830: 4820: 4811: 4807: 4801: 4797: 4791: 4783: 4776: 4775: 4774: 4773: 4755: 4749: 4746: 4743: 4739: 4733: 4729: 4723: 4719: 4713: 4703: 4693: 4690: 4687: 4683: 4677: 4673: 4667: 4663: 4657: 4647: 4638: 4632: 4624: 4619: 4605: 4589: 4588: 4587: 4570: 4551: 4548: 4545: 4541: 4535: 4531: 4525: 4521: 4515: 4505: 4497: 4490: 4479: 4478: 4477: 4475: 4457: 4454: 4451: 4447: 4423: 4404: 4401: 4398: 4394: 4390: 4376: 4358: 4357: 4356: 4339: 4332: 4322: 4318: 4312: 4308: 4304: 4299: 4295: 4289: 4285: 4276: 4271: 4268: 4265: 4262: 4259: 4255: 4249: 4246: 4243: 4238: 4235: 4232: 4228: 4224: 4219: 4207: 4197: 4192: 4170: 4146: 4145: 4144: 4142: 4141:Jones, page 4 4135: 4133: 4132: 4128: 4124: 4120: 4116: 4112: 4107: 4106: 4102: 4098: 4093: 4091: 4087: 4083: 4079: 4075: 4070: 4068: 4064: 4045: 4038: 4028: 4024: 4018: 4014: 4010: 4005: 4001: 3995: 3991: 3982: 3977: 3974: 3971: 3968: 3965: 3961: 3955: 3952: 3949: 3944: 3941: 3938: 3934: 3930: 3925: 3909: 3891: 3890: 3889: 3886: 3883: 3881: 3877: 3858: 3851: 3835: 3821: 3816: 3804: 3794: 3789: 3767: 3743: 3742: 3741: 3724: 3717: 3707: 3703: 3697: 3693: 3689: 3684: 3680: 3674: 3670: 3661: 3656: 3653: 3650: 3647: 3644: 3640: 3634: 3631: 3628: 3623: 3620: 3617: 3613: 3609: 3604: 3592: 3582: 3577: 3555: 3531: 3530: 3529: 3527: 3526:Jones, page 4 3508: 3503: 3493: 3489: 3483: 3479: 3475: 3470: 3466: 3460: 3456: 3447: 3442: 3439: 3436: 3433: 3430: 3426: 3420: 3417: 3414: 3409: 3406: 3403: 3399: 3395: 3390: 3376: 3372: 3366: 3362: 3356: 3351: 3348: 3345: 3341: 3330: 3318: 3313: 3309: 3303: 3298: 3295: 3292: 3288: 3268: 3263: 3259: 3253: 3248: 3245: 3242: 3238: 3223: 3222: 3221: 3218: 3216: 3209: 3207: 3206: 3202: 3198: 3177: 3162: 3149: 3144: 3129: 3116: 3111: 3089: 3065: 3064: 3063: 3061: 3060:cross product 3048: 2681: 2674: 2672: 2664: 2660: 2655: 2649: 2645: 2641: 2637: 2633: 2617: 2611: 2608: 2605: 2599: 2593: 2590: 2587: 2581: 2576: 2568: 2565: 2562: 2556: 2550: 2547: 2544: 2535: 2532: 2529: 2519: 2518: 2516: 2515: 2514: 2510: 2506: 2502: 2498: 2494: 2492: 2488: 2484: 2480: 2476: 2472: 2471: 2454: 2447: 2432: 2419: 2414: 2399: 2386: 2381: 2359: 2335: 2334: 2333: 2332: 2328: 2327: 2310: 2305: 2295: 2291: 2285: 2281: 2277: 2272: 2268: 2262: 2258: 2249: 2244: 2241: 2238: 2235: 2232: 2228: 2222: 2219: 2216: 2211: 2208: 2205: 2201: 2197: 2192: 2178: 2174: 2168: 2164: 2158: 2153: 2150: 2147: 2143: 2132: 2120: 2115: 2111: 2105: 2100: 2097: 2094: 2090: 2070: 2065: 2061: 2055: 2050: 2047: 2044: 2040: 2025: 2024: 2023: 2022: 2018: 2017: 2013: 2009: 2005: 2001: 2000: 1997: 1993: 1988: 1983: 1962: 1958: 1952: 1948: 1944: 1939: 1935: 1929: 1925: 1921: 1916: 1912: 1906: 1902: 1898: 1893: 1889: 1883: 1879: 1875: 1870: 1866: 1860: 1856: 1852: 1847: 1843: 1837: 1833: 1826: 1821: 1817: 1809: 1808: 1803: 1799: 1742: 1741: 1738: 1734: 1733: 1732: 1731: 1727: 1723: 1719: 1712: 1708: 1703: 1695: 1688: 1683: 1682: 1681: 1680: 1676: 1672: 1651: 1636: 1623: 1618: 1603: 1590: 1585: 1563: 1539: 1538: 1537: 1534: 1532: 1525: 1521: 1516: 1511: 1454: 1453: 1448: 1444: 1423: 1419: 1413: 1409: 1405: 1400: 1396: 1390: 1386: 1382: 1377: 1373: 1367: 1363: 1359: 1354: 1350: 1344: 1340: 1336: 1331: 1327: 1321: 1317: 1313: 1308: 1304: 1298: 1294: 1287: 1282: 1278: 1270: 1269: 1264: 1260: 1259: 1258: 1257: 1253: 1249: 1245: 1224: 1209: 1196: 1191: 1176: 1163: 1158: 1136: 1112: 1111: 1110: 1108: 1104: 1100: 1079: 1064: 1051: 1046: 1031: 1018: 1013: 991: 967: 966: 965: 956: 952: 947: 942: 938: 934: 933: 932: 928: 924: 920: 916: 915: 914: 913: 910: 906: 901: 895: 894: 893: 892: 888: 884: 879: 878: 874: 870: 866: 862: 858: 854: 850: 849: 845: 841: 837: 833: 829: 825: 821: 820: 816: 812: 808: 804: 800: 796: 792: 791: 787: 783: 779: 775: 771: 767: 763: 762: 758: 754: 750: 746: 742: 738: 734: 733: 729: 725: 721: 717: 713: 709: 705: 704: 700: 696: 692: 688: 684: 680: 676: 674: 604: 603: 602: 601: 600: 581: 566: 553: 548: 533: 520: 515: 493: 469: 468: 467: 450: 445: 435: 431: 425: 421: 417: 412: 408: 402: 398: 389: 384: 381: 378: 375: 372: 368: 362: 359: 356: 351: 348: 345: 341: 337: 332: 318: 314: 308: 304: 298: 293: 290: 287: 283: 272: 260: 255: 251: 245: 240: 237: 234: 230: 210: 205: 201: 195: 190: 187: 184: 180: 165: 164: 163: 157: 142: 138: 132: 129: 128: 125: 108: 104: 100: 99: 91: 85: 80: 78: 75: 71: 70: 66: 60: 57: 54: 50: 45: 41: 35: 27: 23: 18: 17: 8130: 8110:— Preceding 8107: 8104: 8100: 8081: 8060: 8056: 8052: 8048: 8044: 8040: 8036: 8032: 8029: 8025: 8021: 8019: 7850: 7638: 7506: 7491: 7300: 7190: 7168: 7035: 6978: 6792: 6660:reduces to, 6659: 6355: 6166: 6153: 6107: 6103: 5445: 5441: 5417:as sums of δ 5408: 5394: 5225: 5017: 4585: 4473: 4438: 4354: 4139: 4136:Formal proof 4118: 4114: 4108: 4094: 4089: 4085: 4081: 4077: 4073: 4071: 4066: 4062: 4060: 3887: 3884: 3879: 3875: 3873: 3739: 3523: 3219: 3213: 3194: 3049: 2682: 2675: 2669: 2647: 2643: 2639: 2635: 2500: 2496: 2478: 2474: 1801: 1800:And here is 1736: 1717: 1715: 1693: 1686: 1668: 1535: 1528: 1446: 1262: 1241: 1096: 963: 940: 880: 876: 872: 868: 864: 860: 856: 852: 851: 847: 843: 839: 835: 831: 827: 823: 822: 818: 814: 810: 806: 802: 798: 794: 793: 789: 785: 781: 777: 773: 769: 765: 764: 760: 756: 752: 748: 744: 740: 736: 735: 731: 727: 723: 719: 715: 711: 707: 706: 702: 698: 694: 690: 686: 682: 678: 677: 662: 598: 466:reduces to, 465: 161: 137:Low-priority 136: 96: 62:Low‑priority 40:WikiProjects 8133:Staticgloat 8084:Brews ohare 7179:David Tombe 6336:Brews ohare 6110:Brews ohare 5427:Brews ohare 5397:Brews ohare 4123:Brews ohare 4097:Brews ohare 3197:David Tombe 2505:Brews ohare 2483:Brews ohare 1722:Brews ohare 1671:David Tombe 1248:David Tombe 923:David Tombe 883:David Tombe 112:Mathematics 103:mathematics 59:Mathematics 8149:Categories 6126:synthesis 935:Yes, the 8112:unsigned 7006:,- - - - 6979:for all 6321:} and {b 2481:in 7D. 2012:Octonion 1261:Here is 7173:or the 3878:= 3 or 3015:) + (x 2999:) + (x 2967:) + (x 2951:) + (x 2919:) + (x 2903:) + (x 2871:) + (x 2855:) + (x 2823:) + (x 2775:) + (x 2727:) + (x 1763:) + (x 1475:) + (x 667:----- z 625:) + (x 139:on the 30:C-class 8035:& 8024:& 4439:where 4092:= 7. 3031:) + (x 2983:) + (x 2935:) + (x 2887:) + (x 2839:) + (x 2807:) + (x 2791:) + (x 2759:) + (x 2743:) + (x 2634:In 3D 1779:) + (x 1491:) + (x 871:, and 842:, and 813:, and 784:, and 755:, and 726:, and 697:, and 641:) + (x 36:scale. 8097:Proof 7639:and 6167:a × b 6136:deeds 6095:deeds 4088:when 2699:: + z 2659:deeds 2651:7D.-- 1992:deeds 1707:deeds 1520:deeds 951:deeds 905:deeds 663:for z 8137:talk 8120:talk 8088:talk 8079:}. 8055:are 8051:and 7183:talk 6340:talk 6128:. -- 6114:talk 5646:and 5431:talk 5401:talk 4127:talk 4117:and 4111:here 4101:talk 4084:and 3201:talk 2711:= (x 2642:and 2509:talk 2499:and 2487:talk 2477:and 2008:here 1747:= (x 1726:talk 1675:talk 1459:= (x 1252:talk 927:talk 887:talk 609:= (x 8057:not 6150:ijk 5674:tan 5656:tan 5621:cos 5610:sin 5595:tan 5450:3D. 5446:ijk 5415:pqk 5411:ijk 4474:ijk 4063:yes 3217:: 2707:+ z 2703:+ z 2695:+ z 2691:+ z 2687:+ z 2673:? 1737:not 1718:any 941:not 131:Low 8151:: 8139:) 8122:) 8090:) 7970:− 7921:∑ 7912:− 7894:∑ 7871:× 7801:− 7752:∑ 7743:− 7725:∑ 7704:⋅ 7694:− 7603:× 7574:⋅ 7562:− 7455:ε 7425:Σ 7415:Σ 7399:ε 7369:Σ 7359:Σ 7344:Σ 7321:× 7257:ε 7227:Σ 7217:Σ 7206:× 7185:) 7177:. 7137:× 7104:⋅ 7091:− 6939:− 6893:− 6847:− 6761:× 6728:⋅ 6715:− 6612:− 6563:∑ 6554:− 6536:∑ 6478:∑ 6467:− 6425:∑ 6375:∑ 6342:) 6262:− 6213:∑ 6204:− 6186:∑ 6116:) 6052:⋅ 6040:− 5974:× 5904:− 5855:∑ 5846:− 5828:∑ 5807:⋅ 5797:− 5716:θ 5695:− 5682:θ 5677:⁡ 5662:θ 5659:⁡ 5627:θ 5624:⁡ 5616:θ 5613:⁡ 5601:θ 5598:⁡ 5550:⋅ 5538:− 5472:× 5433:) 5423:jq 5419:ip 5403:) 5345:− 5296:∑ 5287:− 5269:∑ 5246:× 5190:⋅ 5178:− 5087:⋅ 5076:− 5038:⋅ 4981:× 4967:× 4952:⋅ 4883:ε 4873:Σ 4863:Σ 4837:ε 4827:Σ 4817:Σ 4788:Σ 4740:ε 4710:Σ 4700:Σ 4684:ε 4654:Σ 4644:Σ 4629:Σ 4606:× 4542:ε 4512:Σ 4502:Σ 4491:× 4448:ε 4395:ε 4377:× 4305:− 4256:∑ 4247:− 4229:∑ 4208:⋅ 4198:− 4129:) 4103:) 4067:no 4011:− 3962:∑ 3953:− 3935:∑ 3910:× 3836:× 3805:⋅ 3795:− 3690:− 3641:∑ 3632:− 3614:∑ 3593:⋅ 3583:− 3528:: 3476:− 3427:∑ 3418:− 3400:∑ 3342:∑ 3331:− 3289:∑ 3239:∑ 3203:) 3163:× 3130:⋅ 3117:− 3047:) 3039:-x 3023:-x 3007:-x 2991:-x 2975:-x 2959:-x 2943:-x 2927:-x 2911:-x 2895:-x 2879:-x 2863:-x 2847:-x 2831:-x 2815:-x 2799:-x 2783:-x 2767:-x 2751:-x 2735:-x 2719:-x 2646:× 2638:∧ 2609:∧ 2600:⋅ 2591:∧ 2566:⋅ 2557:− 2548:⋅ 2533:⋅ 2511:) 2489:) 2433:× 2400:⋅ 2387:− 2278:− 2229:∑ 2220:− 2202:∑ 2144:∑ 2133:− 2091:∑ 2041:∑ 1945:− 1899:− 1853:− 1787:-x 1771:-x 1755:-x 1728:) 1677:) 1637:× 1604:⋅ 1591:− 1499:-x 1483:-x 1467:-x 1406:− 1360:− 1314:− 1254:) 1246:. 1210:× 1177:⋅ 1164:− 1065:× 1032:⋅ 1019:− 929:) 889:) 863:, 855:= 834:, 826:= 805:, 797:= 776:, 768:= 747:, 739:= 718:, 710:= 689:, 681:= 675:, 649:-x 633:-x 617:-x 567:× 534:⋅ 521:− 418:− 369:∑ 360:− 342:∑ 284:∑ 273:− 231:∑ 181:∑ 8135:( 8118:( 8086:( 8077:m 8075:b 8073:l 8071:b 8069:j 8067:a 8065:i 8053:g 8049:f 8045:g 8041:f 8037:b 8033:a 8026:b 8022:a 8005:. 7998:2 7994:) 7988:i 7984:b 7978:j 7974:a 7965:j 7961:b 7955:i 7951:a 7947:( 7942:n 7937:1 7934:+ 7931:i 7928:= 7925:j 7915:1 7909:n 7904:1 7901:= 7898:i 7890:= 7885:2 7880:| 7874:b 7868:a 7863:| 7836:, 7829:2 7825:) 7819:i 7815:b 7809:j 7805:a 7796:j 7792:b 7786:i 7782:a 7778:( 7773:n 7768:1 7765:+ 7762:i 7759:= 7756:j 7746:1 7740:n 7735:1 7732:= 7729:i 7721:= 7716:2 7712:) 7707:b 7701:a 7697:( 7689:2 7684:| 7678:b 7673:| 7667:2 7662:| 7656:a 7651:| 7624:, 7617:2 7612:| 7606:b 7600:a 7595:| 7591:= 7586:2 7581:) 7577:b 7571:a 7567:( 7557:2 7552:| 7546:b 7541:| 7535:2 7530:| 7524:a 7519:| 7477:, 7471:) 7465:k 7462:q 7459:p 7449:q 7445:b 7439:p 7435:a 7429:q 7419:p 7409:k 7406:j 7403:i 7393:j 7389:b 7383:i 7379:a 7373:j 7363:i 7354:( 7348:k 7340:= 7335:2 7330:| 7324:b 7318:a 7313:| 7286:, 7278:k 7274:e 7267:k 7264:j 7261:i 7251:j 7247:b 7241:i 7237:a 7231:j 7221:i 7213:= 7209:b 7203:a 7181:( 7152:2 7147:| 7141:y 7133:x 7128:| 7124:= 7119:2 7114:| 7108:y 7100:x 7095:| 7086:2 7081:| 7075:y 7070:| 7064:2 7059:| 7053:x 7048:| 7019:7 7015:z 6992:1 6988:z 6962:) 6957:5 6953:y 6947:6 6943:x 6934:6 6930:y 6924:5 6920:x 6916:+ 6911:3 6907:y 6901:7 6897:x 6888:7 6884:y 6878:3 6874:x 6870:+ 6865:2 6861:y 6855:4 6851:x 6842:4 6838:y 6832:2 6828:x 6824:( 6821:= 6816:1 6812:z 6776:2 6771:| 6765:y 6757:x 6752:| 6748:= 6743:2 6738:| 6732:y 6724:x 6719:| 6710:2 6705:| 6699:y 6694:| 6688:2 6683:| 6677:x 6672:| 6645:, 6640:2 6636:) 6630:i 6626:y 6620:j 6616:x 6607:j 6603:y 6597:i 6593:x 6589:( 6584:n 6579:1 6576:+ 6573:i 6570:= 6567:j 6557:1 6551:n 6546:1 6543:= 6540:i 6532:= 6527:2 6521:) 6513:k 6509:y 6503:k 6499:x 6493:n 6488:1 6485:= 6482:k 6472:( 6462:) 6455:2 6450:k 6446:y 6440:n 6435:1 6432:= 6429:k 6419:( 6412:) 6405:2 6400:k 6396:x 6390:n 6385:1 6382:= 6379:k 6369:( 6338:( 6323:j 6319:i 6315:k 6313:b 6311:i 6297:. 6290:2 6286:) 6280:i 6276:b 6270:j 6266:a 6257:j 6253:b 6247:i 6243:a 6239:( 6234:n 6229:1 6226:+ 6223:i 6220:= 6217:j 6207:1 6201:n 6196:1 6193:= 6190:i 6175:k 6173:b 6171:i 6112:( 6071:. 6064:2 6059:) 6055:b 6049:a 6045:( 6035:2 6030:| 6024:b 6019:| 6013:2 6008:| 6002:a 5997:| 5993:= 5988:2 5983:| 5977:b 5971:a 5966:| 5939:, 5932:2 5928:) 5922:i 5918:b 5912:j 5908:a 5899:j 5895:b 5889:i 5885:a 5881:( 5876:n 5871:1 5868:+ 5865:i 5862:= 5859:j 5849:1 5843:n 5838:1 5835:= 5832:i 5824:= 5819:2 5815:) 5810:b 5804:a 5800:( 5792:2 5787:| 5781:b 5776:| 5770:2 5765:| 5759:a 5754:| 5719:2 5709:2 5705:n 5701:a 5698:t 5692:1 5685:2 5671:2 5665:= 5604:= 5569:. 5562:2 5557:) 5553:b 5547:a 5543:( 5533:2 5528:| 5522:b 5517:| 5511:2 5506:| 5500:a 5495:| 5491:= 5486:2 5481:| 5475:b 5469:a 5464:| 5442:ε 5429:( 5421:δ 5413:ε 5399:( 5380:. 5373:2 5369:) 5363:i 5359:b 5353:j 5349:a 5340:j 5336:b 5330:i 5326:a 5322:( 5317:n 5312:1 5309:+ 5306:i 5303:= 5300:j 5290:1 5284:n 5279:1 5276:= 5273:i 5265:= 5260:2 5255:| 5249:b 5243:a 5238:| 5209:. 5202:2 5197:) 5193:b 5187:a 5183:( 5173:2 5168:| 5162:b 5157:| 5151:2 5146:| 5140:a 5135:| 5131:= 5110:, 5104:] 5099:b 5094:) 5090:a 5084:b 5080:( 5072:a 5064:2 5059:| 5053:b 5048:| 5043:[ 5035:a 5031:= 5001:. 4995:] 4990:) 4986:b 4976:a 4972:( 4962:b 4957:[ 4949:a 4945:= 4923:] 4919:) 4913:q 4909:b 4903:p 4899:a 4893:k 4890:q 4887:p 4877:q 4867:p 4857:j 4853:b 4847:i 4844:k 4841:j 4831:k 4821:j 4812:( 4808:[ 4802:i 4798:a 4792:i 4784:= 4756:) 4750:k 4747:q 4744:p 4734:q 4730:b 4724:p 4720:a 4714:q 4704:p 4694:k 4691:j 4688:i 4678:j 4674:b 4668:i 4664:a 4658:j 4648:i 4639:( 4633:k 4625:= 4620:2 4615:| 4609:b 4603:a 4598:| 4571:. 4563:k 4559:e 4552:k 4549:j 4546:i 4536:j 4532:b 4526:i 4522:a 4516:j 4506:i 4498:= 4494:b 4488:a 4458:k 4455:j 4452:i 4424:, 4416:k 4412:e 4405:k 4402:j 4399:i 4391:= 4385:j 4381:e 4372:i 4368:e 4340:, 4333:2 4329:) 4323:i 4319:b 4313:j 4309:a 4300:j 4296:b 4290:i 4286:a 4282:( 4277:n 4272:1 4269:+ 4266:i 4263:= 4260:j 4250:1 4244:n 4239:1 4236:= 4233:i 4225:= 4220:2 4216:) 4211:b 4205:a 4201:( 4193:2 4188:| 4182:b 4177:| 4171:2 4166:| 4160:a 4155:| 4125:( 4119:b 4115:a 4099:( 4090:n 4086:b 4082:a 4074:n 4046:? 4039:2 4035:) 4029:i 4025:b 4019:j 4015:a 4006:j 4002:b 3996:i 3992:a 3988:( 3983:n 3978:1 3975:+ 3972:i 3969:= 3966:j 3956:1 3950:n 3945:1 3942:= 3939:i 3931:= 3926:2 3921:| 3915:b 3905:a 3900:| 3880:n 3876:n 3859:, 3852:2 3847:| 3841:b 3831:a 3826:| 3822:= 3817:2 3813:) 3808:b 3802:a 3798:( 3790:2 3785:| 3779:b 3774:| 3768:2 3763:| 3757:a 3752:| 3725:, 3718:2 3714:) 3708:i 3704:b 3698:j 3694:a 3685:j 3681:b 3675:i 3671:a 3667:( 3662:n 3657:1 3654:+ 3651:i 3648:= 3645:j 3635:1 3629:n 3624:1 3621:= 3618:i 3610:= 3605:2 3601:) 3596:b 3590:a 3586:( 3578:2 3573:| 3567:b 3562:| 3556:2 3551:| 3545:a 3540:| 3509:, 3504:2 3500:) 3494:i 3490:b 3484:j 3480:a 3471:j 3467:b 3461:i 3457:a 3453:( 3448:n 3443:1 3440:+ 3437:i 3434:= 3431:j 3421:1 3415:n 3410:1 3407:= 3404:i 3396:= 3391:2 3385:) 3377:k 3373:b 3367:k 3363:a 3357:n 3352:1 3349:= 3346:k 3336:( 3326:) 3319:2 3314:k 3310:b 3304:n 3299:1 3296:= 3293:k 3283:( 3276:) 3269:2 3264:k 3260:a 3254:n 3249:1 3246:= 3243:k 3233:( 3199:( 3178:2 3173:| 3167:y 3159:x 3154:| 3150:= 3145:2 3140:| 3134:y 3126:x 3121:| 3112:2 3107:| 3101:y 3096:| 3090:2 3085:| 3079:x 3074:| 3045:4 3043:y 3041:5 3037:5 3035:y 3033:4 3029:2 3027:y 3025:6 3021:6 3019:y 3017:2 3013:1 3011:y 3009:3 3005:3 3003:y 3001:1 2997:2 2995:y 2993:7 2989:7 2987:y 2985:2 2981:3 2979:y 2977:4 2973:4 2971:y 2969:3 2965:1 2963:y 2961:5 2957:5 2955:y 2953:1 2949:4 2947:y 2945:7 2941:7 2939:y 2937:4 2933:2 2931:y 2929:3 2925:3 2923:y 2921:2 2917:1 2915:y 2913:6 2909:6 2907:y 2905:1 2901:5 2899:y 2897:7 2893:7 2891:y 2889:5 2885:3 2883:y 2881:6 2877:6 2875:y 2873:3 2869:1 2867:y 2865:2 2861:2 2859:y 2857:1 2853:4 2851:y 2849:6 2845:6 2843:y 2841:4 2837:2 2835:y 2833:5 2829:5 2827:y 2825:2 2821:1 2819:y 2817:7 2813:7 2811:y 2809:1 2805:6 2803:y 2801:7 2797:7 2795:y 2793:6 2789:3 2787:y 2785:5 2781:5 2779:y 2777:3 2773:1 2771:y 2769:4 2765:4 2763:y 2761:1 2757:6 2755:y 2753:5 2749:5 2747:y 2745:6 2741:3 2739:y 2737:7 2733:7 2731:y 2729:3 2725:2 2723:y 2721:4 2717:4 2715:y 2713:2 2709:7 2705:6 2701:5 2697:4 2693:3 2689:2 2685:1 2683:z 2677:z 2648:b 2644:a 2640:b 2636:a 2618:. 2615:) 2612:b 2606:a 2603:( 2597:) 2594:b 2588:a 2585:( 2582:= 2577:2 2573:) 2569:b 2563:a 2560:( 2554:) 2551:b 2545:b 2542:( 2539:) 2536:a 2530:a 2527:( 2507:( 2501:y 2497:x 2485:( 2479:y 2475:x 2455:. 2448:2 2443:| 2437:b 2429:a 2424:| 2420:= 2415:2 2410:| 2404:b 2396:a 2391:| 2382:2 2377:| 2371:b 2366:| 2360:2 2355:| 2349:a 2344:| 2311:, 2306:2 2302:) 2296:i 2292:b 2286:j 2282:a 2273:j 2269:b 2263:i 2259:a 2255:( 2250:n 2245:1 2242:+ 2239:i 2236:= 2233:j 2223:1 2217:n 2212:1 2209:= 2206:i 2198:= 2193:2 2187:) 2179:k 2175:b 2169:k 2165:a 2159:n 2154:1 2151:= 2148:k 2138:( 2128:) 2121:2 2116:k 2112:b 2106:n 2101:1 2098:= 2095:k 2085:( 2078:) 2071:2 2066:k 2062:a 2056:n 2051:1 2048:= 2045:k 2035:( 1968:) 1963:5 1959:y 1953:6 1949:x 1940:6 1936:y 1930:5 1926:x 1922:+ 1917:3 1913:y 1907:7 1903:x 1894:7 1890:y 1884:3 1880:x 1876:+ 1871:2 1867:y 1861:4 1857:x 1848:4 1844:y 1838:2 1834:x 1830:( 1827:= 1822:1 1818:z 1805:1 1802:z 1795:) 1793:3 1791:y 1789:5 1785:5 1783:y 1781:6 1777:3 1775:y 1773:7 1769:7 1767:y 1765:3 1761:2 1759:y 1757:4 1753:4 1751:y 1749:2 1745:1 1743:z 1724:( 1697:1 1694:z 1690:1 1687:z 1673:( 1652:2 1647:| 1641:y 1633:x 1628:| 1624:= 1619:2 1614:| 1608:y 1600:x 1595:| 1586:2 1581:| 1575:y 1570:| 1564:2 1559:| 1553:x 1548:| 1507:) 1505:3 1503:y 1501:5 1497:5 1495:y 1493:6 1489:3 1487:y 1485:7 1481:7 1479:y 1477:3 1473:2 1471:y 1469:4 1465:4 1463:y 1461:2 1457:1 1455:z 1450:1 1447:z 1429:) 1424:5 1420:y 1414:6 1410:x 1401:6 1397:y 1391:5 1387:x 1383:+ 1378:3 1374:y 1368:7 1364:x 1355:7 1351:y 1345:3 1341:x 1337:+ 1332:2 1328:y 1322:4 1318:x 1309:4 1305:y 1299:2 1295:x 1291:( 1288:= 1283:1 1279:z 1266:1 1263:z 1250:( 1225:2 1220:| 1214:y 1206:x 1201:| 1197:= 1192:2 1187:| 1181:y 1173:x 1168:| 1159:2 1154:| 1148:y 1143:| 1137:2 1132:| 1126:x 1121:| 1080:2 1075:| 1069:y 1061:x 1056:| 1052:= 1047:2 1042:| 1036:y 1028:x 1023:| 1014:2 1009:| 1003:y 998:| 992:2 987:| 981:x 976:| 925:( 885:( 877:m 875:× 873:l 869:n 867:× 865:j 861:k 859:× 857:i 853:o 848:o 846:× 844:j 840:l 838:× 836:k 832:m 830:× 828:i 824:n 819:o 817:× 815:l 811:k 809:× 807:j 803:n 801:× 799:i 795:m 790:o 788:× 786:m 782:n 780:× 778:k 774:j 772:× 770:i 766:l 761:n 759:× 757:l 753:m 751:× 749:j 745:o 743:× 741:i 737:k 732:o 730:× 728:n 724:m 722:× 720:k 716:l 714:× 712:i 708:j 703:m 701:× 699:n 695:o 693:× 691:k 687:l 685:× 683:j 679:i 669:7 665:1 657:) 655:3 653:y 651:5 647:5 645:y 643:6 639:3 637:y 635:7 631:7 629:y 627:3 623:2 621:y 619:4 615:4 613:y 611:2 607:1 605:z 582:2 577:| 571:y 563:x 558:| 554:= 549:2 544:| 538:y 530:x 525:| 516:2 511:| 505:y 500:| 494:2 489:| 483:x 478:| 451:, 446:2 442:) 436:i 432:y 426:j 422:x 413:j 409:y 403:i 399:x 395:( 390:n 385:1 382:+ 379:i 376:= 373:j 363:1 357:n 352:1 349:= 346:i 338:= 333:2 327:) 319:k 315:y 309:k 305:x 299:n 294:1 291:= 288:k 278:( 268:) 261:2 256:k 252:y 246:n 241:1 238:= 235:k 225:( 218:) 211:2 206:k 202:x 196:n 191:1 188:= 185:k 175:( 143:. 42::

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Index