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April 25

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Non-standard "additions" on the real numbers

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How many "addition" operations +' are there for which the real numbers R form a field with +' as the addition operation and the usual multiplication as the multiplication operation? Given such a non-standard "addition" operation, must there always be an automorphism of the multiplicative group R× that extends to a field isomorphism between (R, +', *) and (R, +, *) where + is the usual addition and * the usual multiplication? Does work for any rational number r with both the numerator and the denominator positive odd integers? GeoffreyT2000 (talk) 03:12, 25 April 2018 (UTC)[reply]

  • Some thoughts about the first/second question: this is equivalent, in functional form, to asking for a function such that and for all reals a,b,c (see Field_(mathematics)#Classic_definition).
Let us denote . From (1), . Define , and assume it is nonzero in what follows (otherwise it means 0+'(anything)=0, which does not lead to an immediate contradiction that I can see, but surely that is a very different structure from the usual).
Apply (3) with c=0 and b≠0, from the previous it follows that , hence . Apply iteratively this relation, substituting and it leads to .
Define . From above, for all rationals r and integers n, . This also applies to negative n (substitute ). If h is continuous near 0 (which is by no means a given!) the only interesting case if , because otherwise for all r, a, b which must thus be equal to zero for all a,b nonzero.
We also have (even if h is not continuous)
I could not go much further but that puts quite a lot of restrictions on your "addition" definition, especially if it is continuous in zero. TigraanClick here to contact me 08:56, 25 April 2018 (UTC)[reply]
It can be proven much simpler if you note that condition (1) means that is a homogeneous function of the first degree and as such , where is an arbitrary function. Condition (2) means that i.e. the function is defined but its values in interval. So, the problem now is with condition (3). Ruslik_Zero 20:42, 25 April 2018 (UTC)[reply]

A quotient of the symmetric group

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I'm defining an equivalence relation on Sn+1. The motivation is more musical than mathematical so you might find that equivalence relation is pulled out of a hat. In fact I'm really only concerned with S12 but I'll phrase the question for Sn+1. So here's the equivalence relation: take a permutation A of order n+1: A(0),...,A(n). I'm saying that the permutation B is equivalent to A if for some p ≤ n B(0)=A(p),...,B(n-p)=A(n),B(n-p+1)=A(0),...,B(n)=A(p-1). In other words all those permutations equivalent to A for that equivalence relation correspond simply to one of the n+1 "shifts" of the string <A(0),...,A(n)>: <A(0),...,A(n)>,<A(1),...,A(n),A(0)>,...,<A(n),A(0),...,A(n-1)>. I have no idea how well or ill behaved that equivalence relation is with respect to the group law. I can also (set-theoretically) take the quotient of Sn+1 by that equivalence relation. Is that equivalence relation something you recognize, that has been studied? What do you get as an entity when you take that quotient? Just a set without any structure? I thought I'd ask you guys just to get an idea. Thanks. Basemetal 16:59, 25 April 2018 (UTC)[reply]

These are exactly the (left?) cosets of the cyclic (sub)group generated by the standard (n + 1)-cycle (1, 2, ..., n + 1). Equivalently, they are the orbits of the (right?) action of this group on all of S_{n + 1}. Since this subgroup is not normal, there is no group structure on its cosets. --JBL (talk) 12:46, 26 April 2018 (UTC)[reply]
Thanks. Basemetal 18:49, 26 April 2018 (UTC)[reply]

Length and width of a rectangle with only the area

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There is a rectangle and its area 144ft², and its width is 4 times the length, thus the length is X and the width is 4X; but what is the formula to find the exact length and width of this rectangle? Gidev the Dood(Talk) 17:36, 25 April 2018 (UTC)[reply]

, so (fixed per below) , which is easily solved for . --Kinu t/c 19:01, 25 April 2018 (UTC)[reply]
Kinu means . The square on the second is a typo. Basemetal 19:15, 25 April 2018 (UTC)[reply]
Facepalm Facepalm... thanks, fixed. I think my brain was one step ahead and doing the multiplication while I was typing that step. --Kinu t/c 19:26, 25 April 2018 (UTC)[reply]
Alrighty, then the length is 6 and the width 24. . Thank you! Gidev the Dood(Talk) 19:33, 25 April 2018 (UTC)[reply]