Wikipedia:Reference desk/Mathematics

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July 20

Goldman and G-domain

I've seen definitions of both Goldman domain and G-domain, but do those two pages describe formally equivalent structures? I understand that G-domain has been used as a synonym for Goldman domain, but is this the standard use? Klbrain (talk) 21:23, 20 July 2017 (UTC)

Not really an expert on this but the definitions given in the article look equivalent. There's already a merge tag in place, perhaps the articles should be merged regardless since the concepts are certainly similar enough that they could be placed in a single article. --RDBury (talk) 15:15, 24 July 2017 (UTC)
Thanks for your view. That was my feeling too, but note that no-one has been willing to commit an opinion on the merge in almost 3 years. I've done some pure maths, but was rather (25 years ...) too rusty to be sure. Klbrain (talk) 17:27, 24 July 2017 (UTC)

July 22

The Fibonacci-like sequence x(1)=x(2)=1, x(n+1)=2*agm(x(n), x(n-1))

Hi, as you know, the Fibonacci sequence has f(n+1)=f(n)+f(n-1) = 2*(the arithmetic mean of f(n) and f(n-1)), and lim as n-->+oo of the ratios f(n)/f(n-1) is M = Golden mean, and the lim as n-->-oo is -1/M. I've played with sequences like y(n+1) = 2*(harmonic mean [and similarly rms mean] of y(n) and y(n-1)), but I'm really wondering about the arithmetic-geometric mean(agm) variation: what are the numbers that the ratios x(n)/x(n-1) approaches as n-->+-oo, where {x(n)} is defined in the title above? Are the ratios well known in another context, like perhaps pi/2? Numerical approximations would also be helpful. Thanks. 64.134.223.214 (talk) 00:55, 22 July 2017 (UTC)

Writing a quick one-liner (almost) in Haskell, the ratios of consecutive terms of your sequence seem to be converging to around 1.60207620749888.... Checking the OEIS for this number gives no hits, so no idea if it's expressible in terms of known constants. --Deacon Vorbis (talk) 04:07, 22 July 2017 (UTC)
Thanks. Does your program tell what the ratio x(n)/x(n-1) looks like for large negative n?64.134.223.214 (talk) 04:14, 22 July 2017 (UTC)
Never mind. I'm such an idiot!64.134.223.214 (talk) 03:17, 23 July 2017 (UTC)

It is not too difficult to find an equation satisfied by the limit ratio. I think the following gives 100 digits of your constant using Mathematica (assuming I got the argument of the elliptic function right):

In[30]:= FindRoot[(1+x)/x^2 == 2/Pi*EllipticK[(1-x)^2/(1+x)^2],{x,1.6},WorkingPrecision->100]
Out[30]= {x->1.602076207498882854166181178384113469249890744070779427538587439268519411756346198948331083813417404}

No idea whether this number is related to anything else. —Kusma (t·c) 10:01, 24 July 2017 (UTC)

[edit conflict] This looks like a job for isc rather than OEIS. But I didn't find a match there either.
FWIW, I've put more digits of the ratio at https://pastebin.com/qSkERhCF, it should be correct to the sf specified. -- Meni Rosenfeld (talk) 10:23, 24 July 2017 (UTC)
I guess the process should converge for any "reasonable" mean and give a limit ratio between and 2. —Kusma (t·c) 11:23, 24 July 2017 (UTC)

July 24

Associative property and composite functions

I posted a question a bit back. I have developed some conclusions.

Let and , where n is a point in time. Hence, and . Thus, using the substitution property, the composite functions and can be defined, such that and . Finally, due to the associative property, .

I am not certain this line of thinking is correct. I appreciate any insight anyone could provide. Schyler (exquirere bonum ipsum) 17:44, 24 July 2017 (UTC)

I think this doesn't quite work without the assumption that f and g are commutative as well. in general.--Jasper Deng (talk) 18:05, 24 July 2017 (UTC)
Yes, you are confusing associative property and commutative property. Ruslik_Zero 19:55, 24 July 2017 (UTC)
If I am making an argument about human behavior from theory alone, can I assume commutativity? In other words, is it a mathematically sound hypothesis to assume commutativity (I know it is psychologically sound). I am preparing a longitudinal experiment. Schyler (exquirere bonum ipsum) 12:05, 25 July 2017 (UTC)
This is getting to the point where it's very difficult to follow what you're trying to get at unless you're willing to get more specific (didn't I see a guideline or essay or something about this somewhere?). For instance, saying it's "psychologically sound" for two functions to commute with each other just sounds really weird. As for the mathematical validity of assuming it. I mean, sure, but most functions don't actually commute with each other. How about and Then but --Deacon Vorbis (talk) 15:07, 25 July 2017 (UTC)
I think the OP means "intuitive" when (s)he says "psychologically sound". But as we know, intuition will not fly, and seasoned mathematicians know the necessity of proof before taking a theorem for granted.--Jasper Deng (talk) 17:33, 25 July 2017 (UTC)
Sorry for not being specific enough. I guess it's an intellectual property fear. According to Vygotsky, teaching, , and learning, , are one in the same phenomenon, called obuchenie in Russian and often translated as "teaching/learning" due to difficulty in translation. By "psychologically sound," I mean that theory supports the assumption of commutativity. But I now see that commutativity can absolutely not be assumed in these functions. I have an economic analysis (Correa & Gruver, 1987) that shows commutativity using by a Cournot adjustment, so I was hoping to arrive at a similar conclusion in a different way on theory alone. I am feeling resigned to the need for empirical evidence, though. Schyler (exquirere bonum ipsum) 23:24, 25 July 2017 (UTC)
@Schyler: Well you're not talking about mathematical functions then are you? I would think the psychological equivalent of the noncommutativity is path dependence.--Jasper Deng (talk) 01:24, 26 July 2017 (UTC)

July 26

Diophantine Equation

Let a be some natural number. What are the solutions in integers of the equation ? עברית (talk) 16:32, 26 July 2017 (UTC)

Any even difference of 2 squares is divisible by 4, so there are many solutions to that equation! Georgia guy (talk) 16:38, 26 July 2017 (UTC)
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