Wikipedia:Reference desk/Archives/Mathematics/2016 July 29

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July 29

A question related to indices

If a^x=bc , b^y=ca , c^z=ab then show that xyz=x+y+z+2 — Preceding unsigned comment added by Sahil shrestha (talk • contribs) 02:08, 29 July 2016 (UTC)

Hint: Logarithm. Let us know if you are still stuck. -- ToE 05:08, 29 July 2016 (UTC)

Hey, that's cute. —Tamfang (talk) 05:38, 30 July 2016 (UTC)

I don't want the answer using logarithm. Rather I would like to get the answer by the indices rules , exponential equations & Whatever!! But not logarithm!! — Preceding unsigned comment added by Sahil shrestha (talk • contribs) 05:49, 30 July 2016 (UTC)

Well, the rules of logarithms are really just the rules of indices in disguise. But if you want to avoid using logarithms,, start by showing that:

a^(xyz) = ((b^y)^z)((c^z)^y)

Gandalf61 (talk) 07:09, 30 July 2016 (UTC)

Why don't you want to use logarithms? Were you bitten by a logarithm as a young child? -- Meni Rosenfeld (talk) 07:07, 31 July 2016 (UTC)

Or perhaps they were frightened by the sight of adders multiplying on a log table. -- ToE 00:00, 1 August 2016 (UTC)

The question is missing quantifiers, and from some reasonable choices of quantifiers the result is false. In particular, if a = b = c = 1 then x, y, z can be arbitrary. --JBL (talk) 16:47, 30 July 2016 (UTC)

"If, for any a,b,c positive reals, (...)" is the only choice that sounds reasonable to me (given the symmetry of the problem, and what you mentioned). Tigraan^{Click here to contact me} 18:44, 30 July 2016 (UTC)

You don't need this to hold for every $a,b,c$, just for some particular $a,b,c$ with $a\neq 1$. -- Meni Rosenfeld (talk) 07:07, 31 July 2016 (UTC)

It suffices to require that $a,b,c>0$ and that $a\neq 1$. Following Gandalf's hint, we can straightforwardly show that $a^{xyz}=a^{x+y+z+2}$, and then $a\neq 1$ gives $xyz=x+y+z+2$. -- Meni Rosenfeld (talk) 07:07, 31 July 2016 (UTC)

Idempotents generating the same principal right ideal

Does there exist a ring R with two distinct idempotents e and f generating the same principal right ideal (i.e. eR = fR)? GeoffreyT2000 (talk) 04:14, 29 July 2016 (UTC)

Yes. 2x2 matrices, with idempotents ${\begin{bmatrix}1&0\\0&0\end{bmatrix}},{\begin{bmatrix}1&1\\0&0\end{bmatrix}}$. Sławomir Biały (talk) 11:55, 30 July 2016 (UTC)

Sequential simultaneous games

Given the existence of simultaneous games and sequential games, what (distinctive) features has their combination, a sequential simultaneous game?--82.137.9.125 (talk) 15:52, 29 July 2016 (UTC)

See Repeated game and in particular Repeated game#Repeated prisoner's dilemma. Loraof (talk) 20:00, 29 July 2016 (UTC)

Combinations between game and queuing theory

What possibilities of combining game theory and queuing theory are there?--82.137.9.125 (talk) 15:59, 29 July 2016 (UTC)

Here's a paper that approached a problem in queuing theory by formulating it as a game and applying game theoretic techniques [1]. SemanticMantis (talk) 18:29, 29 July 2016 (UTC)

Thanks. How about the other perspective, of applying queuing theory techniques in game theory of multiplayer games? What is the degree of equivalence of the opposite approaches?--82.137.9.125 (talk) 17:47, 29 July 2016 (UTC)