Wikipedia:Reference desk/Archives/Mathematics/2017 October 22

= October 22 =

Pro-centrism / anti-centrism of ranked-preference voting systems
I've read that when the political ideology of candidates is more or less on a one-dimensional scale, some ranked-preference voting systems (Condorcet methods, Borda count, Bucklin voting) tend to favor centrists and disfavor extremists compared to plurality voting, whereas instant-runoff voting does the opposite (as was demonstrated by the Burlington mayoral election, 2009). Have any studies been done to quantify or rank the favorability of different ranked-preference voting systems to a centrist (ideally taking into account that voter ideology will be sometimes unimodal and sometimes bimodal)? Neon Merlin  01:20, 22 October 2017 (UTC)


 * Not a direct answer, but you may also want to look at super delegates used by the Democratic Party in the US. These are establishment players who tend to vote for moderates.  In the last US presidential election, the Democrats did not choose their fringe candidate, Bernie Sanders, while the Republicans, who lack super delegates, did choose their fringe candidate, Donald Trump.  Super delegates are not the only factor, of course, but they do have a measurable effect.  The same effect from the primaries could occur in the general election if super delegates were used there, such as former presidents, VPs, state governors, etc.  Of course, this would require a Constitutional amendment.  StuRat (talk) 01:30, 22 October 2017 (UTC)


 * Don't the Republicans have the equivalent of superdelegates as well, even if their share of the vote is smaller? I get the impression that the biggest problem for the viable Republican moderates (Kasich, Cruz, Rubio) was that the vote was split among the three of them. (This article, and some other polls I read, suggested that Trump would have been neck-and-neck with Sanders for 5th place in a Condorcet merged primary/general Trump/Cruz/Kasich/Rubio/Clinton/Sanders runoff.) Neon  Merlin  01:58, 22 October 2017 (UTC)


 * Yes, those are among the "other factors". According to our article: "Republican Party superdelegates are obliged to vote for their state's popular vote winner...", so, it makes no difference who the actual delegates are, if their hands are tied like that. StuRat (talk) 02:24, 22 October 2017 (UTC)

Equivalence between differential expressions
Hi there! Are the following two differential expressions (encountered recently in a theoretical science textbook) equivalent as it is claimed on the pages of that textbook?

$$ \frac{G}{(1 - x_2)^2} + {1 \over {1 - x_2}} (\frac)_\frac{x_1}{x_3} = \frac{\bar{G_2}}{(1 - x_2)^2}$$ (1)

and

$$ (\mathfrak{d} \frac{G}{\frac{1 - x_2}{\mathfrak{d} x_2}})_{\frac{x_1}{x_3}} = \frac{\bar{G_2}}{(1 - x_2)^2}$$    (2)

I can't figure out what intermediate steps (for encapsulation in a differential quotient) are used! Thanks!--82.137.9.135 (talk) 16:09, 22 October 2017 (UTC)


 * Looks to me like the product rule (applied to a quotient) combined with bad formatting on your part. But this is just a guess because some of the expressions you've written seem very mal-formed. --JBL (talk) 16:17, 22 October 2017 (UTC)


 * It is a good guess (almost certain possibility) and represents what I've missed/lapsed to think of, especially the part with product rule. It seemed that hints related to a quotient derivative occurred to me but they didn't add up something was missing.
 * I admit the bad formatting, previous attempts using \frac{\partial }{\partial x} have not been successful to render the second expresion as in the textbook.--82.137.9.55 (talk) 17:01, 22 October 2017 (UTC)
 * Previous attempts are on Talk:Gibbs-Duhem equation and on that article history.--82.137.9.55 (talk) 17:06, 22 October 2017 (UTC)
 * Perhaps you could post a picture somewhere so that we can see what you're trying to reproduce? --JBL (talk) 20:42, 23 October 2017 (UTC)
 * I'll try. I haven't seen your (second) reply from here until very recent hours.--82.137.14.90 (talk) 00:38, 25 October 2017 (UTC)
 * The left side of (2) may be $$\frac{\partial}{\partial x_2}\left(\frac{G}{1-x_2}\right)$$, but then the first term in (1) is missing a "−" . Any particular reason why you don't identify the textbook you're talking about? --Wrongfilter (talk) 21:11, 23 October 2017 (UTC)
 * "but then the first term in (1) is missing a "−"." No it's not.  But the meaning of the lower subscripts is also problematic (among other things). --JBL (talk) 00:48, 24 October 2017 (UTC)
 * The first term is missing two "−"s? Embarrassed...--Wrongfilter (talk) 02:46, 24 October 2017 (UTC)
 * This notation would be used if one were taking the partial derivative with respect to $$x_2$$ in such a way as to keep the ratio $$x_1/x_3$$ fixed. But there's no canonical way to take this "partial derivative", so obviously we're missing some context if we are to give the equations a precise meaning.   Sławomir Biały  (talk) 01:39, 24 October 2017 (UTC)
 * The context of this nonstandard partial derivative variable to be held constant is indeed that detected by User:Sławomir Biały. It refers to the representations in ternary plot of variables of composition - mole fractions in this case - that add up to a constant.--82.137.14.90 (talk) 23:57, 24 October 2017 (UTC)
 * I propose to all to continue the discussion on talk:Gibbs-Duhem equation.--82.137.14.90 (talk) 00:45, 25 October 2017 (UTC)