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July 3

Virtual Graph

Is the graph for cos(x^y) = csc(y^x) same as made by Desmos or is it largely different? Exclusive Editor _{Notify Me!} 13:15, 3 July 2022 (UTC)[reply]

Is there a way to find out what Desmos thinks the graph is without downloading it? I don't have it on my computer. Without getting into the details, it seems likely the graph consists of discrete points, assuming it's non-empty. (It would be the intersection of cos(x^y) = 1 and sin(y^x) = 1, combined with the intersection of cos(x^y) = −1 and sin(y^x) = −1.) This could cause problems with a graphing calculator. --RDBury (talk) 14:50, 3 July 2022 (UTC)[reply]

It is not hard to show that the graph is a discrete set. The system

x y = 2 π

,

y x = ½ π

has the solution

x \approx 5.425421011388989

,

y \approx 1.086796732660537

, so the set is not empty. I think that, more in general, the system

x y = 2 mπ

,

y x = (2 n + ½) π

has a solution for all integers

m > 0

,

n \geq 0

. --Lambiam 15:54, 3 July 2022 (UTC)[reply]

There is an online version of Demos. On entering

\cos \left(x^{y}\right)=\csc \left(y^{x}\right),

I get a strange plot of way too many scattered points, plus a notice: ⚠This equation contains fine detail that has not fully been resolved. Learn more. See here. --Lambiam 16:42, 3 July 2022 (UTC)[reply]

On further examination, it seems that the separate graphs of

cos(x y) = 1

and

csc(y x) = 1

each consist of a set of closely spaced curves; the solution set of the combined system of equations consists of the points where the curves of these two graphs intersect, and there are very many such intersection points. Given a sufficiently high resolution, you can see that the points forming the graph are not randomly scattered, as they appear to be on the Desmos plot, but form a pattern reminiscent of moiré. --Lambiam 19:10, 3 July 2022 (UTC)[reply]

Cubic formula

There are formulas from the 16th century for solving the general cubic, that were found with lots of work and ingenuity. Suppose I don't know about them, but want to discover the formula today, using modern "technology". I think the basic idea is that the Galois group for the general cubic is the symmetric group S3, and S3 factors into cyclic subgroups that correspond to square and cube root operations in the field extensions. But I'm missing some intermediate steps about turning that into formulas. The same goes for the quartic, while the quintic is unsolvable because A5 is a simple group with no normal subgroups.

Is there some subtlety to the above? Is this a good "project" to try, if I want to get better at this kind of algebra? I took a class once, but don't remember that much and didn't attempt this at the time (though I wondered about it). Thanks.

Inspiration for this question: [1]. 2601:648:8202:350:0:0:0:FD2B (talk) 22:27, 3 July 2022 (UTC)[reply]

You've described a 10,000 ft. view of the relationship between Galois theory and solving equations, and there are a lot of details to be filled in before you get what might be a described as a recipe. For me, the crucial observation for solving the cubic is that (a+b+c)|(a³+b³+c³-3abc). I'm not sure if Galois theory is really helpful to arrive at this observation, although I think Galois theory does, in some sense, "explain" it. Galois theory may be more useful for finding a similar factorization of 4th degree polynomials, but I'm not familiar with the details. In any case, I think Galois theory is most useful for the 5th degree equation since it's used in the other direction; you're starting from a proposed solution in radicals and deducing facts about the Galois group which turn out not to be true. Meanwhile for the cubic and quartic you're starting with the group structure and trying to derive the solution in radicals. If you want to explore the theme more, you might want to investigate Guass' construction of the regular 17-gon. It may be worthwhile to note that Ruffini's incomplete proof of the Abel–Ruffini theorem was published in 1799, more than a decade before Galois was born. --RDBury (talk) 07:05, 6 July 2022 (UTC)[reply]