User talk:Hugo Spinelli

Nice picture in Collatz's conjecture article

Hello, I'm a student in mathematics and beside of my studies like working on complexe extention of Collatz's function. I came out with the same pictures as you uploaded on Wikipedia and was wondering if you were doing some research on it or just did some nice visuals about it. If so, this could be interresting to exchange some knowledge about. I was also wondering how you generated these nice pictures.

Looking forward to your reply :)

Thank you, Sacha Yersl (talk) 18:58, 10 December 2023 (UTC)[reply]

@Yersl You mean the fractals? The codes are in the description of each file:

I'm not doing any research, I just plotted the fractals as described in the papers cited by the Wikipedia article. The only original aspect of the images was the technique of smoothing the divergence count for smooth transitions of colors. For this, I adapted some of the ideas described in Plotting algorithms for the Mandelbrot set § Continuous (smooth) coloring. Is there anything more specific you'd like to know? —Hugo Spinelli ^(talk) 10:33, 11 December 2023 (UTC)[reply]

Your treatment of X^Y = Y^X

Hi Hugo. Patrick Conroy here, the guy who first contributed the material in this Wikipedia page drawing attention to the idea that every point P=(x,y) in R+ can be viewed as the intersection of two curves, y=mx and y=x^n; I referred to this as a 'coverage' characteristic. It was one of your posts that removed that formulation, and I think that was a mistake (read on for the reason). I am not a trained mathematician. But. In both your Lambert treatment as the preferred solution strategy, and in your modification of my treatment, you enter the assumptive phrase "let y=x*v". I humbly suggest that this is unjustified without knowing the nature of the actual solution, whereas my 'coverage' treatment doesn't require any assumptions on the form of the solution. Please reconsider your Wikipedia posts to reflect this - I am not going to ride in with any adjustments myself. I acknowledge the added value of using the Lambert function, i.e. to identify (e,e) as the point at which the solution curve crosses the curve y=x. In addition, I may have flubbed my computation of dy/dx=-n^2 - your thoughts on this are welcome. I consider this discussion a pleasant diversion, and definitely not a challenge. Have a good day. PFConroy (talk) 02:40, 28 February 2024 (UTC)[reply]

@PFConroy In y=x*v, v is simply y/x, which is well-defined, but different for each pair of (positive) solutions x and y. Perhaps the explanation could be improved, but I see no problem with the derivation. As I mentioned in Talk:Equation xy = yx § Removal of redundant proof, the "coverage" discussion seems unnecessary, but feel free to edit the article if you disagree. —Hugo Spinelli ^(talk) 12:35, 28 February 2024 (UTC)[reply]

Ah, that helps. Thanks, Hugo. PS: I still think my solution is simpler and more accessible to more readers (doesn't need any knowledge of the Lambert equation), but that's not a mathematical issue. As I said, I'm not going to try to push things in this direction. Thanks for your time. PFConroy (talk) 21:16, 28 February 2024 (UTC)[reply]