What algorithm might one use to find the global min of the Weierstrass function numerically?
Every algorithm I’ve seen in my optimization course, Newton’s, grad descent, sub gradient descent, would not work. Perhaps Monte Carlo methods?
Hello, I want to see how the Weierstrass p function and its derivative can define the curve
y^2=4x^3-g_2 x-g_3
I think I can simulate the p function and its derivative properly, but I don't understand how could they define the curve mentioned above. Here is the link to my worksheet: https://www.geogebra.org/m/eydcwzzt
As you can see I am not a master of this topic, but I am very interested and I want to understand these relations.
Thank you for your help! :D
https://www.desmos.com/calculator/iv4ec9hmcv
https://preview.redd.it/1z5dwat0g2z61.png?width=238&format=png&auto=webp&s=dd172ed980a2e28de75604461094ff2f0edcca4f
I'm doing a Math Analysis Class final project on Weierstrass' Function, and while I was researching on the history I saw this line:
>" Riemann had suggested in 1861 that such a function could be found, but his example failed to be non-differentiable at all points. "
Does anyone know what exactly the function that Riemann had suggested? And why his function didn't work? Any link to the function & counter-example would be appreciated!
Additional question: What is the intuition behind Weierstrass' Function and why it works? why use cosine? I get that corners and cusps and things with slopes that approach infinity are examples of something that is continuous but nondifferentiable, but would building the function using just those would be hard, and that's why he used trigonometry?
"You've got a bad amplitude."
Thank you. Thank you. Tip your waitresses, please. They're working hard out there tonight.How about 3blue1brown makes a video about weierstrass functions? I think it'll be pretty cool.
I think that's pretty cool.
It's exactly the same as the one on this page. I was inspired by Perlin noise, adding together sinusoidal functions at increasing frequency and proportionally decreasing amplitudes.
As the title states I am looking to understand how I can use the Weierstrass elliptic "p" function to solve this ODE:
y'(x)^2 = 4(c_1-x)(c_2-x)(c_3-x)
Where c_1, c_2 and c_3 are constants.
Image to avoid ambiguity as well:
https://i.imgur.com/YLZjZYY.png
Because you've got no tan lines.
http://en.wikipedia.org/wiki/Weierstrass_approximation_theorem
The title really says it all, can I take an arbitrary function of three dimensions or so (f(x,y,z)) and approximate it using polynomials such as c + ax + by +dz ...?
If you cannot what are the functions break it?
Thank you in advance
specifically this sentence from the Wikipedia article: "The ℘ functions constitute branched double coverings of the Riemann sphere by the torus, ramified at four points" - what does this mean?
Feel free to tell me if this is stupid...
How's that possible? I can't comprehend the paradox in this concept.
Can you help me?
Maybe I don't quite understand the Weierstrass function, but I was under the impression that the reason it wasn't differentiable anywhere was due to its recursive or fractal (sorry if that's the wrong terminology) nature. If that's the case, it feels to me like it's kind of cheating to just use that and say "hey look! we found a function that's continuous everywhere but nowhere differentiable! told you we could do it!" How about just a regular function whose output isn't approaching some limit at every point (like the Weierstrass function does, to the best of my understanding)? Is it possible? Is there anything that doesn't exhibit this "fractal" nature but is also continuous and not differentiable?
My intuition tells me that clearly the answer must be no... but my intuition also tells me that clearly the answer must be yes, but overall I'm leaning toward no.
No: The arc length formula you would normally integrate over isn't even defined. Furthermore its a fractal with an unknown hausdorff dimension that I believe can be constrained depending on the construction to be strictly greater than one. General intuition just tells me that it simply must be infinite.
Yes: Its Riemann integrable... and while this is super hand wavy and arguably strictly wrong, it would seem that from any element of the riemann sum, the length of the arc is proportional to the infinitesimal step and in some sense seems to me like this constant of proportionality must be bounded in order to maintain continuity. ...I apologize for the disgusting abuse of mathematical terms in this above paragraph, but I lack the tools of measure theory and a proper background in analysis to make a rigorous argument.
I tried to differentiate it term by term to convince myself it is nowhere differentiable, but I got this:
[;f(x) = \sum_{n=0}^{\infty} 0.9^{n}\cos(7^{n}\pi x) ;]
[;f'(x) = \sum_{n=0}^{\infty} \frac{d}{dx} [0.9^{n}\cos(7^{n}\pi x)] ;]
[;= \sum_{n=0}^{\infty} 0.9^{n} \frac{d}{dx} [\cos(7^{n}\pi x)] ;]
[;= -\sum_{n=0}^{\infty} 0.9^{n}7^{n}\pi\sin(7^{n}\pi x);]
But doesn't this series vanish at even integers, in particular zero? What am I missing here?
