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Weierstrass functions: continuous but not differentiable anywhere

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Re: Weierstrass functions: continuous but not differentiable anywhere

#31

Post by gohrt »

gballan wrote: post #20
crashandburn4 wrote: post #16

I was finding it hard to understand why it wasn't differentiable (guess my maths is getting a bit rusty) until I read the below[0] which helped it make sense to me: >The function has detail at every level, so zooming in on a piece of the curve does not show it getting progressively closer and closer to a straight line. Rather between any two points no matter how close, the function will not be monotone. [0] http://en…

Good point, thanks. I added the quote from wikipedia, and moved the link from the CoffeeScript to the HTML.

Also note that your code is for a differentiable finite approximation, unless you implement unlimited zoom.

Re: Weierstrass functions: continuous but not differentiable anywhere

#33

Post by trhway »

psykotic wrote: post #22

Fractal curves like the Koch snowflake are likewise continuous and nowhere differentiable. John McCarthy (yes, the inventor of Lisp) has a simpler to analyze construction of a Weierstrass-like function from a triangle wave: http://www-formal.stanford.edu/jmc/weierstrass/weierstrass.h... Here's the big picture. When you analyze variations in f at the kth scale by setting |dx| = 2^(-2^k), the only term that really cont…

>Fractal curves like the Koch snowflake are likewise continuous and nowhere differentiable.

they nowhere "differentiable" in classic, smooth sense - ie. no tangent affine bundle of _whole_ dimension exist, ie. nowhere a tangent line (less tangent plane, etc...) exists. That isn't surprising of course and obviously calls for extended notion of differentiability - like a tangent "bundle" of non-whole, fractal dimension. One can even imagine a Taylor/power series using such differentiability...

Re: Weierstrass functions: continuous but not differentiable anywhere

#34

Post by rcthompson »

Also fun is the Dirichlet function[1], which is discontinuous at every point yet is integrable everywhere. There's also a similar function that is discontinuous at all rational numbers but continuous at all irrational numbers[2].

[1] http://en.wikipedia.org/wiki/Dirichlet_function [2] http://en.wikipedia.org/wiki/Thomae%27s_function

Re: Weierstrass functions: continuous but not differentiable anywhere

#35

Post by mturmon »

yummyfajitas wrote: post #30
mturmon wrote: post #28

Earlier quoted context omitted.

As long as you're nit picking, I'd suggest you re read carefully what you wrote, because it is wrong. What you meant to say is that on a set of measure zero, the paths can have points where continuity is not satisfied. But you did not say that. In any case, the version of Brownian motion I studied was the version that excludes this null set. ;-)

Yes I meant to say differentiable, brownian motion is always continuous. But with probability zero, brownian motion can generate a differentiable curve. Take the typical random walk construction as the limit of a piecewise continuous curve. I.e., 50% chance of being y=x, 50% chance of being y=-x on [0,1], then subdivide. With zero probability, the walk can converge to a straight line x=t, which is certainly different…

You have used a particular construction of B.M. that happens to admit a differentiable function (on a set of measure zero). But other constructions need not have this property; in fact, you could make a version (in the technical sense of that term, see http://almostsure.wordpress.com/2009/11/03/stochastic-proces...) of the B.M. that uses your construction, but removes the null set of differentiable sample paths.

In other words, this property ("differentiability on null sets") is not characteristic of a B.M., it is incidental to your construction. I can't believe you bothered to fuss about these hypothetical properties on null sets when we are dealing with cadlag processes on a separable index set. Sheesh.

Re: Weierstrass functions: continuous but not differentiable anywhere

#36

Post by Sharlin »

Florin_Andrei wrote: post #10
oakwhiz wrote: post #3

This function produces an audio signal that sounds like an organ. https://gist.github.com/improv32/8414717 http://vocaroo.com/i/s13ZO9QOMULL

I am quite surprised it does not sound like white noise. Anyway, I played the output of that script through some filters, and this sounds nice: play -c 2 weierstrass.wav gain -12 loudness chorus 0.5 0.9 50 0.4 0.25 2 -t 60 0.32 0.4 2.3 -t 40 0.3 0.3 1.3 -s reverb gain 12

In white noise the power spectral density (power per Hz) is flat, that is, every frequency contributes equally. A Weierstrass frequency distribution is very far from that.

Re: Weierstrass functions: continuous but not differentiable anywhere

#38

Post by LeFuneste »

mturmon wrote: post #35
yummyfajitas wrote: post #30

Earlier quoted context omitted.

Yes I meant to say differentiable, brownian motion is always continuous. But with probability zero, brownian motion can generate a differentiable curve. Take the typical random walk construction as the limit of a piecewise continuous curve. I.e., 50% chance of being y=x, 50% chance of being y=-x on [0,1], then subdivide. With zero probability, the walk can converge to a straight line x=t, which is certainly different…

You have used a particular construction of B.M. that happens to admit a differentiable function (on a set of measure zero). But other constructions need not have this property; in fact, you could make a version (in the technical sense of that term, see http://almostsure.wordpress.com/2009/11/03/stochastic-proces... ) of the B.M. that uses your construction, but removes the null set of differentiable sample paths. In…

I agree, no one cares in probability theory on the behaviour of BM on this null set. It could be as well a Weierstrass function that it will not change anything to the law of your process.

Re: Weierstrass functions: continuous but not differentiable anywhere

#39

Post by alowe »

rcthompson wrote: post #34

Also fun is the Dirichlet function[1], which is discontinuous at every point yet is integrable everywhere. There's also a similar function that is discontinuous at all rational numbers but continuous at all irrational numbers[2]. [1] http://en.wikipedia.org/wiki/Dirichlet_function [2] http://en.wikipedia.org/wiki/Thomae%27s_function

In a similar vein, the cantor function [1] (also known as the devil's staircase) is continuous and has derivative 0 almost everywhere, but still manages to increase in value.

[1] http://en.wikipedia.org/wiki/Cantor_function#Properties

Re: Weierstrass functions: continuous but not differentiable anywhere

#40

Post by [deleted] »

trhway wrote: post #33
psykotic wrote: post #22

Fractal curves like the Koch snowflake are likewise continuous and nowhere differentiable. John McCarthy (yes, the inventor of Lisp) has a simpler to analyze construction of a Weierstrass-like function from a triangle wave: http://www-formal.stanford.edu/jmc/weierstrass/weierstrass.h... Here's the big picture. When you analyze variations in f at the kth scale by setting |dx| = 2^(-2^k), the only term that really cont…

>Fractal curves like the Koch snowflake are likewise continuous and nowhere differentiable. they nowhere "differentiable" in classic, smooth sense - ie. no tangent affine bundle of _whole_ dimension exist, ie. nowhere a tangent line (less tangent plane, etc...) exists. That isn't surprising of course and obviously calls for extended notion of differentiability - like a tangent "bundle" of non-whole, fractal dimension…

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