Re: [math-fun] less lumpy fun function
In differential topology it's crucial that there's a C^oo function f: R -> R that's 0 for x <= 0, 1 for x >= 0, and strictly increasing in between. That implies each derivative is bounded. (This often allows for glueing together local things on a manifold to create a global thing. There's can't be such a function that's real analytic (C^w) everywhere, which is why it's often much harder to show that global C^w things exist on manifolds, and often they don't.) The standard example f is given as follows: { 0, x <= 0 Let g(x) = { { exp(-1/x), x > 0 Then define f(x) = g(x) / (g(x) + g(1-x)). It would be interesting to know if such an example is possible such that all its derivatives are bounded above by a single constant. In a similar vein (I've seen the answer), it's an interesting question as to whether there exists a C^oo function h: R -> R such that h(x) = 0 for x <= 0, and all its derivatives h^(n)(x) > 0 for all n >= 0 and x > 0 (where h^(0)(x) := h(x)). --Dan Fred wrote: << I didn't want to get to involved in detail at this stage, since I'm not sure how relevant it might be. But the essence is in the rider, that "it's desirable (in some sense) to minimise the growth of the derivatives with n". As Dan points out, it's easy to take some sigmoid function such as tanh(y), then rescale both dimensions to get say f(x) = (1 + tanh(y))/2 with y = (x - 1/2)/(x(1-x) . But it runs straight into the "lumpiness" problem: rescaling introduces --- apparently inevitably --- wild oscillations in the higher derivatives near the endpoints. Other simplistic approaches based on polynomial or trigonometric approximants fall into the "top-hat" trap: the limit f(x) always turns out to be constant, except at x = 1/2 where there is a step of height unity. My suggestion is that problems of this nature demand an unconventional approach, involving the articulation and (numerical) solution of some functional equation. I wrote: << Fred wrote: << I haven't been paying much attention to the discussion, so this stuff may well be hopelessly off-message --- in which case, apologies in advance. But some of the features here remind me of my investigation in 2004 into blending pairs of functions for graphics purposes. The problem boils down to finding a blending function f(x) such that [1] f(0) = 0, f(1) = 1; [2] (d^n/dx^n) f(x) = 0 at x = 0,1, for all integer n > 0; and f(x) has continuous derivatives of all orders n throughout the unit interval --- and for all x, as things turned out. In addition, it's desirable (in some sense) to minimise the growth of the derivatives with n. What may be relevant is that the eventual construction is everywhere non-analytic, and its effective numerical computation involves a completely unconventional (though efficient) algorithm. As usual, I never got around even to submitting it for publication --- but if anybody is interested, I'm happy to forward an account!
I am puzzled as to what question this function is the answer to. (There are certainly C^oo functions f satisfying [1] and [2] that fail to be real analytic only at x=0 and x=1.)
________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx
On 4/30/10, Dan Asimov <dasimov@earthlink.net> wrote:
In differential topology it's crucial that there's a C^oo function f: R -> R that's 0 for x <= 0, 1 for x >= 0, and strictly increasing in between. That implies each derivative is bounded.
I think this should have read " 0 for x <= 0, 1 for x >= 1 "; and given its stated purpose, and the subsequent construction, presumably my constraints on the derivatives vanishing at the end-points are also involved.
(This often allows for glueing together local things on a manifold to create a global thing. There's can't be such a function that's real analytic (C^w) everywhere, which is why it's often much harder to show that global C^w things exist on manifolds, and often they don't.)
The standard example f is given as follows:
{ 0, x <= 0 Let g(x) = { { exp(-1/x), x > 0
Then define f(x) = g(x) / (g(x) + g(1-x)).
This looks essentially similar to the naive construction I mentioned earlier, and suffers from the same (practical) defect: for example, its first derivative has x^2 (1-x)^2 in the denominator.
It would be interesting to know if such an example is possible such that all its derivatives are bounded above by a single constant.
The $64,000 question. For what it's worth, I'm pretty sure the answer is a resounding "no": the best behaviour I managed to achieve was |(d^n/dx^n) f(x)| <= 2^( n(n+1)/2 ). Can anybody improve on this?
In a similar vein (I've seen the answer), it's an interesting question as to whether there exists a C^oo function h: R -> R such that h(x) = 0 for x <= 0, and all its derivatives h^(n)(x) > 0 for all n >= 0 and x > 0 (where h^(0)(x) := h(x)).
Steve --- hope we haven't gone and hijacked your thread --- it wasn't deliberate, honestly! Fred Lunnon
--Dan
Fred wrote: << I didn't want to get to involved in detail at this stage, since I'm not sure how relevant it might be. But the essence is in the rider, that "it's desirable (in some sense) to minimise the growth of the derivatives with n".
As Dan points out, it's easy to take some sigmoid function such as tanh(y), then rescale both dimensions to get say f(x) = (1 + tanh(y))/2 with y = (x - 1/2)/(x(1-x) .
But it runs straight into the "lumpiness" problem: rescaling introduces --- apparently inevitably --- wild oscillations in the higher derivatives near the endpoints.
Other simplistic approaches based on polynomial or trigonometric approximants fall into the "top-hat" trap: the limit f(x) always turns out to be constant, except at x = 1/2 where there is a step of height unity.
My suggestion is that problems of this nature demand an unconventional approach, involving the articulation and (numerical) solution of some functional equation.
I wrote:
<<
Fred wrote: << I haven't been paying much attention to the discussion, so this stuff may well be hopelessly off-message --- in which case, apologies in advance. But some of the features here remind me of my investigation in 2004 into blending pairs of functions for graphics purposes.
The problem boils down to finding a blending function f(x) such that
[1] f(0) = 0, f(1) = 1;
[2] (d^n/dx^n) f(x) = 0 at x = 0,1, for all integer n > 0;
and f(x) has continuous derivatives of all orders n throughout the unit interval --- and for all x, as things turned out. In addition, it's desirable (in some sense) to minimise the growth of the derivatives with n.
What may be relevant is that the eventual construction is everywhere non-analytic, and its effective numerical computation involves a completely unconventional (though efficient) algorithm. As usual, I never got around even to submitting it for publication --- but if anybody is interested, I'm happy to forward an account!
I am puzzled as to what question this function is the answer to. (There are certainly C^oo functions f satisfying [1] and [2] that fail to be real analytic only at x=0 and x=1.)
________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx
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Fred lunnon