April 2010 - mathfun@mailman.xmission.com

Re: [math-fun] less lumpy fun function
by Dan Asimov 30 Apr '10

30 Apr '10

Fred wrote: << I 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. >> Since the function f is C^oo, and f(x) = 0 for x <= 0 and f(x) = 1 for x >= 1, it immediately follows that all derivatives f^(k)(0) = f^(k)(1) = 0 for all k >= 1. So, no additional conditions are required. I continued: << (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. >> I'm not sure what this means, but the proof that this function has the stated properties can be found as example 4.1b on this web page: < http://www.math.lsu.edu/~lawson/Chapter4.pdf > (though Lawson's definitions of f and g are the reverse of mine.) --Dan ________________________________________________________________________________________ "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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Re: [math-fun] less lumpy fun function
by Dan Asimov 30 Apr '10

30 Apr '10

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

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Re: [math-fun] less lumpy fun function
by Dan Asimov 30 Apr '10

30 Apr '10

Frede 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.) --Dan ________________________________________________________________________________________ "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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Re: [math-fun] less lumpy fun function
by Fred lunnon 29 Apr '10

29 Apr '10

On 4/29/10, Steve Witham <sw(a)tiac.net> wrote: > So, my superexponential function turned out like the > Minkowski question mark function, i.e. fractally lumpy. I wanted > a flattish area around zero (super negative exponentials) but no > other lumps. > ... 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 f(0) = 0, f(1) = 1; (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! Fred Lunnon

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[math-fun] less lumpy fun function
by Steve Witham 29 Apr '10

29 Apr '10

So, my superexponential function turned out like the Minkowski question mark function, i.e. fractally lumpy. I wanted a flattish area around zero (super negative exponentials) but no other lumps. Including (successive approximations to) the inverse function in the feedback loop started to reduce the lumps very slowly for a lot of work. Bah. But in the process I got the idea of an "index function" i(x), where something like f(x) = exp_like_function( f( i( x ) ) ). So here is the current hack, in Python: def f( x ): if abs( x ) < 2**-27: return x * 1.1390199403775014 else: return 3.0 * sinh( f( ( 2.0 * x**3 + x ) / 3.0 ) ) i(x) = ( 2 x^3 + x ) / 2 -- i(+/- 1) = +/- 1 and i'(+/- 1) = 2. The 3.0 before the sinh is necessary so that f'(0) = f'(0). The 2**-27 is approximately where asinh() and i() stop being linear to double precision. The 1.1390199403775014 is f'(0) such that f(.5) = 1. To get the inverse function I need to solve the cubic in i(x). f(x) looks like an tangent function except it does a superexponential thing on the way up and down. It's not flat near zero, to get that you can do def g( x ): if x < 0: return -g( -x ) elif x == 0: return 0 else: return exp( f( 2.0 * x - 1.0 ) ) Except for catching infinities near g(0). There is a wobble between g(0) and g(1), which can be fixed to make a prettier plot, but I'm not sure it's a better function for it. Plot: http://www.tiac.net/~sw/2010/03/Superbola/superbolav.pdf Code: http://www.tiac.net/~sw/2010/03/Superbola/superbolav.py --Steve

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[math-fun] Logicomix
by Thane Plambeck 29 Apr '10

29 Apr '10

Anyone have an opinion about this? I just bought it. **************** http://www.logicomix.com/ Covering a span of sixty years, the graphic novel *Logicomix* was inspired by the epic story of the quest for the Foundations of Mathematics. This was a heroic intellectual adventure most of whose protagonists paid the price of knowledge with extreme personal suffering and even insanity. The book tells its tale in an engaging way, at the same time complex and accessible. It grounds the philosophical struggles on the undercurrent of personal emotional turmoil, as well as the momentous historical events and ideological battles which gave rise to them. The role of narrator is given to the most eloquent and spirited of the story’s protagonists, the great logician, philosopher and pacifist Bertrand Russell. It is through his eyes that the plights of such great thinkers as Frege, Hilbert, Poincaré, Wittgenstein and Gödel come to life, and through his own passionate involvement in the quest that the various narrative strands come together. -- Thane Plambeck tplambeck(a)gmail.com http://thaneplambeck.typepad.com/

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[math-fun] *^ commutators
by rcs＠xmission.com 28 Apr '10

28 Apr '10

I've been playing with commutators lately, and the following puzzle turned up ... Suppose A and B are integers > 1. When is A^B / B^A an integer? What values are possible? Clearly A and B must have the same prime divisors, but the primes can occur to different powers. A = 45, B = 75 is a suggestive example. Rich

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[math-fun] Help with a PowerPowerMod
by Hans Havermann 27 Apr '10

27 Apr '10

A large number n = 689*10^((5889*10^((578*10^196-218)/81)-165)/81)-1 and I would like to know the value of n mod 81. Is this doable?

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[math-fun] Conditional probability using "natural frequencies"
by Jason 27 Apr '10

27 Apr '10

This approach seems like a good way to teach people to handle garden-variety conditional probabilities: http://opinionator.blogs.nytimes.com/2010/04/25/chances-are/

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[math-fun] Bloopy math counting problem
by Dan Asimov 26 Apr '10

26 Apr '10

Define an n-bloop to be any set of any n disjoint geometric circles (of possibly unequal radii) in the plane. If B is an n-bloop, then by continuously varying the centers and radii of its circles C_k we can get a family {B_t | 0 <= t <= 1} of n-bloops with B_0 = B . . . that is, AS LONG AS at no time t do any two of the n circles of B_t intersect. If B = B(0) can be so varied to B' = B(1), we say B and B' are *equivalent*. Let f(n) be the number of equivalence classes of n-bloops. What is an expression for f(n) in closed form? (I just thought of this problem and have a quick guess, but don't know the answer at the moment.) --Dan ________________________________________________________________________________________ "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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