Re: [math-fun] computing non-rigorously defined quantities
Some wags would say that many (most?) large scale, long term computer simulations fall into this category, although you might have a hard time getting their proponents to agree with this assessment. (You know what I'm talking about!) On the other hand, Chris Anderson's now-famous rant "The End of Theory: The Data Deluge Makes the Scientific Method Obsolete" argues that data trumps theory: http://www.wired.com/science/discoveries/magazine/16-07/pb_theory "All models are wrong, but some are useful." So proclaimed statistician George Box 30 years ago, and he was right. Peter Norvig, Google's research director, offered an update to George Box's maxim: "All models are wrong, and increasingly you can succeed without them." At 08:41 AM 7/2/2013, James Propp wrote:
For some thoughts about the enterprise of computing quantities one hasn't rigorously defined, see http://mathoverflow.net/questions/135536/procedure-based-as-opposed-to-defin... (which I hope will elicit interesting responses over the next few days).
Jim Propp
Some of the funsters' flow work fits here: Giving a numerical definition of the half-flow, or the t-th flow, for a function, assuming it's close to an analytically flowable function. Simple examples: model f(x) = x^2 + x as the formal power series near x=0, with half-flow F(.5,x) = x + .5x^2 + ... (the rest of the power series can be computed from commutative composition, F(.5,f(x))=f(F(.5,x))). And also near x=infinity, model f(x) as near g(x) = x^2, with half flow G(.5,x) = x^sqrt2. For intermediate values like x=1, iterate f forward or backward to get close to 0 or infinity. The numerical results of the two methods almost match, but not quite. Another, even more surprising example, is f(x) = sqrt2^x. This has fixed points at x = 2 & 4, and is well-behaved (nearly linear, smooth, etc.) near the fixed points and in the interval between. Using the procedure "iterate forward or backward to get near the fixed point, model F(.5,-) with the square-root of the derivative, uniterate", evaluate the half-flow at x=3. Which fixed point to use? The two results agree to about 25 decimals, but then differ. Third example: Modular Dilogarithms (see http://richard.schroeppel.name:8015/dilog-paper-020402) "defines" a modular version of the dilog function Li2(z), (with power series z + z^2/4 + z^3/9 + ... + z^n/n^2 + ..., Li2(1)=zeta(2)=pi^2/6), as "something that matches the dilog functional equations", and does a computer search to find examples. Each of these defines a function as the outcome of a calculation, and the first two examples can even prove the limits exist. But the failure to be consistent over the choice of fixed points, or to be unique, undermines the believability of the answer. Rich -----
At 08:41 AM 7/2/2013, James Propp wrote:
For some thoughts about the enterprise of computing quantities one hasn't rigorously defined, see http://mathoverflow.net/questions/135536/procedure-based-as-opposed-to-defin... (which I hope will elicit interesting responses over the next few days).
Jim Propp
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The first example is especially perplexing, since it has been proved (see Erdos & Jabotinsky, 1960) that f(x) = x^2 + x has no non-integer analytic iterates. --Dan On 2013-07-02, at 1:16 PM, rcs@xmission.com wrote:
Some of the funsters' flow work fits here:
Giving a numerical definition of the half-flow, or the t-th flow, for a function, assuming it's close to an analytically flowable function. Simple examples: model f(x) = x^2 + x as the formal power series near x=0, with half-flow F(.5,x) = x + .5x^2 + ... (the rest of the power series can be computed from commutative composition, F(.5,f(x))=f(F(.5,x))). And also near x=infinity, model f(x) as near g(x) = x^2, with half flow G(.5,x) = x^sqrt2. For intermediate values like x=1, iterate f forward or backward to get close to 0 or infinity. The numerical results of the two methods almost match, but not quite.
Another, even more surprising example, is f(x) = sqrt2^x. This has fixed points at x = 2 & 4, and is well-behaved (nearly linear, smooth, etc.) near the fixed points and in the interval between. Using the procedure "iterate forward or backward to get near the fixed point, model F(.5,-) with the square-root of the derivative, uniterate", evaluate the half-flow at x=3. Which fixed point to use? The two results agree to about 25 decimals, but then differ.
Third example: Modular Dilogarithms (see http://richard.schroeppel.name:8015/dilog-paper-020402) "defines" a modular version of the dilog function Li2(z), (with power series z + z^2/4 + z^3/9 + ... + z^n/n^2 + ..., Li2(1)=zeta(2)=pi^2/6), as "something that matches the dilog functional equations", and does a computer search to find examples.
Each of these defines a function as the outcome of a calculation, and the first two examples can even prove the limits exist. But the failure to be consistent over the choice of fixed points, or to be unique, undermines the believability of the answer.
Rich
-----
At 08:41 AM 7/2/2013, James Propp wrote:
For some thoughts about the enterprise of computing quantities one hasn't rigorously defined, see http://mathoverflow.net/questions/135536/procedure-based-as-opposed-to-defin... (which I hope will elicit interesting responses over the next few days).
Jim Propp
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I'm wondering whether it's possible to rigorize some of the intuitions underlying this discussion. We have some mathematical operator that is well-defined over some domain. Over that domain, we observe that the operator obeys certain formal algebraic laws; let's say that we have proven these laws over the well-defined domain. Now, we seek to extend the domain of the operator, feeding it "illegal" inputs. Although the result is undefined in the original formulation, the formal laws we discovered earlier can serve as a sort of life-raft. If we assume these laws continue to hold, we can sometimes deduce putative values for the operator outside its original domain. It feels to me as if there are two cases: in one, the values thus deduced are of the same "data type" as the old ones. This is the case for a lot of Euler's "summations" of divergent series. In the other case, one has to invent a new data type, usually a generalization of the original range of the operator. This is, essentially, the origin of complex numbers. Note, in particular, the history of the cubic formula; at a certain stage, algebraists were willing to use the formula even in cases that required them to take square roots of negative reals, because all the problematic expressions canceled, leaving a real result. On Tue, Jul 2, 2013 at 5:21 PM, Dan Asimov <dasimov@earthlink.net> wrote:
The first example is especially perplexing, since it has been proved (see Erdos & Jabotinsky, 1960) that f(x) = x^2 + x has no non-integer analytic iterates.
--Dan
On 2013-07-02, at 1:16 PM, rcs@xmission.com wrote:
Some of the funsters' flow work fits here:
Giving a numerical definition of the half-flow, or the t-th flow, for a function, assuming it's close to an analytically flowable function. Simple examples: model f(x) = x^2 + x as the formal power series near x=0, with half-flow F(.5,x) = x + .5x^2 + ... (the rest of the power series can be computed from commutative composition, F(.5,f(x))=f(F(.5,x))). And also near x=infinity, model f(x) as near g(x) = x^2, with half flow G(.5,x) = x^sqrt2. For intermediate values like x=1, iterate f forward or backward to get close to 0 or infinity. The numerical results of the two methods almost match, but not quite.
Another, even more surprising example, is f(x) = sqrt2^x. This has fixed points at x = 2 & 4, and is well-behaved (nearly linear, smooth, etc.) near the fixed points and in the interval between. Using the procedure "iterate forward or backward to get near the fixed point, model F(.5,-) with the square-root of the derivative, uniterate", evaluate the half-flow at x=3. Which fixed point to use? The two results agree to about 25 decimals, but then differ.
Third example: Modular Dilogarithms (see http://richard.schroeppel.name:8015/dilog-paper-020402) "defines" a modular version of the dilog function Li2(z), (with power series z + z^2/4 + z^3/9 + ... + z^n/n^2 + ..., Li2(1)=zeta(2)=pi^2/6), as "something that matches the dilog functional equations", and does a computer search to find examples.
Each of these defines a function as the outcome of a calculation, and the first two examples can even prove the limits exist. But the failure to be consistent over the choice of fixed points, or to be unique, undermines the believability of the answer.
Rich
-----
At 08:41 AM 7/2/2013, James Propp wrote:
For some thoughts about the enterprise of computing quantities one hasn't rigorously defined, see
http://mathoverflow.net/questions/135536/procedure-based-as-opposed-to-defin...
(which I hope will elicit interesting responses over the next few days).
Jim Propp
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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