[math-fun] Wilson's 1D gun problem, reply to Kleber objection.
From: Gareth McCaughan <gareth.mccaughan@pobox.com> Warren Smith wrote:
The bullet that annihilates the leader, follows a line segment in the time-space plane. This line segment totally separates what came before & what follows. The new stuff is a total restart, totally independent of all the previous stuff.
Really?
If "before" means "before the time at which that bullet is fired" or something of the kind: the line segment doesn't separate "before" from "after" because it's at an angle.
--before/after means more precisely on one side vs other of that line.
If "before" means "on the earlier side of the line segment": the line segment does separate "before" from "after" but that doesn't make "before" and "after" independent, because e.g. if the-bullet-that-annihilates-the-leader leaves at time t then a superfast bullet at time t+1 would hit *that* bullet, making it not annihilate the leader after all. So conditional on that bullet actually annihilating the leader, there are speed constraints on bullets fired shortly after that one that weren't there at time t=0.
Am I missing something here?
--ok, you (Gareth) are correct, it is not independent because a superfast bullet "after" would not be permitted. However, intuitively speaking, it seems like this kind of lack of independence only helps my argument, not hurts it. So the proof seems no longer to be a proof, I agree, but the "flaw" in the proof at least *seems* to only be helping the proof... i.e. it does not (or hardly at all) diminish my faith that my answer is correct. Further, perhaps this intuition can be made rigorous? Modify Wilson's problem so that the permitted velocity range is not [0,1] but rather [0,X] for some positive constant X, or allow X to be an increasing function of time trapped in [0,1]. Obviously this with X constant is an equivalent problem (just rescale time) and with X varying in trapped-monotone-increasing manner as defined, it seems to me all previous arguments still work. Now in my proof, Gareth's complaint basically causes us to have the [0,X] version of the problem for various X with 0<X<1 during period after each "reset" where X increases back toward 1. This would seem only to make my argument more true?
I put a little empirical data at https://sites.google.com/site/michaelkleber/wilson-s-gun/rightmost-velocity-... In particular there's a picture there of the CDF of velocities of the rightmost bullet at time 100 and 1000, from 20K random trials. Warren, I'm not sure whether you were arguing that these two distributions ought to have the same shape, but certainly they do not. --Michael On Wed, Jun 13, 2012 at 1:34 PM, Warren Smith <warren.wds@gmail.com> wrote:
From: Gareth McCaughan <gareth.mccaughan@pobox.com> Warren Smith wrote:
The bullet that annihilates the leader, follows a line segment in the time-space plane. This line segment totally separates what came before & what follows. The new stuff is a total restart, totally independent of all the previous stuff.
Really?
If "before" means "before the time at which that bullet is fired" or something of the kind: the line segment doesn't separate "before" from "after" because it's at an angle.
--before/after means more precisely on one side vs other of that line.
If "before" means "on the earlier side of the line segment": the line segment does separate "before" from "after" but that doesn't make "before" and "after" independent, because e.g. if the-bullet-that-annihilates-the-leader leaves at time t then a superfast bullet at time t+1 would hit *that* bullet, making it not annihilate the leader after all. So conditional on that bullet actually annihilating the leader, there are speed constraints on bullets fired shortly after that one that weren't there at time t=0.
Am I missing something here?
--ok, you (Gareth) are correct, it is not independent because a superfast bullet "after" would not be permitted.
However, intuitively speaking, it seems like this kind of lack of independence only helps my argument, not hurts it. So the proof seems no longer to be a proof, I agree, but the "flaw" in the proof at least *seems* to only be helping the proof... i.e. it does not (or hardly at all) diminish my faith that my answer is correct.
Further, perhaps this intuition can be made rigorous? Modify Wilson's problem so that the permitted velocity range is not [0,1] but rather [0,X] for some positive constant X, or allow X to be an increasing function of time trapped in [0,1].
Obviously this with X constant is an equivalent problem (just rescale time) and with X varying in trapped-monotone-increasing manner as defined, it seems to me all previous arguments still work.
Now in my proof, Gareth's complaint basically causes us to have the [0,X] version of the problem for various X with 0<X<1 during period after each "reset" where X increases back toward 1. This would seem only to make my argument more true?
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participants (2)
-
Michael Kleber -
Warren Smith