Message from discussion
Calculus XOR Probability
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From: Tony Orlow <a...@cornell.edu>
Newsgroups: sci.math
Subject: Re: Calculus XOR Probability
Date: Wed, 26 Apr 2006 16:55:50 -0400
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cbr...@cbrownsystems.com said:
> Tony Orlow wrote:
> > cbr...@cbrownsystems.com said:
> > > Tony Orlow wrote:
> > > > cbr...@cbrownsystems.com said:
> > >
> > > <snip>
> > >
> > > > > What do you think the limiting curve of "y = sin(n*x)/sqrt(n)" is, as
> > > > > n->oo?
> > > >
> > > > y=0, or more specifically, some indeterminate infinitesimal value between -
> > > > 1/sqrt(oo) and 1/sqrt(oo).
> > > >
> > >
> > > So in other words, you don't know or can't say what the limit is,
> > > besides that it is some indeterminate value?
> >
> > No more than you know the limit of y=sin(n*x) as n->oo. You divide by a term,
> > sqrt(n), which goes to oo as n->oo, thus reducing the value at all points to
> > standard 0, but this ignores the infinitesimal differences over infinitesimal
> > distances that make it a curve instead of a straight line, and account for the
> > difference in length.
>
> So you doubt that the limit of {1/2, 1/4, 1/8, ..., 1/2^n,...} = 0?
No, what logic led you to that conclusion?
>
> >
> > >
> > > Do you also think that "sqrt(2)*sqrt(2)" is "2", or more specifically,
> > > some indeterminate value around 2?
> >
> > It's 2. That's not a compressed oscillating function.
> >
>
> So you also doubt that the limit of {-1/2, 1/4, -1/8, ... (-1^n)/(2^n),
> ...} = 0?
No, but I doubt that the length of the curve defined by (-1/2)^x from 1 to any
n is equal to n, just because the limiting curve appears to be y=0, expecially
because it travels around in the complex plane and is far from straight.
>
> > >
> > > Just like the value of sqrt(2) is obtained by a /definition/ of sqrt,
> > > the limit of "y = f_n(x)" is obtained by a /definition/ of limit. You
> > > don't have to guess.
> >
> > And yet, your definition of limit is insufficient for measuring the arc length,
> > since it's pointwise and nondirectional. That's the problem, not inductive
> > proof of equality in the infinite case.
>
> Why do you think I called it "the limit of" and not "the accurate
> method of measuring arc length of"? Do you think that was an accidental
> typo, or do you think I actually meant only what I wrote?
I think that "limit" is used in a number of ways, as you yourself stated, and
that if your example was meant to demonstrate that assuming an equality holds
in the infinite case is unsound, that it shouldn't also include a separate
obviously unsound assumption that fully accounts for the error demonstrated in
the proof. You try to attribute the error to one thing, but it's obviously due
to the other.
>
> >
> > >
> > > > >
> > > > > Is the formula "y = 0" differentiable?
> > > >
> > > > Yes, as so stated.
> > > >
> > > > >
> > > > > Given n, what is the length of the curve "y = sin(n*x)/sqrt(n)" between
> > > > > x = 0 and x = 2*pi?
> > > >
> > > > Given y=0 as the limit of the curve as n->oo being a pointwise limit, and not a
> > > > parallel element limit, the equation between the length of the curve with the
> > > > length of the line is invalid. You cannot measure length with dimensionless
> > > > points, as I've pointed out.
> > > >
> > >
> > > So you claim that the length of "y = sin(16*x)/4" between 0 and 2*pi is
> > > undefined?
> >
> > No, I claim that y=sin(n*x)/4 does not have the same length as y=0, for any n,
> > finite or infinite. Do you claim that the length of y=sin(n*x) from 0 to 2 pi
> > is 2 pi, for ANY x?
>
> To nitpick: For any x, y = sin(n*x) is a point; so of course it has 0
> length. If you mean any n, then yes - when n = 0, the length of y =
> sin(n*x) = sin(0*x) = sin(0) = 0 is 2*pi.
Huh? We're talking about arc length, and you're claiming 0=2*pi??? That's only
true for angular measure, not linear. Are you just trying to test to see if I'm
a total moron? Hmmm....
Apparently, what you meant to say is that, for n=0 and finite x, y=sin(n*x)=sin
(0)=0, and the length of y=0 from 0 to 2 pi is 2 pi. So, I will concede that
for n=0, the length is 2 pi. For any nonzero n, then, is the length 2 pi?
>
> But of course you're right. However, I'm not claiming (and never have
> been as part of premise A) that the length of y = sin(n*x)/sqrt(n) gets
> closer to 2*pi as n -> oo. In fact, in this case the length gets
> arbitrarily large, even as the curve gets closer to y = 0.
It gets arbitrarily large as n->oo within [0,2 pi]? Ummm... yes I can see that.
While the division by sqrt(n) MIGHT make it closer in length to the straight
line, the multiplication of x by n in the sine function compresses the curve
into a 2D space completely filled with the curve, indicating an infinite curve.
So, yes, I agree 1000%. That's a good example where the limiting curve has very
different length from the curve in the limit. But, are limit curves really
analogous to the type of limit Han was using? It seems rather different.
>
> Instead, I note that at any point x (say x = 1), the /value/ of y(1) =
> sin(n*1)/sqrt(n) gets arbitrarily close to 0 as n-> oo; with the limit
> being y(1) = 0; and in fact, in the limit, y(x) = 0 for all x. We
> usually write the resulting function as "y = 0"
Yes, I can see that's a limit curve in the sense you're using. I don't disagree
with that. I still wonder why the example was brought up, interesting as it may
be. But, interesting as it is, I guess that's neither here nor there. :)
>
> >
> > >
> > > > >
> > > > > Is the limit of that length, as n -> oo, the same as the length of "y =
> > > > > 0" between x = 0 and x = 2*pi?
> > > >
> > > > No, it's not, for the reasons reiterated. That's not how one approximates the
> > > > length of the curve. One uses straight elements which are parallel. Do you
> > > > argue with that?
> > >
> > > No; but then I never said that the limit of (the length of the curves
> > > "y =f_n(x)") is supposed to neccessarily be the same as length of (the
> > > limit of the curves "y = f_n(x)").
> > >
> > > In fact, it's clear from the definitions of these two things that this
> > > assumption is false.
> >
> > It's clear, given your notion of limits of curves, that they are inadequate for
> > calculating measure, that's true. However, that notion of limits does not
> > include direction, but only location, of points on the curve, and that's why it
> > fails.
>
> You're right. And that's how we see that the claim "some property of
> (the limit of a sequence) is always the same as the limit of (some
> property of the nth element of the sequence)" (i.e., premise B) is
> false in general.
>
> In order to draw conclusions such as these, we need to know exactly
> what the definition of "limit" being used is; and then we can draw
> conclusions from that.
Okay, I think we agree. Now, I would pose this suggestion again. Inductive
proof is considered generally invalid "in the limit", that is, in the infinite
case. The method of inductive proof is only considered to prove a property for
all FINITE n. However, in this sense of a limit, my suggestion is that an
equality between expressions proven inductively holds for all n, finite or
infinite. I don't think that causes any problems such as what you've been
suggesting, do you?
Oh, time to get outta here. Have a nice evening.
>
> Cheers - Chas
>
>
--
Smiles,
Tony
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