Re: Calculus XOR Probability

In article <MPG.1ede39f76ea10d98ad21@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Matt Gutting said:

Oh. Then the symbol doesn't necessarily mean anything. Can you take a
limit as
n approaches soemthing that doesn't exist?

No, and I'm not. It's not true that "n approaches infinity"; n increases
without
bound. And one can certainly take a limit as n increases without bound.

Right, but you can't say what the curve IS in the limit without considering
having REACHED the limit.

What definition of limit process is TO using in which the limit is
actually "reached", in the sense of the relevant variable actually
taking the "limit" value?

Consider, for example, lim_{x -> 0} sin(x)/x.
That limit is well known to exist even though x = 0 is specifically
prohibited in the limit process.

TO is still off in his never-never land, full of childish notions, and
afraid to grow up.


The staircase approaches the diagonal, location-wise, but in the end, it's
not
the same object, as I've demonstrated.

TO's "demonstrations" are invalid in normal space, and hold, if at all,
only in his own dream world.


You haven't defined "infinitesimal" to anyone's satisfaction, certainly not
to mine.

Any finite divided by any infinite yields an infinitesimal.

As an "infinite" in TOmatics has not yet been shown to be a number
capable of being used in arithmetic, dividing by it is not yet possible.


No; you're assuming that the limit is some construct involving
infinitesimals.

I am assuming that when n is a specific infinity, 1/n is a specific
infinitesimal, and that's part of the problem we're discussing.

WE are not discussing any such thing. TO is pontificating, but not being
the Pope, nor being able to speak ex cathedra, he fails at it.

If an infinitesimal is larger than 0, one can distinguish between one
endpoint
and the other one an infinitesimal distance away, no?

Yes, on the infinitesimal scale, not on the finite scale, as in standard
mathematics.

In rings with the Archimedean property, one does not have infinitesimals.

http://planetmath.org/encyclopedia/ArchimedeanProperty.html

Let x be any real number. Then there exists a natural number, n,
such that n > x.


Corollary 1 If x and y are real numbers with x > 0, there exists a
natural, n, such that n*x < y.

Corollary 2 If x is a real number greater than 0, there exists a
natural, n, such that 0 < 1/n < x.

Corollary 3 If x and y are real numbers with x < y , there exists a
rational number, r, such that x < r < y.




Generally, my understanding of curves is that they're defined almost
everywhere.
I'll check up on this.

That's the common notion of a curve, but a general definition may be adopted
that distinguishes between continuous and discrete curves. What exactly do
you
call a series of points separated by finite space?

A sequence

It can be considered a
discrete curve, and Archimedean principle may be applied to get the common
notion of curves as continuous.

What principle does TO want to apply to "continuous" curves that are not
continuous? A "curve" with a gap in it is not continuous across the gap.




By "the limit as n->oo", I assume that you mean "the limit *of a
sequence
of curves C_n* as n->oo". I'm trying to figure out what you mean by the
sentence though. The n *could* refer to an indexed curve in the sequence
of curves, or to an indexed point on a specified curve. Or, I suppose,
it's possible it might refer to something else. Clarification?

My pleasure. The variable n here denotes the number of segments in the
curve.

Except that a staircase of n steps is made up of 2*n segments.


Each of those segments has a position in the sequence, m, from 1 through
n
(including the initial offet). Each n, or number of segments, denotes a
different curve, and as n->oo and the number of segments increases
without
bound, we have the "curve in the limit". What groups all these curves
together
as one family is the pair of formulas that give the offsets in each
segment of
the curve, f_x and f_y. Did that help clarify things? :)


To an extent. I was indexing curves by their position in the sequence,
which
was coincidentally the number of segments; so that agrees with what you're
saying. But I'm still unclear about what you mean by "we have the 'curve
in the limit'." Is the "curve in the limit" one of the curves in your
sequence of curves?

Sure, if we consider n from 1 through Big'un,

Who is this "we"? So far the only person who considers "Big'un"
anything but a bad joke is TO.



If so, at what position can it be found?

Big'un.

Where is that?

Are there curves beyond it in the sequence? What do they look like?

Well, where the staircase becomes infinitesimally close to the diagonal

Which it never does in any limiting process. There are NO limit
definitions in Cartesian geometry which even hint at infinitesimal
anything.
.

Relevant Pages

  • Re: Calculus XOR Probability
    ... but you can't say what the curve IS in the limit without considering ... without bound" without having to talk about "reaching infinity". ... and the object is no longer a staircase. ... The sequence of staircases "approaches the diagonal" arbirtarily ...
    (sci.math)
  • Re: EC-PRNG
    ... That's why you compute the group order already taking the point at infinity ... If you don't, then you get a curve of order n-1, not n as you ... No matter how small the probability of the step 3c is; ... Donald Knuth's quote about not using random methods to generate random ...
    (sci.crypt)
  • Quantum Gravity 166.3: Comparing dy/dt = ky and y = exp(kt) As Linear vs Exponential
    ... a bounded linear operator or bounded linear transformation T has ... rather Birkhoff Causation which approximates Probable ... To try to piece together how they relate, look at the curve y = exp ... is not only a Singularity but an Infinity! ...
    (sci.physics)
  • Quantum Gravity 136.1: Non-Tunneling Condition Yields 1 + y - x
    ... terminates on the boundary of the unit square except at. ... then the y coordinate of the curve would have reached infinity ... Substitute for gfrom into: ...
    (sci.physics)
  • Re: Infinitly many squares for y rational?
    ... seek rational points on the curve. ... elliptic curves, and Silverman's first book on elliptic curves. ... s=0 (and the point at infinity). ... no solution to x+1/x=0 in the rationals, so the only points on the ...
    (sci.math.research)