Re: Calculus XOR Probability

Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx said:



What is missing is a statement of /exactly what you mean/ by "the
length of (the limit of the staircases) is {whatever you propose}".

The limit of the staircases is the series Sum(n->oo: {1/n,0},{0,1/n}). That's n repetitions of a step with length 2/n, for a total length of 2.

You're assuming again that you can interchange the sum and the limit process.
The length of the limit of staircases need not equal the limit of the length
of staircases with the standard definition.

Do you mean to say that the limit of the staircases is a series? That's how
your sentence is phrased, but it doesn't seem to make sense. You're apparently
making a sequence of geometric figures (staircases), then stating that the
limit is an infinite series, presumably evaluated in the same way that
infinite series typically are - although you need to be clearer about the
meaning of "n->oo (1/n,0),(0,1/n)". How does the limit of a sequence of
geometric figures get to be a sequence of real numbers? Or is that what you
meant?


In order for me to understand your answer, you must first state
/exactly what you mean/ by "the limit of the staircases"; which you
have not done in the above paragraphs. Is "the limit of the staircases"
a function? Is it a real number? A set of line segments? A set of pairs
of pairs in R^2 x R^2? A white elephant?

I stated already it's a sequence of line segments. See above, "defined as a pair of reals which represent the x and y coordinate differences between subsequent points". Each of those pairs represents a line segment.

So, each staircase is a sequence of line segments. How do you decide
that the limit is also a sequence of line segments?


The closest you get is this cryptic comment: "Because of the difference
in vector direction, even at the infinitesimal scale, the staircase is
longer than the diagonal." But this doesn't tell me what "the limit of
the staircases" is; it simply mentions several (undefined) properties
you propose it to have.

It's a staircase with oo stairs, each 2/oo long, given riser and tread. What is your question?

*My* question is, since you haven't actually defined oo, how can you tell
whether oo or 2/oo exist?


For example, presumably there is some point p = (a,b) in R^2 that is in
the limit of the staircases. Does that point satisfy b = 1 - a, or does
it not?

The tread of one step meets the riser of the next at a point on the diagonal. Where the riser meets its tread, that corner is NOT on the diagonal, even if it may be only an infinitesimal difference away, and consider coincident with the line according to stard finitist limits.

Given that point p, what is the "vector direction, at the infinitesimal
scale" associated with it? Can we deduce it from the values of a and b?
For example, how do I determine the "vector direction, at the
infinitesimal scale" at the point (1/2,1/2) (which I presume is in the
"limit of the staircases")?

The point (1/2,1/2) is in every staircase for n>1, for sure. The direction of the tread before it is horizontal, and the diretcion of the riser after that point is vertical. Remember, directions are not defined for points, but for segments. That point has not direction of its own, hence the need to look at the limit, not of the points, but of the segments.

How do you know that the limit of the segments exists, and that it is a
segment?


Given two points p and q in R^2 which are in the limit, how do I
determine whether p and q have the same or different "vector
directions, at the infinitesimal scale"?

Points do not have directions, ultimately. The segment {1/2,0} is horizontal, and {0,1/2} is vertical.

Okay, so how about the infinitesimal scale?


Once you have addressed these questions, we can suppose that your
definition of "the limit of the staircases" is a mathematical object
called "L". /Then/ I can evaluate a statement you might make of the
form "the length of L is {whatever you propose}".

Are you sure you won't ask the alreayd answered questions, again?

I still have questions about your answers to the questions.


Until then, you haven't defined what you mean by "the length of (the
limit of the staircases)"; all you have defined is "the limit of (the
length of the staircases)"; and at least in its result, we are all in
agreement: the limit of the length of the staircases is 2, and the
length of the diagonal is sqrt(2).

But you disagree that the limit of the staircases is anything other than the diagonal, whereas I have demonstrated a form of limit which shows clearly that there's a difference, and which accounts precisely for the error.

I don't see a clear definition of limit. Can you fill in the blanks here:

DEFINITION: The limit of a _____ (insert name of mathematical object)
is a ___ (insert name of a mathematical object) satisfying the following
criteria: ______.

Both blanks have to be filled with terms which either are agreed upon
generally, or are defined in turn according to the template provided.

Once you can fill in those blanks, then we have something we can talk
about. Until then, your definition is not sufficiently well-formed to
be able to discuss anything related to it.

*** Posted via a free Usenet account from http://www.teranews.com ***
.

Relevant Pages

  • Re: Calculus XOR Probability
    ... By defining the curve as a sequence of segments, rather than a set of locations, the difference is quite detectable, because the segment definition preserves the notion of direction IN THE LIMIT. ... the 1/n's have a limit of 0 as n->oo, but what that essentially means is that, for any given actual infinite n, 1/n is infinitesimal, and larger than absolute 0. ... definition of "the limit of the staircases" is a mathematical object ... A curve is defined as a series of pairs, the first denoting the x and y offset of the first point from the origin in R^2, and each subsequent pair being the offset of the next point from the last. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... The limit of the staircases is the series Sum(n->oo: ... presumably evaluated in the same way that infinite series typically ... actually curve, such as a section of circular arc. ... The segments all have endpoints on the curve, ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... sequence of curves "staircases". ... are where the risers meet the treads, but then there are infinitesimal segments ... the length of the limit of the set of staircases D is not ... You postulate something magical happening in the infinite case ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... sequence of curves "staircases". ... are where the risers meet the treads, but then there are infinitesimal segments ... the length of the limit of the set of staircases D is not ... You postulate something magical happening in the infinite case ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... as sequences of segments .... ... staircases is 2; which no one is disagreeing with; and that the length ... I stated already it's a sequence of line segments. ... Definition 2 says that the endpoints of these line segments are real ...
    (sci.math)