Science > Physics > Kronecker Delta, Delta Function, Green's Function, Probable Influence 3: Correction
| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
15 Feb 2006 09:52:25 PM |
| Object: |
Kronecker Delta, Delta Function, Green's Function, Probable Influence 3: Correction |
From Osher Doctorow
Part 1 of this thread in its proof is partly correct and partly wrong,
though the result comes out the same up to the case x = 1 (see below).
Let's define:
1) C = 1 + y - x - y/x
Since y < = x, it follows that C < = 1. To prove that C > = 0, we
write:
2) C = 1 + y(1 - 1/x) - x = (1 - x) + (y/x)(x - 1) = (1 - x)(1 - y/x) >
= 0
which is true since x < = 1 and y < = x.
From the factorization in (2), to find out where C = 0, we solve:
3) (1 - x)(1 - y/x) = 0
which holds iff y = x or x = 1 or both. So, up to (except for) the
case x = 1 and y < x, it follows that C = 0 iff y = x.
Next, let's see where C = 1 if anywhere. Notice from (1) that if y =
0, which doesn't constrain x (because y < = x), then we have:
4) C = 1 - x (for y = 0)
As x --> 0+, C --> 1- (that is to say, as x > 0 gets closer and closer
to 0, C < 1 gets closer and closer to 1).
So the notation "C = 1- iff x = 0+" or to rewrite (5) of Part 1, "C =
1- iff x = 0+", expresses this result correctly. Formally:
5) C = 0 iff x = y except for the case x = 1 and y < 1
C = 1- iff x = 0+
From here on, part 2 then follows without corrections with the
qualification "except for the case x = 1 and y < 1".
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Kronecker Delta, Delta Function, Green's Function, Probable Influence 3: Correction |
15 Feb 2006 10:09:37 PM |
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From Osher Doctorow
Could we have chosen y > 0 instead of y = 0 in the C = 1- scenario?
In other words, although the proof holds for y = 0, could the "iff" (if
and only if) for that part actually be "if"? If we choose y > 0, then
since y < = x, we have that x --> 0+ will eventually constrain y -->
0+. So this choice of y > 0 gives redundant information regarding the
limit or one-sided limit for C. There is, however, a well-known
difficulty concerning the limit or one-sided limit of y/x as x --> 0+
with y < = x, namely that y can be chosen as rx for 0 < = r < = 1 and
therefore y/x = r is multiple-valued and so has no limit or even
one-sided limit at 0. That is the difficulty with conditional
probability (which is y/x) at or near x = 0. Therefore, just as we
constrained the previous result up to x = 1 with y < x, so we have to
constrain the present result up to y = 0 (the "opposite end of the
scale), that is to say we impose the condition y = 0 on C when taking
the C = 1- one-sided limit. And with these 2 constraints, the results
of part 2 hold (but subject to these 2 constraints).
Osher Doctorow
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| User: "John" |
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| Title: Re: Kronecker Delta, Delta Function, Green's Function, Probable Influence 3: Correction |
16 Feb 2006 09:46:55 AM |
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QUACK !!
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