mm-99
===
lim delta->0 [f(x+delta)-f(x)]/(delta)^2 exists.>
wondering if there are functions that the following limit does
exist:> lim delta->0 [f(x+delta)-f(x)] / [(delta)^alpha]
exists. where alpha>1> Any references would be nice
too.f(x+delta) = f(x) + delta f'(x) + delta^2/2 
f''(x) +
o(delta^2)[f(x+delta) - f(x)]/(delta)^2 = f'(x)/delta +
f''(x)/2 + o(1)Therefore, you must have 
f'(x) = 0 ie f
constant as it has been said. Ifthis is not for every x, then
you only have to have a null derivative atthe considered
points.For your second question, you must have the n-th ?st
derivatives withn = ?lpha) equal to zero.-- Nicolas
===
lim delta->0 [f(x+delta)-f(x)]/(delta)^2 exists.> >
wondering if there are functions that the following limit
does exist:> > lim delta->0 [f(x+delta)-f(x)] /
[(delta)^alpha] exists. where alpha>1> > Any references
would be nice too.> f(x+delta) = f(x) + delta f'(x) +
delta^2/2 f''(x) + o(delta^2)> [f(x+delta) -
f(x)]/(delta)^2 = f'(x)/delta + f''(x)/2 + 
o(1)> Therefore,
you must have f'(x) = 0 ie f constant as it has been said. If>
this is not for every x, then you only have to have a null
derivative at> the considered points.> For your second
question, you must have the n-th ?st derivatives with> n =
?lpha) equal to zero.Careful, careful; he didn't assume
the function was twicedifferentiable, or even once
differentiable for that matter. (Althoughit follows.)If the
intent is for the limit of (f(x+delta)-f(x))/delta^2 to
existfor ALL x, then of course f(x+t) - f(x) f(x+t)-f(x) lim
------------- = lim ----------- * t = 0 t->0 t t->0 t^2for
all x, so f is differentiable and f'(x) = 0 identically: so f
isconstant.If the intent is to ask whether it is possible for
this limit to existfor SOME x, of course it is. Take f(x) =
x^3, for example, at x = 0.--Ron Bruck
===
About 2 months ago
I ran across a problem on MIT's website to ?d theinde?ite
integral of a certain function. It was from some
problemsolving seminar, and I think it was rated as the
hardest problem ofall the problems during the semester. I
posted about a while backabout this integral but everybody
just said that since Mathematicacan't do it it's 
probably not
integrable in terms of elementaryfunctions. Well, after about
two months of toiling over this problem,I think I might
?ally have a solution. However, there's a fewplaces where,
at least to me, things were a little iffy about whetheror not
I could do what I was doing. So I want to put my solution
andsee if there's any ?o, the problem is to ?d in
closed formIntegral[f(x) dx], where f(x) =
x/Sqrt[x^4+4x^3-6x^2+4x+1]So, ?st note two things:1) f(x) =
f(1/x) (assuming of course that x is not 0)2) the polynomial
in the bottom is a reciprocal polynomialReciprocal
polynomials generally can be solved by making asubstitution
of the form t = x+1/x. So I use this to my advantagelater.So
anyway, ?st we use obvservation 1 to say that if I
=Integral[f(x) dx], then 2I = Integral[f(x) dx] +
Integral[f(1/x) dx]In the second integral, let u = 1/x, and
it transforms toIntegral[f(x) dx] - Integral[f(u)/u^2
du]Since x is just a placeholder, I replace the x's with 
u's
in the ?stintegral and get2I = Integral[(1 - 1/u^2)f(u)
du]Now, we use the second observation. We already have2I =
Integral[(1 - 1/u^2)*(u/Sqrt[u^4+4u^3-6u^2+4u+1]) du]Bringing
that u from the numerator into the denominator, we get2I =
Integral[(1 - 1/u^2)*(1/Sqrt[u^2+4u-6+4/u+1/u^2]) du]Now we
are set up to write the part under the square root as
apoloynomial in x+1/x. Indeed, we get2I = Integral[(1 -
1/u^2)*(1/Sqrt[(u+1/u)^2+4(u+1/u)-8]) du]Let v = u+1/u. Then
dv = (1-1/u^2) dv, which is perfect since there'sa (1-1/u^2)
sitting right there! So the integral transforms to:2I =
Integral[1/Sqrt[v^2 + 4v - 8] dv]Now this integral is easily
evaluated as 2I = Log[2 + v + Sqrt[v^2 + 4v - 8]]Substituting
v gives 2I = Log[2 + u + 1/u + Sqrt[u^4 + 4u^3 - 6u^2 + 4u +
1]]And substituting x givesI = 0.5*Log[2 + x + 1/x +
(1/x^2)Sqrt[x^4 + 4x^3 - 6x^2 + 4x + 1]]So, is this
right?
===
Nobody <toolshed37@yahoo.com> escribi.97 en el
mensaje> About 2 months ago I ran across a problem on MIT's
website to ?d the> inde?ite integral of a certain function.
It was from some problem> solving seminar, and I think it was
rated as the hardest problem of> all the problems during the
semester. I posted about a while back> about this integral
but everybody just said that since Mathematica> can't do it
it's probably not integrable in terms of elementary>
functions. Well, after about two months of toiling over this
problem,> I think I might ?ally have a solution. However,
there's a few> places where, at least to me, things were a
little iffy about whether> or not I could do what I was
doing. So I want to put my solution and> see if there's any
?> So, the problem is to ?d in closed form>>
Integral[f(x) dx], where f(x) = x/Sqrt[x^4+4x^3-6x^2+4x+1]>>
So, ?st note two things:>> 1) f(x) = f(1/x) (assuming of
course that x is not 0)> 2) the polynomial in the bottom is a
reciprocal polynomial>> Reciprocal polynomials generally can
be solved by making a> substitution of the form t = x+1/x. So
I use this to my advantage> later.>> So anyway, ?st we use
obvservation 1 to say that if I => Integral[f(x) dx], then 2I
= Integral[f(x) dx] + Integral[f(1/x) dx]>> In the second
integral, let u = 1/x, and it transforms to>> Integral[f(x)
dx] - Integral[f(u)/u^2 du]>> Since x is just a placeholder,
I replace the x's with u's in the ?st> integral and 
get>> 2I
= Integral[(1 - 1/u^2)f(u) du]>> Now, we use the second
observation. We already have>> 2I = Integral[(1 -
1/u^2)*(u/Sqrt[u^4+4u^3-6u^2+4u+1]) du]>> Bringing that u
from the numerator into the denominator, we get>> 2I =
Integral[(1 - 1/u^2)*(1/Sqrt[u^2+4u-6+4/u+1/u^2]) du]>> Now
we are set up to write the part under the square root as a>
poloynomial in x+1/x. Indeed, we get>> 2I = Integral[(1 -
1/u^2)*(1/Sqrt[(u+1/u)^2+4(u+1/u)-8]) du]>> Let v = u+1/u.
Then dv = (1-1/u^2) dv, which is perfect since there's> a
(1-1/u^2) sitting right there! So the integral transforms
to:>> 2I = Integral[1/Sqrt[v^2 + 4v - 8] dv]>> Now this
integral is easily evaluated as>> 2I = Log[2 + v + Sqrt[v^2 +
4v - 8]]>> Substituting v gives>> 2I = Log[2 + u + 1/u +
Sqrt[u^4 + 4u^3 - 6u^2 + 4u + 1]]Actually is2I = Log[2 + u +
1/u + Sqrt[u^2 + 4u - 6 + 4/u + 1/u^2]]I = (1/2)Log[2 + x +
1/x + Sqrt[x^2 + 4x - 6 + 4/x + 1/x^2]]I = (1/2)Log[2 + x +
1/x + Sqrt[x^4 + 4x^3 - 6x^2 + 4x + 1]/x]ButdI/dx = (x^2 -
1)/(2xSqrt[(x^4 + 4x^3 - 6x^2 + 4x + 1]) = (1/2)(1 -
1/x^2)f(x) ???There are something more wrong ...I think that
it is in> Integral[f(x) dx] - Integral[f(u)/u^2 du] (#1)>>
Since x is just a placeholder, I replace the x's with 
u's in
the ?st> integral and get>> 2I = Integral[(1 - 1/u^2)f(u)
du]It would be true if the integrals would be de?ites. But
not if they areinde?ites. The ?st integral in #1 is a
function of x, while the secondis a function of u, and x =/=
u, actually is x = 1/u. If you replace x by uin the ?st
integral, you must to replace dx by -du/u^2, and you get2I =
-2*Integral[f(u)/u^2 du]true, but useless ...-- Ignacio
Larrosa Ca.96estroA Coru.96a
(Espa.96a)ilarrosaQUITARMAYUSCULAS@mundo-r.com
===
[snip]> So
anyway, ?st we use obvservation 1 to say that if I =>
Integral[f(x) dx], then 2I = Integral[f(x) dx] +
Integral[f(1/x) dx]>> In the second integral, let u = 1/x,
and it transforms to>> Integral[f(x) dx] - Integral[f(u)/u^2
du]>> Since x is just a placeholder, I replace the x's with
u's in the ?st> integral and get>> 2I = Integral[(1 -
1/u^2)f(u) du][snip]This looks wrong. x is in fact not just
a placeholder, because you aredoing an inde?ite integral. Try
writing the integral with explicit limits,say from 1 to
X.However, is this step really necessary? You should be able
to get by withoutit.-Michael.
===
> So, the problem is to
?d in closed form> Integral[f(x) dx], where f(x) =
x/Sqrt[x^4+4x^3-6x^2+4x+1]MR1798560 (2002b:11093)van der
Poorten, Alfred J.(5-MCQR-NT); Tran, Xuan
Chuong(5-MCQR-NT)Quasi-elliptic integrals and periodic
continued fractions. (English. English summary)Monatsh.
Math. 131 (2000), no. 2, 155--169.11J70 (11Y65)Summary: In
this report we detail the following story. Several centuries
ago, Abel noticed that the well-known elementary integral
$$intfrac{dx}{sqrt{x^2+2bx+c}}=logleft(x+b+sqrt{x
^2+2bx+c}right)$$ is just a presage of more surprising
integrals of the form
$$intfrac{f(x)dx}{sqrt{D(x)}}=logleft(p(x)+q(x)
sqrt{D(x)}right).$$ Here $f$ is a polynomial of degree $g$
and the $D$ are certain polynomials of degree $deg
D(x)=2g+2$. Speci?ally, $f(x)=p'(x)/q(x)$ (so $q$ divides
$p'$). Note that, morally, one expects such integrals to
produce inverse elliptic functions and worse, rather than an
innocent logarithm of an algebraic function.Abel went on to
study abelian integrals, and it was Chebyshev who
explained---using continued fractions---what is going on with
these `quasi-elliptic' integrals. Recently, the second author
computed all the polynomials $D$ over the rationals of degree
4 that have an $f$ as above. We explain various contexts in
which the present issues arise. These contexts include
symbolic integration of algebraic functions, the study of
units in function ?lds and, given a suitable polynomial $g$,
the consideration of the period length of the continued
fraction expansion of the numbers $sqrt{g(n)}$ as $n$
varies over the integers. But the major content of this
survey is an introduction to period continued fractions in
hyperelliptic---thus quadratic---function ?lds.Reviewed by
M. Mend?s France-- Robin Chapman,
www.maths.ex.ac.uk/~rjc/rjc.htmlNeedless to say, I had the
last laugh. Alan Partridge, _Bouncing Back_ (14 times)
===
>
About 2 months ago I ran across a problem on MIT's website to
?d the> inde?ite integral of a certain function. It was from
some problem> solving seminar, and I think it was rated as the
hardest problem of> all the problems during the semester. I
posted about a while back> about this integral but everybody
just said that since Mathematica> can't do it it's 
probably
not integrable in terms of elementary> functions. Well, after
about two months of toiling over this problem,> I think I
might ?ally have a solution. However, there's a few> places
where, at least to me, things were a little iffy about
whether> or not I could do what I was doing. So I want to put
my solution and> see if there's any ? So, the problem
is to ?d in closed form> Integral[f(x) dx], where f(x) =
x/Sqrt[x^4+4x^3-6x^2+4x+1]> So, ?st note two things:> 1)
f(x) = f(1/x) (assuming of course that x is not 0)> 2) the
polynomial in the bottom is a reciprocal polynomial>
Reciprocal polynomials generally can be solved by making a>
substitution of the form t = x+1/x. So I use this to my
advantage> later.> So anyway, ?st we use obvservation 1 to
say that if I => Integral[f(x) dx], then 2I = Integral[f(x)
dx] + Integral[f(1/x) dx]> In the second integral, let u =
1/x, and it transforms to> Integral[f(x) dx] -
Integral[f(u)/u^2 du]> Since x is just a placeholder, I
replace the x's with u's in the ?st> integral and get> 
2I
= Integral[(1 - 1/u^2)f(u) du]> Now, we use the second
observation. We already have> 2I = Integral[(1 -
1/u^2)*(u/Sqrt[u^4+4u^3-6u^2+4u+1]) du]> Bringing that u
from the numerator into the denominator, we get> 2I =
Integral[(1 - 1/u^2)*(1/Sqrt[u^2+4u-6+4/u+1/u^2]) du]> Now
we are set up to write the part under the square root as a>
poloynomial in x+1/x. Indeed, we get> 2I = Integral[(1 -
1/u^2)*(1/Sqrt[(u+1/u)^2+4(u+1/u)-8]) du]> Let v = u+1/u.
Then dv = (1-1/u^2) dv, which is perfect since there's> a
(1-1/u^2) sitting right there! So the integral transforms
to:> 2I = Integral[1/Sqrt[v^2 + 4v - 8] dv]> Now this
integral is easily evaluated as > 2I = Log[2 + v + Sqrt[v^2
+ 4v - 8]]> Substituting v gives > 2I = Log[2 + u + 1/u +
Sqrt[u^4 + 4u^3 - 6u^2 + 4u + 1]]> And substituting x
gives> I = 0.5*Log[2 + x + 1/x + (1/x^2)Sqrt[x^4 + 4x^3 -
6x^2 + 4x + 1]]> So, is this right?> Well, I de?itely made
one mistake in that substituting v should notgive you what I
put. I think the ?al result should beI = 0.5*Log[(x+1)^2 +
Sqrt[(x+1)^4-12x^2]] - 0.5*Log[x]If someone could
differentiate this using Mathematica and simplify itI would
appreciate it.
===
> About 2 months ago I ran across a
problem on MIT's website to ?d the> inde?ite integral of
a certain function. It was from some problem> solving
seminar, and I think it was rated as the hardest problem of>
> all the problems during the semester. I posted about a
while back> about this integral but everybody just said
that since Mathematica> can't do it it's probably not
integrable in terms of elementary> functions. Well, after
about two months of toiling over this problem,> I think I
might ?ally have a solution. However, there's a few>
places where, at least to me, things were a little iffy about
whether> or not I could do what I was doing. So I want to
put my solution and> see if there's any ? > So,
the problem is to ?d in closed form> > Integral[f(x)
dx], where f(x) = x/Sqrt[x^4+4x^3-6x^2+4x+1]> > So, ?st
note two things:> > 1) f(x) = f(1/x) (assuming of course
that x is not 0)> 2) the polynomial in the bottom is a
reciprocal polynomial> > Reciprocal polynomials generally
can be solved by making a> substitution of the form t =
x+1/x. So I use this to my advantage> later.> > So
anyway, ?st we use obvservation 1 to say that if I =>
Integral[f(x) dx], then 2I = Integral[f(x) dx] +
Integral[f(1/x) dx]> > In the second integral, let u =
1/x, and it transforms to> > Integral[f(x) dx] -
Integral[f(u)/u^2 du]> > Since x is just a placeholder, I
replace the x's with u's in the ?st> integral and get> 
>
> 2I = Integral[(1 - 1/u^2)f(u) du]> > Now, we use the
second observation. We already have> > 2I = Integral[(1 -
1/u^2)*(u/Sqrt[u^4+4u^3-6u^2+4u+1]) du]> > Bringing that u
from the numerator into the denominator, we get> > 2I =
Integral[(1 - 1/u^2)*(1/Sqrt[u^2+4u-6+4/u+1/u^2]) du]> >
Now we are set up to write the part under the square root as
a> poloynomial in x+1/x. Indeed, we get> > 2I =
Integral[(1 - 1/u^2)*(1/Sqrt[(u+1/u)^2+4(u+1/u)-8]) du]> >
> Let v = u+1/u. Then dv = (1-1/u^2) dv, which is perfect
since there's> a (1-1/u^2) sitting right there! So the
integral transforms to:> > 2I = Integral[1/Sqrt[v^2 + 4v
- 8] dv]> > Now this integral is easily evaluated as >
> 2I = Log[2 + v + Sqrt[v^2 + 4v - 8]]> > Substituting
v gives > > 2I = Log[2 + u + 1/u + Sqrt[u^4 + 4u^3 - 6u^2
+ 4u + 1]]> > And substituting x gives> > I =
0.5*Log[2 + x + 1/x + (1/x^2)Sqrt[x^4 + 4x^3 - 6x^2 + 4x +
1]]> > So, is this right?> > Well, I de?itely made
one mistake in that substituting v should not> give you what
I put. I think the ?al result should be> I =
0.5*Log[(x+1)^2 + Sqrt[(x+1)^4-12x^2]] - 0.5*Log[x]> If
someone could differentiate this using Mathematica and
simplify it> I would appreciate it.The derivative is x^2 - 1
---------------------------- 2 x
sqrt(x^4+4x^3-6x^2+4x+1)--Ron Bruck
===
Try
this:x^4+4x^3-6x^2+4x+1=(x-a)*(x-b)*(x-c)*(x-d),wherea=-1+1/2
*sqrt(12)+sqrt(3-2*sqrt(3))b=-1+1/2*sqrt(12)-sqrt(3-2*sqrt(3)
)c=-1-1/2*sqrt(12)+sqrt(3+2*sqrt(3))d=-1-1/2*sqrt(12)-sqrt(3+
2*sqrt(3))Thusx/(x^4+4x^3-6x^2+4x+1)=x/((x-a)*(x-b)*(x-c)*(
x-d))Try expressing this as a sum of integrable polynomial
fractions in x, thenintegrate.> About 2 months ago I ran
across a problem on MIT's website to ?d the> inde?ite
integral of a certain function. It was from some problem>
solving seminar, and I think it was rated as the hardest
problem of> all the problems during the semester. I posted
about a while back> about this integral but everybody just
said that since Mathematica> can't do it it's probably 
not
integrable in terms of elementary> functions. Well, after
about two months of toiling over this problem,> I think I
might ?ally have a solution. However, there's a few> places
where, at least to me, things were a little iffy about
whether> or not I could do what I was doing. So I want to put
my solution and> see if there's any ?> So, the problem is
to ?d in closed form>> Integral[f(x) dx], where f(x) =
x/Sqrt[x^4+4x^3-6x^2+4x+1]>> So, ?st note two things:>> 1)
f(x) = f(1/x) (assuming of course that x is not 0)> 2) the
polynomial in the bottom is a reciprocal polynomial>>
Reciprocal polynomials generally can be solved by making a>
substitution of the form t = x+1/x. So I use this to my
advantage> later.>> So anyway, ?st we use obvservation 1 to
say that if I => Integral[f(x) dx], then 2I = Integral[f(x)
dx] + Integral[f(1/x) dx]>> In the second integral, let u =
1/x, and it transforms to>> Integral[f(x) dx] -
Integral[f(u)/u^2 du]>> Since x is just a placeholder, I
replace the x's with u's in the ?st> integral and 
get>> 2I =
Integral[(1 - 1/u^2)f(u) du]>> Now, we use the second
observation. We already have>> 2I = Integral[(1 -
1/u^2)*(u/Sqrt[u^4+4u^3-6u^2+4u+1]) du]>> Bringing that u
from the numerator into the denominator, we get>> 2I =
Integral[(1 - 1/u^2)*(1/Sqrt[u^2+4u-6+4/u+1/u^2]) du]>> Now
we are set up to write the part under the square root as a>
poloynomial in x+1/x. Indeed, we get>> 2I = Integral[(1 -
1/u^2)*(1/Sqrt[(u+1/u)^2+4(u+1/u)-8]) du]>> Let v = u+1/u.
Then dv = (1-1/u^2) dv, which is perfect since there's> a
(1-1/u^2) sitting right there! So the integral transforms
to:>> 2I = Integral[1/Sqrt[v^2 + 4v - 8] dv]>> Now this
integral is easily evaluated as>> 2I = Log[2 + v + Sqrt[v^2 +
4v - 8]]>> Substituting v gives>> 2I = Log[2 + u + 1/u +
Sqrt[u^4 + 4u^3 - 6u^2 + 4u + 1]]>> And substituting x
gives>> I = 0.5*Log[2 + x + 1/x + (1/x^2)Sqrt[x^4 + 4x^3 -
6x^2 + 4x + 1]]>> So, is this right?>
===
> Try this:>
x^4+4x^3-6x^2+4x+1=(x-a)*(x-b)*(x-c)*(x-d),> where>
a=-1+1/2*sqrt(12)+sqrt(3-2*sqrt(3))>
b=-1+1/2*sqrt(12)-sqrt(3-2*sqrt(3))>
c=-1-1/2*sqrt(12)+sqrt(3+2*sqrt(3))>
d=-1-1/2*sqrt(12)-sqrt(3+2*sqrt(3))> Thus>
x/(x^4+4x^3-6x^2+4x+1)=x/((x-a)*(x-b)*(x-c)*(x-d))> Try
expressing this as a sum of integrable polynomial fractions
in x, then> integrate.I wish it were so simple.
Unfortunately, you need to take the squareroot of that
quartic polynomial on the bottom.
===
>Try
this:>x^4+4x^3-6x^2+4x+1=(x-a)*(x-b)*(x-c)*(x-d),>where>a=-1+
1/2*sqrt(12)+sqrt(3-2*sqrt(3))>b=-1+1/2*sqrt(12)-sqrt(3-2*
sqrt(3))>c=-1-1/2*sqrt(12)+sqrt(3+2*sqrt(3))>d=-1-1/2*sqrt(12
)-sqrt(3+2*sqrt(3))>Thus>x/(x^4+4x^3-6x^2+4x+1)=x/((x-a)*(x-b
)*(x-c)*(x-d))>Try expressing this as a sum of integrable
polynomial fractions in x, then>integrate.... except that
it's x/sqrt(x^4+4x^3-6x^2+4x+1) that he wants to integrate.
According to Maple,> int(x/sqrt((x-a)*(x-b)*(x-c)*(x-d)),x);
/(a - c) (x - d)1/2 2 /(c - d) (x - b)1/2 2 (d - a)
|---------------| (x - c) |---------------| (a - d) (x -
c)/ (b - d) (x - c)/ /(c - d) (x - a)1/2 /
|---------------| | (a - d) (x - c)/ /(a - c) (x -
d)1/2 /(c - b) (d - a)1/2 c EllipticF(|---------------| ,
|---------------| ) + (a - d) (x - c)/ (d - b) (c - a)/ (d
- c) /(a - c) (x - d)1/2 a - d /(c - b) (d - a)1/2
EllipticPi(|---------------| , -----, |---------------| )
(a - d) (x - c)/ a - c (d - b) (c - a)/ / 1/2 | / ((a
- c) (c - d) ((x - a) (x - b) (x - c) (x - d)) ) / /Robert
Israel israel@math.ubc.caDepartment of Mathematics
http://www.math.ubc.ca/~israel University of British Columbia
Vancouver, BC, Canada V6T 1Z2
===
> About 2 months ago I ran
across a problem on MIT's website to ?d the> inde?ite
integral of a certain function. It was from some problem>
solving seminar, and I think it was rated as the hardest
problem of> all the problems during the semester. I posted
about a while back> about this integral but everybody just
said that since Mathematica> can't do it it's probably 
not
integrable in terms of elementary> functions.Mathematica 5
gives an answer -- terribly messy -- in closed form in
termsof elliptic integrals. I have no idea if that answer is
correct or not.> Well, after about two months of toiling over
this problem,> I think I might ?ally have a solution.
However, there's a few> places where, at least to me, things
were a little iffy about whether> or not I could do what I
was doing. So I want to put my solution and> see if there's
any ?> So, the problem is to ?d in closed form>>
Integral[f(x) dx], where f(x) =
x/Sqrt[x^4+4x^3-6x^2+4x+1][snip]> I = 0.5*Log[2 + x + 1/x +
(1/x^2)Sqrt[x^4 + 4x^3 - 6x^2 + 4x + 1]]>> So, is this
right?Have you forgotten how easy such a thing is to check???
Just differentiatewhat you think to be the antiderivative, and
see if that result is equalto the integrand!Unfortunately, it
seems that you did something wrong...David
===
> About 2
months ago I ran across a problem on MIT's website to ?d
the> inde?ite integral of a certain function. It was from
some problem> solving seminar, and I think it was rated as
the hardest problem of> all the problems during the
semester. I posted about a while back> about this integral
but everybody just said that since Mathematica> can't do it
it's probably not integrable in terms of elementary>
functions.> Mathematica 5 gives an answer -- terribly messy
-- in closed form in terms> of elliptic integrals. I have no
idea if that answer is correct or not.> > Well, after about
two months of toiling over this problem,> I think I might
?ally have a solution. However, there's a few> places
where, at least to me, things were a little iffy about
whether> or not I could do what I was doing. So I want to
put my solution and> see if there's any ?> So, the
problem is to ?d in closed form>> Integral[f(x) dx],
where f(x) = x/Sqrt[x^4+4x^3-6x^2+4x+1]> [snip]> I =
0.5*Log[2 + x + 1/x + (1/x^2)Sqrt[x^4 + 4x^3 - 6x^2 + 4x +
1]]>> So, is this right?> Have you forgotten how easy
such a thing is to check??? Just differentiate> what you
think to be the antiderivative, and see if that result is
equal> to the integrand!> Unfortunately, it seems that you
did something wrong...> DavidI don't have Mathematica, and
it's a painful function to differentiateand simplify. Of
course I could have done it but there's alot of roomfor
error, and if I make a mistake differentiating I'm likely to
thinkthat my answer is wrong, when in fact it's not. However,
you said itis wrong, so I assume you differentiated the result
using Mathematicaand didn't get the same thing. Unfortunate
indeed.Is it possible someone can point me to the error?
===
>
> Have you forgotten how easy such a thing is to check??? Just
differentiate> what you think to be the antiderivative, and
see if that result is equal> to the integrand!> >
Unfortunately, it seems that you did something wrong...> >
> David> > I don't have Mathematica, and it's a 
painful
function to differentiate> and simplify. Of course I could
have done it but there's alot of room> for error, and if I
make a mistake differentiating I'm likely to think> that my
answer is wrong, when in fact it's not. However, you said it>
is wrong, so I assume you differentiated the result using
Mathematica> and didn't get the same thing. Unfortunate
indeed.BTW a simple calculator is a great way to check
complicated algebra.Just pick a random value for X and
compute the numericalresult at each stage. You can quite
quickly track down whichstep contained the error. Numerical
differentiation also works quiteeffectively if you have
suf?ient precision on the calculatorand use common sense in
selecting delta-X.
===
theorems. Not being able to ?ure out
that proof was really buggingme, I'll sleep better now.> >>
>If F is given consider the sequence of functions (de?ed on
|z| = 1)>F[r](z) = F(rz), where r is a monotone sequence of
reals approaching 1, say>r = 1/(1+n) with n a natural
number. Since F is continuous F[r](z) uniformly>approaches
F(z) as r approaches 1. [This is the point I was missing
before ->it holds because the unit circle is a closed
set.] > Or more precisely: This works because the closed
_disk_ is a _compact_> set, which implies that f is uniformly
continuous.> >Let E/2 be the worst case>error value of
|F(z) - F[r](z)|. Since rz lies within the radius of>
>convergence of F, F converges uniformly there, and we can
take a partial sum>P[r](z) that is uniformly within E/2 of
F[r](z). This partial sum is a>polynomial in (z) as well as
in (rz). For all z on the unit circle |F(z) ->P[r](z)| <=
|F(z) - F[r](z)| + |F[r](z) - P[r](z)| < E which goes to zero
as>r approaches 1. So the P[r](z) are the desired
polynomials in z. [My only>nagging doubt here is that this
argument would seem to apply to an F de?ed>on any circle,
not just the unit circle.]> Well you can relax, it's true
for any closed disk. (In fact it's true> in much greater
generality, although it's not so easy to prove; > 
Runge's
theorem, included in most complex books, gives a>
generalization, and Mergelyan's theorem gives the result
under> weaker hypotheses yet.)> >If the sequence of
polynomials is given they converge and therefore form a>
>Cauchy sequence. The difference between any two of them is
an analytic>function and therefore reaches its maxium
modulus on the boundary of the>unit disk. So the
polynomials form a uniform Cauchy sequence on the closed>
>unit disk and therefore converge uniformly. [Interestingly
none of the math>books I had included Cauchy's criterion
for uniform convergence, just>Cauchy's criterion for
convergence - too obvious an extension to state>explicitly
no doubt. But I did ?d explicit mention of it on the web.]>
>>-The polynomials converge uniformly on the closed unit
disk and are>continuous, therefore they converge to a
continuous function there.>-The polynomials converge
uniformly on the open unit disk and are analytic,>
>therefore they converge to an analytic function there.>>
>Ian>> >>Been staring at this for hours and can't
seem to ?d a proof that>>seems needlessly complex or
just plain doesn't work... can someone see>>a simple
argument? I would be much appreciated - I don't even need>>
>this result for anything but somehow I just can't let it go.>
>>>>Let f be a continuous function on the unit circle T
= {|z| = 1}. Show>>that f can be approximated uniformly
on T by a sequence of polynomials>>in z if and only if f
has an extension F that is continuous on the>>To
approximate such an F, consider dilates F[subscripted r](z)
=>>F(rz).>>>>I can't get either direction to
work... if F is given then the hint>>seems to imply that
you can obtain the polynomial sequence from the>>power
series expansion of F, but how to show uniform convergence in
a>>simple way?> >> The power series need not converge
to F at points of the boundary.>> But that's not what the
hint suggests - read the hint again...> >>If the
polynomial sequence is given then it's easy to use the
Cauchy>>integral formula on the polynomials, take the
limit, and get F>>analytic (limit of the integral =
integral of the limit because of the>>uniform
convergence, and analyticity follows from the continuity of>
>>the limit funciton f). But how to show that F is
continuous at points>>on the unit circle?> >> Don't
use the Cauchy formula. Use the Maximum Modulus Theorem,>>
and look up uniform convergence and Cauchy sequence in a
book>> on adbanced calculus.> >>I found this in
Complex Analysis by Gamelin, section V.4, problem 14.>
>> David C. Ullrich>> > David C. Ullrich
===
Two
thoughts on why e is good, and a limerick:(1) If we could
choose an irrational number as ourbase (ie instead of the
usual 10 or 2 for binary),then e would be the most ef?ient.
Why?An n-symbol string in base m can represent m^n
differentnumbers, and to do so we use n*m different symbols(m
letters, but each letter can be n differentpositions). So if
we write the size of therepresented space (m^n) as a function
of the numberof different symbols k=m*n, we getf(k) = m^(k/m)
= (m^(1/m))^k.I like to think of the number (m^(1/m))
ascapturing the ef?iency of thestandard base-m type
encoding. In the integers,3 maximizes this as 1.4422... But
if we usesome calculus, the global maximum is actually ate,
with a value of 1.4446...In other words, our ef?iency is
maximized at e.This argument is my own, but I'm sure it has
beendone better elsewhere, because everything I thinkI have
that's even slightly original turns out that way...(2) One of
the most beautiful theorems of modernmathematics: the central
limit theorem. Usuallypeople think of it as only applying to
probability, sinceit is easiest to interpret in that realm.
But itview, you could declare the presence of e in
thestatement of CLT to be a consequence of Fourieranalysis
(and sort of from Euler's formula e^(ix) =cos(x) + i sin(x)),
but this is a bit of an over-simpli?ation.limit theorem; so
the best elementary idea of it thatI can give is that there
is a single function,something like e^(-x^2/2) (times a
constant), whichrepresents the limiting behavior of virtually
_any_ functionwhich is randomly sampled and then averaged. I
knowthat's a bad description, but it's a 
start.Finally, the
limerick (written by me :)My favorite number is eit compounds
continuously;if you raise it to pimultiplied by i,negative one
it will be.-Tyler
===
Part IINote this is all constructive
criticism of Hal's ideas on metric engineering.Excerpts from
Puthoff & Ibison?H.E. PUTHOFF*, S.R. LITTLE AND M.
IBISONInstitute for Advanced Studies at Austin, 4030 West
Braker Lane, Suite 300, Austin, Texas 78759-5329, USA.with my
comments.3. The Quantum Vacuum3.1 Zero-Point Energy (ZPE)
BackgroundQuantum theory tells us that so-called empty
space is not truly empty, but is the seat of myriad
energetic quantum processes. Speci?ally, quantum ?ld theory
tells us that, even in empty space, ?lds (e.g., the
electromagnetic ?ld) continuously ?te about their zero
baseline values.So far, so good. However IMHO it is the QED
PV virtual electron-positron zero point vacuum ?tions
that dominate the low energy effective macro-quantum ?ld
theory out of which Einstein's gravity together with exotic
vacuum uni?d dark energy/matter ?lds co-emerge like love
and marriage all together as phase and amplitude modulation
patterns, respectively, of the vacuum coherence ?ld that is
formed by the attractive BCS exchange of virtual photons
between the virtual electron-positron pairs all inside the
vacuum. Note that Hal makes no reference at all to the new
discovery in precision cosmology of dark energy as being
relevant to his Quixotic Ahabian quest in search of for the
vacuum wind. :-)The energy associated with these ?tions
is called zero point energy (ZPE), re?g the fact that
such activity remains even at a temperature of absolute zero.
Such a concept is almost certain to have profound implications
for future space travel, as we will now discuss.Agreed.When
a hypothetical ZPE-powered spaceship strains against gravity
and inertia ...That's a strange way of putting it. The local
net random micro-quantu zero point stress-energy density
tensor ?ld isTuv(Exotic Vacuum) = (String
Tension)/ZPEguvusing Ed Witten's natural units h = c =
k(Boltzmann) = 1Tuv(Exotic Vacuum) = (String
Tension)^2[(String Tension)^3/2|Vacuum Coherence|^2
-1]guvWhat is wrong with Puthoff's paradigm is that he has no
idea of Vacuum Coherence in his informal thinking nor
formalized in his mathematics.The zero point energy density
is Too and I use the 3 GR sign conventions in whichToo(Exotic
Vacuum) > 0 means strongly anti-gravity negative pressure
dark energy since the general equation of state(pressure) =
w(energy density)reduces to w = -1 for ALL zero point quantum
?lds of all spins.Too(Exotic Vacuum) < 0 means strongly
gravitating positive pressure dark matter.Therefore dark
energy needs(String Tension)^3/2|Vacuum Coherence|^2 > 1Dark
matter needs(String Tension)^3/2|Vacuum Coherence|^2 < 1Where
the critical Vacuum Coherence corresponding to zero Einstein
Cosmological Constant of the non-exotic vacuum in the
large-scale FRW metric is(String Tension)^3/2|Vacuum
Coherence|^2 = 1Or|Vacuum Coherence|*^2 = (String
Tension)^-3/2The Einstein exotic vacuum local geometrodynamic
?ld equation is thenTuv(Induced Time and Space Warps) +
Tuv(Exotic Vacuum) ~ 0The total covariant 4-divergence
vanishes conserving total local stress-energy Einstein
current density.Unlike 1915 geometrodynamics, the Bianchi
identities break down so that the two pieces of the
divergence are not separately zero!Only the sum is zero. This
is necessary for practical soft metric engineering of the
Tech Gnostic Underground Stream inside the vacuum.That
is,Tuv(Induced Space and Time Warps)^;v + Tuv(Exotic
Vacuum)^;v = 0This is the ?st small step for Mankind leading
to The Right Stuff to Make Star Trek Real.See my lecture in
Time Travel: The Art of the Possible Disk 2 of Paramount
Pictures DVDSpecial Collector's Edition of Star Trek IV:
The Voyage Home indeed! ;-)*Before you read on click on
this! Turn the volume up on your sound.Sarfatti pitches to
Puthoff at bat in the World Series
:-)http://www.niehs.nih.gov/kids/lyrics/ballgame.htmthere
are three elements of the equation that the ZPE technology
could in principle address: (1) a decoupling from
gravity,NO! We want just the opposite! Big error of
strategic thinking!The Question is: What is The
Question?Here Hal & Co ask a wrong question IMHO.STRIKE
ONE!(2) a reduction of inertiaNO! Ditto. That's like
preventing your car from being stolen by blowing it up if it
is broken into!STRIKE TWO!or (3) the generation of energy to
overcome both.NO, NO, NO a zillion times NO! ;-)That's the
Brute Force approach.That is notThe Tao Chi of ET!STRIKE
THREE!You're Out!Take me out to the ball game ....3.2
GravityWith regard to a ZPE basis for gravity, the Russian
physicist Andrei Sakharov was the ?st to propose that in a
certain sense gravitation is not a fundamental interaction at
all, but rather an induced effect brought about by changes in
the quantum-?tion energy of the vacuum when matter is
presentYes, this is essentially correct and is precisely
what my theory is all about. The details are
inhttp://qedcorp.com/APS/EmergentGravity.doc or
http://qedcorp.com/APS/EmergentGravity.doc (smaller ?e)to be
continued.Engineering the Zero-Point Field and Polarizable
Vacuum for Interstellar Flight?. IntroductionThe concept of
?ngineering the vacuum?found its ?st expression in the
mainstream physics literature when it was introduced by T. D.
Lee in There he stated:?he experimental method to alter the
properties of the vacuum may be called vacuum
engineering.... If indeed we are able to alter the vacuum,
then we may encounter some new phenomena, totally
unexpected.?his legitimization of the vacuum engineering
concept was based on the recognition that the vacuum is
characterized by parameters and structure that leave no doubt
that it constitutes an energetic medium in its own right.
Foremost among these are its properties that (1) within the
and ?ld ?tions, and (2) within the context of general
relativity the vacuum is the seat of a spacetime structure
(metric) that encodes the distribution of matter and energy.
Indeed, on the ? of a book of essays by Einstein and
others on the properties of the vacuum we ?d the statement
?he vacuum is fast emerging as the central structure of
modern physics?[3].Given the known characteristics of the
vacuum, one might reasonably inquire as to why it is not
immediately obvious how to catalyze robust interactions of
the type sought for space-?applications. To begin, in
the case of quantum ?tions there are uncertainties that
remain to be clari?d regarding global thermodynamic and
energy constraints. Furthermore, the energetic components of
potential utility involve very small-wavelength,
high-frequency ?lds and thus resist facile engineering
solutions.?his last remark may not be correct in general,
although it is correct within the framework Puthoff is
pursuing i.e. using the stress-energy density of the
electromagnetic ?ld to attempt to directly warp space-time
geometry with practical utility. Such a brute-force approach
is hopeless and a waste of effort IMHO because Einstein?
equation in this case is simplyInduced Space-Time Warp =
(Applied EM Stress-Energy Density)/(G-String Tension)The
G-String Tension is 10^19Gev per 10^-33 cm, which is simply
too stiff! One would need a way to lower the
G-String-Tension, for example an ? Theory?that yields
something like:G*-String Tension = e^-(r*/Lp)^?(G-String
Tension)Where r* = compacti?ation scale of extra-dimensions
of hyperspace beyond 4-D space-time andLp^2 = hG/c^3 =
1/(G-String Tension) for h = c = 1Puthoff alludes to this
barrier in his next remark:?ith regard to perturbation of
the space-time metric, the required energy densities exceed
by many orders of magnitude values achievable with existing
engineering techniques. Nonetheless, we can examine the
constraints, possibilities and implications under the
expectation that as technology matures, felicitous means may
be found that permit the exploitation of the enormous,
as-yet-untapped potential of so-called ?mpty space?This is
the problem that Hal goes into denial and wishing about.
There is no way that will brute-force approach will work in
any practical way. The UFOs do not use high-energy EM ?ld
densities and Hal is primarily interested in the UFOs. That
should tell him right away that he is pursuing the wrong path
to solve the problem.?. Propellantless Propulsion2.1 Global
ConstraintRegardless of the mechanisms that might be
entertained with regard to ?ropellantless?or ?eld?
ropulsion of a spaceship, there exist certain constraints
that can be easily overlooked but must be taken into
consideration. A central one is that, because of the law of
conservation of momentum, the center of mass-energy (CM) of
an initially stationary isolated system cannot change its
position if not acted upon by outside forces. This means that
propellantless or ?ld propulsion, whatever form it takes, is
constrained to involve coupling to the external universe in
such a way that the displacement Engineering the Zero-Point
Field and Polarizable Vacuum for Interstellar Flight?t has
been known since the late 1950? how to circumvent this
problem provided one has ?xotic stuff?with a strong enough
?egative pressure?to provide what Herman Bondi called
?egative matter?in the late 1950? and Kip Thorne called
?xotic matter?in the late 1980? in his ?raversable
wormhole time travel?papers with additional work by Igor
Novikov and others. Robert Forward summarized Bondi? early
work in a paper written in the early 1990?. More details
are to be found inWe now know since 1999 from Type 1a
supernova data showing the acceleration of the expansion
speed of the Universe that approximately 3/4 of all the
large-scale stuff of the Universe is exotic vacuum
anti-gravitating ?ark energy?with a strong negative zero
point pressure that is equal and opposite to the positive
zero point energy density. This fact is the key to practical
metric engineering of the vacuum that Puthoff et-al
completely misses. The modi?d vacuum Einstein equation
isTuv(Geometry) + Tuv(Exotic Vacuum) = 0Where Tuv is the
local stress-energy density tensor under the Einstein DIFF(4)
symmetry group.Tuv(Geometry) = (String
Tension)Guv(Geometry)WhereGuv(Geometry) = Ruv [CapitalEth]
(1/2)RguvAndTuv(Exotic Vacuum) = (String
Tension)/zpfguv/zpf = (String Tension)^-1[(String
Tension)^3/2|Vacuum Coherence|^2 [CapitalEth] 1]/zpf > 0
is negative pressure ?ark energy?exotic vacuum that is ~
3/4 of large-scale stuff of the Universe./zpf < 0 is
positive pressure ?ark matter?exotic vacuum that is ~ 1/4
of the large scale stuff of the Universe.Gravity effect of
exotic vacuum ?tuff?~ c^2/zpfWhere + sign on RHS is
universally repelling anti-gravity and a [CapitalEth] sign
is universally attracting gravity with effective short scale
coupling strengths much larger than Newton?.The key idea not
found in any of Puthoff? papers on the subject of metric
engineering is the local Einstein stress-energy tensor
current density? covariant divergence Tuv^;v where ;v is the
Diff(4) covariant derivative relative to the metric torsion
free connection ?ld for parallel transport of tensors along
tangent vector ?lds in curved space-time.When there is no
exotic vacuum, i.e., /zpf = 0 corresponding to optimal
vacuum coherence, the Bianchi identities work and we have
closed current conservation of the pure geometrodynamic
stress-energy tensor local currents, i.e.Tuv(Geometry)^;v =
0This prevents any practical metric engineering! In contrast,
when there is exotic vacuum thenTuv(Geometry)^;v =/=
0InsteadTuv(Geometry)^;v + Tuv(Exotic Vacuum)^;v = 0Where the
common string-tension factor cancels out of this current
conservation equation. This essentially solves the problem
for practical metric engineering bypassing the string tension
problem completely! One uses the Bohm-Aharonov-Josephson
effect to tweak Tuv(Exotic Vacuum) in a ?oft?way.To be
continued.The Physical Principles of Metric Engineeringby
Jack Sarfatti5th draftexcerpt from the book Super CosmosThe
term metric engineering was coined by Dr. Harold Puthoff.
Hal, as he prefers to be called, was a US Naval Of?er who
then worked for the National Security Agency before going to
the Stanford Research Institute where he conducted the famous
Remote Viewing experiments with Russell Targ testing Uri
Geller, Ingo Swann, Pat Price and other psychics in a project
paid for by the CIA and Department of Defense Intelligence
Agencies. I ?st met Hal and Russell at SRI in 1973 and that
story is told in my book Destiny Matrix. Hal has held very
high USG security clearances and it is well known that he is
obsessed with the UFO mystery. Hal, like his co-worker Bernie
Haisch, who is an editor of the Astrophysical Journal, both
strongly believe in the physical reality of ?saucers.
They, with Jacques Vallee, have been active in the NASA
Breakthrough Propulsion Program and in UFO groups some
?anced by Laurance Rockefeller and the Howard Hughes clone
Las Vegas Hotel Tycoon Robert Bigelow who owns Bigelow
Aerospace Corporation. Since they take the reality of the
?saucers seriously so do I. The two lines of theoretical
physics research that Hal has pursued for several decades now,
zero point energy and a polarized vacuum model of gravity, is
primarily motivated by the quest to understand how the
saucers ?e alleged reality of such advanced alien
technology is clearly of immense importance to US National
Security and beyond. The recent developments in physics shown
in NOVA's Elegant Universe with Brian Greene, in Stephen
Hawking's The Universe in a Nutshell, Michio Kaku's
Hyperspace and Igor Novikov's The River of Time makes
time traveling alien interference in our history more
probable not less probable. This is all an aspect of metric
engineering de?ed as the practical control of space and
time warps. I have discussed this in Paramount Pictures DVD
Special Collector's Edition of Star Trek IV: The Voyage
Home in Time Travel: The Art of the Possible and in
Learning Channel's Ultra-Science: Time Travel. The ultimate
form of metric engineering is seen in Q in Star Trek where
a Super Mind is able to warp space-time. We also need a
physics of consciousness to see if that ?tion can be
realized in fact. British Astronomer Royal, Sir Martin Rees
who runs the laboratory where Stephen Hawking works and who
is the new Master of Trinity College, Cambridge has added the
dimension of Doomsday WMD to the quest for a metric
engineering breakthrough in his Chapter 9 of his pessimistic
Our Final Hour on the Ice Nine rip-in-space propagating
at the speed of light that could literally destroy our
universe. We are now at that turning point in our history
where, like Mickey Mouse in Walt Disney's Fantasia we
could, in our incompetence, destroy the entire Universe just
as we are now surely destroying Earth's Biosphere. Therefore,
any aliens out there with Star Gate Time Travel and Warp Drive
Super Technology are already here in order to stop us from
killing them as well as ourselves. The American Christian
Fundamentalist Right has a strong belief in Apocalypse Now
and there is real danger that this is a self-ful?ling
prophecy in a precognitive remote viewing of our future by
Saint John in The Bible's Revelations. All the more reason
for me, like Paul Revere, to alert the public to these new
developments in physics because only real knowledge can
possibly save us and even then there is no guarantee.What
does mainstream physics tell us about the possibility of
metric engineering? The relevant parts of mainstream physics
for metric engineering are Einstein's general theory of
relativity and micro-quantum mechanics including the More is
Different emergence of macro-quantum super?rder in
ground state and vacuum instabilities developed by condensed
matter Princeton physicist P.W. in the in?ary chaotic
cosmology of A. Linde consistent with the issue of Scienti?
American.Let's start with Einstein's gravity ?ld 
equation of
~ 1915. Einstein viewed pure space-time geometry like the
marble in a statue by Michelangelo with gross matter and
radiation as wood. Einstein's equation, in the 21st Century
language, isSpace-Time Curvature = (Stress-Energy
Density)/(String Tension)Curvature has the physical
dimensions of 1/Area, String Tension has Energy/Length and
Stress-Energy Density has Energy/Volume. Simple algebra
con?ms that this way of writing Einstein's equation is
correct dimensionally. The String Tension is the basic
parameter that Ed Witten of the Princeton Institute of
Advanced Study uses in his discussions of M theory that
uni?s the ?e limiting cases of super string theory in
which the elementary leptons, quarks and gauge force bosons
are vibrating strings of pure energy pulsating in the extra
space dimensions of Calabi-Yau hyperspace. The ordinary
matter and radiation on the right hand side of Einstein's
equation are open strings whose ends are stuck to 3Dim brane
worlds of which our universe is one. You can picture a string
as a tiny wormhole whose two ends need not be on the same
brane world because only then can you not have equal numbers
Another equivalent way to look at Einstein's 1915 equation
from our Cosmic Justice peeking through the blindfold. That
is, the stress-energy density tensor of pure marble
geometry isStress-Energy Density Tensor of Pure Geometry =
(String Tension)(Space-Time Curvature) (2)Einstein's equation
is then like the static equilibrium balance of forces in
architecture in which the sum of all the contributions to the
stress-energy density tensor add up to a perfect zero! In this
simplest of casesStress-Energy Density Tensor of Geometry +
Stress-Energy Density of Matter etc. = 0 (3)All physically
real objects are either tensors, spinors or twistors in the
theory of relativity. This is because the coordinate map is
not the physical territory. The form of the laws of physics
must be the same locally no matter how the coordinate map is
morphed. To this we must add the Einstein Equivalence
Principle or EEP which says that even in a curved space-time,
we can use the special theory of relativity locally for a
special class of LIF observers to describe what is happening
to a good approximation. The EEP is only meant to apply in
this approximate way and it will break down if one is falling
into a black hole singularity or if the microscope
magni?ation is so large that quantum gravity zero point
energy density ?tions in the space-time geometry itself
get large. There may also be torsion ?lds not included in
Einstein's 1915 geometrodynamics that may require a
modi?ation of the EEP. The spinor is a square root of a
tensor and the Penrose twistor is a spinor in a complex
space-time. The physics of point matrix space-time is in
reality the physics of extended strings and membranes in real
space-time. Hawking's imaginary time and its connection to
inverse temperature is part of that same story.The key
concept for metric engineering is the Einstein current!
Einstein's ?ld equation is always the statement that the sum
of all the stress-energy density tensors when put on the same
side of the equation all balance out to exactly zero. We then
take a covariant divergence of the equation to get the
Einstein currents, which are conserved when added together.
The covariant divergence depends on something called a
connection ?ld, which tells us how to parallel transport
tensors and spinors along paths or histories in space-time
and beyond including the extra dimensions of hyperspace.
Every time a gauge force ?ld is added, like a torsion ?ld,
there is an additional piece added to this connection ?ld.
Einstein's 1915 theory is a degenerate case of the bigger
uni?d ?ld theory just like a circle is a degenerate case of
an ellipse. An ellipse has two centers or foci. When the two
foci merge together the ellipse degenerates into a circle.
Einstein's 1915 theory in the form of the Bianchi
identities forbids practical metric engineering because it
has an impenetrable barrier completely separating the marble
geometric current from the wood matter current. Both kinds of
currents must be able to intermingle to transform into each
other in order to have soft practical metric engineering of
Star Gate Time Travel traversable wormholes and weightless
superluminal warp drives using the anti-gravitating exotic
vacuum dark energy with negative zero point exotic vacuum
pressure that is equal in magnitude, but opposite in sign, to
the zero point energy density. The new Type Ia super novae
data showing that our brane world Universe is accelerating in
its rate of expansion, i.e. speeding up rather than slowing
down. Indeed approximately 3/4 of all the stuff of our
Universe on the large scale is made out of dark energy.
Therefore, the idea of ancient traversable Star Gate
wormholes stabilized by dark energy is not so far fetched.
The brute force approach to metric engineering taken by Hal
Puthoff now for at least two decades can never work because
space-time is too stiff to bend directly with electromagnetic
?ld energy density to do anything worth doing because the
string tension is too large. In contrast, we can used the
quantum interference of the Einstein stress-energy density
currents via the Bohm-Aharonov-Josephson effect for practical
metric engineering provided that the pure geometric currents
are not separately conserved like they are in Einstein's
original theory. One way to do that is to bottle or
harness the dark energy as General Douglas Mac Arthur
precognitively remote viewed in his Duty, Honor, Country
Farewell Speech to the Cadets at West Point in 1962 in a
speech made even more remarkable for its reference to the
coming war in space with extra-terrestrials. Michael Turner,
a professor of physics at the impossible to bottle dark
energy and if Einstein's 1915 theory is the ?al theory he
is correct. However, the real UFO evidence that drives Hal
Puthoff like Captain Ahab after The White Wale, Moby Dick, is
the evidence that suggests Professor Turner will be proved
wrong in that
prediction.ref.http://qedcorp.com/APS/StarGate1.movhttp://
qedcorp.com/APS/EmergentGravity.pdf
===
I'm attempting to
connect two 3D squares together. They are of 16points each
and I?ve run them through the Bernstein equation.Now, I?m
quite new to 3 dimensional mathematics and new to
theBernstein formula, but its proving to be very
interesting.I've been trying for days now to get the squares
to match up smoothly.There is a curvature in the surface of
the one square/patch and I?mattempting to keep the same
curvature through the second square (tomake half a ?al).
I've heard about the tangents method, abouthaving tangents
from the control points on the one square to thecontrol
points on the other square, which should create a
smoothsurface. And here in is the problem, there is a de?ite
line, aninward slope as the two squares/patches meet. As far I
can tell thetangents line up along the join, I just don't have
a clue why the joinisn't perfectly smooth.Many people must
have run into this problem. If you have any thoughts,ideas or
possible directions to pursue, they would all be very
muchappreciated.T. Overton
===
I'm attempting to connect two
3D squares together. They are of 16points each and I?ve run
them through the Bernstein equation.Now, I?m quite new to 3
dimensional mathematics and new to theBernstein formula, but
its proving to be very interesting.I've been trying for days
now to get the squares to match up smoothly.There is a
curvature in the surface of the one square/patch and
I?mattempting to keep the same curvature through the second
square (tomake half a ?al). I've heard about the
tangents method, abouthaving tangents from the control points
on the one square to thecontrol points on the other square,
which should create a smoothsurface. And here in is the
problem, there is a de?ite line, aninward slope as the two
squares/patches meet. As far I can tell thetangents line up
along the join, I just don't have a clue why the 
joinisn't
perfectly smooth.Many people must have run into this problem.
If you have any thoughts,ideas or possible directions to
pursue, they would all be very muchappreciated.T.
Overton
===
I am guessing, if we take an (n_1 by n_2 by n_3 by
...n_m) m-dimensional box (where the n's are positive
integers),and we let the number of lattice points of positive
integer coordinates(k_1,k_2,k_3, ...k_m), where NO common
primes divide EACH k, be q(n),then:limit{n_1->oo, n_2->oo,
n_3->oo, ...n_m->oo}q(n)/(n_1 * n_2 * n_3 * ...n_m) =
1/zeta(m).IE; the fraction of all ?ite sequences of m
?uniformly randomly chosen' positive integers, which are so
that no common prime divides every integer in these
sequences,is 1/zeta(m).(example of counted 4-tuple: (4,3,2,4)
*is* counted because 3 is not divided by 2.)But...what is
meant here by taking the limits of the n's??I assume that
each n must approach in?ity ?independently' of the
others.(Would the limit be the same if n_k was a
monotonically increasing function of n_{k-1}, for
example?)And, what is meant exactly by ?uniformly randomly
chosen'?I am *not* quoting from a book, but rather have
myself UNRIGOROUSLY derived the above well-known result
today, and wish to know if the result has a more rigorous and
technically-correct statement.thanks,Leroy Quet
===
> Because
Khinchtine's constant exists and is ?ite, we know that the>
distribution of the integers among the terms of the simple
continued> fraction of a suf?ently random (whatever is meant
by this) positive> real is not normal.> In other words, 1 is
much more likely to occur than, say, 1000 in the> continued
fraction expansion of most positive reals.> > So, let us
say we have a real x where the geometric average of x's>
continued fraction terms approaches Khinchtine's constant,
where these> terms are {a(k)}.> > So, what can be said
about the number theoretical aspects of {a(k)},> such as:>
1) What is the likelyhood that any a(k) will be prime, as k
-> oo.> ie. If pcf(m) is the number of primes among a(k)
for 1 <= k <= m, then> what is pcf(m) asymptotical towards?>
> 2) Same question as (1), but replace ?primes' with
?squarefree> integers'.> 3) What is likelihood a(k) and
a(k+1) will be coprime?> 4) What are the expected number of
divisors for all a(k) where 1 <= k> <= m?> > Etc etc etc
....> Of course, the number of primes among the ?st m CF
terms of mostpositive reals would exceed that, on average,
among the ?st mpositive integers in general, on
average.Right? > thanks,> Leroy> Quet
===
> For some even
positive integer m,> we have a m-by-m grid.> In this 2
player game, each player has (m^2/2) counters, > each counter
numbered with a distinct integer from 1 to (m^2/2).> The
players take turns placing the counters into the grid's
squares in> any order the players wish.> (We do not
actually need the counters, for players can simply write> the
numbers in the grid's squares. But the counters make it easy
to> know which numbers each player has already used, for each
integer is> to be used one per player.)> (Or we can simply
play this on a computer.)> > Scoring: > One player is
rows, the other player is columns.> For, say, rows, every
set of adjacent integers, where each immediately> adjacent
(to left/right) pair is coprime, is multiplied, then these>
groups of multiplied integers are all added up to get the>
row-player's score.> For columns, we do the same, but we
consider immediately adjacent> pairs which are adjacent
above/below for multiplication if coprime.> As to help
explain what I mean, here is an example (of a game played>
against myself without using any strategy):> > (Who plays
which number is unimportant.)> 8 5 8 3> 6 1 2 6> 7 3 1 5> 2
4 7 4 > Rows gets:> 8*5*8*3 +> 6*1*2 + 6 > + 7*3*1*5> + 2
+ 4*7*4> Columns gets: > 8 + 6*7*2 +> 5*1*3*4 +> 8 +
2*1*7 +> 3 + 6*5*4> > I would guess that higher m than 4
would be more interesting.We can use other criteria other
than coprimality when determiningwhich integers to
multiply.(One advantage of coprimality is that if 2 positive
integers arelower, thenthey are more likely to be coprime
than if they had been higher, which{I feel} improves this
game's strategy.)Any interesting variations on this
game???thanks,Leroy Quet
===
Does anyone have any idea how to
solve the following:x+y+z=axy+yz+xz=bxyz=cI have a hunch that
x,y,z should be the roots of the cubic equations
t^3-at^2+bt-c=0
===
Brian Troutwine
<goofy_headed_punk@msn.com> grava .88 la saucisse et au
marteau:> Does anyone have any idea how to solve the
following:> x+y+z=a> xy+yz+xz=b> xyz=c> I have a hunch
that x,y,z should be the roots of the cubic equations
t^3-at^2+bt-c=0That's true. Now you've got many 
methods to
solve 3rd degree'sequations, such as Cardan or Ferrari.--
Nicolas
===
Could anyone help me solve the following:Find six
different nondegenerate triangles with integer sides forwhich
a=16 and A=60 degrees; where the angles of the triangles
arelabeled A,B,C and the sides opposite them a,b,c.
===
>Could
anyone help me solve the following:>Find six different
nondegenerate triangles with integer sides for>which a=16 and
A=60 degrees; where the angles of the triangles are>labeled
A,B,C and the sides opposite them a,b,c.The law of cosines
says a^2 = b^2 + c^2 - 2 b c cos(A), i.e.256 = b^2 + c^2 - b
c.Sorry: the only positive integer solution is b=c=16. I
think the least value of a for which there are six different
triangles(if you count (a,b,c) and (a,c,b) as different when
b<>c) is 49, for whichyou have the triangles[49, 16, 55],
[49, 21, 56], [49, 35, 56], [49, 39, 55], [49, 49, 49], [49,
55, 16], [49, 55, 39], [49, 56, 35], [49, 56, 21]Robert
Israel israel@math.ubc.caDepartment of Mathematics
http://www.math.ubc.ca/~israel University of British Columbia
Vancouver, BC, Canada V6T 1Z2
===
First my original post
copy/pasted, then the post I am replying to byFred the Wonder
Worm, before my reply below.>We have a regular hexagon whose
sides are lables, clockwise from top,>A through F.>>the
hexagon's inner walls as if the walls were mirrored, but it
also>affects the direction of *itself*; for whenever it
crosses its own>path (as already drawn), it passes through
the path, but is re?>as if a mirror has been placed
perpendicularly to the previous pathat>the point of
intersection.>know>if such a particular path actually exists,
because the path's ?al>direction is heavily dependent upon
the accuracy in which the path is>drawn and the accuracy of
the angles re?.>>But I will give the order of the
hexagon's surfaces as visited by the>path (as drawn at the
time of each particular crossing) below.>>(I know this, if my
by-hand approximation was not too ?)>>(I have no idea.
You best use exact-rational arithetic to get this,if>it is
possible to ?ure out at all.)>>E, B , D, 2 crossings, F,
E, 1 crossing, F, 3 crossing, D, B, 3>crossings, F, 7
crossings, and back to its staring point.>>(If no crossings
are listed between letters, than no crossings occur>between
them.)>>(If someone solves this, they are going to have to
post some link toa>webpage with the answer, I am
afraid.)>>thanks,>Leroy Quet> If anyone wishes to check my
angle chase, here's one description of it.> I've tried 
it
three times now, so either it's right or I'm hitting 
a> blind
spot. Start by drawing in a representative diagram up to that>
point. (Make it big -- this is important!) Let the vertices
in> clockwise order be labelled L, M, N, O, P, Q, so that LM
is side A, MN> is side B, etc. Let the photon start at point
R, and successive points> where it changes direction be S,
T,..., Z. (Z is the point on the last> D-F segment, and I aim
to show that Z does not actually exist.)> That is, R is on
side A (LM), S is on side E (PQ), T is on B (MN),> U is on D
(OP), V is on ST, W is on RS, X is on F (QL), Y is on E
(QS),> and Z should be on WX.> Let angle LRS = theta. Then
we can chase angles (all degree signs> omitted) as follows:>
> QSR = 120 - theta> PST = 120 - theta> RST = 2*theta - 60>
MTS = 120 - theta> NTU = 120 - theta> TUO = theta> PUV =
theta> UVS = 120> SVW = 120> VWS = 120 - 2*theta> SWX = 120 -
2*theta> RWX = 60 + 2*theta> LXW = 180 - 3*theta> QXY = 180 -
3*theta> WXY = 6*theta - 180> XYQ = 3*theta - 120> SYZ =
3*theta - 120> XYZ = 420 - 6*theta> But then in triangle
XYZ, angles XYZ + ZXY sum to 240 degrees, which is>
impossible. We conclude that the ray is actually heading at
60 degrees> _away_ from the last segment from D-F. So that
ray must actually cross> segment WS rather than WX. I believe
it then crosses VS and hits E> again, possibly then going via
UV, TV and maybe back to the starting> point. (I think, but
am not sure, that there is enough leeway for this.)> So a
modi?d version of the problem might have the pattern as>
A, E, B, D, 2, F, E, 2, E, 2, start> Geoff.I wonder if the
ray, after crossing VS does *not* then head to E,but rather
heads more upwards towards UV, then (perhaps) spiralsinwardly
and in?itely towards point V.But a number of other things
could perhaps happen, for my diagram(hand-drawn) is
ambiguous.(Around points V and Z, there are a few segments
that could eachthanks,Leroy Quet
===
sum[k^2/k!] from k=1 to
k=infSurely there are some cute algebraic tricks to get 2e
out of this, Ijust don't know them - lil help
please?thx,cdj
===
>>sum[k^2/k!] from k=1 to k=inf>>Surely
there are some cute algebraic tricks to get 2e out of this,
I>just don't know them - lil help please?>>thx,>>cdjWrite e^x
= SUM(n=0 t =oo) (x^n/n!)Differentiate both sides and then
multiply both by x Again, differentiate both sides and
multiply both bu x. Put x = 1. BINGO!
===
> sum[k^2/k!] from
k=1 to k=infThat equals sum_(k=1,oo) k/(k-1)! = sum_(k=0,oo)
(k+1)/k! = d(xe^x)/dx at x = 1.
===
> > sum[k^2/k!] from
k=1 to k=inf> That equals sum_(k=1,oo) k/(k-1)! =
sum_(k=0,oo) (k+1)/k! = d(xe^x)/dx at x > = 1.well aren't I
just a genius now? lolthanks all,cdj
===
> >>sum[k^2/k!]
from k=1 to k=inf> > That equals sum_(k=1,oo) k/(k-1)! =
sum_(k=0,oo) (k+1)/k! = d(xe^x)/dx at x > = 1.Or, in the
middle,sum_(k=0,oo) (k+1)/k! = sum(k>0,k/k!) +
sum(k>=0,1/k!)= sum(k>0, 1/(k-1)!) + e= 2eAs I recall,
sum(k>=0, k^n/k!) is an integer multiple of e.The Bell
numbers come to mind.
===
cdj>> sum[k^2/k!] from k=1 to
k=inf>> Surely there are some cute algebraic tricks to get 2e
out of this, I> just don't know them - lil help please?Write
k^2 = k(k-1) + kand get two simpler series.LH
===
>
sum[k^2/k!] from k=1 to k=inf>> Surely there are some cute
algebraic tricks to get 2e out of this, I> just don't know
them - lil help please?e^(e^x) = 1 + e^x + e^2x/2! + e^3x/3!
+ ...Differentiate twice, then set x = 0.
===
Here is 2 player
game. (Inspired by math and other preexistingcommonly-played
games.) Each player has an identical set of 15 square tiles
colored asfollowed:Y = yellowR = redB = blueP = purple*Y **
*Y **** R* R* *B*Y ** *Y P* *B RB RB **PY P* PY P*** R* R*
*BPY P* PY *B RB RB Players take turns making a row of tiles,
one tile added at each turn.A player can ONLY add a tile if
the immediately previous tile she/heis to add shares at least
one color with the new tile.Each player get one point for each
color-combination not among his/heropponent's PREVIOUSLY
placed pieces.(In other words, if the lines of tiles are
expanding to the right andare both aligned, then a player
gets a point for every piece ofhis/hers where the
corresponding piece of her/his opponent is NOT tothe LEFT of
the player's own tile in question.)And if any player cannot
use all of his/her pieces (because ofcolor-matching rule),
the player with the most played pieces gets 2points for each
additionally played piece beyond the number played byhis/her
opponent.(We could also just give each player 2 points for
each played piece,since the score-difference would be the
same anyway.)Sample game beginning (played without strategy
here)----------------------------------------------------
Player 1 Player 2-------- ---------*Y **RB RB*Y point **
point** R*PY point *Y point** R*PY point *Y point*B *B**
point P* point*B *BPY point P* pointRB **** P* pointR* RB**
**RB *Betc... etc...(And the points become less frequent as
play progresses.)I would suggest a greater # of colors, such
as 5, 6, 7, or 8 (8 for'advanced' play). If we have n 
colors,
we would have (2^n -1) piecesto get all combinations of colors
(except no colors).thanks,Leroy Quet
===
I was looking through
my ?es the other day and found a simple Fermatme, but that's
not saying much since I am not an expert mathematicianso I
thought I would post it here for comment by the
researchcommunity.Note: I am posting this out of mere
curiosity, so please be gentle :-)Fermat's Proof
Below:--------------------------The is a simple proof that
for the equation x^n+y^n=z^n there are nointeger solutions
for n>2.BackgroundFermat must have been considering the
Pythagorean Theorem x^2+y^2=z^2and observed the integer
solution sets x=3I, y=4I, z=5I and mused, dointeger solutions
exist for x^n+y^n=z^n (and had decided no).Restate the
equation: x^n (x^n+y^n)^(n-1) + y^n (x^n+y^n)^(n-1)
=(x^n+y^n)^nNotice the factor (x^n+y^n) has the exponent
(n-1) on the left side ofthe equation and the exponent (n) on
the right side.To solve this equation we must introduce some
exponent factor so thatf(scaler, n-1) and f(scaler, n) have a
common denominator (n).Fermat's proof simply observes (n-1)
and (n) are sequential. Theexponent operators are
A^xA^y=A^(x+y) (addition) or A^x/A^y=A^(x-y)(subtraction).
Sequential numbers do not have common denominatorsgreater
than 1. There is no number that can be added to, orsubtracted
from, sequential numbers that will change the fact thatthey
are sequential.Proof-->Sequential numbers do not have common
denominators greater than 1-->If two numbers are sequential,
one is ODD and one is EVEN and theirdifference is
1-->Factor the
numbers-ODD=ODDa*ODDbEVEN=EVENa*EVENb=EVENa*ODDb-->There are
2 equations:1=ODD-EVEN 1=EVEN-ODDReplace with factors and
solve.-->1=(ODDa*ODDb)-(EVENa*ODDb)
1=(EVENa*ODDb)-(ODDa*ODDb)1=ODDb(ODDa-EVENa)
1=ODDb(EVENa-ODDa)SolveODDb=1 ODDb=1ODDa=EVENa+1
EVENa=ODDa+1Sequential numbers do not have common
denominators greater than 1.
===
The Fermat crackpot position
is ?led by James Harris and has beenfor a decade or more.
You'll have to stand in line. Thousands of professional
mathematicians worked on the problem for acentury or so. And
your friend waltzed in with a trivial solution??Truly
amazing!>I was looking through my ?es the other day and
found a simple Fermat>me, but that's not saying much since I
am not an expert mathematician>so I thought I would post it
here for comment by the research>community.>>Note: I am
posting this out of mere curiosity, so please be gentle
:-)>>Fermat's Proof Below:>-------------------------->The
is a simple proof that for the equation x^n+y^n=z^n there are
no>integer solutions for n>2.>>Background>Fermat must have
been considering the Pythagorean Theorem x^2+y^2=z^2>and
observed the integer solution sets x=3I, y=4I, z=5I and
mused, do>integer solutions exist for x^n+y^n=z^n (and had
decided no).>>Restate the equation: x^n (x^n+y^n)^(n-1) + y^n
(x^n+y^n)^(n-1) =>(x^n+y^n)^n>>Notice the factor (x^n+y^n) has
the exponent (n-1) on the left side of>the equation and the
exponent (n) on the right side.>>To solve this equation we
must introduce some exponent factor so that>f(scaler, n-1)
and f(scaler, n) have a common denominator (n).>>Fermat's
proof simply observes (n-1) and (n) are sequential.
The>exponent operators are A^xA^y=A^(x+y) (addition) or
A^x/A^y=A^(x-y)>(subtraction). Sequential numbers do not have
common denominators>greater than 1. There is no number that
can be added to, or>subtracted from, sequential numbers that
will change the fact that>they are
sequential.>>Proof>>-->Sequential numbers do not have common
denominators greater than 1>-->If two numbers are sequential,
one is ODD and one is EVEN and their>difference is
1>>-->Factor the
numbers->ODD=ODDa*ODDb>EVEN=EVENa*EVENb>=EVENa*ODDb>>-->There
are 2 equations:>1=ODD-EVEN 1=EVEN-ODD>Replace with factors
and solve.>>-->1=(ODDa*ODDb)-(EVENa*ODDb)
1=(EVENa*ODDb)-(ODDa*ODDb)>1=ODDb(ODDa-EVENa)
1=ODDb(EVENa-ODDa)>Solve>ODDb=1 ODDb=1>ODDa=EVENa+1
EVENa=ODDa+1>>Sequential numbers do not have common
denominators greater than 1.
===
(some snips)> Fermat must
have been considering the Pythagorean Theorem x^2+y^2=z^2>
and observed the integer solution sets x=3I, y=4I, z=5I and
mused, do> integer solutions exist for x^n+y^n=z^n (and had
decided no).> Restate the equation: x^n (x^n+y^n)^(n-1) +
y^n (x^n+y^n)^(n-1) => (x^n+y^n)^nYou might notice that the
restated equation is always true for any value of x, y, and
n. Eliminating z in that way removes the dependency of the
truth of the equation on the existence of a counterexample to
FLT. Thus, any further proof attempt that focuses on the
restated equation can't possibly prove FLT.-- Mark
Thornquist
===
> (some snips)> > Fermat must have been
considering the Pythagorean Theorem x^2+y^2=z^2> and
observed the integer solution sets x=3I, y=4I, z=5I and
mused, do> integer solutions exist for x^n+y^n=z^n (and had
decided no).> > Restate the equation: x^n (x^n+y^n)^(n-1)
+ y^n (x^n+y^n)^(n-1) => (x^n+y^n)^n> You might notice
that the restated equation is always true for > any value of
x, y, and n. Wait a second... He then proves that this
equation has no solution! This is absolutely brilliant. >
Eliminating z in that way removes the > dependency of the
truth of the equation on the existence of a > counterexample
to FLT. Thus, any further proof attempt that > focuses on the
restated equation can't possibly prove FLT.
===
> To solve this
equation we must introduce some exponent factor so that>
f(scaler, n-1) and f(scaler, n) have a common denominator
(n).Depending on what this is supposed to mean, it is either
irrelevant orsatis?d by scaler[sic]^(n-1) .-- Daniel W.
Johnsonpanoptes@iquest.nethttp://members.iquest.net/~panoptes
/039 53 36 N / 086 11 55 W
===
Unless you are a crank (which I
will assume you arenot (for now)), you should avoid at all
costs posting proofsof FLT. Even if you got it right, people
will alljust assume you're crazy. Statistically, 
it'spretty
much true, and hardly any seasoned mathiewill take the time
to ?d the ? your proof.If you want to learn math, and
interact withmore experienced mathies, I suggest ?ding
otherquestions to approach and try to solve - howmany
polyominoes are there? Does the 3n+1sequence always
terminate? If you _insist_ onstudying FLT, then it is most
instructive to?d the mistakes in your proof for
yourself.Believe me, unless you really suck at math(or are a
crank),you truly are capable of ?ding your
ownmistakes.Finally, and a bit hypocritically, I tried to?d
the ? your proof but right away Ican't tell what you
mean. What is your overallform of argument? I assume it is to
arriveat a contradiction -- that is, I assume that youare
assuming that x^n+y^n = z^n for some ?edpositive integer
triple x,y,z and ?ed n>2; andthat you want to derive a
contradiction from this.So why did you stop using z in your
equation? Youwill have to use that z is a positive
integer_somewhere_. Also, what is this functionf you
introduce? What is an exponent factorand how do you plan to
use it to solve theequation? Why must f(--) and f(---) havea
common denominator of n? Finally, scalar isspelled scalar not
scaler.Tyler> I was looking through my ?es the other day and
found a simple Fermat> me, but that's not saying much since I
am not an expert mathematician> so I thought I would post it
here for comment by the research> community.> Note: I am
posting this out of mere curiosity, so please be gentle :-)>
> Fermat's Proof Below:> --------------------------> The is
a simple proof that for the equation x^n+y^n=z^n there are no>
integer solutions for n>2.> Background> Fermat must have
been considering the Pythagorean Theorem x^2+y^2=z^2> and
observed the integer solution sets x=3I, y=4I, z=5I and
mused, do> integer solutions exist for x^n+y^n=z^n (and had
decided no).> Restate the equation: x^n (x^n+y^n)^(n-1) +
y^n (x^n+y^n)^(n-1) => (x^n+y^n)^n> Notice the factor
(x^n+y^n) has the exponent (n-1) on the left side of> the
equation and the exponent (n) on the right side.> To solve
this equation we must introduce some exponent factor so that>
f(scaler, n-1) and f(scaler, n) have a common denominator
(n).> Fermat's proof simply observes (n-1) and (n) are
sequential. The> exponent operators are A^xA^y=A^(x+y)
(addition) or A^x/A^y=A^(x-y)> (subtraction). Sequential
numbers do not have common denominators> greater than 1.
There is no number that can be added to, or> subtracted from,
sequential numbers that will change the fact that> they are
sequential.> Proof> -->Sequential numbers do not have
common denominators greater than 1> -->If two numbers are
sequential, one is ODD and one is EVEN and their> difference
is 1> -->Factor the numbers-> ODD=ODDa*ODDb>
EVEN=EVENa*EVENb> =EVENa*ODDb> -->There are 2 equations:>
1=ODD-EVEN 1=EVEN-ODD> Replace with factors and solve.>
-->1=(ODDa*ODDb)-(EVENa*ODDb) 1=(EVENa*ODDb)-(ODDa*ODDb)>
1=ODDb(ODDa-EVENa) 1=ODDb(EVENa-ODDa)> Solve> ODDb=1 ODDb=1>
ODDa=EVENa+1 EVENa=ODDa+1> Sequential numbers do not have
common denominators greater than 1.
===
Hash: SHA1examine the
statement before posting it, and deemed it worthy
ofdiscussion. With regard to whether or not it quali?s as a
proof,we must ?st agree on a de?ition of the word.My
understanding is that a mathematical proof is a
demonstrationthat, given certain *axioms* (widely accepted
truths), a presentedtheory is necessarily true. Since widely
accepted is a subjectiveassessment, it often becomes
necessary for a mathematician (or otherproblem solver, for
that matter) to break proposed problems down intosmaller
parts that are more easily reconciled with reality.If the
above understanding is correct, then proving or
disprovingFLT involves a demonstration of how existing
axioms eithernecessitate its truth or untruth.Thus it appears
that the submitted work stands upon following axioms:1) The
Axiom of Variable Substitution - used here to restate
theequation to:x^n(x^n+y^n)^(n-1) + y^n(x^n+y^n)^(n-1) =
(x^n+y^n)^n2) In order for this equation to be solved, its
writer states that(n-1) and (n)--sequential numbers--must be
shown to have a commondenominator greater than 1. As a
layperson, I can not con?m theaxiomatic nature of this
statement, perhaps someone else can con?mor deny it.3)
Sequential numbers can not share a common denominator greater
than1.In short, the attempted proof reveals that the root of
the problem isthat sequential numbers can not be shown to
have common denominatorsgreater than 1, explaining why the
equation breaks at n > 2, butworks for 1 and 2.In light of
the above, there are only 3 ways to disprove this theory:1)
The de?ition of proof as proposed is incorrect.2) One of
the three (3) axioms can be shown to be false (2 lookslike
a candidate from my seat, 1 & 3 appear rather obvious).3) An
error in deductive reasoning was made at one of the three
steps(e.g., the variable substitution was not performed
correctly).With all of the brainpower in this forum, I am
con?ent that we caneither con?m or rapidly dispense with (I
agree this is more likely)this theory and perhaps emerge with
a greater level of understandingas well.As iron sharpens
iron,so one man sharpens another. -
ProverbsiQA/
AwUBP8O79KDLMcpPQeVjEQLBpQCfXmBOKPS7Z5gwAogY332Z2P8jmkAAoMYOlI
bArqOy/03cGqLxG2ljzlel=IpQk> Unless you are a crank (which I
will assume you are> not (for now)), you should avoid at all
costs posting proofs> of FLT. Even if you got it right,
people will all> just assume you're crazy. Statistically,
it's> pretty much true, and hardly any seasoned mathie> will
take the time to ?d the ? your proof.> If you want to
learn math, and interact with> more experienced mathies, I
suggest ?ding other> questions to approach and try to solve
- how> many polyominoes are there? Does the 3n+1> sequence
always terminate? If you _insist_ on> studying FLT, then it
is most instructive to> ?d the mistakes in your proof for
yourself.> Believe me, unless you really suck at math> (or
are a crank),> you truly are capable of ?ding your own>
mistakes.> Finally, and a bit hypocritically, I tried to>
?d the ? your proof but right away I> can't tell what
you mean. What is your overall> form of argument? I assume it
is to arrive> at a contradiction -- that is, I assume that
you> are assuming that x^n+y^n = z^n for some ?ed> positive
integer triple x,y,z and ?ed n>2; and> that you want to
derive a contradiction from this.> So why did you stop using
z in your equation? You> will have to use that z is a
positive integer> _somewhere_. Also, what is this function> f
you introduce? What is an exponent factor> and how do you plan
to use it to solve the> equation? Why must f(--) and f(---)
have> a common denominator of n? Finally, scalar is> spelled
scalar not scaler.> Tyler> > I was looking through my
?es the other day and found a simple Fermat> me, but
that's not saying much since I am not an expert
mathematician> so I thought I would post it here for
comment by the research> community.> > Note: I am
posting this out of mere curiosity, so please be gentle :-)>
> > Fermat's Proof Below:> -------------------------->
> The is a simple proof that for the equation x^n+y^n=z^n
there are no> integer solutions for n>2.> > Background>
> Fermat must have been considering the Pythagorean Theorem
x^2+y^2=z^2> and observed the integer solution sets x=3I,
y=4I, z=5I and mused, do> integer solutions exist for
x^n+y^n=z^n (and had decided no).> > Restate the
equation: x^n (x^n+y^n)^(n-1) + y^n (x^n+y^n)^(n-1) =>
(x^n+y^n)^n> > Notice the factor (x^n+y^n) has the
exponent (n-1) on the left side of> the equation and the
exponent (n) on the right side.> > To solve this equation
we must introduce some exponent factor so that> f(scaler,
n-1) and f(scaler, n) have a common denominator (n).> >
Fermat's proof simply observes (n-1) and (n) are
sequential. The> exponent operators are A^xA^y=A^(x+y)
(addition) or A^x/A^y=A^(x-y)> (subtraction). Sequential
numbers do not have common denominators> greater than 1.
There is no number that can be added to, or> subtracted
from, sequential numbers that will change the fact that>
they are sequential.> > Proof> > -->Sequential
numbers do not have common denominators greater than 1>
-->If two numbers are sequential, one is ODD and one is EVEN
and their> difference is 1> > -->Factor the numbers->
> ODD=ODDa*ODDb> EVEN=EVENa*EVENb> =EVENa*ODDb> >
-->There are 2 equations:> 1=ODD-EVEN 1=EVEN-ODD> Replace
with factors and solve.> > -->1=(ODDa*ODDb)-(EVENa*ODDb)
1=(EVENa*ODDb)-(ODDa*ODDb)> 1=ODDb(ODDa-EVENa)
1=ODDb(EVENa-ODDa)> Solve> ODDb=1 ODDb=1> ODDa=EVENa+1
EVENa=ODDa+1> > Sequential numbers do not have common
denominators greater than 1.
===
> examine the statement
before posting it, and deemed it worthy of> discussion. With
regard to whether or not it quali?s as a proof,> we must
?st agree on a de?ition of the word.No, we already have a
de?ition. If you wish to use the wordproof, you are well
advised to use its current meaning and not makeup some new
meaning. > If the above understanding is correct, then
proving or disproving> FLT involves a demonstration of
how existing axioms either> necessitate its truth or
untruth.Yes, and the axioms in question are Peano's axioms,
which are the veryde?ition of the natural numbers.
Thomas
===
> My understanding is that a mathematical proof is
a demonstration> that, given certain *axioms* (widely
accepted truths), a presented> theory is necessarily true.
Since widely accepted is a subjective> assessment, it often
becomes necessary for a mathematician (or other> problem
solver, for that matter) to break proposed problems down
into> smaller parts that are more easily reconciled with
reality.Oh, and axioms are statements which are too *simple*
to be proved.Example: If a = b and b = c, then a = c. Your
axioms have nobusiness being claimed as axioms, although
axiom 1 has the virtue ofbeing close to the relevant
axioms. (You might as well add the FLTitself as axiom 4 and
really shorten your proof.)You don't get to pick out the holes
in your proof and call themaxioms.-- Daniel W.
Johnsonpanoptes@iquest.nethttp://members.iquest.net/~panoptes
/039 53 36 N / 086 11 55 W
===
> Oh, and axioms are statements
which are too *simple* to be proved.> Example: If a = b and b
= c, then a = c. What makes that too simple to be
proved?
===
> 1) The Axiom of Variable Substitution - used
here to restate the> equation to:> x^n(x^n+y^n)^(n-1) +
y^n(x^n+y^n)^(n-1) = (x^n+y^n)^nYou've restated the equation
as a tautology. This is true for allconceivable values of x,
y, z, and n. > 2) In order for this equation to be solved,
its writer states that> (n-1) and (n)--sequential
numbers--must be shown to have a common> denominator greater
than 1. As a layperson, I can not con?m the> axiomatic
nature of this statement, perhaps someone else can con?m> or
deny it.The equation as stated above certainly does not
require any particularcommon DIVISOR for (n-1) and (n). And
I'm still not sure what thisalleged axiom is trying to
say.Would this axiom assert that a solution for a * k^2 + b
* k^2 = k^3requires a common divisor for 2 and 3?-- Daniel W.
Johnsonpanoptes@iquest.nethttp://members.iquest.net/~panoptes
/039 53 36 N / 086 11 55 W
===
[.snip.]>My understanding is
that a mathematical proof is a demonstration>that, given
certain *axioms* (widely accepted truths), a presented>theory
is necessarily true. No. Mathematical axioms are NOT widely
accepted truths; they aremerely formal statements. Truth
don't enter into it.>2) In order for this equation to be
solved, its writer states that>(n-1) and (n)--sequential
numbers--must be shown to have a common>denominator greater
than 1. This is nonsense as written. You mean a common
DIVISOR.>As a layperson, I can not con?m the>axiomatic
nature of this statement, perhaps someone else can con?m>or
deny it.The statement is unsupported. The fact that n-1 and n
are relativelyprime (have no common divisors other than 1 and
-1) was not shown toimply Fermat's Last Theorem. Don't 
know
why he thinks this is the case.>3) Sequential numbers can not
share a common denominator greater than>1.DIVISOR, not
denominator. The ?denominator' of an integer isusually
taken to be 1, period.>In short, the attempted proof reveals
that the root of the problem is>that sequential numbers can
not be shown to have common denominators>greater than 1,
explaining why the equation breaks at n > 2, but>works for 1
and 2.No: the attempted proof ->ASSERTS<- that Fermat's Last
Theorem has ton and n-1 being relatively prime, but that
assertion is notjusti?d. As for 1 and 2, 1 and 2 have no
common divisors other than 1 and -1, so why would the
assertions about n not work for n=2?>In light of the above,
there are only 3 ways to disprove this theory:>>1) The
de?ition of proof as proposed is incorrect.Your
understanding of what an axiom is is certainly
incorrect.>2) One of the three (3) axioms can be shown to
be false (2 looks>like a candidate from my seat, 1 & 3 appear
rather obvious).2 is not and was not asserted to be, an axiom.
Neither is 3, for thatmatter. 3 is a conclusion derived from
the basic properties of theintegers, while 2 was asserted and
not proven:>> Notice the factor (x^n+y^n) has the exponent
(n-1) on the left side of>> the equation and the exponent
(n) on the right side.>> >> To solve this equation we
must introduce some exponent factor so that>> f(scaler,
n-1) and f(scaler, n) have a common denominator (n).The last
paragraph is (a) nonsense as written; what is scaler?
Whatis f(scaler,n-1)? What is f(scaler, n)? What common
denominator?Does he mean common factors? In short:
imprenetable
nonsense.
===
===========================

===
=========It's not denial. I'm just very selective
about what I accept as reality. --- Calvin (Calvin and
Hobbes)
===
============================

===
========Arturo
Magidinmagidin@math.berkeley.edu
===
Hash: SHA1Geez...I tried
to preface my post with the fact that I am NOT
amathematician, I was just looking for a little peer review
of an textI received from an associate. In any event:>My
understanding is that a mathematical proof is a
demonstration>that, given certain *axioms* (widely accepted
truths), a presented>theory is necessarily true. > No.
Mathematical axioms are NOT widely accepted truths; they
are> merely formal statements. Truth don't enter into it.-
- -snip-> Your understanding of what an axiom is is
certainly incorrect.I appreciate you attempting to polish up
my de?ition, but fail tosee how any formal statement is
axiomatic? You may also want toclearthis up with both Merriam
Webster (www.m-w.com) and the MITmathematics lab since they
share the same de?ition:http://tinyurl.com/wl0m (Section 3:
Axioms)All progress and greater knowledge originate from this
process ofseparating the wheat from the chaff, so I do
greatly appreciate theconstructive comments and illumination
that others have contributedand hope that in some small way
the post may have stimulated
theiQA/AwUBP8QWvaDLMcpPQeVjEQI9sQCgvlW+
v5MIEIMXmn8Xk0mnz6rEiL0AoMYqU53LXg0i2RFQ+cICbmaO/2h8=Nc/3
===

> Finally, and a bit hypocritically, I tried to> ?d the ?yo
ur proof but right away I> can't tell what you mean.He
seems to be tacitly assuming that if m and n are coprime, so
are k^mand k^n. (And the second half of the proof simply
establishes thatn-1 and n are coprime.)-- Daniel W.
Johnsonpanoptes@iquest.nethttp://members.iquest.net/~panoptes
/039 53 36 N / 086 11 55 W
===
> Unless you are a crank (which
I will assume you are> not (for now)), you should avoid at all
costs posting proofs> of FLT. Even if you got it right, people
will all> just assume you're crazy.I have never assumed anyone
is crazy for posting a FLT proof;just optimistic. And if it
even is right, all the better.> Statistically, it's> pretty
much true, and hardly any seasoned mathie> will take the time
to ?d the ? your proof.That's strange - every single FLT
proof I have seen here overthe years has been shot down by
someone fairly quickly.(...)---J K
Hauglandhttp://www.neutreeko.com
===
> I was looking through
my ?es the other day and found a simple Fermat> me, but
that's not saying much since I am not an expert
mathematician> so I thought I would post it here for comment
by the research> community.With many simple proofs of hard
theorems, there is a very simple way to see that is something
?hy, and it works with this proof as well. It is well known
that a^n + b^n = c^n has plenty of solutions in positive
integers a, b, c if n = 2 and none have ever been found if n
> 2. So whatever proof you have that a^n + b^n = c^n _must_
fail if n = 2. If it doesn't fail for n = 2 then it proves
something wrong, so even if you don't understand the proof
you _know_ it cannot be correct. The proof you posted never
requires that n > 2.
===
> The proof you posted never requires
that n > 2.It never even requires that n > 1.-- Daniel W.
Johnsonpanoptes@iquest.nethttp://members.iquest.net/~panoptes
/039 53 36 N / 086 11 55 W
===
> With many simple proofs
of hard theorems, there is a very simple way > to see that is
something ?hy, and it works with this proof as well. >
(SNIP)> The proof you posted never requires that n > 2.Care
should be taken, though. The proof of the irrationality
ortranscendance of pi requires that pi be the ratio of a
circle'scircumference to it's diameter in a rather 
subtle,
not-obvious way.
===
fuffy> I was looking through my ?es the
other day and found a simple Fermat> me, but that's not
saying much since I am not an expert mathematician> so I
thought I would post it here for comment by the research>
community.>> Note: I am posting this out of mere curiosity,
so please be gentle :-)>> Fermat's Proof Below:>
--------------------------> The is a simple proof that for
the equation x^n+y^n=z^n there are no> integer solutions for
n>2.>> Background> Fermat must have been considering the
Pythagorean Theorem x^2+y^2=z^2> and observed the integer
solution sets x=3I, y=4I, z=5I and mused, do> integer
solutions exist for x^n+y^n=z^n (and had decided no).>>
Restate the equation: x^n (x^n+y^n)^(n-1) + y^n
(x^n+y^n)^(n-1) => (x^n+y^n)^n>> Notice the factor (x^n+y^n)
has the exponent (n-1) on the left side of> the equation and
the exponent (n) on the right side.>> To solve this equation
we must introduce some exponent factor so that> f(scaler,
n-1) and f(scaler, n) have a common denominator (n).>>
Fermat's proof simply observes (n-1) and (n) are
sequential. The> exponent operators are A^xA^y=A^(x+y)
(addition) or A^x/A^y=A^(x-y)> (subtraction). Sequential
numbers do not have common denominators> greater than 1.
There is no number that can be added to, or> subtracted from,
sequential numbers that will change the fact that> they are
sequential.>> Proof>> -->Sequential numbers do not have
common denominators greater than 1> -->If two numbers are
sequential, one is ODD and one is EVEN and their> difference
is 1>> -->Factor the numbers-> ODD=ODDa*ODDb>
EVEN=EVENa*EVENb> =EVENa*ODDb>> -->There are 2 equations:>
1=ODD-EVEN 1=EVEN-ODD> Replace with factors and solve.>>
-->1=(ODDa*ODDb)-(EVENa*ODDb) 1=(EVENa*ODDb)-(ODDa*ODDb)>
1=ODDb(ODDa-EVENa) 1=ODDb(EVENa-ODDa)> Solve> ODDb=1 ODDb=1>
ODDa=EVENa+1 EVENa=ODDa+1>> Sequential numbers do not have
common denominators greater than 1.
===
>> > fuffy>> >
>> > fuffy>> >> Fuffy has re?ed its tactics. First,
all that Fuffy did was>> to keep the name of the most
current poster in a thread>> from appearing on Google.>>>
Free clue for the clueless: If you want to see the>> threads
have lots of sub-threads, and the default>> order makes it
dif?ult to ?d recent comments>> in more than one.>>>
What is to be done about Fuffy?>> >> Duh-h-h. Don't ask
me.>>> Well, I suspect virtually every Google user but
you>> already knew about sort by date and how to ?d >>
recent posters.>>> I'm a Mathie.>>> <snort>People
want to know who the last poster in a thread is,>without
having to do a Google Advanced Search. Second clue for the
clueless:Sort by date is on every thread with more than one
message. Lookaround. Nothing to do with Advanced Search.>
For>example, if Google shows Justin Van Twinkie as the
last>poster, I won't bother to read the last postingThen
you'll have no idea whether or not 15 people have posted
sincethe last time you read the thread.
===
> >> >
fuffy>> > >> > fuffy>> >> Fuffy has re?ed
its tactics. First, all that Fuffy did was>> to keep the
name of the most current poster in a thread>> from
appearing on Google.>> >> Free clue for the clueless: If
you want to see the>> threads have lots of sub-threads, and
the default>> order makes it dif?ult to ?d recent
comments>> in more than one.>> >> What is to be done
about Fuffy?>> >> Duh-h-h. Don't ask me.>> >>
Well, I suspect virtually every Google user but you>>
already knew about sort by date and how to ?d >> recent
posters.>> >> I'm a Mathie.>> >> <snort>>>
>People want to know who the last poster in a thread is,>
>without having to do a Google Advanced Search. > Second
clue for the clueless:> Sort by date is on every thread
with more than one message. Look> around. Nothing to do with
Advanced Search.> > For>example, if Google shows Justin
Van Twinkie as the last>poster, I won't bother to read the
last posting> Then you'll have no idea whether or not 15
people have posted since> the last time you read the
thread.All right, now I see what you mean. Now I know how to
keep McCallum from wasting my time. On the otherhand, I'll
bet there are Google Users other than myself who don'tyou
seem to want to believe, and for people who don't know how
itworks, McCallum fucks things up in a way that he has no
right to.
===
Suppose you have some random variables X_j that
satisfythe conditions of the central limit theorem
(identialmeans and variances). We all know that the SUM of
suchrandom variables tends toward a Gaussian distribution.Now
suppose that you have some function f(X) that maps each of
these random variables X_j. Suppose for simplicitythat the
function f(X) is invertible, at least in some
restricteddomain, but perhaps nonlinear.Is there any
statement that can be made about the distributionof the sum
of f(X_j)?References would be greatly appreciated, also.Much
thanks,JB
===
>Suppose you have some random variables X_j that
satisfy>the conditions of the central limit theorem
(idential>means and variances). We all know that the SUM of
such>random variables tends toward a Gaussian
distribution.>Now suppose that you have some function f(X)
that maps >each of these random variables X_j. Suppose for
simplicity>that the function f(X) is invertible, at least in
some restricted>domain, but perhaps nonlinear.>Is there any
statement that can be made about the distribution>of the sum
of f(X_j)?Well, obviously you want the f(X_j) to satisfy
conditionson means and variances. They don't have to be
identical, though. Lindeberg's Theorem says if Y_j are
independent with means mu_jand variances sigma_j^2, M_n =
sum_{j=1}^n mu_j and S_n^2 = sum_{j=1}^n sigma_j^2, with
1/S_n^2 sum_{j=1}^n E[(Y_j - mu_j)^2 I(|Y_j - mu_j| >= S_n
epsilon) -> 0as n -> in?ity for all epsilon > 0 (I(A) being
the indicator function of A, i.e. 1 if A is true and 0
otherwise), then 1/S_n sum_{j=1}^n (Y_j - mu_j) converges in
distribution to a standard normal random variable.A condition
that may be easier to apply is that 1/S_n^3 sum_{j=1}^n E
|(Y_j - mu_j)^3| -> 0 as n -> in?ity.Robert Israel
israel@math.ubc.caDepartment of Mathematics
http://www.math.ubc.ca/~israel University of British Columbia
Vancouver, BC, Canada V6T 1Z2
===
>Suppose you have some
random variables X_j that satisfy>the conditions of the
central limit theorem (idential>means and variances). We
all know that the SUM of such>random variables tends toward
a Gaussian distribution.Actually we don't know that: assuming
your X_j are independent, but not identically distributed,
identical means and variancesare not enough for a Central
Limit Theorem: you need the Lindebergcondition. An amusing
example isX_j = 0 with probability 1-1/j^2 +/- j with
probability 1/(2 j^2) eachso all X_j have mean 0 and variance
1. But since sum_j 1/j^2 < in?ity,with probability 1 all but
?itely many X_j are 0, and thus 1/n sum_{j=1}^n X_j -> 0
almost surely. >Now suppose that you have some function
f(X) that maps >each of these random variables X_j. Suppose
for simplicity>that the function f(X) is invertible, at
least in some restricted>domain, but perhaps nonlinear.>
>Is there any statement that can be made about the
distribution>of the sum of f(X_j)?> Well, obviously you
want the f(X_j) to satisfy conditions> on means and
variances. They don't have to be identical, though. >
Lindeberg's Theorem says if Y_j are independent with means
mu_j> and variances sigma_j^2, M_n = sum_{j=1}^n mu_j and >
S_n^2 = sum_{j=1}^n sigma_j^2, with > 1/S_n^2 sum_{j=1}^n
E[(Y_j - mu_j)^2 I(|Y_j - mu_j| >= S_n epsilon) -> 0I left
out a bracket: that should be 1/S_n^2 sum_{j=1}^n E[(Y_j -
mu_j)^2 I(|Y_j - mu_j| >= S_n epsilon)] -> 0> as n -> in?ity
for all epsilon > 0 (I(A) being the indicator function of >
A, i.e. 1 if A is true and 0 otherwise), then > 1/S_n
sum_{j=1}^n (Y_j - mu_j) converges in distribution to a
standard > normal random variable.E.g. suppose for simplicity
all X_j and f(X_j) have mean 0, f(0) = 0, and there are
positive constants a and b such that a(y-x) < f(y) - f(x) <
b(y-x) whenever x < y. Thena^2 Var(X_j) <= Var(f(X_j)) <= b^2
Var(X_j). I'll write S_n^2 = sum_{j=1}^n Var(X_j) and 
S'_n^2 =
sum_{j=1}^n Var(f(X_j)),so a S_n <= S'_n <= b S_n.In
Lindeberg's conditionf(X_j)^2 I(|f(X_j)| > S'_n 
epsilon) <=
b^2 X_j^2 I(|X_j| > S_n epsilon a/b)so if X_j satisfy the
condition, so do f(X_j). Robert Israel
israel@math.ubc.caDepartment of Mathematics
http://www.math.ubc.ca/~israel University of British Columbia
Vancouver, BC, Canada V6T 1Z2
===
OK I see where I went wrong
now, I think. Let me try again.Consider the Category, (I use
capital C to indicate it may have the cardinality of all
classes, thus have to be a super-category or whatever);whose
objects are all ordinary categories, and whose arrows are
functors.This Category has natural transformations de?ed
within it, which are(I think?) mappings from functors to
functors. So they are a sort of2nd-order Arrow in this
Category.So, given their properties, what do Natural
Transformations (in this Category) correspond to in the way
of mappings-of-arrows in an ordinary category?Does this make
sense now? What special named sort of arrow-transformation do
they correspond
to?----------------------------------------------------------
-------------------- Bill Taylor
W.Taylor@math.canterbury.ac.nz-------------------------------
----------------------------------------------- I shot an
arrow in the air And down it came, I know not where. I looked
for it up in a tree Which was, though, just a
categ'ry!----------------------------------------------------
--------------------------
===
>A and B have different
dimensions,>but AB and BA are both square.>Can anyone provide
a proof showing the eigenvalues are the same for AB and BA
?Side question: What's your favorite linear algebra text for
answeringthis sort of question? I pulled out my Horn and
Johnson but realized Ihad no idea where to look to ?d a
proof, if there is one.Eigenvalue was too broad an index
entry.Related question: Do you have a favorite book
coveringspecially-structured matrices? That would include,
for example, fastToeplitz inversion methods? Or could easily
answer the question whensomebody asks, is there a name for
this kind of matrix? - Randy
===
0072>> > I've got a list
of points (in x,y form) that are the corners of a> >
polygon, in order. It's not necessarily a convex polygon. Is
there a> > standard algorithm for determining whether a
given point is inside> > or outside the polygon? All
this takes place in the plane... For> > practical
purposes, an algorithm is ok if it can handle up to 7
points> > decently.> >> > Many thanks -Adrian> >>>
> This is probably in the faq at
comp.programming.games.algorithms, or> something like that.
Google will probably also help.>> I know it's in the
comp.graphics.algorithms FAQ.>>
http://www.faqs.org/faqs/graphics/algorithms-faq/>> Check out
question 2.03.>> - Randy
===
>> > I have looking at a few
web pages dealing with the largest known> > calculated
primes and a great deal of computational time is taking> >
into searching for these numbers and verifying they are
primes. I have> > seen the Euclid's proof of in?itude
primes and it occurs that me> > that super-large prime
numbers can be calculated using the following:>> <snip>>
>> Others have already given you an example of why your
idea won't ?tIthere> is no largest prime (and
therefore an in?ite number of primes) is as> follows:>>
> If there is a largest prime, P, then the number of primes
is ?ite. Wecan> therefore form a new number, N, which is
one greater than the product of> all primes. Therefore, N
> P (and, indeed, it would be very much greater> than P).>
>> When divided by any prime in our list, it leaves
remainder 1. Therefore,> /either/ it is prime /or/ it is
the product of two or more primes, atleast> one of which is
greater than P.>> In /either/ case, P has been shown not
to be the largest prime number.>> Nope. Nowhere have you
assumed that the list of primes known is> complete up to P.
You've simply proved that the ?ite list of> known primes is
not the list of all primes.>> If you use Euclid's proof as
constructive, then the sets you generate0079> {2,3,7}>
{2,3,7,43}> {2,3,7,13,43,139}> {2,3,7,13,43,139,3263443}> and
thence you ?d the new primes {547,607,1033,31051}, all of
which> are less than the currently largest known prime
3263443.>> Don't forget the second case! :-)>> Don't 
forget
that if you're pretending that you don't know what> 
the primes
are, you can't suddenly pull otherwise well-known> features of
the prime numbers out of a hat.>> Phil> --> Unpatched IE
vulnerability: ADODB.Stream local ?e writing> Description:
Planting arbitrary ?es on the local ?e system> Exploit:
http://ip3e83566f.speed.planet.nl/eeye.html> (but unrelated
to the EEye exploit)
===
>Well duh!>All of _what_,
though.Primes.John
===
> Nope. That is prior art. However
taxing folks on their exusions and > bodily productions is
not far in the future. The jism tax. > It's the coming
thing.HOWWWWWWWWWWLL !!!!!! Pun of the week!But seriously, we
already have, here in NZ, proposed taxes on emissions,not of
people but cows. In order to comply with the perceived
obligationsunder the Kyoto protocol, (hey there's a song line
there... Kyo-to pro-to..),our PC government proposed a new
cattle emission tax on farmers for the allegedharm to the
ozone layer due to emitted methane!Naturally, this was
immediately dubbed the fart tax by the predatory media,and
attracted so much derision, joking, and protest from the
farming communitythat it's unlikely to see the light of day;
which is in some ways a pitybecause a fart tax would have been
a wondrous thing! (Even if smellinga little
?hy!)-------------------------------------------------------
----------------------- Bill Taylor
W.Taylor@math.canterbury.ac.nz-------------------------------
----------------------------------------------- Camou?ndo
ms: So they won't see you
coming.------------------------------------------------------
------------------------
===
>I should note that the above
carries no attribution. It is>>copyrighted material
originally published in The Onion,>>but I couldn't locate any
of the original Onion material>>online anywhere. >>I've
already posted the Onion material (actually, I've posted the
link>to the material fron the Onion's own website) elsewhere
in this thread.Huh. I didn't realize their archive went back
that far. I once triedand eventually gave up. It's the one
with several eminent physicistsdiscussing Big Bang theory.
Like man, what if the whole universe is,like, a big Atom in
another Universe? Wow, heavy, man. That sort ofthing. -
Randy
===
When a Riemann surface is constructed by joining
copies of the cutplane the arbitrariness in the choice of
cuts carries over into anarbitrariness in the Riemann
surface. And where there is no uniquedomain there is no
unique function so it is incorrect to speak ofthe function
w(z) de?ed by an initially given relation f(w,z)=0.It is
rigor not pedantry to remark that the derivable functions
aremany and that the initially given relation is just what
they have incommon. When Riemann introduced his Riemann
surfaces it may be that he didso to eliminate the
multiplicity that comes with branches. But themultiplicity
reappears in this altered form. To understand it one hasto
stay with analysis and not move on to a more abstract
geometricaltreatment. For other notes on this topic visit my
websitewww.riemannsurfaces.info.
===
>When a Riemann surface
is constructed by joining copies of the cut>plane the
arbitrariness in the choice of cuts carries over into
an>arbitrariness in the Riemann surface.Lucky that that's not
the only way to construct a Riemann surfacethen...> And where
there is no unique>domain there is no unique function so it
is incorrect to speak of>the function w(z) de?ed by an
initially given relation f(w,z)=0.Look up analytic
continuation in an advanced book on complexanalysis: the
standard construction of _the_ Riemann surface heredoes not
have anything to do with cuts, and there's nothingarbitrary
about it. The surface is just the set of all functionelements
which can be reached by continuation along a path,
startingwith a given function element.>It is rigor not
pedantry to remark that the derivable functions are>many and
that the initially given relation is just what they have
in>common.> When Riemann introduced his Riemann surfaces it
may be that he did>so to eliminate the multiplicity that
comes with branches. But the>multiplicity reappears in this
altered form. To understand it one has>to stay with analysis
and not move on to a more abstract geometrical>treatment.>
For other notes on this topic visit my
website>www.riemannsurfaces.info.> David C.
Ullrich
===
>Message-id:
<2410d7e.0311241320.5b02c68e@posting.google.com>Can someone
provide with a mathematical algorithm which explains
the>horrendous pixel-crime perpetred by the infamous Mr.
Mensanator?>>http://www.thequantummachine.com/images.phpHey
, the new image is a big improvement over the original. A
littleconstructive criticism helps doesn't it?>>I thought
that my TFT screen had been degraded by some air
pollutant,>but then I realized that it was only his image
which had some fungus>on it... :-)--MensanatorAce of
Clubs
===
> >> Given an integral domain R, can we not
construct a ?ld in the same >> way we construct Q from Z?
I.e., equivalence classes on R x >> (R{0}), where (a, b) ~
(c, d) iff a d = b c . (Integral domains >> are by
de?ition commutative, no? So says Herstein.)>>If R has a
1, then the mapping f(r) = (r,1) gives the desired>
>embedding; but without the 1, how do we embed R?> You are
not really mapping to (r,1), you are mapping to the class of>
(r,1). So map r to the class of (r^2,r) if r is nonzero, and
map 0 to> the class of (0,r) for arbitrary r different from
0. > Doh! Of course; because cancellation holds in R (follows
fromintegrality), if we map r,s to (r^2,r) and (s^2,s), resp.,
then r^2.s= s^2.r iff r = s...>

===
=================================

===
===> It's not denial. I'm just very selective about>
what I accept as reality.> --- Calvin (Calvin and Hobbes)>

===
=================================

===
===> Arturo Magidin>
magidin@math.berkeley.edu
===
Please help with
equations/sketches for a circular torus intersectionwhen the
cutting plane is not parallel to its axis of
symmetry.Equations may be expressed in terms of torus tube
radius, tubedisplacement from axis,inclination/length of
perpendicular from torusaxis/center to the intersecting
plane. References requested preferablyfrom the internet.
TIA
===
> Please help with equations/sketches for a circular
torus intersection> when the cutting plane is not parallel to
its axis of symmetry.> Equations may be expressed in terms of
torus tube radius, tube> displacement from
axis,inclination/length of perpendicular from torus>
axis/center to the intersecting plane. References requested
preferably> from the internet. TIAjust take a
three-dimensional parametrisation of the torus like ( cos s *
(cos t + 2) )f(s,t) = ( sin s * (cos t + 2) ) ( sin t )This is
for a torus of displacement 2 and tube radius one. With
vectorcalculus you now can easily deduce the cutting with any
plane.Rene.-- Ren.8e MeyerStudent of Physics &
MathematicsZhejiang University, Hangzhou, China
===
An
interesting integral just appeared in alt.math, and there
should bea neater solution than I found. The problem is to
evaluate int_0^1 int_0^{1-x} 1/(1-axy) dy dx,where |a| <
4.My solution was to expand into a geometric series (justi?d
since |a|< 4, 0 <= y <= 1-x ==> |4axy| < 1), obtaining
int_0^1 a^n x^n (1-x)^(n+1) dx,and then use the standard
identity int_0^1 x^a (1-x)^b = Gamma[a+1]
Gamma[b+1]/Gamma[a+b+2].This yields the integral as a power
series, sum_{n=0}^infty a^n n!^2/(2n+2)!,which is then
recognizable as 2 arcsin(sqrt(a)/2)^2 --------------------- .
aBut there ought to be a direct way to see this. Any
takers?--Ron Bruck
===
Two minor corrections: that should read
|a|<4, 0 <= y <= 1-x ==> |axy| < 1,and we obtain for the value
of the integral (1-x)^(n+1) sum_{n=0}^infty int_0^1 a^n
x^n ----------- dx. n+1But the ?al answer remains
unchanged.--Ron Bruck> An interesting integral just appeared
in alt.math, and there should be> a neater solution than I
found. The problem is to evaluate> int_0^1 int_0^{1-x}
1/(1-axy) dy dx,> where |a| < 4.> My solution was to
expand into a geometric series (justi?d since |a|> < 4, 0 <=
y <= 1-x ==> |4axy| < 1), obtaining> int_0^1 a^n x^n
(1-x)^(n+1) dx,> and then use the standard identity>
int_0^1 x^a (1-x)^b = Gamma[a+1] Gamma[b+1]/Gamma[a+b+2].>
> This yields the integral as a power series,>
sum_{n=0}^infty a^n n!^2/(2n+2)!,> which is then
recognizable as> 2 arcsin(sqrt(a)/2)^2>
--------------------- .> a> But there ought to be a direct
way to see this. Any takers?> --Ron Bruck
===
Set R_1 = x+y,
R_2 = xyIs there an easy way of writing the polynomial x^n +
y^n in terms ofR_1 and R_2? I know it can be done by
expanding out with R_1^n, butthis is obviously not practical
for large n. Is there some sort ofidentity that allows you
to write the answer down immediately,similar to the binomial
theorem?
===
> Set R_1 = x+y, R_2 = xy> Is there an easy way
of writing the polynomial x^n + y^n in terms of> R_1 and R_2?
I know it can be done by expanding out with R_1^n, but> this
is obviously not practical for large n. Is there some sort
of> identity that allows you to write the answer down
immediately,> similar to the binomial theorem?The buzzword is
Dickson polynomial.Ley u = x + y and x = xy. Form the
generating functionF(t) = sum_{n=0}^in?ity (x^n +
y^n)t^n.ThenF(t) = 1/(1-xt) + (1-yt) = (2 - ut)/(1 - ut +
vt^2) = (2 - ut)sum_{m=0}^in?ity (u + vt)^m t^m = (2 -
ut)sum_{n=0}^in?ity t^n sum_m {m choose n-m}u^{2m-n}
v^{n-m}etc. Pulling out the coef?ient of t^n gives
D_n(u,v)the Dickson polynomial of the ?st kind with D_n(u,v)
= x^n + y^n.-- Robin Chapman,
www.maths.ex.ac.uk/~rjc/rjc.htmlNeedless to say, I had the
last laugh. Alan Partridge, _Bouncing Back_ (14 times)
===
>
Set R_1 = x+y, R_2 = xy>> Is there an easy way of writing the
polynomial x^n + y^n in terms of> R_1 and R_2? I know it can
be done by expanding out with R_1^n, but> this is obviously
not practical for large n. Is there some sort of> identity
that allows you to write the answer down immediately,>
similar to the binomial theorem?Here's my attempt:Write f_n =
x^n+y^n. Then expand (x+y)*(x^n+y^n) to get the
recurrencerelation:f_(n+1) - R_1 * f_n + R_2 * f_(n-1) =
0with the initial conditions: f_0 = 2 and f_1 = R_1.The
solution to the recurrence relation may be written as:f_n = A
* u^n + B * v^n, whereu and v are roots of the quadratic
equation t^2 - R_1 * t + R_2 = 0.The coef?ients A and B may
be found by considering the initial
conditions.-Michael.
===
> > Set R_1 = x+y, R_2 = xy>
Here's my attempt:> Write f_n = x^n+y^n. Then expand
(x+y)*(x^n+y^n) to get the recurrence> relation:> f_(n+1) -
R_1 * f_n + R_2 * f_(n-1) = 0> with the initial conditions:
f_0 = 2 and f_1 = R_1.That is a special case of the Newton's
identities. > The solution to the recurrence relation may be
written as:> f_n = A * u^n + B * v^n, where> u and v are
roots of the quadratic equation t^2 - R_1 * t + R_2 = 0.Don't
you think that the solutions to this equation would be x and
y.... > The coef?ients A and B may be found by considering
the initial conditions.... with A=1 and B=1? So this
approach won't give much information, as thesolution you get
will be (drums, please)f_n=x^n+y^n :)Jyrki
===
>> > Set
R_1 = x+y, R_2 = xy>> Here's my attempt:>> Write f_n =
x^n+y^n. Then expand (x+y)*(x^n+y^n) to get the recurrence>
relation:> f_(n+1) - R_1 * f_n + R_2 * f_(n-1) = 0> with
the initial conditions: f_0 = 2 and f_1 = R_1.>> That is a
special case of the Newton's identities.>> The solution to
the recurrence relation may be written as:> f_n = A * u^n +
B * v^n, where> u and v are roots of the quadratic equation
t^2 - R_1 * t + R_2 = 0.>> Don't you think that the solutions
to this equation would be x and y....Indeed....>> The
coef?ients A and B may be found by considering the
initialconditions.>> ... with A=1 and B=1? So this approach
won't give much information, as the> solution you get will
be (drums, please)>> f_n=x^n+y^n :)Oh well, it seemed like
such a good idea at the time.... :-)-Michael.
===
> Set R_1
= x+y, R_2 = xy> Is there an easy way of writing the
polynomial x^n + y^n in terms of> R_1 and R_2? I know it can
be done by expanding out with R_1^n, but> this is obviously
not practical for large n. Is there some sort of> identity
that allows you to write the answer down immediately,>
similar to the binomial theorem?Search for Newton's
identities (a useful recurrence relation) andWaring's
formula. They are also helpful for a larger number of
variables(x,y,z,etc.). I don't know what would be the
simplest formula for twovariables. Wouldn't any formula for
Dickson's polynomials do? Searchfor any of these key
phrases!Jyrki Lahtonen, Turku, Finland
===
> Ok, I apologize.>
> I did not say it was fraud, I _asked_ if it was.> You say
it was not, that answers the question.Ok. I accept your
apologies. I am little tired of this controversy,and seems
stupid to get angry with people I don't personally meet.>
It's just that extraordinary claims require extraordinary
evidence. What you > present on your web site cannot even be
considered evidence, let alone> extraordinary evidence. I'm
supposed to take your word that if presented> correctly,
the lines would be parallel? That's not how science is
done.I agree with that. As I said, the main investigation was
done byCrater&McDaniel. I didn't reproduce step by step, but
made my owncalculations and simulations that were consistent.
At that time thecomputer simulation was very cost, so I made
some simulations thatshowed the same level that theirs.Also,
revised the protocol, and the pattern can be ?ted with
thatmean deviation inside of the mounds, the problem is
trying to do byhand. But it is easy, just using least squares
(if you insist, asCrater, in a coordinate ?).I will prepare
when I get some time to do it by computer, you mustnotice
that at the time I did the diagram, as it is rotated, it
wouldhave took me a considerable time to locate the
correspondingcoordinates and there was the issue of the
non-recti?d image.What I have thought is to use a
superimposed image, with the centersclearly marked, so it is
visually discernible how far is the deviationin each mound.>
And let's not forget that you solicited comments. Do you
think that I am the> only one who sees problems with what
you presented? You should be happy to > see such comments so
that you can correct the situation. The problem is that the
critics here and other posts were far fromgentle. I am not
accustomized to this. Also notice that I am littletired of
writing some statements, and getting them ignored.For
example, I stated that the simulations (and calculation by
handusing area ratios will do) show that you need about 75 or
more moundsto get reasonable levels of that pattern being by
chance.And a poster in my forum came with 106 dots and showed
some nicealignments. Perhaps I loose patience very fast, but I
had to explainthat not only those nice alignments were
present, but very likely themounds pattern. Other thing was
to extract it, which would likelyrequire a computer.> I
will clarify also in my webpage soon, > Good.> > givin
links to your webpage.> Don't you have your priorities
wrong? Shouldn't you focus on what _you_ are> presenting? It
is now closed. I think the best test would be to perfom a
completescan of the martian surface. It could be used to
detect crater chainsand make crater population counts also
(they are similar to mounds,but with lightning inverted).
However, in order to avoid bias, Ishould make it independent
from the mounds, and when I detectautomatically, this may be
somehow indicative of non-randomness.> You will be acussed
of dishonesty. > Dishonesty? I just presented what I saw.
You admit that the original image> was not orthorecti?d, so
I was honest in saying that the lines do not appear>
parallel. Now if I made a mistake, it was an honest one,
because I used the> image that _you_ provided. Rather than
explaining in the text why the image> was incorrect,
wouldn't it have been better to post a correct image?But that
was said in the webpage. I retire the accusation ofdishonesty
here and in my webpage, accepting that you did not notice.And
yes, I will do that. I will also publish the source code I use
andthe link to the original image in NSSDC website.> You can
sue me also, if you want. I am wishing it.> Sue you for
what? Calling me dishonest? Calling me pathetic? Calling me a
slow> thinker? Calling me stupid? These comments won't make
you look good.I retire those comments, but you must notice
that I have heardeverything, from being a pseudoscientist, to
not knowing probabilitynotions, fraud, and countless
others.The problem is a mathematical one. For example my
claim that this isthe only pattern (after a square-based one)
with the greater number ofDIFFERENT parallel/perpendicular
directions for 5 points (BAGED, plusrepeating P) goes
unchallenged. And this is what irritates me. Thatclaim is
just a very de?ed one, testable with analitical geometry.But
based on a diagram, so... could I have missed some
con?uration?I recommend to read most of the material I
present, at leastsuper?ially, before making comments, and
let the childish I am moreintelligent than you issues out
of the conversation.
===
> >Message-id:
<2410d7e.0311220235.6c366ff9@posting.google.com>> >> You
are one of those who confuse the probability of an event
occuring> with the event itself. Duh...It is you who confuse
it. You still believe that the probability ofthat event has
turned to 1. This is a claim I have heard countlesstimes. Yet
a math PhD laughed at it.see how they do that.Also the crater
chains are analysed probabilistic, but still they arethere
since some million years ago... > In the mounds problem, the
probabilistic experiment is having 6 mounds> formed by some
process. The process has a great number of pausible>
outcomes, depending on its particular physical nature. If you
assume> some type of a random formation process, then the
probability of> getting the formation shown is extremely
low.> Instead of wasting your time uploading gifs and
accusing people, why> don't you program a simulation of 6
mounds forming at random, set your> accuracy range for
parallel and perpendicular lines and see of you can> get that
or similar formations and what the frequency of those are.>
That will convince you the probability of getting that
formation or> similar ones is extremely low. That's all there
is to it.I have done that particular experiment. I used
parallel lines only fora computer simulation of a number of
points, not just 6 but greater.Still, for such a number of
parallel relationships I got 1/1000 for 6points. I draw some
of them at hand (I got numbers in the screen onlybecause at
that time I didn't get any notion of image processing).Most
were unremarkably, and there was a tendency to form what I
calleddegenerated lines (a bug).You can do by yourself and
check. Otherwise you are talking withoutfoundation, just as
counting sheeps, and you know. DO THE SIMULATIONAND WHEN YOU
HAVE A FIGURE, WE TALK AGAIN. If not, you are
makingpseudoscience.
===
>Message-id:
<1f366ae.0311240955.b79503f@posting.google.com>>
>Message-id:
<2410d7e.0311220235.6c366ff9@posting.google.com>
>[snip]>> You just can't _prove_ it's not random.
Probability has no meaning>> for something that has already
happened. You claim the odds are>> 200,000,000,000 to 1 and
then you say that no other con?uration>> has this
property.>>> Duh.>>>[snip]>>You are one of those who
confuse the probability of an event occuring>with the event
itself. Duh...>>In the mounds problem, the probabilistic
experiment is having 6 mounds>formed by some process. The
process has a great number of pausible>outcomes, depending on
its particular physical nature. If you assume>some type of a
random formation process, then the probability of>getting the
formation shown is extremely low.The formation? Don't you
mean a formation?>>Instead of wasting your time uploading
gifs and accusing people, why>don't you Why doesn't 
the OP
have a mathematically correct drawing on his web
site?>program a simulation of 6 mounds forming at random,
What, exactly, is the geological process I am supposed to
simulate? Or should Ijust sprinkle random pixels onto an
image?>set your accuracy rangeAccuract range? This is math,
not physics. There is no range allowed indetermining parallel
and perpendicular.>for parallel and perpendicular lines and
see of you can>get that or similar formations and what the
frequency of those are.>That will convince you the
probability of getting that formation or>similar ones is
extremely low. That's all there is to it.You're wrong. 
There
is a world of difference between that formation andsimilar
formations.>>By the way, has you ever been to a casino? Have
you ever looked at the night sky and noticed all the polygons
formed bythe stars?Do you think there is some mysterious
reason for this?>The number you see the>roullette ball
sitting on of course just happended but you can't bet>on it.
Have you thought about _why_ they won't let you bet on it? Do
you know what theprobability is of an event that has
already happened? >If you think talking about its probability
has no meaning>because it just happened, Whatever the
probability was before the event, after the event the
probabilityis simply 1.Possibly you didn't see this example,
so I'll repeat it:Put 64 coins in a box and shake. Note the
pattern of heads and tails. Theprobability of that pattern
occuring is1/18446744073709551616and yet it happened! Shake
the box again. The pattern you see now also had
a1/18446744073709551616chance of occuing and it happened
also. The chance of BOTH those patternsoccuring back to back
is1/340282366920938463463374607431768211456What conclusion do
you draw from this?>then if you have just won, why should
they>pay you back 36 times your bet? They shouldn't,
roullette payouts are 35 to 1. Of course, you would know
thisif you've ever been to a casino.>Just take your money
back and that's>it.:)--MensanatorAce of Clubs
===
> Whatever
the probability was before the event, after the event the
probability> is simply 1.The probability is related to the
chances of that event happening. Iwould not apply to events
that have already happened. The p value doesnot get altered,
because p is predictive, and you are talking of avalue which
is not predictive.I would think that it is more a semantic
question.I would assign 0 to event which are impossible and 1
to event that aresurely TO HAPPEN (in the
future).
===
>Message-id:
<2410d7e.0311250420.2e0c3caf@posting.google.com>> Whatever
the probability was before the event, after the event
the>probability>> is simply 1.>>The probability is related to
the chances of that event happening. I>would not apply to
events that have already happened. The p value does>not get
altered, because p is predictive, and you are talking of
a>value which is not predictive.>>I would think that it is
more a semantic question.>>I would assign 0 to event which
are impossible and 1 to event that are>surely TO HAPPEN (in
the future).FYI, I have heard it said probabilty of 1 and
certainty do not mean the samething for a future event, but
I can't explain it. Put a coin in a box and shake it. Without
looking, what is the probability ofshowing heads? This isn't
quantum mechanics. It does not depend on whether youlook or
not. It is either heads or it isn't. Before you shake the
box, you canask what is the probability that it will show
heads after the shake. After theevent, the probability of
heads is either 0 (if it actually came up tails) or 1(if it
did come up heads). The trick is you don't know which case it
is.You can try to guess what the coin is showing, but in that
case the probabilityapplies to the guess, not the event. In
the real world (such as footballgames), a coin is called in
the air while the probability function is
stillvalid.--MensanatorAce of Clubs
===
> Put a coin in a box
and shake it. Without looking, what is the probability of>
showing heads? This isn't quantum mechanics. It does not
depend on whether you> look or not. It is either heads or it
isn't. Before you shake the box, you can> ask what is the
probability that it will show heads after the shake. After
the> event, the probability of heads is either 0 (if it
actually came up tails) or 1> (if it did come up heads). The
trick is you don't know which case it is.But the probability
is 1/2 BEFORE the trial. I say that AFTER thetrial, it is
still 1/2, because you cannot apply to that alreadyhappened
case. Anyway, it is just semantic.
===
> Put a coin in a box
and shake it. Without looking, what is the probability of>
showing heads? This isn't quantum mechanics. It does not
depend on whether you> look or not. It is either heads or
it isn't. Before you shake the box, you can> ask what is
the probability that it will show heads after the shake.
After the> event, the probability of heads is either 0 (if
it actually came up tails) or 1> (if it did come up heads).
The trick is you don't know which case it is.> > But the
probability is 1/2 BEFORE the trial. I say that AFTER the>
trial, it is still 1/2, because you cannot apply to that
already> happened case. Anyway, it is just semantic.If it was
just semantic, people wouldn't get into heated arguments about
it.Smarter people than me have saidone should not make
probability statements about past actualities, but only about
future possibilitieswhich I tend to agree with. But let's say
you want to use it anyway. What isthe probability that you
will get heads on two consecutive coin ?efore we start,
we know that the probability of both is the probabilitiesof
the individual ?ultiplied together: 1/2 * 1/2 = 1/4.You
?e coin and it comes up tails. NOW what is the
probability thatboth ?ill be heads? We know the answer
is 0, we know the probabilityof the second ? 1/2. The
probability of the past event being headsmust therefore be
0.Similarly, had the ?st ?en heads, we succeed if and
only if thesecond ? heads. Thus, the chance of success
(at this stage) is 1/2.But that's the probability of the
second individual ?herefore, theprobability of the past
event must be 1.The probability of the past event is not the
same as a future event.The function has collapsed from sample
space of two possible outcomes toa sample space of one actual
outcome.I may be stepping onto thin ice here, but it seems to
me that once you havean event, the only thing you can prove is
that the sample space of thesingle actual outcome must be a
subset of the sample space of multiplepossible outcomes prior
to the event.Suppose instead of a coin, our box contains a
regular polyhedron whosefaces are numbered sequentially form
1 to n (depending on the polyhedron).We are not told which
polyhedron it is, but when rolled, the result is 13.This
actual outcome must be a subset of the possible outcomes and
onlyan icosahedron has a set of outcomes that includes 13, so
we can safelysay that the probability of that outcome prior to
the roll was 1/20. Now had the outcome been 8, we cannot say
what the probability was because it could have been 1/8 for
an octahedron, 1/12 for or dodecahedron or 1/20 for the
icosahedron.We do not know how large that sample space was of
the forces that generatedthe mounds and it is possible that
the real sample space is much smaller than your analysis
indicates. If you say it is extremely unlikely that these
mounds would form this way, but they did, then there are
threepossibilities:1: they did NOT form that way (let's not
re-argue the geometry issue)2: they DID form that way and
your estimate of likelyhood is wrong3: coincidenceThat's
where statistics comes into play, to help distinguish case 2
from can't prove anything.
===
Hey, Mensanator, that post is
from other poster.
===
>Message-id:
<2410d7e.0311241350.632214f9@posting.google.com>>> Note
that the original red line G-P also misses mound P by>> a
mile and any claims of parallelism to the other red lines>>
is just a ?ment of the imagination.>>>You are simply a
LIAR. Even in the by hand adjusted image, the>deviation is of
7 pixels. It has been enlarged 40%, which corresponds>to less
than 5 pixels in the original image.>>With a resolution of
less than 50 meters/pixel, that gives:>>5 * 50 = 250
meters.>>That is several times less than 1 mile, and also
must be noticed that>the adjust by computer has more
precission, being the deviation of>less than 3.5 pixels, AS
CLEARLY stated in my webpage.>>You are simply a LIAR, and
even so, you CRIMINALLY acusse me of fraud.>I may be wrong,
or not, but you are a disgusting dishonest person.Did you see
my earlier post? You asked for a retraction and you got it.
Why areyou still belaboring the point?To reiterate, I did
not accuse you of fraud, I asked if the image (that youadmit
does not represent what you claim it does) was intended to
mislead thereader. I'll accept your answer that you did not
intend to mislead.As far as being a liar, perhaps you are
unaware of the phrasea miss is as good as a mileLet me
broaden your knowledge of English from The New Dictionary of
CulturalLiteracy, Third Edition. 2002<quote>A miss is as
good as a mile A near miss is still a miss and therefore no
better than missing by a greatmargin. </quote>When someone
says misses by a mile, it does not mean literally 1 mile.
Itsimply means that unless the measurement is EXACT, it
matters not whether thedeviation is 3.5 pixels or 50 meters.
The de?ition of parallel does not allowfor ANY deviation, no
matter how small. You do remember that you were askingfor
math commentary? If you need help presenting your data in a
clear, mathematically unambiguousformat, just ask. Flying
off the handle isn't going to get you
anywhere.--MensanatorAce of Clubs
===
> Did you see my earlier
post? You asked for a retraction and you got it. Why are> you
still belaboring the point?Yes, I saw and replied. I use
webposting, it takes more time than fromOutlook.> To
reiterate, I did not accuse you of fraud, I asked if the
image (that you> admit does not represent what you claim it
does) was intended to mislead the> reader. I'll accept your
answer that you did not intend to mislead.That is right now.
As I provide the maximum information for thewebpage it is
seen that I do not mislead. Notice that I haveboldered the
rebuttals, that would be far from a pseudoscientist ofhoaxer
technique.> As far as being a liar, perhaps you are unaware
of the phrase> a miss is as good as a mile> Let me
broaden your knowledge of English from The New Dictionary of
Cultural> Literacy, Third Edition. 2002> <quote>> A miss
is as good as a mile > A near miss is still a miss and
therefore no better than missing by a great> margin. >
</quote>> When someone says misses by a mile, it does not
mean literally 1 mile. It> simply means that unless the
measurement is EXACT, it matters not whether the> deviation
is 3.5 pixels or 50 meters. The de?ition of parallel does
not allow> for ANY deviation, no matter how small. You do
remember that you were asking> for math commentary? Ok. I am
not english native speaker as it is obvious, so I took
yourcommentary literally.> If you need help presenting your
data in a clear, mathematically unambiguous> format, just
ask. Flying off the handle isn't going to get you
anywhere.
===
The paper I refer was published in Journal of
Scienti? Exploration,not Statistical Science. I wanted to
mean that I would have preferedSSc.
===
> If you need help
presenting your data in a clear, mathematically unambiguous>
> format, just ask. Flying off the handle isn't going to get
you anywhere.Notice that the paper by
Dr.Horace&Prof.McDaniel is presented thatway. It was
submitted to a peer-review journal, though I had prefereda
mainstream publication. (Statistical Science, in spite of
theirBible code publication).You can ?d a link to the paper
and a rebuttal by Dr. Ralph Greenbergin my website. Notice
that the issue is complicated. I just showed thedata, I have
not published nothing in peer-review journals, notconducted
any proper scienti? test, except a lot of
helpfulcommentaries with Crater & McDaniel, and some personal
replications ofpart of their work.I personally veri?d my
model of parallel lines (which differs fromtheirs though the
ideal polygon is completely identical), getting a?ure of
about 1/1000 for parallel lines alone (I mean, notreplicating
their pattern, but ANY pattern of 6 mounds with a highnumber
of parallel relations).The perpendicular relations lower the
ocurrences (not conducted. Imade other tests for 4 points,
but they just showed coincidence inmagnitude with their
work).It must be noticed that I use a model which is not
?-coordinated,i.e., took simply the deviations in pixels
from the point (Crater usedvirtual mounds, I used points).
That corresponds to a diagram in whichyou join the estimated
center of the mounds, and then calculate howmuch the ?ure
deviates from an ideal model. I didn't use thatbecause I
prefer noticing the mound separations from the lines,
thanhaving to interpret visually how much a line difers from
parallelismwith other.I cannot claim scienti?ally anything
on my tests, as they were justamateur (I am a physicist, and
a mathematics graduate student helpedme to make the program
in C), but seem to give support to C&Manalysis, performed
with not 6 but 12 (+4) mounds. As you see, I donot talk of my
work in the site, and that wouldn't be honest. I justtalk here
about it as anecdote.I encourage the readership of their
work.I will replicate my work and publish all the data online
so you cancriticize properly. But as my only interest is
scienti? curiosity,and do not seek money, fame, attention,
etc, etc, I would appreciate aproper dialoge with no bold
claims.
===
> Why is P seven pixels below, and one pixel to
the right of the > mound you claim it's on. Why is D six
pixels below and 3 pixels > to the right of the mound you
claim it's on?That is greater deviation from an adjusted ?
by computer. Anyway,there is a deviation for an ideal ?ure.
As explained clearly in thetext, there must be room for some
deviation, because1) We are in a km scale, not crystal-scale.
Geological events are notprecise nor perfect on that scale.2)
You don't know if the ? is better (or worst) when
measuringground distances, instead of aerial.3)
Orthorecti?ation is not present in the image. That said,
thedeviations are slightly corrected.4) A crater chain
generally presents worst alignments.5) The chances of
randomness drop with the square of the averagedeviation (that
is valid for a non-coordinate ?, a coordinate one issomehow
more strigent). Would not be deviation, the chances would
bezero. In the actual scenario they are very small FOR THAT
PARTICULARPATTERN (and here enters directly the issue of
uniqueness).> Mansanator doesn't need the likes of me
sticking up for his > character, but the evidence is that you
are spouting gibberish > and displaying ?t disregard for
principles of probability,> and Mensanator is displaying a
perfectly suf?ient knowledge > of geometry to interpret and
invalidate your illusions, a likwise> suf?ient knowledge of
probability to puncture your probabilistic > posturing, and
to be perfectly frank, is being perfectly honest> and
reasonable with you when he criticises your nonsense. More >
honest than you appear to deserve.> *_PLONK_*He has
apologized and I have accepted the apologies. I am
perfectlyhonest, and be sure I don't get money, fame or
whatever with thisissue. My quest is completely a
curiosity-based one.I am a physicist, I have talked with
PhD's about the issue. The mostimportant objections are
these:1) Mound selection.2) Uniqueness of pattern and the
issue of a priori selection.The issue of the deviation is
accepted as perfectly normal. Somesimply asked for what was
the average deviation, directly. Not thatthere was one and
that invalidated all (reasons above).
===
> Suppose you've got
a stick of length 1. Now, following happens.> 1) The stick is
broken at X (where X is the distance between the> point of
impact and the left end of the stick).> 2) The left part of
the stick gets hit again and brakes at Y (where> Y is the
distance between the point of impact and the left end of> the
stick).> Now, provided that the both hits are uniformly
distributed, what> is the conditional distribution function
for Y given that X = x?> [...]This sounds like Y|X ~
Uniform(0,X), for which P(Y<=y|X=x) = y/x.Did I miss
something?
===
> This sounds like Y|X ~ Uniform(0,X), for
which P(Y<=y|X=x) = y/x.> Did I miss something?I think i
treated the problem as more dif?ult than what it was.You
didn't miss anything. On the contrary, that got me to the --
KindlyKonrad-------------------------------------------------
--May all spammers die an agonizing death; have no burial
places;their souls be chased by demons in Gehenna from one
room toanother for all eternity and more.Sleep - thing used
by ineffective people as a substitute for coffeeAmbition - a
poor excuse for not having enough sence to be
lazy---------------------------------------------------
===
To
my best knowledge this is neither proven nor disproven -
sometimesyou hear : conjugate a+i*b to a - i*b, by this you
get the dot-productand so you have (R2, +, r.s.m., dot), the
euclidian vectorspace. Theconjugate is widely used in C, but
how do You get it, whenC is by de?ition the commutative ?ld
(R2, +, r.s.m., * ) and * is given in
(a,b)*(u,v)=(au-bv,av+bu) ?The pleasure of ?ding out (and to
be the ?st to publish) i leaveto yourself. My opinion i'll
tell You in about a month time, in themeantime You might ?d
pleasure in reading a fairytale:Once upon a time - no, in
our times, there is a little dot-product,somehow skew in its
dimensions. A big-booster-genius, C , master ofhis own
universe, is boosting of his richness and intellectuality,
nosimple mind can dare to comprehend.They live on the vast
plain, where arrow-?lds have to be worked on.C was strolling
around with his apologist Riemann sur Face, so called'cause he
could give a broad smile on his face- as the re?n ofthe
sun in the hair of little Dot caught his eyes. C envied her
forher intricating abilities - she was just casting a shadow
of an arrowonto the iron bars of the coordinate-grid, behind
which he kept hisellbow ticked the ?f sur Face as he
shouted :Hey, Dotty, comehere, let me have a look into your
mirror! She thought :This mighttake an interesting twist.
He never asked me that before. and whensur Face added with a
grin from one ear to the other:Your algebraiccompleteness is
calling upon you - come here ! she went over to them.With an
obidient gesture, but with the words :Underneath Your
clothsYou are naked. she handed the mirror to C.The
continuation of this You can ?d under: z
inhttp://i-is-no-longer-imaginary.gmxhome.de and
http://i-z.eu.ttHave funHero
===
Ummm... are we suppose to
guess what that means?> 0038> >> >> >> A trillion seems
a little high. (I hear that billion isn't a word> >> in
the UK? Does that mean that trillion isn't either?)> >>
>In the UK, these words do exist, but with different meanings
than those> to> >which you might be accustomed. Billion
means 10^12 (a million squared),> >trillion means 10^18
(a million cubed), quadrillion means 10^24 (a> >million
to the power 4), etc etc (except in the hands of ignorant
people> >such as politicians, businessmen, and the
media).>> And then for the other numbers, they say
thousand billion, thousand> trillion, etc.?>>
never sounded right to my ears that a million squared was a
trillion,> not a billion.> -->
dgates@spamfreelinkline.com
===
> In message
<bp4jq1$91$1@sparta.btinternet.com>, Richard Heath?ld>>>
A quote from:>> The New Fowler's Modern English Usage, Third
Edition, Edited by R. W.>> Burch?ld, The acknowledged
authority on English usage>> [all of that from the front of
the dust jacket...]0074>> Under the topic billion:>> It
is best now to work on the assumption that the word means ?a>
>> thousand millions' in all English-speaking areas...>>
>But that is a false assumption. My home is an
English-speaking area inwhich>the word does not mean that.>
>>If the book suggests that it is best to work on false
assumptions, then>perhaps it isn't that good a book.>> at
least a county, rather than an individual dwelling.>> Nick>
--> Nick Wedd nick@maproom.co.uk
===
As I vaguely understand
it, the Nash embedding theorem applies to the problemof
embedding one (compact) Riemannian manifold isometrically
into Euclideanspace. Suppose instead you have a mapping M->T
of one Riemannian manifold ontoanother so that the ?res are
all Riemannian manifolds. Can one concludethat there is a
vector bundle V with metric over T and a bundle map of
M->Tinto V which is ?rewise an isometric embedding? If so,
what is a referenceto this result?Ignorantly,Allan
Adlerara@zurich.ai.mit.edu***********************************
****************************************** ** Intelligence
Lab. My actions and comments do not re?* in any way on
MIT. Moreover, I am nowhere near the Boston ** metropolitan
area. **
************************************************************
*****************
===
>>As I vaguely understand it, the Nash
embedding theorem applies to the problem>of embedding one
(compact) Riemannian manifold isometrically into
Euclidean>space. Suppose instead you have a mapping M->T of
one Riemannian manifold onto>another so that the ?res are
all Riemannian manifolds. Can one conclude>that there is a
vector bundle V with metric over T and a bundle map of
M->T>into V which is ?rewise an isometric embedding? If so,
what is a reference>to this result?>>Ignorantly,>Allan
Adler>ara@zurich.ai.mit.edu>> ...I'm not sure, but you may be
able to show this for compact M bychoosing a trivial bundle
over T of suf?iently large dimension andusing the parametric
h-principle for isometric immersions (see 3.1.2(H) in Partial
Differential Relations, M. Gromov, 1980Springer-Verlag)
together with the construction of isometricembeddings given
in 3.1.1 on page 223.John Mitchell
===
Let V be space of
smooth functions on R(real number) satisfying alinear
homogeneous ODE with Polynomial Coef?ients , and the
leadingcoef?ient of the ODE MUST BE 1.ie. f(x) in V if f
satisfys fn(x) + P(n-1) fn-1(x) + P(n-2) fn-2(x) +...
+P(0)f(x) = 0fk represents the kth dervative of f.the
coef?ient of fn must be 1.e.g e to the power x^2 is in V
because it satis?sf'(x)-2xf(x)=0prove that V is a ring. I
thought for weeks but still do not think of the solution.
Anyone canhelp me?
===
> Let V be space of smooth functions on
R(real number) satisfying a> linear homogeneous ODE with
Polynomial Coef?ients , and the leading> coef?ient of the
ODE MUST BE 1.> ie. f(x) in V if f satisfys > fn(x) +
P(n-1) fn-1(x) + P(n-2) fn-2(x) +... +P(0)f(x) = 0> fk
represents the kth dervative of f.> the coef?ient of fn must
be 1.> e.g > e to the power x^2 is in V because it satis?s>
f'(x)-2xf(x)=0> prove that V is a ring. > I thought for
weeks but still do not think of the solution. Anyone can>
help me?it might make your life easier if you used the fact
that ?the leading coef?ient of the ODE MUST BE 1'is
equivalent to the more reasonable statementthe leading
non-zero coef?ient is constant. (just divide through; it's
homogenous)
===
>>> at 04:13 PM, Mitch Harris
<harrisq@tcs.inf.tu-dresden.de> said:>>topology is
really just a fancy word for geometry, right?>Wrong.
There are many issues in Geometry beyond the scope of
Topology.>Before the last few centuries, hardly any work in
Geometry had>anything to do with Topology.>>I meant by my
short comment about topology and geometry that, by >>whatever
route one is lead to topology (category theory, point-set
>>analysis, differential manifolds, etc) that one is doing
generalized >>geometry.> Point-set topology is in no way
generalized geometry, nor> even related to it.I agree that
they seem to be quite different, but I am sure there could be
some connection made between them. Couldn't the
incommensurability of the diagonal of a square to its side
(is that geometrical?) be seen as a precursor to point-set
topological ideas like continuity, openness, closure,
connectedness, etc.? Aren't points and sets of points and the
aformentioned ideas abstractions of spatial concepts?I get
your (and Shmuel's) point. I apologize for my ?t,
tendenciously presented relationship between geometry and
topology.Philosophically -everything- is distinct. But then
also everything has similarities with everything else. All
I'm trying to point out is that in the nebulous cloud of
intuitively described mathematical areas, some areas are more
similar than others. I think I've watered this down enough to
say absolutely nothing. Or possibly not.
Maybe.Mitch
===
[snip]> Yes, the gist of it is single
register you only have to remember one> number, all along
the process.>> fact.>> But is _that aspect_ of the method
Vedic? The expression crosswise vertical> doesn't say
much about it, does it?>> Oh yes, it does. That is the beauty
of Sanskrit. It is not verbose;> it just de?es the heart of
things, as brie?possible, and> leaves the rest to those
with vision. Vedic means secret - only the> seer can
understand the Vedas. They are meaningless to others.Modern
mathematics likes to make everything as lucidand explicit as
possible. It doesn't need to appeal to mystery,or to ?old
books with forgotten knowledge, rediscoveredbut clear only to
the chosen ones'. Everything open and clear.And you know
what... all in all, modern maths is _less_ verbose.After all,
it doesn't need 16 books to explain a trivial thing
likeelementary arithmetic <grin>... (*)Herman Jurjus(*):(It
is said that the original discovery, around 1910, of
Vedicmaths was written down in 16 books.)
===
> [snip]> >
Yes, the gist of it is single register you only have to
remember one> > number, all along the process.>>
fact.>> > But is _that aspect_ of the method Vedic? The
expression crosswise vertical> > doesn't say much about
it, does it?>> Oh yes, it does. That is the beauty of
Sanskrit. It is not verbose;> it just de?es the heart of
things, as brie?possible, and> leaves the rest to
those with vision. Vedic means secret - only the> seer can
understand the Vedas. They are meaningless to others.>
Modern mathematics likes to make everything as lucid> and
explicit as possible. It doesn't need to appeal to mystery,>
or to ?old books with forgotten knowledge, rediscovered> but
clear only to the chosen ones'. Everything open and clear.No,
it is not. It is a bunch of dogma, the way they teach them
inschool these days. I have two daughters, one who has given
VCE (year12) and has now completed her graduate studies. The
other will giveVCE next year. I helped them in their Maths,
and was totallyhorri?d by the course material, and the way
the are taught. Theydid not do any of your precious geometry
- Euclid was totallyavoided. Then, they did not learn how to
derive any formula from?st principles. Even the formula was
Geometric Progression was notderived! It was just given in
the book like some divine truth. Thechildren are expected to
put it in their calculator and use theformula to get some
answers to problems. They are *not* given thederivation! When
I derived it for my daughter, she was impressed, andsaid it
was so cool.Basically, they want the kids to become morons
and believe whateverthe top authorities say. This is nothing
but a step to pure tyranny.> And you know what... all in all,
modern maths is _less_ verbose.> After all, it doesn't need 16
books to explain a trivial thing like> elementary arithmetic
<grin>... (*)But the most elementary long multiplication has
been well beyond thebrains of the most advanced mathematical
minds of our time, as thisthread shows... that happens when
you stop to think, and are expectedonly to believe. Which, of
course, results from the way you aretaught in school.>
Herman Jurjus> (*):> (It is said that the original
discovery, around 1910, of Vedic> maths was written down in
16 books.)At any rate, this proves that the book on Vedic
maths is not a fraud. Whether in 16 books or 1600, it took me
but half a page of that bookin the English language to ?d out
a new and better multiplicationmethod. Who knows what will
happen to the ?ld of maths when all the16 books are fully
understood... But then, if all mathematicians aretrained to
be dogmatic morons, how can that ever happen?Arindam
Banerjee.
===
[snip]> It is certainly true that the
stupidest mistakes are often> made by the smartest
people.>> No, this is illogical.It's called a 
?paradox'. That
is something that is deliberatelyworded to seem illogical, but
has a hidden message.I could explain it in more length, but i
won't. It would feellike explaining a joke.Herman Jurjus
===
>
[snip]> > It is certainly true that the stupidest mistakes
are often> > made by the smartest people.>> No, this is
illogical.> It's called a ?paradox'. You are 
confusing
learned people with smart people. Learned peopleusually are
not smart. So, those who are useless at worthwhile jobsor
business, or not strong enough to take up a trade like
plumbing,take to learning and live off the public via
university funding. Theworst among the learned people are
also not keen to learn new thingsthat will upset their hard
earned knowledge. So, they try their bestto crush new ideas.
Smart people, such as those who are paid to ?dout and try
out new things for more competive advantage, usually havevery
little regard for such learned people. They just do their
ownthing, and when they have proved what they want to prove,
let thelearned people absorb their new ideas and propagate
them as divinetruths with their usual pomposity. In the
meantime, the smart peoplehave to endure the hostility of the
learned people, but if they haveindependent means to pursue
their work, they do not particularly care.Arindam
Banerjee.That is something that is deliberately> worded to
seem illogical, but has a hidden message.> I could explain it
in more length, but i won't. It would feel> like explaining a
joke.> Herman Jurjus
===
>>[snip]>>>>It is
certainly true that the stupidest mistakes are often>>made
by the smartest people.>>No, this is illogical.>>It's
called a ?paradox'. > > You are confusing learned people
with smart people. Learned people> usually are not smart. So,
those who are useless at worthwhile jobs> or business, or not
strong enough to take up a trade like plumbing,> take to
learning and live off the public via university funding.
The^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^^^^^^^^^^^^^^^^This distinction between smartness and
learnedness is interesting,but, somewhat super?ial. Like
in India, anyone looking western orspeaking unaccented
english used to be called smart. In north america,since
most people look western and their mother tongue is
english,de?ition of smart is somewhat different. In
general, the acceptedidea is related to
innovativeness/creativeness in your ?ld of work.In that
sense, a plumber can be smart, just as much as a
universityresearcher or a professor, or a corporate guy. Even
a learned fellowcan be smart too. Smart folks bring
innovative apporaches toproblem solving, as distinct from the
others.Arjoe> worst among the learned people are also not keen
to learn new things> that will upset their hard earned
knowledge. So, they try their best> to crush new ideas. Smart
people, such as those who are paid to ?d> out and try out new
things for more competive advantage, usually have> very little
regard for such learned people. They just do their own> thing,
and when they have proved what they want to prove, let the>
learned people absorb their new ideas and propagate them as
divine> truths with their usual pomposity. In the meantime,
the smart people> have to endure the hostility of the learned
people, but if they have> independent means to pursue their
work, they do not particularly care.> Arindam Banerjee.>
That is something that is deliberately>>worded to seem
illogical, but has a hidden message.>>I could explain it in
more length, but i won't. It would feel>>like explaining a
joke.>>Herman Jurjus>
===
If A can be diagonalized as PAP',
then computing A^n is as simple ascomputing (PAP')^n = P A^n
P' where A^n is just composed of entries witheigenvalues
raised to n. Similarly, functions of A such as sin(A)
areequivalent to P sin(A) P' where the middle term is a
series that convergesto a matrix with entries on the diagonal
that are the sin of theeigenvalues. Consider the special case
of the Jordan matrix where J cannotbe diagonalized, i.e. (not
necessarily 5x5):[L 1 0 0 0][0 L 1 0 0][0 0 L 1 0][0 0 0 L
1][0 0 0 0 L](L are eigenvalues). My question is, what is
f(J) generally? For example, ifall L = Pi, then sin(J) is A
- (A^3)/3! + (A^5)/5! - ... What does thisconverge to? I can
see what happens to the matrix when taking successivepowers,
but I can't ?ure out what happens to the series. Also, does
thismatrix have a special name? I think I've seen it before,
but don't knowwhere.JR
===
>If A can be diagonalized as PAP',
then computing A^n is as simple asI think one of these A's is
supposed to be a D. >computing (PAP')^n = P A^n P' 
where A^n
is just composed of entries with>eigenvalues raised to n.
Similarly, functions of A such as sin(A) are>equivalent to P
sin(A) P' where the middle term is a series that converges>to
a matrix with entries on the diagonal that are the sin of
the>eigenvalues. Consider the special case of the Jordan
matrix where J cannot>be diagonalized, i.e. (not necessarily
5x5):>[L 1 0 0 0]>[0 L 1 0 0]>[0 0 L 1 0]>[0 0 0 L 1]>[0 0 0
0 L]>(L are eigenvalues). My question is, what is f(J)
generally? For example, if>all L = Pi, then sin(J) is A -
(A^3)/3! + (A^5)/5! - ... What does this>converge to? I can
see what happens to the matrix when taking successive>powers,
but I can't ?ure out what happens to the series. Also, does
this>matrix have a special name? I think I've seen it before,
but don't know>where.If this matrix (called a Jordan block) is
n x n, note that J - L I = N where N^n = 0. For 1 <= j <= n-1,
N^j has 1's on thej'th superdiagonal and 
0's elsewhere. If
f(z) = sum_{j=0}^in?ity c_j (z-L)^j, then f(J) =
sum_{j=0}^in?ity c_j (J-L)^j = sum_{j=0}^{n-1} c_j N^jThus
f(J) is an upper triangular Toeplitz matrix with entries
a_{j,k} = f^(k-j)(L)/(k-j)! for k >= j. Robert Israel
israel@math.ubc.caDepartment of Mathematics
http://www.math.ubc.ca/~israel University of British Columbia
Vancouver, BC, Canada V6T 1Z2
===
> If A can be diagonalized as
PAP', then computing A^n is as simple as> computing 
(PAP')^n =
P A^n P' where A^n is just composed of entries with>
eigenvalues raised to n. Similarly, functions of A such as
sin(A) are> equivalent to P sin(A) P' where the middle term
is a series that converges> to a matrix with entries on the
diagonal that are the sin of the> eigenvalues. Consider the
special case of the Jordan matrix where J cannot> be
diagonalized, i.e. (not necessarily 5x5):>> [L 1 0 0 0]> [0 L
1 0 0]> [0 0 L 1 0]> [0 0 0 L 1]> [0 0 0 0 L]>> (L are
eigenvalues). My question is, what is f(J) generally? For
example, if> all L = Pi, then sin(J) is A - (A^3)/3! +
(A^5)/5! - ... What does this> converge to? I can see what
happens to the matrix when taking successive> powers, but I
can't ?ure out what happens to the series. Also, does this>
matrix have a special name? I think I've seen it before, but
don't know> where.Simpler case:A = [a 1] [0 a]A^k = [a 1]^k =
[a a^k + ak] [0 a] [0 a ]Hence exp(A) = [ exp(a)
sum(k=0,..,infty: (a^k + ak)/k! )] [ 0 exp(a) ] = [ exp(a)
exp(a) + a*exp(1) ] [ 0 exp(a) ]In general, one expects that
the entries above the main diagonalof J^n are given by linear
recursion relations, derived by evaluatingJ^n * J = J^(n+1)and
imposing suitable initial conditions for n = 1.Substituting
this form of J^n into the series expansion and summingelement
by element gives the result (if J has a zero eigenvalue,
theseexpansions may terminate.)-- P.A.C. SmithThe vast
majority of Iraqis want to live in a peaceful, free world.And
we will ?d these people and we will bring them to
justice.
===
>> If A can be diagonalized as PAP', then
computing A^n is as simple as> computing (PAP')^n = P A^n
P' where A^n is just composed of entries with> eigenvalues
raised to n. Similarly, functions of A such as sin(A) are>
equivalent to P sin(A) P' where the middle term is a series
that converges> to a matrix with entries on the diagonal
that are the sin of the> eigenvalues. Consider the special
case of the Jordan matrix where J cannot> be diagonalized,
i.e. (not necessarily 5x5):>> [L 1 0 0 0]> [0 L 1 0 0]>
> [0 0 L 1 0]> [0 0 0 L 1]> [0 0 0 0 L]>> (L are
eigenvalues). My question is, what is f(J) generally? For
example, if> all L = Pi, then sin(J) is A - (A^3)/3! +
(A^5)/5! - ... What does this> converge to? I can see what
happens to the matrix when taking successive> powers, but
I can't ?ure out what happens to the series. Also, does
this> matrix have a special name? I think I've seen it
before, but don't know> where.>> Simpler case:>> A = [a 1]>
[0 a]>> A^k = [a 1]^k = [a a^k + ak]> [0 a] [0 a ]This should
read [a^k a^k + ak] [0 a^k ]> Hence exp(A) = [ exp(a)
sum(k=0,..,infty: (a^k + ak)/k! )]> [ 0 exp(a) ]>> = [ exp(a)
exp(a) + a*exp(1) ]> [ 0 exp(a) ]>> In general, one expects
that the entries above the main diagonal> of J^n are given by
linear recursion relations, derived by evaluating>> J^n * J =
J^(n+1)>> and imposing suitable initial conditions for n =
1.>> Substituting this form of J^n into the series expansion
and summing> element by element gives the result (if J has a
zero eigenvalue, these> expansions may terminate.)>> -->
P.A.C. Smith>> The vast majority of Iraqis want to live in a
peaceful, free world.> And we will ?d these people and we
will bring them to justice.>>-- P.A.C. SmithThe vast
majority of Iraqis want to live in a peaceful, free world.And
we will ?d these people and we will bring them to
justice.
===
>Can we embed every G without adding or
removing >edges into the graph of some 0/1-polytope?I don't
see the point of this added requirment as the ?alembedded
result preserves edges.----
===
> I am a science teacher but
I have taught math for a few years. I too> agree tht it
could be possible that people have a math gene. Through>
out my school years I have always struggled with math, while
some> students just seemed to get it right off. No matter
how hard I> studied it was a constant battle.>> While
teaching math to my students I see the same types of
behaviors I> think this makes me a better math teacher
cause I know what it feels> like not to understand, which
in turn makes me more patient with my> students. I think
the book is worth reading and considering. However> you
have to be careful when sharing this information with your>
students cause it could turn into a self pity issue.> In
fact, there is a book by that title. Listed on Amazon.com
(search Books> with the words math gene.) I read it a few
years ago, and found it> interesting. The same part of the
brain involved in higher math (not basic> arithmatic) does
language processingMaths is one of the most interesting
subjects that i have ever took,you would have to be stupid
not to be able to understand it once it isexplained to
u!!
===
> Maths is one of the most interesting subjects that i
have ever took,> you would have to be stupid not to be able to
understand it once it is> explained to u!!Congratulations, you
are a bearer of the math gene. |-}}}Joking aside, I think
that everybody can understand the topics in mathsbut the
deepness of understanding usually depends on the quality of
theexplanation.Karl
===
>> Ok. But if a set is connected,
then any two points should have a polygonal> path between
them, right?>> No. But if the set is open and connected in
R^2 as you said earlier, then> yes.>Theorem: locally path
connected S, open connected U subset S ==> U path
connectedProof relies upon connected chain theoremShow
subspace U is locally path connected, etc...Indeed R^2 is
locally path connected with path connected balls.So in R^2 a
connected open set can be path connected by a ?ite number
ofend to end line segments all contained in a ?ite chain of
overlappingballs which surely gives an analysist enuf wriggle
room to smooth thebroken line into a polynomial; thus I
wriggle out of proceeding withfurther details.
===
>
Example:> > The last event time was 15. Here are some
events and the times at> which they occured:> Event A: 3,
4, 5, 9, 13> Event B: 1, 2, 6> Event C: 7, 8, 10, 11> If
I were assuming independence, I'd just note down the>
interarrival times, the times between events:> A: 1, 1, 4,
4. Mean = 2.5> C: 1, 4. Mean = 2.5> B: 1, 2, 1 Mean = 1.3>
But are they independent? Does event A depend on when B or> C
occurred, or how many times A has occurred?Let's assume they
are independent.> > Intuitively, I would expect to order
the events A, C, B in decreasing> order of expected
soon-ness.> Why? Look at the inter-event spacings. Why do
you have this> expectation? Do you have some prior knowledge
that is> relevant?Because although the inter-event spacing
for event C has mean 2.5,note that the event last occured at
time ?4', and we are now at time'15'. So 
the mean inter-event
measure is rather out-of-date. It looksto me as though this
event C's cluster of occurences has probablyended, and I have
little expectation that the event will occur againin the near
future.Toby.
===
> > Example:> > > The last event
time was 15. Here are some events and the times at> > which
they occured:> > Event A: 3, 4, 5, 9, 13> > Event B: 1, 2,
6> > Event C: 7, 8, 10, 11> > > Intuitively, I would
expect to order the events A, C, B in decreasing> > order
of expected soon-ness.> > Why? Look at the inter-event
spacings. Why do you have this> expectation? Do you have
some prior knowledge that is> relevant?> Because although
the inter-event spacing for event C has mean 2.5,> note that
the event last occured at time ?4', and we are now at time>
?15'.Ah. Excellent point. What that means statistically, 
I'd
say, isthat you have more information about the
number-of-events pertime interval rvs than about the
interarrival time rvs. Theoretically the same information is
contained in both. Butas you point out, time gaps give you
more sample informationabout one than the other.One statistic
you can get at easily from your data isP0(t) = the probability
of no events in a time intervalof length t. For instance, for
t = 1- (just less than 1),P0_A(t) is the probability that a
random interval [a,a+1)in [0,15] excludes 3, 4, 5, 9 and 13.
That means a can'tfall in (2,3], (3,4], (4,5], (8,9] or
(12,13], i.e.,(2,5], (8,9], (12,13]. Since a can range from 0
to 14,and you are excluding a total range of 5, the
probabilitythat a random interval of size 1- will contain no
eventsof type A is (14-5)/14 = 9/14 = 0.64.Conversely, the
probability of getting at least oneevent of type A in an
interval of length 1 is 0.36.For event B, the probability of
no events is(14-3)/14 = 0.79, and the probability of seeing
anevent is 0.21.For event C, the probability of no events
is(14-4)/14 = 10/14 = 0.71, and the probability of seeingan
event is 0.29.All this gets back to the fact that your
intuitionis correct: You saw more events of type A than ofany
other, so you can expect that one most often.You saw very few
of type B, so you can expect thatone least often. I got lost
in the math and missedthe obvious.However, you can get more
than that from yourdata. You can estimate as I did the
probability ofseeing an event of each type in any given
1-secondinterval. But you can estimate other
probabilitiestoo, which will re?he clustering
behavior.What is the probability of seeing at least one event
ina 5 second interval?Event A: There is no 5-second interval
in [0,15] whichis free of events. Probability of at least one
event = 100%.Event B: The interval (6,15] is free of events.
If Ichoose a random interval [a,a+5) with 0 <= a <= 10,then
the probability of no events is the probabilitythat a > 6,
which is (10-6)/10 = 40%. The probabilityof at least one
event is 60%.Event C: Similarly, the probability of seeing no
eventsin [a,a+5) is the probability that a is in [0,2],
i.e.20%. The probability of at least one event is
80%.Summary:Estimate of P(at least one event in the next T
sec): 1 sec 5 sec A 36% 100% B 21% 60% C 29% 80%And you can
do this for any interval size.> So the mean inter-event
measure is rather out-of-date. It looks> to me as though this
event C's cluster of occurences has probably> ended, and I
have little expectation that the event will occur again> in
the near future.Yes, you're right. Somewhere along the line I
got confused between B's data and C's data. The 
analysisabove
re?your original data as quoted above.And you can repeat
it for any other convenienttime interval. - Randy
===
right
term in english but was too lazy to look it up. Regarding
yourown experiments and patents, I hope we don't get
ourselves a legal?ht, wait for part III of my journal and
you'll see what I mean.Happy however to have my own
observations totally con?med.see you around, Guenther>
However there is something else I wan't to share with you, I
carefully> observed the trajectory of the device from the
time I released it to> the time it landed and found it was
a parable. I'm going to try and> ?ure the implicatins this
has, but I already consider it a very> interesting and
revolutionary fact in itself.> similar results with the
Action Device that I have built. I'm not going> to go into
detail, because my patent applications are still pending.>
However, I sincerly doubt that the trajectory you observed
was a> parable, or a usually short ?titious story that
illustrates a moral> attitude or a religious principle
[Merriam-Webster]. More likely, it was> a parabola.> See
<http://www.m-w.com/mw/art/parabola.htm> for details.> I'm
quite convinced that my Action Device works, because,
although it> didn't ? into space immediately, I left it
by the curb overnight in> the hopes that it was just getting
warmed up, and sure enough, the next> morning it was gone!
I'm left to conclude that it actually did ?> with uniform
acceleration into space.> Good luck,> -Mike
===
loathe as i
am to use the jsh thing in the subject, i'm aware that
itwould only offend jsh if i didn't, and that isn't 
the point
of this.sorry, james, but i am one of those dreaded
mathematicians. i hope youdon't jump to any unfounded
conclusions though.reading your postings, it seems that you
have little idea of what wedo with our time. you almost
appear to say that we sit at our deskswrite something down,
which is worthwhile maths (by ?t because weare
mathematicians) and deny anyone's claim that disagrees with
ournarrow viewpoint. i wish that i could just write something
down andhave it instantly enter the annals, but it doesn't
work like that.instead, you ,hopefully, ?d something
interesting to think about.then when you've ?ured out why
the ?st three attempts went wrong,you talk to some other
people. then when you've possibly got somethingthat 
isn't
garbage you show it off to someone else and they point
outwhere you went wrong, or that it was published 5 years
ago, orsomething. then after you've changed what needs to be
changed youmight get it submitted to some journal and the
referee points out somemore errors, where you've used phi for
two different things in thesame equation and all the typos.at
each stage you take it with good grace. at least, if
you'rereasonable. perhaps there is something to learn
there?so, james, there are adequate commentaries on your work
that showthere are many errors in your algebraic integer
argument. my summarywould be this: look, i've assumed i can
invert 7 here, and it implies1/7 is in the ring of algebraic
integers. of course it isn't (you canprove that i hope, 
it's
a simple exercise most undergrads should beable to do), but
instead of checking your assumtion you're off on acrusade. i
hope you've read enough of the commentaries to understandwhy
you can't divide by seven. it would then seem reasonable
tocorrect these aspects. of course as has been pointed out
you can't.and i mean that as an impersonal you, rather than a
personalcriciticism.however, here's some advice for you when
writing more ?proofs' there is nothing special in 7,7,22 in
your proof. so why did youchoose them? for clarity's sake
wouldn't 2,2,3 have been much moreilluminating?this is
written in plain text. there are therefore lots of
constraintson the symbols you can use. try avoiding
unnecessary subscriptnotation. start with a simple statement
of what it is that is wrong. it takes alot of effort to
decode mathematical arguments, and a simple ?theproblem is
foo' goes along way. at best you say you've uncovered
ahundred year old problem, but don't state what it is, just
launch intosome unmotivated reasoning and say, ?voila, that's
wrong.'your prime counting algorithm is at best a party trick
i'm afraid. ifyou could ?d some new primes for us, 
that'd be
interesting.if you wish to demonstrate your skills, then how
about ?ding somedemonstrably silly argument, say one of the
anti-cantor ones, and deconstructing it. it might earn some
respect. my favourite would bethe one showing the reals are
in 1-1 correspondence with some subsetof the p-adic integers.
if you don't know what those are, tryexplaining why the
arguments about two expansions representing thesame number is
junk.there have been many counter-intuitive results found in
mathematics,and more will be discovered. no one would
begrudge the discovereranything.
===
> loathe as i am to use
the jsh thing in the subject, i'm aware that it> would only
offend jsh if i didn't, and that isn't the point of 
this.>
sorry, james, but i am one of those dreaded mathematicians.
[...]Oh, I thought that you were the ghost of e. e.
cummings.-- G.C.
===
> loathe as i am to use the jsh thing in
the subject, i'm aware that it> would only offend jsh if i
didn't, and that isn't the point of this.JSH in the 
subject
line isn't due to James's vanity. It is includedso 
that other
readers can kill?e threads without having them clutterthe
newsgroup.-- Jesse HughesLike the ski resort full of girls
hunting for husbands and husbands hunting for girls, the
situation is not assymmetrical as it might seem. -- Alan
MacKay
===
> It is not in general possible to obtain analytic
solutions to anarbitrary> differential equation. This is not
simply because ingenuity fails, but> because the repertory of
standard functions (polynomials, exp, sin and so> on) in terms
of which solutions may be expressed is too limited to>
accommodate the variety of differential equations encountered
inpractice.One could expand on this and state that the usual
elementary functions mayindeed be *de?ed* to be speci?
solutions to speci? linear differentialequations. The fact
that an ODE is linear basically implies that we onlyhave to
solve a small number of ODE's (read: de?e a small number
ofelementary functions), and this allows us to -
essentially - write down asolution to *all* linear ODE's in
terms of these functions.One could try the same with
nonlinear ODE's, but here a solution to a someODE can not be
generalized to other non-linear ODE's. One basically has
tostart from scratch each time one encounters a new nonlinear
ODE.-Michael.
===
Robert Israel>Firstly what exactly are
NONLINEAR differential equations?>> A linear differential
equation (with dependent variable> y and independent variable
x) can be written as>> a_n(x) d^n y/dx^n + ... + a_1(x) dy/dx
+ a_0(x) y = F(x)>> A nonlinear differential equation is a
differential equation that is> not linear. In particular, any
terms involving some more complicated> function of y and/or
its derivatives makes the equation nonlinear.>>Secondly
why are they considered so dif?ult to solve?>> Because
they are not easy to solve. There are very simple-looking>
?st-order nonlinear differential equations such as>> dy/dx =
x y^2 + x + 1>> for which AFAIK no closed-form solution is
known, even involving> an integral. By contrast, there is a
well-known formula (involving> integrals) for solving
?st-order linear differential equations.Just to add 2, if f
is a solution of a homogeneous linear DE, then so is kffor
any constant k. If g is another solution, so is
f+g.Inhomogeneous linear eq'ns have a related property, found
in all thetextbooks.LH
===
I'd like to write n as a fucntion of
N, ie n = f(N). How do I do this easily?
===
race which would
end up in lots of isolated and often persecuted
groupscompeting for livelyhoods provided exactly the
conditions under whichevolution works best.-- Bruce
Harveybruce@bearsoft.co.ukThe centre for inside out
thinking.>> >> > >> >>> >> >>>
>>... [fkasner] snipped incredible nonsense> >>from a
dedicated anti-semite ...[Tom Potter]> >>> >>
>Careful, Kaz, these days the ones who always cry
anti-Semite> >are now perceived to be the real bigots and
racists at heart.> >> >Re?n that and ask yourself
why this came about.> >Jewish moral and intellectual
superiority, ......perhaps?> >All good things do come to
an end, Kaz,.......ahahahaha....>> >>>One has to wonder
why some people do this.> >>>Are they brainwashed,>
>>>or do they support the people who instigate>
>>>con?nd war for power and riches?????> >>>Tom
Potter> >> >> >I don't know, nor care, what
post preceded this one,> >but since ever mankind existed
some people of/in any> >tribe have always and will always
instigate con?nd> >war for power and riches......
or what other reasons are> >there for killing each
other?......and if the conditions were> >right, Tom,
you'd be in the forefront doing the same, given> >the
personality you exhibit on the Usenet......ahahahaha...>
>> >>Gee, I don't get this. How do I get into this
power elite? I'musually> >>considered quite bright and I
am Jewish. But I have never been inthe> >>company of any
Jew who was in this power elite. My latebrother-in-law>
>>was rich, Jewish, but not one of the power elite. How do
I getthere.> >>Why have the other Jews kept this a secret
from me and all my> relatively> >>poor ancestors? Even the
atomic bomb had those who blabbed about it.> Why>
>>haven't I heard about this grand scheme? I want in. I
couldcontribute> a> >>lot. I have lots of Christian
relatives who would provide cover for> me.> >>I know a
few fairly powerful politicians and could in? them if>
it> >>were worth my while. Why has all this been hidden
from me for somany> >>years?> >>> >> >Very
astute and SEEMINGLY factual, Kaz, BUT.....> >See, Kaz,
apparently you really don't get it. It's very 
simple.>
>It is not so-called riches and power that gets Jews into
deep shit> >periodically and epically. A lot of goys have
far more of each, yet> >it normally does not get them into
trouble. Most have learned how> >to handle it: Discretely,
smoothly and QUIETLY.> >> >But not the Jews. With the
Jews it is the opposite. It is their> fucking,>
>incessant **loudness about themselves**, advocating that
they are> >richer, better, smarter and more lovable
BECAUSE THE ARE JEWISH,> >from private events to
general media stuff, (like i.e. reporting onthe> >school
> bus accident report that killed 20 US-goy kids into the>
>background......)> >that is what makes the goyim,
which outnumber Jew almost a million> >fold, so irate and
belligerent towards them. --- It is a simple as> that.>
>We won't even have to touch that for 50 years US taxpayers
got> >forced to pay billions, annually, to the Jewish
State... with NOTHING> to> >show for in return!> >>
>Will it change? No. Why not?> >Because that would
mean for Jews to become Un-Jewish....ahahahaha....> >Of
course Potter will come along now and give recital of his>
>Not all Jews are... song>
.........aahahahaha........ahahahaha.........> >>
>>Could it be that Tom Potter is a troll who makes this all
up out> >>of whole cloth? Naw, not Tom Potter.>
>>Everybody knows he is honest to a fault.> >>FK>
>>> >> >ahahahaha........fault, Kaz, what
fault?.........honesty and morality> >is Potterian by
DEFAULT.....ahahahahaha........but then it becomes> >more
evident with each post of yours that you, Kaz, are the Jewish>
>mirror image of Potter> > >>How so?> >>Tom
Potter> >> >> >
AHAHHAHAH..........ahahahahaha........Tom, Tom.......> >
until you answer this yourself, to yourself, about yourself
and for> yourself> > you are as blind to the solution of
the problem as are yourinstigators> of> > con?nd war
for power and riches.......ahahahha.....ahahaha> >> >>
>>Are you asserting that fkasner> >>is against the
instigation of con?nd war> >>for power and riches, by
non-Jews like Bush, Blair and Rumfeld,> >>and that
fkasner thinks that the root causes of how con?>com
es about should be discussed, researched, and something>
>>done to eliminate the root causes?> >> > Suit yourself
Tom, but let me repeat for your bene?, Tom:> >> >Of
course Potter will come along now and give recital of his>
>Not all Jews are... song>
.........aahahahaha........ahahahaha.........> >> > which
he just did with a variation thereof.> >> >>In other
words, are you asserting that fkasner> >>thinks that the
root cause of con?nd war like> >>instigation for power
and riches,> >>should be treated, rather than symptoms like>
>>boys throwing rocks at tanks demolishing their> >>homes
and neighborhoods, and girls becoming human bombers,>
>>because family members were murdered by soldiers?> >>Tom
Potter>> > Suit yourself, Tom, whether in your, mine or
other words.> > You are reading different things off/in
my post to Kaz then I do.> > You come across to me here
just as arguing for argument's sake.> > Let me repeat for
your bene?, Tom:> >> >Of course Potter will come along
now and give recital of his> >Not all Jews are... song>
.........aahahahaha........ahahahaha.........> >> > which
he just did with another variation thereof.>> Forget it.
He's a dumb-ass troll who thinks that he can continue to
ring> the bells of those who as a point of ethics and
morality see the grand> Jewish conspiracy as a mean
spirited attempt to pick on a small minority> who as
emancipation has left them free to succeed are a wonderfully>
> handy scapegoat. Why even in countries that have almost no
Jews they are> now having their prime ministers engage in
blatant anti-semitism. What a> deal for a troll. But
probably in point of fact Tom Potshard is an> anti-semite.
But then again he probably hates almost anyone who he is>
convinced won't punch out his lights for showing his hate. So
he is what> is remarkable an anti-semitic troll. PLUNK.>
FK>> It is interesting to see that when someone points out>
that the Bolsheviks.instigated the class wars of the 1900's,>
and are instigating the religious wars of the 2000's,> that
fkasner implies that all instigators of con?nd war>
are Jews, and that the folks who are opposed to war,> and the
instigation of war, are anti-semitic trolls.>> All Jews are
not instigators of con?nd war for pro?,> any more than
all Muslims are terrorists,> and all Gypsies are fortune
tellers.>> And as can be seen in the cases of Bush, Blair and
Rumfeld,> all instigators of war for pro? are not Jews.>> It
is interesting to see that fkasner uses the same>
boilerplate tactic that the war-for-pro? gang> of attacking
the messenger, rather than addressing> the message in an
intelligent, rational, MORAL way.>> I suggest that folks
should think about what it means to> them, their families, to
their country, and to society,> to give in to people who
instigate con?nd war for power and riches,> and to wimp
out because they are so aggressive in> attacking the folks
who question their motives and tactics.>> Giving in to this
kind of behavior encourages and rewards it.>> Would you let a
whining, abusive, destructive child have his way?>> --> Tom
Potter http://tompotter.us>
===
Is there a version of the
spectral theorem for symmetric matrices withentries in F_2
(?ld with two elements). Certainly one would need towork in
the algebraic completion of F_2, but can we
necessarilydiagonalize symmetric matrices?TIA!
===
> Is there
a version of the spectral theorem for symmetric matrices
with> entries in F_2 (?ld with two elements). Certainly one
would need to> work in the algebraic completion of F_2, but
can we necessarily> diagonalize symmetric matrices?Try
diagonalizing (0 1 // 1 0) :-)-- Robin Chapman,
www.maths.ex.ac.uk/~rjc/rjc.htmlNeedless to say, I had the
last laugh. Alan Partridge, _Bouncing Back_ (14 times)
===
>
> Is there a version of the spectral theorem for symmetric
matrices with> entries in F_2 (?ld with two elements).
Certainly one would need to> work in the algebraic
completion of F_2, but can we necessarily> diagonalize
symmetric matrices?> Try diagonalizing (0 1 // 1 0)
:-)Rats...shoulda thought of that myself.This shows that
there's no Jordan form, either: letting A = [0 1; 1 0],we
havechar poly A = x^2 - 1 = (x-1)^2so 1 is a repeated
eigenvalue. There is only one eigenvector v=[1;1]but no
solution to (A-I)x=v.Does anything work? Are there conditions
that one can check to seewhen one has diagonalizability or
Jordan form?
===
>>> Is there a version of the spectral
theorem for symmetric matrices with>> entries in F_2 (?ld
with two elements). Certainly one would need to>> work in
the algebraic completion of F_2, but can we necessarily>>
diagonalize symmetric matrices?>>> Try diagonalizing (0 1
// 1 0) :-)> Rats...shoulda thought of that myself.> This
shows that there's no Jordan form, either: letting A = [0 1; 1
0],> we have> char poly A = x^2 - 1 = (x-1)^2> so 1 is a
repeated eigenvalue. There is only one eigenvector v=[1;1]>
but no solution to (A-I)x=v.?What about (1 0)?-- Robin
Chapman, www.maths.ex.ac.uk/~rjc/rjc.htmlNeedless to say, I
had the last laugh. Alan Partridge, _Bouncing Back_ (14
times)
===
> What about (1 0)?Boy, I'm braindead today.So, is
Jordan form OK over F_2?
===
>So, is Jordan form OK over
F_2?Jordan form works for a matrix A over any splitting ?ld
for the characteristic polynomial of A.Or if you want to stay
within the original ?ld, you can use rationalcanonical
form.Robert Israel israel@math.ubc.caDepartment of
Mathematics http://www.math.ubc.ca/~israel University of
British Columbia Vancouver, BC, Canada V6T 1Z2
===
>>Let E
be a subset of [a,b] with mE > 0, where m is the Lebesgue>>
>measure. Then there exists a closed subset F of E with mF >
0.>>> If you are assuming that E is measurable, this is
true. Otherwise,>> it is not.>You're saying there's a
non-measurable set E with m(E) > 0?Depends somewhat on your
de?itions. If m(E) is the outer measure,then yes. If it is
only de?ed for E measurable, then no. Booksdiffer on this
point.In fact, any non-measurable set has measure >0.--Dan
Grubb
===
>>Let E be a subset of [a,b] with mE > 0, where
m is the Lebesgue>>measure. Then there exists a closed
subset F of E with mF > 0.>> >> If you are assuming that
E is measurable, this is true. Otherwise,>> it is not.>
>You're saying there's a non-measurable set E with 
m(E) > 0?>
> Depends somewhat on your de?itions. If m(E) is the outer
measure,> then yes. If it is only de?ed for E measurable,
then no. Books> differ on this point.> In fact, any
non-measurable set has measure >0.The point is, the OP
EXPLICITLY CALLED m(E) the Lebesgue measure.Do books differ
in calling Lebesgue outer measure Lebesgue measure?I think
not. There's too much room for confusion.If you mean, some
books use m(E) to denote the Lebesgue outer measure ofE, and
also the Lebesgue measure of E, then yes, I agree.
(Althoughmost would use m^*.) But I cannot ?d a single
example where outermeasure is CALLED Lebesgue measure.--Ron
Bruck
===
[...]> > >You're saying there's a non-measurable
set E with m(E) > 0?> > Depends somewhat on your
de?itions. If m(E) is the outer measure,> then yes. If it
is only de?ed for E measurable, then no. Books> differ on
this point.> > In fact, any non-measurable set has
measure >0.> The point is, the OP EXPLICITLY CALLED m(E)
the Lebesgue measure.> Do books differ in calling
Lebesgue outer measure Lebesgue measure?> I think not.
There's too much room for confusion.> If you mean, some
books use m(E) to denote the Lebesgue outer measure of> E,
and also the Lebesgue measure of E, then yes, I agree.
(Although> most would use m^*.) But I cannot ?d a single
example where outer> measure is CALLED Lebesgue measure.>
Here's one...H. Federer, GEOMETRIC MEASURE THEORY
(1969).2.1.2 (page 53), de?ition of measure: The domain is
2^X, thecollection of all subsets of a set X. Of course then
only countablesubadditivity can be required, and some special
subsets are calledmeasurable. Later (page 111) Lebesgue
measure is a measure.
===
> In some texts, the term Lebesgue
measure means what I call Lebesgue> outer measure. The OP
did not say what text he was using.Which texts?
===
The
Russell's paradox and the Barber of Seville share a
comminality.The assumption of the existence of non-existent
things leads tocontradiction!Neither the ?Russell Class' nor
the ?Barber of Seville' exist!The essence of these
contradictions resides in the notion that:~ER(xRy <->x
~(xRx)) or ~ER(yRx <->x ~(xRx)).Both are valid.That the x's
and the y's are not of the same type is not
relevant.Russell's theory of types does not resolve the
paradox of the Barber ofSeville!Although, his ?Types' does
eliminate the possibility of expressing (x e x).(ix: (x e y)
<->x ~(x e x )) is just as contradictory as is (ix: x=y
<->x~(x=x)).~E!(iy: (x e y <->x ~(x e x))~E!(iy: x=y <->x
~(x=x))In general: ~E!(iy: xRy <->x ~(xRx)), and,~E!(yRx <->x
~(xRx)) for every R.It is the case that: Ax(xRy <-> ~(xRx))
and Ax(yRx <-> ~(xRx)) are bothcontradictions.for all
R's.Witt
===
> There's a puddle of water in front of my house
where the main water> line runs about 6.94 below grade. In
that area is a pipe around the> shut-off valve that straddles
the inlet/outlet pipes on the valve and> the bottom is resting
on a couple ?e bricks.>What's 6T mean?> The puddle covers
about 5 square feet and didn't get smaller last> night, but
no larger either. I don't know the relative humidity or> how
well water will perk, and the water meter isn't moving with>
everything off. There is also some plastic sheeting on the
ground> under the water, but it is not complete.>> It is
possible water was .93spilled.94 there by local children
getting a> drink, and water under our home (during a
previous leak) took MONTHS> to evaporate with the vents wide
open.>> Here are some ?st results pouring water into a
quart measuring> cup while timing it:>> A) ?tes that
were not suf?ient to move the water meter> 1) *1.24 gph> 30
gpd> 2) 1.77 gph> 42.5 gpd>> B) ?tes that make the water
meter move very slowly> 1) 2.63 gph> 63 gpd> 2) 1.11 gph> 27
gpd>As A1&2 con?ith B2, data is suspect.Typo?
Measurement or recording mistake?Inaccuraces of meter?Perhaps
mininum useful data is 5 gph.
===
> There's a puddle of water
in front of my house where the main water> line runs about
6?elow grade. In that area is a pipe around the>
shut-off valve that straddles the inlet/outlet pipes on the
valve and> the bottom is resting on a couple ?e bricks.>
>> What's 6T mean?Comparison with the following paragraph
suggests that ?^T' indicates aclose quote or double quote; so
either 6 inches or 6 feet.> It is possible water was
?illed?here by local children getting a> drink, and
water under our home (during a previous leak) took MONTHS>
to evaporate with the vents wide open.-- P.A.C. SmithThe
vast majority of Iraqis want to live in a peaceful, free
world.And we will ?d these people and we will bring them to
justice.
===
Wow, just as I was about to cross-post
toRocketSci.LittleOldLadies.BasketWeaving an answer
appears...LIKE I SAID ORIGINALLY, the meter isn't moving
during the leak, if it is aleak.With the water off (and
Grandma and children relocated for the night andfollowing
day) the puddle was gone but came back quickly when water was
turnedback on (cut off at street and house cut-off to prevent
gravity sourcing fromthe house).What I wanted to know from
all the supposed math gods was if there was enoughinfo in
the data to extrapolate the constant leak (which was not
enough tomove the meter) when the threshold of meter movement
was passed and several?easured. I did not think there was
suf?ient data, but (assumingreading skills) I thought someone
here was at least bright enough to know ifthere was an
obtainable answer...If you didn't glean this from the OP, 
I'm
the plumber and I didn't want to digup the pipes for nothing
while I'm dealing with the ?bviously now, Ihave to, but
-Is there an answer to the original post?>> This ain't math,
it's plumbing psychology detective work.> At bed time,
securely tape all the toilet and faucet handles with duck>
tape. Take ?ght, read water meter and note time. Go to
bed and> don't have any wet dreams. ;-) Upon waking or
whenever you get around to> it, such as after work, read
water meter and note time.>> If in the mean time the water
meter has been spinning its wheels, you know> its your leak
to entertain or aggravate you plumber who ?ed the privious>
leak. Otherwise its water company's leak, who before letting
you pin the> leak on them, will repeat the test.>>
Furthermore, be there any doubt about the source, such as
being ground> water, the water company can take water samples
and test for clorine,> ? or whatever other locally
popular water pollutants the water> company is found of
making you drink and bath in.>>There's a puddle of water in
front of my house where the main water>line runs about 6??
below grade. In that area is a pipe around the>shut-off
valve that straddles the inlet/outlet pipes on the valve>
>and the bottom is resting on a couple ?e bricks.> 6 what?
Inches, feet, meters?>>The puddle covers about 5 square
feet and didn't get smaller last>night, but no larger
either. I don't know the relative humidity or>how well
water will perk, and the water meter isn't moving with>
>everything off. There is also some plastic sheeting on the
ground>under the water, but it is not complete.>>It is
possible water was ^?illed^?here by local children
getting>a drink, and water under our home (during a
previous leak) took>MONTHS to evaporate with the vents wide
open.> You could bail the water and see what happens.> Is the
water tap for public use? If not, get a locking faucet.>
Otherwise complain to the water department or other lible
agency.>> ----
===
> LIKE I SAID ORIGINALLY, the meter isn't
moving during the leak, if it is a> leak.>Didn't seem you
got any results. Rate of ?t important, what'simportant
is what volumn is needed for meter to register
noticeablechange. Take a shower and see how much meter moves.
If much, then ?liet a time or two and see how much meter
moves.> With the water off (and Grandma and children
relocated for the night and> following day) the puddle was
gone but came back quickly when water was turned> back on
(cut off at street and house cut-off to prevent gravity
sourcing from> the house).>Thus puddle comes from water
line. Is not puddle suf?ient volume tomove meter? What to
check? Water faucet or what ever where childrenplay? Fittings
near and around said location? Pipes underneath puddle?How far
down is the pipe? Six inches? That'd make for easy spot digs
to?d what direction leak is moving.Did watching puddle ?l
give any hint entry point into puddle of water?When my sewer
was ?ed, contractor dug 3 ft thru dry hard packed dirt
tohave miss his target. Eventually he ?ured out where to
dig. Happy thisdig was within access of back hoe that found
pipe underneath sidewalk 12 ftdown. Yea, 100 year old line
with no city records as to location.> What I wanted to know
from all the supposed math gods was if there was enoughWhat
feelings does an appellation ?plumer god' or ?pipe 
god'
impart?> info in the data to extrapolate the constant leak
(which was not enough to> move the meter) when the threshold
of meter movement was passed and several> ?easured. I
did not think there was suf?ient data,>> If you didn't glean
this from the OP, I'm the plumber and I didn't want to 
dig>
up the pipes for nothing while I'm dealing with the ?vio
usly now, I> have to, but ->Even feeling well, nobody
would want to. What a nuisance to have people begone
overnight to assure accurate test.> Is there an answer to the
original post?>I've made some comment thereto in other
post.
===
One thing that simply can't be denied is that
Legendre's Method isugly.Besides being ugly it's 
clunky.It's
a really ugly, clunky *algorithm* like consider.To count the
composites up to 10, you need to already know that 2 and3 are
primes. The method is too stupid to know that 2 and 3
areprimes, so you have to tell it.Then you take ?0/2) -
1 = 4, to get the composites with 2 as afactor, and those
composites are 4, 6, 8, and 10. Next you take?0/3) - 1
to get the composites with 3 as a factor, and thoseare 6 and
9.OOPS! What the fuck!!! You've DOUBLE COUNTED as you have 6
in bothfreaking lists!!!So now what to do, what to do, oh,
you subtract ?0/6) = 1, sothat you get the correct count
which is 5 for the 5 composites4,6,8,9, and 10.That's the
single big idea that mathematicians managed to come up
withover a hundred years for counting primes.Then they
tweaked it, ?dled with it endlessly, so there are allthese
names for different algorithms.Now I come along, over a
hundred years after Legendre came up with his*crappy* method,
and starting from scratch, I come up with a MUCH,much cleaner
idea.Then stupid mathematicians get on *my* case because
there are allthese dumb algorithms lying around from tweaking
the crappy, uglymethod.Mathematicians are so fucking stupid. I
hate mathematicians.It's not my fault you dimwits just came
close to the elegantfunction!!!I found it, so play fair!!!
Put my partial difference equation in thetextbooks!You stupid
fucking twit mathematicians.James HarrisMy math discoveries,
found for pro?http://mathforpro?.blogspot.com/
===
> One
thing that simply can't be denied is that Legendre's 
Method
is> ugly.>> Besides being ugly it's clunky.>> It's a 
really
ugly, clunky *algorithm* like consider.>> To count the
composites up to 10, you need to already know that 2 and> 3
are primes. The method is too stupid to know that 2 and 3
are> primes, so you have to tell it.>> Then you take
?0/2) - 1 = 4, to get the composites with 2 as a>
factor, and those composites are 4, 6, 8, and 10. Next you
take> ?0/3) - 1 to get the composites with 3 as a
factor, and those> are 6 and 9.>> OOPS! What the fuck!!!
You've DOUBLE COUNTED as you have 6 in both> freaking
lists!!!>> So now what to do, what to do, oh, you subtract
?0/6) = 1, so> that you get the correct count which is 5
for the 5 composites> 4,6,8,9, and 10.>> That's the single big
idea that mathematicians managed to come up with> over a
hundred years for counting primes.>> Then they tweaked it,
?dled with it endlessly, so there are all> these names for
different algorithms.>> Now I come along, over a hundred
years after Legendre came up with his> *crappy* method, and
starting from scratch, I come up with a MUCH,> much cleaner
idea.>> Then stupid mathematicians get on *my* case because
there are all> these dumb algorithms lying around from
tweaking the crappy, ugly> method.>> Mathematicians are so
fucking stupid. I hate mathematicians.>> It's not my fault
you dimwits just came close to the elegant> function!!!>> I
found it, so play fair!!! Put my partial difference equation
in the> textbooks!>> You stupid fucking twit
mathematicians.>If I knew someone was evil, out to get
me, stupid, etc., I wouldn'tcontinuecorrespondence with
them. Why haven't you ?ured this out yet? Isthere some sort
of problem you have that you need to have acceptance?
Maybesome father issues ??(I'm assuming of course that you
know who your fatheris.)We'll see...MB
===
> If I knew someone
was evil, out to get me, stupid, etc., I wouldn't>
continuecorrespondence with them. Why haven't you ?ured this
out yet? Is> there some sort of problem you have that you
need to have acceptance?Maybe> some father issues ??(I'm
assuming of course that you know who your father> is.)Mr.
Harris has a condition known as NPD. (See below and this will
answeryour questions, this egomaniac has NPD and is totally
deranged with ?s ofgrandeur)!Here is a clinical
de?ition:<http://www.behavenet.com/capsules/disorders/
narcissisticpd.htm>***Diagnostic criteria for 301.81
Narcissistic Personality Disorder (cautionarystatement)A
pervasive pattern of grandiosity (in fantasy or behavior),
need foradmiration, and lack of empathy, beginning by early
adulthood and present ina variety of contexts, as indicated
by ?e (or more) of the following:(1) has a grandiose sense
of self-importance (e.g., exaggerates achievementsand
talents, expects to be recognized as superior without
commensurateachievements)(2) is preoccupied with fantasies of
unlimited success, power, brilliance,beauty, or ideal love(3)
believes that he or she is special and unique and can only
beunderstood by, or should associate with, other special or
high-status people(or institutions)(4) requires excessive
admiration(5) has a sense of entitlement, i.e., unreasonable
expectations ofespecially favorable treatment or automatic
compliance with his or herexpectations(6) is interpersonally
exploitative, i.e., takes advantage of others toachieve his
or her own ends(7) lacks empathy: is unwilling to recognize
or identify with the feelingsand needs of others(8) is often
envious of others or believes that others are envious of him
orher(9) shows arrogant, haughty behaviors or
attitudesReprinted with permission from the Diagnostic and
Statistical Manual ofMental Disorders, fourth Edition.
Copyright 1994 American PsychiatricAssociation***
===
that's a
nice explanation, if it's true of Legendre's crappola, 
butwhat
does ?ean -- greatest integer part? so, then, what is
different about your method? > To count the composites up to
10, you need to already know that 2 and> 3 are primes. The
method is too stupid to know that 2 and 3 are> primes, so you
have to tell it.> Then you take ?0/2) - 1 = 4, to get
the composites with 2 as a> factor, and those composites are
4, 6, 8, and 10. Next you take> ?0/3) - 1 to get the
composites with 3 as a factor, and those> are 6 and 9. > So
now what to do, what to do, oh, you subtract ?0/6) = 1,
so> that you get the correct count which is 5 for the 5
composites> 4,6,8,9, and 10. > I found it, so play fair!!!
Put my partial difference equation in the> textbooks!>
http://mathforpro?.blogspot.com/
===
> I found it, so play
fair!!! Put my partial difference equation in the>
textbooks!Text books take quite a bit of time to write. Since
your method is sonew I wouldn't expect to see it in the books
for another year or two.Why don't you go away in the
meantime.V.-- homepage: cs utk edu tilde lastname
===
> One
thing that simply can't be denied is that Legendre's 
Method
is> ugly.>> Besides being ugly it's clunky.> You stupid
fucking twit mathematicians.> James HarrisJimmy Baby!You
are friggin' beautiful! I am rolling on the ?aughing my
ass off!
===
> You stupid fucking twit mathematicians.Mr.
Harris,your language is quite colorful during holiday
seasons.As for us twits, please provide a greater estimate (x
= 4*10^22) for pi(x)that beats this one.
See:http://numbers.computation.free.fr/Constants/Primes/
countingPrimes.htmlIn fact, lets make it simpler for you,
please duplicate this result usingyour magnanimous
method!Come on, these twits certainly surpassed anything you
have done, haven'tthey? Do you believe your method is
superior, then prove it (and we allknow the answer don't we
JSH?).You are such a joke!Get a life you ignoramus!
===
Let
X,Y,X',Y' be Z-modules (that is, abelian groups). 
Suppose X+Y
isisomorphic to X'+Y' (where + should mean the 
ordinary
direct sum asZ-modules). When (and how) can one conclude that
X is isomorphic to X'and Y to Y'?Now, I 
don't think it's true
in general, so, what if Y=Y'=Z ? What ifX is known to be
free/projective ? I'd be grateful about any hint inthat
direction, or counterexamples to such a
claim...Philipp
===
>>Let X,Y,X',Y' be Z-modules (that is,
abelian groups). Suppose X+Y is>isomorphic to X'+Y' 
(where
+ should mean the ordinary direct sum as>Z-modules). When
(and how) can one conclude that X is isomorphic to X'>and Y
to Y'?As written, almost never: pick any two nonisomorphic
Z-modules X andY, and let X'=Y, Y'=X. >Now, I 
don't think
it's true in general, so, what if Y=Y'=Z ?For 
->?itely
generated<- Z-modules, it is true that if X+Y = X+Y',then
Y=Y'. For non-?itely generated Z-modules, it need not
betrue. E.g., let Y be the direct sum of a countable number
of copies ofZ, and let X=Z and X'=0, or any direct sum of a
?ite or in?itecountable number of copies of Z.> What if>X
is known to be free/projective ?Still not true for free: the
example above has both Y and X free,Y=Y', but X need not be
isomorphic to
X'.
===
===============================

===
=====It's not denial. I'm just very selective about
what I accept as reality. --- Calvin (Calvin and
Hobbes)
===
============================

===
========Arturo Magidinmagidin@math.berkeley.edu===I
have one question regarding Goldbach's Conjecture.It states
that ?every even integer greater than 2 issum of two prime
numbers'.I'm curious whether it says about two 
different
primesor whether it allows situation where even numberis sum
of two equal (that means the same) primes.As an example take
6 = 3+3.Of course, it can be also written as 6=1+5,but, does
the hypothesis say that always, for everyeven number, must
exist TWO different primes,whoose sum give that number,or
does it allow situation as speci?d above?--Krzysiek
===
> I
have one question regarding Goldbach's Conjecture.> It states
that ?every even integer greater than 2 is> sum of two prime
numbers'.> I'm curious whether it says about two 
different
primes> or whether it allows situation where even number> is
sum of two equal (that means the same) primes.> As an
example take 6 = 3+3.That's a perfectly good example. It is
in fact the only way todecompose 6 into two primes.> Of
course, it can be also written as 6=1+5,> but, does the
hypothesis say that always, for every> even number, must
exist TWO different primes,> whoose sum give that number,> or
does it allow situation as speci?d above?First: 1 is not
prime.Second: The conjecture (obviously) means the sum of two
primes--whichstatement does not say two different primes, so
it doesn't mean twodifferent primes.Thomas
===
Krzysztof Olczyk
<> grava .88 la saucisse et au marteau:> As an example take 6
= 3+3.> Of course, it can be also written as 6=1+5,> but,
does the hypothesis say that always, for every> even number,
must exist TWO different primes,> whoose sum give that
number,> or does it allow situation as speci?d above?As 1 is
not considered as prime, I guess the two primes may be equal
:)-- Nicolas <kZhwb.6937$9O5.66@fed1read06>X-Cise:
tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate:
admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose:
george_cox@btinternet.comX-Punge: Micro$oft
===
at 10:57 PM,
Austrolopithecus Afarensis <fossil.lucy@cox.net>quoted:>The
problem is this same brush is used to tar *anyone* outside
the>establishment who attempts to advance fundamental
objections to or >criticisms of the canonical theory, no
matter how rational and articulateThat claim, of course, is
precisely the sort of statement that gets[people labeled as
cranks. There have been people who publishedcriticisms and
fundamental objections to GR without being tarred bythat
brush, but they published papers on Physics instead of
polemicsagainst Einstein or against the
establishment.Follow-up set.-- Shmuel (Seymour J.) Metz,
SysProg and JOATnot reply to
spamtrap@library.lspace.org
===
the Stanford low-orbit thing
sounds interesting, buthow could it be de?itive, resting
upon the same-old interpretationof Herr E.? > That claim, of
course, is precisely the sort of statement that gets> [people
labeled as cranks. There have been people who published>
criticisms and fundamental objections to GR without being
tarred by> that brush, but they published papers on Physics
instead of polemics> against Einstein or against the
establishment.--ils duces d'Enron!
<kZhwb.6937$9O5.66@fed1read06>
<1f366ae.0311240654.1bd2d91f@posting.google.com>X-Cise:
tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate:
admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose:
george_cox@btinternet.comX-Punge: Micro$oft
===
at 06:54 AM,
undeniablefact@yahoo.com (Undeniable) said:>At any rate,
whatever happened>to the ultimate test of GR involving gyros
in low orbit? There is no ultimate test of any theory. The
test you're thinking ofseems to be back on track.-- Shmuel
(Seymour J.) Metz, SysProg and JOATnot reply to
spamtrap@library.lspace.org
===
> This is the ?st time
someone is going through same thought process> through which
I have gone. I was just scratching my head day and night> for
nine months. The mistake I made that I was using BOLT(!!!)>
instead of SPRING(or any source of restoring force). If you
use bolt> AB and CB instead of stretched springs and try to
push ends A and C of> both arms in direction towards point B,
same thing will happen which> you have shown in the ?ure.>
Just imagine that AB and CB are two shock-ups in which
there is> tremendous restoring force directed towards point
B, is acting at> point A and C. Now just put your both hands
on end A and end C of> these shock-ups,What do you mean by
tremendous restoring force? Will the springswant to stretch
or shrink? More precisely, which way does yourrestoring
force point to?'> (1)will the vertex point B move in
downward direction?(As, by this> time you know very well,
angle ABC does not change.)Depends on the direction of the
force of the springs.> (2)if your hands are feeling
restoring force in direction towards> point B ONLY, then why
both ends of left and right arm will not feel> same force in
direction towards point B ONLY?They will feel the whole thing
twitching until it reaches equilibrium.And they will feel the
distance AC change.> (3)The biggest trick involved in
this drama is that if both ends of> left and right arm are
feeling restoring force towards point B, these> arms just
can't move in downward direction AT EXACTLY SAME MOMENT.As
long as AD and CD can move freely and independently of each
otherthey can.> I am going to recover myself psychologically
and physically.Great idea.Greetings!Volker
===
>> This device
does work.>>Did you build it?More importantly, did you use it
with no other assists.One of the earliest lessons that our
development grouplearned was to run their own code _all the
time_. There'snothing like having a system crasher in?
developers'decisions about whether they'll write new 
code or
? whatthey just broke so they can use a working machine to
write the newcode./BAH
===
> I am going to recover myself
psychologically and physically.(This is> one of the trick
He trained me to do in extreme situation, PULL> BACK!. This
is no less than a military operation for me).Military
operations aren't successful if the attacker retreats as soon
asfatigue sets in.People here haven't even been overly nasty
with you, compared to certainothers. Perhaps because they
sense that you are probably a very unbalancedand vulnerable
person underneath your veneer of, dare I say it,
holywarrior.However....He trained you in how to protect
yourself, but it is also He who hurlsall the attacks your
way.Explain how that isn't a contradiction.> I am going> back
to my workplace, take some rest. As I will not have any access
to> internet, I will not be able to see or post here.This is
exactly why the key ingredient of deprogramming is to take
theperson into an isolated situation, and refuse to stop the
arguments thatinvalidate the logic of their delusions.>> But,
I Will Be Back!As soon as you, once more, believe your own
story fully, I'm sure.A tip: Stop posting to this kind of
newsgroup if you can't handle rationalarguments against your
point of view, without perceiving yourself as beingunder
attack.
===
euan===> > >>Right on. The people who say
another thing coming are probably the >>same ones who
say I could care less.> >>Gib> > > I could care
less is just sarcasm isn't it? Makes perfect sense then.>
I don't know. What do you mean by sarcasm in this context?>
> GibI mean, I could care less said sarcastically means the
same thingas I couldn't care less meant literally. Whenever
I've heard peoplesay the former (which is not too often)
they've been careful to say itin a _very_ sarcastic tone -
but maybe where you are it's caught on tosuch an extent that
people no longer do this.David
===
>>>>>>Right on.
The people who say another thing coming are probably the
>>same ones who say I could care
less.>>>>Gib>>>I could care less is just sarcasm
isn't it? Makes perfect sense then.>>I don't know. 
What do
you mean by sarcasm in this context?>>Gib> > I mean, I
could care less said sarcastically means the same thing> as
I couldn't care less meant literally. Whenever I've 
heard
people> say the former (which is not too often) they've been
careful to say it> in a _very_ sarcastic tone - but maybe
where you are it's caught on to> such an extent that people
no longer do this.Where I am (New Zealand) nobody says
could when they mean couldn't - it is a usage that 
I've
noticed on the web and on US media. It seems that you are
suggesting that some people say I _could_ care less,
implying but not much less. But I don't think this is way
I've usually heard it, rather I get the impression that the
speaker just doesn't realize what he's saying.To bring 
us
back to mathematics, I'm reminded of the mad hatter's
tea-party, where Alice is offered more tea, when she hasn't
yet had any. She protests, but is corrected (I have to give
the full quotation):Take some more tea, the March Hare said
to Alice, very earnestly.I've had nothing yet, Alice replied
in an offended tone: so I can't take more.You mean you
can't take less, said the Hatter: it's very easy to 
take
more than nothing.Nobody asked your opinion, said
Alice.Who's making personal remarks now? the Hatter asked
triumphantly.It's amazing how much entertainment JSH
provides, isn't it?Gib
===
>>>>>>Right on. The
people who say another thing coming are probably the
>>same ones who say I could care
less.>>>>Gib>>>I could care less is just sarcasm
isn't it? Makes perfect sense then.>>I don't know. 
What do
you mean by sarcasm in this context?>>Gib> > I mean, I
could care less said sarcastically means the same thing> as
I couldn't care less meant literally. Whenever I've 
heard
people> say the former (which is not too often) they've been
careful to say it> in a _very_ sarcastic tone - but maybe
where you are it's caught on to> such an extent that people
no longer do this.> DavidMy experience is that, by far, the
vast majority of users of this formof the expression (I could
care less) fail to recognize the sarcastic(or even the
ironic) tone of that version, as contrasted to the
literaltone of the other version (I couldn't care less).
Whether they'reirony-de?ient or simply don't hear the 
words
any longer, due to theircountless years of repetition, I
could never determine.As for the identity of users of the
phrase I could care less andthe users of the phrase
another thing coming, I grew up in a familythat used
another thinG comingas well as I couldn't care lessWe
(that is, I) always understood it in the sense that the
previous poster (Jesse F. Hughes), who interpreted it
something like if youthink X, you're going to ?d out that
the truth is Y, another thing.Well, he said it much better:>
> When one says, If you think X, you've got another thing
coming, X is> typically a future event or a regularity. The
word think here is> used synonymously with expect or
believe, but I want to stress the> expect meaning.> So
the ?st thing is not the thought about X, but X itself (very>
loosely). If you expect X, you have another thing coming. If
you> think I'll give you $5, you have another thing coming
(an experience> contrary to the actual giving of the $5).
This is the interpretation> that I have always had for the
expression.
===
I really *could* care less, you smouldering
piece o'****, butthen I'd just have to ignore you.
please,dyspose of yourself at once -- and don't start a
forest ?e! >I could care less is just sarcasm isn't
it? Makes perfect sense then.> >>I don't know. What do
you mean by sarcasm in this context?--ils duces
d'Enron!
===
>> Right. It's that deliberately bad grammar that
sounds like>> think is the word intended there, in an
attempt to be>> affectedly folksy. It sounds like one of
those made-up>> Texas proverbs in other words.>> Now I'm
gonna hear from more Texans than you can shake a>> stick
at.>>As a technical Texan[1], I claim that thing is the
commonest Texas>version. At least in my family.Well then
you'll know what I'm talking about with Texas 
language.
Ididn't really mean proverb. Probably simile. Or is it
metaphor. That boy is slower than .... This newsgroup is
slicker ?n....You know, this
stuff:http://uts.cc.utexas.edu/~samoht/texassimiles.htmlSo
tell me. Does anyone actually talk this way?I'm a damn 
lib'rl
east-coast yankee. For all I know, the whole damncountry west
of Pittsburgh talks this way. - Randy
<87k75qmyqh.fsf@phiwumbda.org> <87r7zy3cvu.fsf@phiwumbda.org>
<bprnef$3jo$1@news.fas.harvard.edu>
<8765hatedb.fsf@phiwumbda.org>
<df76407e.0311240732.7c82261b@posting.google.com>
<87smkdvi03.fsf@phiwumbda.org>
<adm5svc6vcokof2i0n0frpckcrd0ifp9r1@4ax.com>
===
>>> Right.
It's that deliberately bad grammar that sounds like>
think is the word intended there, in an attempt to be>
affectedly folksy. It sounds like one of those made-up>
Texas proverbs in other words.>> Now I'm gonna hear from
more Texans than you can shake a> stick at.>>As a
technical Texan[1], I claim that thing is the commonest
Texas>>version. At least in my family.>> Well then you'll
know what I'm talking about with Texas language. I> 
didn't
really mean proverb. Probably simile. Or is it
metaphor.>> That boy is slower than ....> This newsgroup
is slicker ?n.... [owl shit on a brass> doorknob]At least
that's how I ?ish #2. #1 has more endings.>> You know, this
stuff:> http://uts.cc.utexas.edu/~samoht/texassimiles.html>>
So tell me. Does anyone actually talk this way?I do, but it's
a bit affected. I think it's funny, so I picked up thehabit on
purpose. My family uses plenty of Southern and Texasphrasing,
but not so much comparisons like those. More things likeI'll
be on you like ugly on an ape, and if it was a snake,
itwould've bit you, and That was bigger'n all get 
out.>>
I'm a damn lib'rl east-coast yankee. For all I know, 
the
whole damn> country west of Pittsburgh talks this way.For
eight years, at least a small part of Pittsburgh talked that
way,too.-- Jesse F. HughesTruth is common stuff, ready to
your hand, but lies you have to makeyourself, and you can't
be sure they are any good until you'veused them --- and then
it's too late. John Steinbeck
===
> I'm a damn lib'rl
east-coast yankee. For all I know, the whole damn> country
west of Pittsburgh talks this way.> For eight years, at
least a small part of Pittsburgh talked that way,> too.OK,
I'll bite. What eight years? Is this a reference tosome
presidential administration? - Randy
<87k75qmyqh.fsf@phiwumbda.org> <87r7zy3cvu.fsf@phiwumbda.org>
<bprnef$3jo$1@news.fas.harvard.edu>
<8765hatedb.fsf@phiwumbda.org>
<df76407e.0311240732.7c82261b@posting.google.com>
<87smkdvi03.fsf@phiwumbda.org>
<adm5svc6vcokof2i0n0frpckcrd0ifp9r1@4ax.com>
<87y8u4gb9c.fsf@phiwumbda.org>
<df76407e.0311251042.7db5196a@posting.google.com>
===
>> I'm
a damn lib'rl east-coast yankee. For all I know, the whole
damn>> country west of Pittsburgh talks this way.>>> For
eight years, at least a small part of Pittsburgh talked that
way,>> too.>> OK, I'll bite. What eight years? Is this a
reference to> some presidential administration?No, it's the
eight years I spent in Pittsburgh doing my damnedest
toin? speech patterns. My damnedest was essentially
limited toteaching a German or two a few key phrases and the
?er points of theone-syllable vs. the two-syllable versions
of shit.My main German student was willing to learn and got
the usage patternsdown pretty well, but Hoo-doggies! That
puppy was slicker'n owlshiton a brass doorknob! just 
doesn't
?en your accent is similar toSchwarzenegger's[1].It
probably doesn't help that I never made a very persuasive
Okie mydamn self. Even when I still lived there, the locals
would regularlyask me where I was from. Blame too many Monty
Python records during amisspent youth.Footnotes: [1] Yes,
yes, I'm sure that my buddy's accent was nowhere
nearSchwarzenegger. It was close enough for my crudely
trained ear.-- 'Every man who has ever lived holds tight to
the belief that for himalone the laws of probability are
canceled out by love[...] Therefore,you will marry Guinevere.
You do not want advice --- only agreement.'Merlin sighed...
-- John Steinbeck
===
> What is this crap?Repunctuate. What!
... Is this crap?The answer may be left as an exercies.--
Chris HenrichDrag people to events to invite them. -- A tool
tip in Apple's iCal program.
===
Assume an n x n Go board. We
place k stones on this board so that theyare well-connected
(i.e. for every stone there is a stone in anadjacent square,
not counting diagonals squares). What is the maximumnumber of
liberties (empty squares with at least one stone in anadjacent
square) achievable and what is the minimum number of stoneswe
can achieve it with?I know the maximum upper bound for the
liberties on a board n x n is:4 + (n^2 / 3 - 1) * 2 = 2 +
2n^2 / 3I have a possible optimal placing for a 19x19 board
but would like toprove it can't be improved.
===
> Assume an n
x n Go board. ... for every stone there> is a stone in an
adjacent square, not counting ...Ooops? Stones are placed on
the intersection points,not within the
squares.+---+---+---+---+---+---+---+---+| | | | | | | |
|+---+---+---+---+---+---+---+---+ A nice little| | | | | | |
| | 9x9 version for+---+---+---+---+---+---+---+---+
exercising.| | | | | | | | | It's
possible+---+---+---+---+---+---+---+---+ to play terribly| |
| | | | | | | exciting games+---+--( )--+---+---+---+---+---+
even on such a| | | | | | | | | small board!+--( )-(#)-(
)--+---+---+---+---+| | | | | | | |
|+---+---+---+---+---+---+---+---+ ( ) = white stones| | | |
| | | | | (#) = black
stones+---+---+---+---+---+---+---+---+| | | | | | | |
|+---+---+---+---+---+---+---+---+Black stone (#) with one
liberty left. (No not leftbut below :-)RR
===
Please stop
wasting your time and mine.Witt> G. Frege, apparently,
would agree with your #2.>> For Frege:> If, x/y is
de?ed (the z: x*z = y), then 0/0 = 0, and 1/0 = 0.>
Frege surely wouldn't agree with that. The z: x*z = y> is
commonly called y/x, even by Frege and all your other>
authorities included :-)>> RR>>
===
> >>> the Riemann
Hypothesis follows from the existence >> of a prime between
n and n + sqrt(n)....>>Really? Seems quite unlikely. >
Really? Why does that seem unlikely?There are numerous
reasons.If the statement were true, there would be a very
powerful heuristic argument in favor of RH, which I'venever
heard about.It is easy to create a set of integers with the
samedensity distribution as the primes, and having the
aboveproperty, but such that the Zeta function has roots
withreal part not = 1/2.It looks like a confusion with
another result concerningthe difference between pi(n) and
li(n)The result is startling enough that if it were known
tobe true, everyone, including me and you, would have
heardabout it.
===
>> >> the Riemann Hypothesis
follows from the existence >>> of a prime between n and n +
sqrt(n)....>> >Really? Seems quite unlikely. >>> Really?
Why does that seem unlikely?>>There are numerous reasons.>>If
the statement were true, there would be a very >powerful
heuristic argument in favor of RH, which I've>never heard
about.>>It is easy to create a set of integers with the
same>density distribution as the primes, and having the
above>property, but such that the Zeta function has roots
with>real part not = 1/2.>>It looks like a confusion with
another result concerning>the difference between pi(n) and
li(n)>>The result is startling enough that if it were known
to>be true, everyone, including me and you, would have
heard>about it.All excellent reasons, except for the last
half of the lastone: Doubtless I would have heard of it, but
there's noreason I would remember it - I know that there are
preciserelations between the distribution of primes and RH
butI have no idea what they are, never having had any
reasonto try to put that into long-term memory (hence I
wasn'tcertain that I had not heard of it...)David C.
Ullrich
===
> I assumed that what he thought was unlikely was
the statement> that RH _follows_ from the existence of a
prime between n> and n + sqrt(n) (for large n, of course). Do
you have a reference> for that?Alas, no. It may be that I
misremembered something I heard/read long ago, and that the
implication only goes in the other direction.--
===
> >>
>> > Real science is data ?st.> >> > I agree with
this.> >> > For example the tens of millions of measured
redshifts that agree> > perfectly with the Big Bang model.
That's DATA for you, in capital> > letters.>> ROTFLMAO!
How do they agree?>> They ?d that objects which are more
redshifted tend to be farther> away.>> Thats how.1) Assume
objects with larger redshift are farther away.2) Claim that
observing larger redshift proves it is farther away.That's
one of the shortest circular arguments I've seen in awhile.>>
> Because we simply *assume* that redshift> gives us a
distance.>> You would assume such a thing, yes.>> A scientist
wouldn't.>> That's why you'll never be a 
scientist.Too bad so
many have. That is the basis for the Big Bang. But we
agreethat Big Bang cosmologists aren't scientists.> (There
are a few dozen *measurements* of distance> versus
redshift.>> Given that I have done a whole lot more than a
few dozen myself, I> can con?ently state that you are
lying.Fascinating. Citation, please. (I'll settle for 100,
independent distancemeasurements that do not depend directly
or indirectly on the Hubble law)> There are *tens* *of*
*millions* of measurements of distance versus> redshift in
perfectly ?e agreement with Big Bang cosmology.Yawn. Since
you get to adjust parameters both on distance and on the
BBside, I'm not impressed.> Besides, it only takes one real
case to disprove the assumption.>> There is no assumption
anywhere.Sure there is. You are assuming that the *ONLY*
contribution to redshift isdoppler (or doppler and expansion
if you want to be picky). No ?tiredlight', no ?change in
physical parameters', no nothing.> You pick a light source --
say a> particular type of galaxy or a particular type of
supernova or a> particular type of variable star (it doesn't
really matter, just pick> some object you can recognize) and
you measure their brightness and> some are bright and some
are dim and there's some distribution of> brightness between
the brightest and the dimmest and when you look at> objects
that are farther away then you'll see the exact same>
distribution of brightness except that all objects will
appear dimmer> because they're farther away.Sure, with the
adjustible parameter of extinction to help you out. And
youcan't know that the distribution of brightness is the same
without assumingthe hubble relation that gave you the distance
-- or vice versa.> And then you measure redshift and you plot
apparent brightness against> redshift and you ?d that by and
large redshift increases as> brightness decreases.Ah! But the
?by and large' is the assumption. The point being that
theremay be exceptions.> The Big Bang does NOT claim that all
objects must lie on a perfect> redshift/distance curve, merely
that there is an *additional* linear> term in any
distance-redshift relation. Namely the Hubble ?d that is
the assumption. That redshift is only and always due to
motion(physical or expansion).> Whatever population of things
you measure the redshift distribution> of, you ?d an
additional redshift the farther things are away -- by>
whatever means the distance is determined.But that's the
whole point! The whatever means is always the redshift.All
?direct' measurements of distance end with parallax. Beyond
that istheory -- adjustible to meet the BB postulate. UNLESS
one ?ds a singleconnected group of objects with signi?antly
different redshift.> And it doesn't have to be determined
terribly precisely: if your> distance determination has a
factor two uncertainty, then you get a> spread of measured
points around the linear term. If it has a factor> *ten*
uncertainty then you get a *wider* spread around the EXACT
SAME> linear term.We all agree there are uncertainties. So
why are you so ?certain' you arecorrect?> Besides, it only
takes one real case>> And in these tens of millions of
observation there is not a single> real case that disproves
anything whatsoever.Even if there were 10 million independent
determinations of distance, thesimple exclusion of contrary
data makes your claim meaningless.> If you (or Mr. Arp or>
anybody) want to claim that any two galaxies are in
interaction with> each other then the burden of proof is on
you (or Mr. Arp or whoever> makes the claim).Precisely. And
Dr. Arp was well on the way to providing just that. Whichis
--of course -- why Dr. Arp was kicked off telescopes. True
science in themodern sense. The burden of proof is on Arp --
so we'll simply remove fromhim the ability to collect proof
(data). Problem solved.> But JUST to make sure we're not
missing something somewhere, we> actually look at the stuff
Mr. Arp hands us and we ?d that the> galaxies in question
are NOT in fact in interaction with each other.How did you
manage to determine this fact? Quite simply, you cannot
haveany basis for your claim -- except theory (which is not
fact.) Quitesimply the ONLY argument that can be used on
Arp's objects is the wordcoincidence. (So your claiming
fact is a bold-faced lie.)Let's take an example. An
(apparently) physically connected group of threegalaxies. Two
which appear to be ejected from the main one. The outer
twohave signi?antly higher redshift than the main one. The
connectionappears to be gas.How did you ?disprove' the
connection? AH, yes! The redshifts aredifferent THEREFORE,
the objects cannot be connected BECAUSE the distance
isdifferent, because we assume redshift determines distance
(assuming ourconclusion). The ONLY valid objections are that
this ONE case is simply acoincidence. There is no way to
PROVE or DISPROVE one case. But a claim ofcoincidence can
only be made from a statistical viewpoint.So ... obviously
... one must do a number survey of such possiblecoincidences.
Which is what Arp was doing when he was kicked off
thetelescopes.> So he hands us another pair and we con?m
AGAIN that he has no point.> Then we do the same thing AGAIN
and AGAIN and after the tenth or> hundredths time we can
con?ently state that we have given the Arps> of the world
their due hearing and all their points simply evaporate> upon
inspection.>> There is NOT ONE observation in the universe in
contradiction with the> Big Bang. There are a few dozen
observations that the Arps of the> world can't ?ure out
(dif?ult observations, all of them) and it is> simply a
*lie* to proclaim that they pose some kind of contradiction>
to the Big Bang. They don't. No matter how long we stare into
the> heavens, there will always be some observation at the
sheer limits of> observability that is dif?ult to interpret
-- but that's all they> are: dif?ult to interpret. Not any
kind of disproof.And you are a pathetic liar, yourself. For
you obviously understand theissue. You claim that each single
case of Arps is not proof. But notbecause it is not true
-- but because you can claim each case is purecoincidence
(which is why you weaseled and never mention *HOW*
eachobservation of Arp's is wrong). You slime over to
claiming they aredif?ult to interpret, and only a few
dozen can't be ?ured out. Youevade the central issue:
That you claim the burden of proof is on Arp, andthen you
deliberately prevent Arp from gathering data that could
providethat proof.If you had an ounce of honesty or
professional integrity, you would beagitating for Arp's
reinstatement to viewing time. Then you could evaluateArp's
data and the statistics would tell all. And -- if you are
correct --the statistics would support you. But your actions
(and those of the restof the BB heirarchy) is simply to avoid
the possibility of disproof, byterminating data collection.>
> Not to mention the CMB (and in particular the
*anisotropies*> > in the CMB in perfect agreement with what
you'd expect from the Big> > Bang). And so forth.>> To
which version of the BB are you referring?>> I am referring
to the mathematical model that the word Big Bang has>
referred to for at lest 50 years now.There are a dozen
versions of the BB, over the past 50 years. To whichversion,
speci?ally, are you referring? With in?, or without?
Withlight elements or without? With dark matter, or without?
Etc...> Certainly not the predictions> (which were 1 part
in 100, prior to COBE).>> Ah, the predictions.>> Let me
submit that you have no clue what you're talking about.I
?ured you'd dodge the issue. :)> Let me further submit that
COBE was in perfect agreement with BB> cosmologyLOL!> and
that WMAP has pushed the same agreement an order of>
magnitude further. As Mike Turner tends to say cosmology is
becoming> boring, every measurement only con?ms and re?es
the previous one.ROTFLMAO!!! Those are pretty major
?re?ings!'> The BB constantly adds new> epicycles every
time something is observed.>> Nothing new has been added to
the Big Bang since the original FRW> metric was been worked
out unless you're willing to consider in?> to be a part
of the Big Bang Model now in which case noting has been>
added to the Big Bang since in?. That's it. Everything
else> follows from what was there all along.ROTFLMAO!!! Do
you really believe this? How old are you? And can I sellyou a
bridge?> To ?st order (and to second one) the CMB should be
isotropic and> thus we build a satellite to measure that and
it con?ms what we> expect to within its margin of
observational error.LOL! COBE was designed to measure the
(predicted) 1-part-in-100 to1-part-in-1000 variation in the
background. So COBE was designed to go downto
1-part-in-10000, just to be safe. There was NO signal at
1-part-in-10000(a factor of 10 to 100 below BB predictions).
Data enhancement has beenused to ?d a signal at 1 part in
100000 level. But that is a factor of100 to 1000 below the
actual predictions of the BB.NOW you claim that the BB
?predicted' 1 part in 100000. Yeah, that's theticket!> 
Then
someone says> well, if we're really nitpicky about the
equations then there should> actually be this tiny, most
minute little anisotropy in there, but way> beyond COBE's
capability. So we scratch our heads, build a satellite> more
than an order of magnitude more sensitive (WMAP) and sure
enough> it ?ds these minute anisotropies *exactly* at the
level at which> they were expected.COBE would NEVER have been
built too coarse to identify the BB predictions.It's just that
the BB ?predictions' change (as always) as soon as
anothercontradiction is found.> Expected from the exact same
Big Bang model people referred to before> COBE existed.LOL!
This is too good to parody.{snip}>> > And while they're at
it, they squeeze these few oddball cases into> > roles they
never had: taking an object who's redshift seems to place> >
it outside the hubble ? pretend that there IS NO hubble
?d> > the entire theory explaining it must be wrong.>
>> Yes. That is the inescapable logic of science.>> No, it
isn't. You are merely clueless what you are talking about.>>
The Big Bang makes no statement whatsoever about the velocity
of any> one single object. None. Period. It never has, it
never will. Any odd> object can have any odd velocity with
respoce to any odd other object> (within the observed
constraints of relativity).>> All it does is ADD a small,
(essentially) linar term to the overall> velocities of
objects. At any distance you'll ?d stuff moving> towards us
and other stuff moving away from us, but the farther out> you
look the more the overall distribution of velocities is going
to> be shifted in the direction *away* from us, by something
like 70 km/s> per Mpc ditance.>> It does NOT say (has never
said, will never say) that all objects at> 1Mpc distance must
move away from us at 70km/sec. Merely that if you> measured
all the velocities of objects at 1Mpc distance that their>
velocities would be centered around 70km/s. And that's
exactly what is> observed.>> The radial velocity of any one
single object doesn't say *squat* about> the Big Bang. It
merely means that there is one single object that> happens to
move at some velocity relative to the hubble ?ve always
understood your theory. And I've not in any fashion
disagreedthat your above statement describes the theory and
assumptions behind thebig bang. The problem is that you don't
appear to understand the scienti?method, or even minimal
history of your own theory and it'sconstantly-changing
content and ?predictions.'--greywolf42ubi dubium ibi
libertas
===
> >> > >> > > Real science is data
?st.> >> > I agree with this.> >> > For
example the tens of millions of measured redshifts that
agree> > perfectly with the Big Bang model. That's DATA
for you, in capital> > letters.> >> > ROTFLMAO! How
do they agree?>> They ?d that objects which are more
redshifted tend to be farther> away.>> Thats how.> 1)
Assume objects with larger redshift are farther away.> 2)
Claim that observing larger redshift proves it is farther
away.You would assume such a thing, yes.A scientist
wouldn't.That's why you'll never be a 
scientist.> That's one
of the shortest circular arguments I've seen in awhile.Of
course it is circular: it is YOUR argument, not anybody
else's.>> > Because we simply *assume* that redshift> >
gives us a distance.>> You would assume such a thing,
yes.>> A scientist wouldn't.>> That's why 
you'll
never be a scientist.> Too bad so many have. Zero. You are
a disgusting lying pig.> That is the basis for the Big Bang.
The basis for the Big Bang is the correlation of
*independently*measured redshift and distance.You are a
disgusting lying pig.> But we agree> that Big Bang
cosmologists aren't scientists.More that you'll ever 
be.>
There are *tens* *of* *millions* of measurements of distance
versus> redshift in perfectly ?e agreement with Big Bang
cosmology.> Yawn. Since you get to adjust parameters both
on distance and on the BB> side, I'm not impressed.Yes, you
would do such a thing.A scientist wouldn't.That's why 
you'll
never be a scientist.> > Besides, it only takes one real
case to disprove the assumption.>> There is no
assumption anywhere.> Sure there is. You are assuming
that the *ONLY* contribution to redshift is> doppler (or
doppler and expansion if you want to be picky). No ?tired>
light', no ?change in physical parameters', no 
nothing.Both
of the two you mentioned require that redshift is correlated
withdistance. Which you are denying.(?tired light' has been
squarely disproven multiple times in this NGalone).> You
pick a light source -- say a> particular type of galaxy or
a particular type of supernova or a> particular type of
variable star (it doesn't really matter, just pick> some
object you can recognize) and you measure their brightness
and> some are bright and some are dim and there's some
distribution of> brightness between the brightest and the
dimmest and when you look at> objects that are farther away
then you'll see the exact same> distribution of brightness
except that all objects will appear dimmer> because they're
farther away.> Sure, with the adjustible parameter of
extinction to help you out. And you> can't know that the
distribution of brightness is the same without assuming> the
hubble relation that gave you the distance -- or vice
versa.brightness is a *measured* quantity.You are a
disgusting lying pig.> And then you measure redshift and
you plot apparent brightness against> redshift and you ?d
that by and large redshift increases as> brightness
decreases.> Ah! But the ?by and large' is the assumption.
No, it is what everybody ?ds who's done the observation.
There'snothing assumed here.You are a disgusting lying
pig.> The point being that there> may be exceptions.The Big
Bang does not say *squat* about the velocity of any one
singleobject. Is says something about a general matter ?
ADDITION TOany local relative velocities objects may have due
to any physicalprocesses they damn well please to have.This
has been explained to you before.> The Big Bang does NOT
claim that all objects must lie on a perfect>
redshift/distance curve, merely that there is an *additional*
linear> term in any distance-redshift relation. Namely the
Hubble ?And that is the assumption. Absolutely nothing
whatsoever is assumed when a plot of apparentbrightness
versus redshift shows that by and large the brighterobjects
have less redshift.> That redshift is only and always due to
motion> (physical or expansion).What does this have to do
with the *observed fact* that objects thatare far away have
greater redshift? The fact that you deny?1) There is an
*observed* redshift-distance relation that isindependent of
any cosmological model.You LIE that this isn't the case.2)
There is a cosmological model that says that such a thing
should beobserved *on the largest scales* (in broad,
universal terms, butNOT!!! locally) if the universe is
expanding in some fashion.You LIE that some object with a
peculiar velocity can somehowdisprove that model.>
Whatever population of things you measure the redshift
distribution> of, you ?d an additional redshift the
farther things are away -- by> whatever means the distance
is determined.> But that's the whole point! The whatever
means is always the redshift.It isn't. You are a disgusting
lying pig.I PRESENTED TO YOU an obvious means of getting at
distance throughapparent brightness.> All ?direct'
measurements of distance end with parallax.They don't. You
are a disgusting lying pig.> Beyond that is> theory --
adjustible to meet the BB postulate. UNLESS one ?ds a
single> connected group of objects with signi?antly
different redshift.1) Any small group of objects can have any
local relative velocitiesand the Big Bang wouldn't care -- it
is a *global* theory of theuniverse as a whole.2) No single
connected group of objects with signi?antly
differentredshift has been found, ever. If you want to claim
such, thentheburden of proof is on you to *show* that any
group of objects isconnected.You are a disgusting lying pig.>
> And it doesn't have to be determined terribly precisely: if
your> distance determination has a factor two uncertainty,
then you get a> spread of measured points around the linear
term. If it has a factor> *ten* uncertainty then you get a
*wider* spread around the EXACT SAME> linear term.> We
all agree there are uncertainties. So why are you so
?certain' you are> correct?I wasn't saying anything
whatsoever about uncertainties. You areTens of millions of
astronomical observations ?d themselves inperfectly ?e
agreement with the Big Bang. NONE ?d themselves
incontradiction with it. A buncha disgusting lying pigs tries
to worshipa few dozen that are simply hard to ?ure out.> Even
if there were 10 million independent determinations of
distance, the> simple exclusion of contrary data makes your
claim meaningless.There is no exclusion of contrary data
here anywhere.You are a disgusting lying pig.> If you (or
Mr. Arp or> anybody) want to claim that any two galaxies
are in interaction with> each other then the burden of
proof is on you (or Mr. Arp or whoever> makes the claim).>
> Precisely. And Dr. Arp was well on the way to providing
just that. Which> is --> of course -- why Dr. Arp was kicked
off telescopes.He wasn't.You are a disgusting lying pig.>
But JUST to make sure we're not missing something somewhere,
we> actually look at the stuff Mr. Arp hands us and we ?d
that the> galaxies in question are NOT in fact in
interaction with each other.> How did you manage to
determine this fact? In other words: you know absolutely
nothing whatsoever about the issueat all -- you haven't read
even a single paper on the subject, you aremerely insulting
millions of people who are performing routinelyobservations
that you have demonstrated repeatedly are too complicatedfor
you to even grasp conceptually.> Quite simply, you cannot
have> any basis for your claim Says the ignorant moron whos
only basis has always been the *lies*he's so fond of
posting.> An (apparently) physically connected group of three
galaxies. if all it takes for a couple things to be
connected is to look likethey're close together then
you've already lost right here.> Two which appear to be
ejected from the main one. The outer two> have signi?antly
higher redshift than the main one. The connection> appears to
be gas.> How did you ?disprove' the connection? AH, yes! The
redshifts are> different THEREFORE, the objects cannot be
connected BECAUSE the distance is> different, because we
assume redshift determines distance (assuming our>
conclusion). *You* *yourself* *assumed* that redshift is
related to distance whenyou brought up tired light or change
of the physical constants.A scientist, of course, would never
do such a thing.Which is why you're never going to be a
scientist.A scientist would simply point to the *measured*
correlation between distance and redshift.> And you are a
pathetic liar, yourself. For you obviously understand the>
issue. Contrary to you I do. You keep *demonstrating* that
you don't.> You claim that each single case of Arps is not
proof. But not> because it is not true -- but because you
can claim each case is pure> coincidence The word
coincidence does not appear once in my post.YOU would do
such a thing.A scientist, of course, wouldn't.Which is why
you're never going to be a scientist.> which is why you
weaseled and never mention *HOW* each> observation of Arp's
is wrongThere's hardly weaseling anywhere involved in not
answering aquestion that nobody asked.But here's a chance for
you to learn what the fuck you're talkingabout for the ?st
time in your worthless life: why don't you read upon one or
two of the papers where people observed some of Arpsobjects?
Then you could learn for yourself how one measures
whetherthings are in connection or not.But of course you'd
never actually poison your retarded brain withactual DATA.>
You slime over to claiming they are> dif?ult to interpret,
They are.It is dif?ult to interpret observations whenever
objects happen tolie close to the same line of sight. That's
it. There's nothingslime about it.Are you claiming
otherwise?> That you claim the burden of proof is on Arp, The
burden of proof is *always* on the positive claim.> and then
you deliberately prevent Arp from gathering data > that could
provide that proof.I don't.You are a disgusting lying pig.> If
you had an ounce of honesty or professional integrity,I'd
advise that you look less ridiculous when you don't use such
wordsright after lying publicly.> There are a dozen versions
of the BB, over the past 50 years.One.> To which> version,
speci?ally, are you referring? With in?, or without?
With> light elements or without? With dark matter, or
without? Etc...*THE* Big Bang model describes a bunch of
observations. Whereverpeople leave out certain aspects from
the computations to make themeasier to perform then that
doesn't magically produce a new versionof the model. It
merely means that the results of the computations aregood to
within some approximation.> > Certainly not the
predictions> > (which were 1 part in 100, prior to COBE).>
>> Ah, the predictions.>> Let me submit that you have
no clue what you're talking about.> I ?ured you'd 
dodge the
issue. :)There is no issue in the sentence you posted and
that I quoted thatone could dodge even if one wanted to.You
are making reference to the predictions without any further
cluewhat the fuck you're referring to. Which would have been
quite trivialif you had the slightest idea what you keep
blathering on and onabout.> Let me further submit that COBE
was in perfect agreement with BB> cosmology> LOL!If you
imagine that LOL somehow constitutes an argument, let
meadvise you that you are as wrong on the issue as you are on
allmatters astronomy.If you imagine that there is any kind of
disagreement between COBEdata and Big Bang cosmology, then
the burden of proof is on you to*present* that disagreement.
Bu that would require, of course, thatyou learn for the ?st
time in your life what the term Big Bangactually *means*.>
> and that WMAP has pushed the same agreement an order of>
magnitude further. As Mike Turner tends to say cosmology is
becoming> boring, every measurement only con?ms and re?es
the previous one.> ROTFLMAO!!! Those are pretty major
?re?ings!'Yes. Re?ements. Your pretending that WMAP 
didn't
re?e the COBEresults doesn't somehow change reality and make
it so. COBE WMAPH0 72 +/- 2 +/- 7 70 +/- 4t0 13 +/- 1.5 13.7
+/- 0.3Omega0 1.03 +/- 0.03 1.02 +/- 0.02OmegaB 0.04 +/-
0.008 0.044 +/- 0.004OmegaM 0.33 +/- 0.035 0.27 +/-
0.04OmegaX 0.78 +/- 0.06 0.73 +/- 0.04PWL-n 1.05 +/- 0.09
0.93 +/- 0.03dn/dlnk -0.02 +/- 0.04 -0.03 +/- 0.02WMAP
con?med and re?ed the COBE results. That's it. 
There's
justsimply nothing else to be added. This is the DATA that
you disgustinglying pigs try to lie out of existence..> >
The BB constantly adds new> > epicycles every time
something is observed.>> Nothing new has been added to
the Big Bang since the original FRW> metric was been worked
out unless you're willing to consider in?> to be a
part of the Big Bang Model now in which case noting has
been> added to the Big Bang since in?. That's it.
Everything else> follows from what was there all along.>
ROTFLMAO!!! Do you really believe this?I believe nothing
whatsoever. I examine the evidence and grant the oneor other
theory to be the one most consistent with the
observableuniverse. Belief is not a mode I engage in.
Contrary to you, of course.> How old are you?I have been
doing astronomy professionally when you were nothing but
alustful glint in some hispanic taxi-drivers eye.> And can I
sell> you a bridge?You are making the mistake of projecting
from yourself onto others allthe time. Including here. > It's
just that the BB ?predictions' change (as always) as soon as
another> contradiction is found.If you had the slightest clue
about science, you'd know that aprediction cannot change
after the fact. What is predicted ispredicted. And then we go
and see if we can con?m the predictionobservationally. And
sometimes we can and sometimes we can't. Andsometimes we were
too optimistic in what might be detectable andsometimes we're
too pessimistic.Nobody has a monopoly on the magnitude of
*any* observed effect UNTILit has been observed. At that
point, there's a degree of freedomremoved from a model that
is simply not available any more. That doesnnot change the
model. It does not produce a new version of a model.It
simply puts an observational contraint on one or more
parameters ofthe model.> Expected from the exact same Big
Bang model people referred to before> COBE existed.> LOL!
This is too good to parody.As they say: truth is stranger than
?tion.>> > And while they're at it, they squeeze these
few oddball cases into> > roles they never had: taking an
object who's redshift seems to place> > it outside the
hubble ? pretend that there IS NO hubble ?d> >
the entire theory explaining it must be wrong.> >> > Yes.
That is the inescapable logic of science.>> No, it isn't.
You are merely clueless what you are talking about.>> The
Big Bang makes no statement whatsoever about the velocity of
any> one single object. None. Period. It never has, it
never will. Any odd> object can have any odd velocity with
respoce to any odd other object> (within the observed
constraints of relativity).>> All it does is ADD a small,
(essentially) linar term to the overall> velocities of
objects. At any distance you'll ?d stuff moving> towards
us and other stuff moving away from us, but the farther out>
> you look the more the overall distribution of velocities is
going to> be shifted in the direction *away* from us, by
something like 70 km/s> per Mpc ditance.>> It does NOT
say (has never said, will never say) that all objects at>
1Mpc distance must move away from us at 70km/sec. Merely that
if you> measured all the velocities of objects at 1Mpc
distance that their> velocities would be centered around
70km/s. And that's exactly what is> observed.>> The
radial velocity of any one single object doesn't say *squat*
about> the Big Bang. It merely means that there is one
single object that> happens to move at some velocity
relative to the hubble ?I've always understood your
theory. You didn't in the past and you don't now. 
Otherwise
you wouldn'tcontinue to assert that the redial velocity of a
single observedobject can somehow disprove the Big Bang.You
are merely a disgusting lying pig.But I'm repeating
myself.
===
>> Whatever population of things you measure the
redshift distribution>> of, you ?d an additional redshift
the farther things are away -- by>> whatever means the
distance is determined.>>> But that's the whole point! The
whatever means is always the redshift.[...]>Tens of
millions of astronomical observations ?d themselves
in>perfectly ?e agreement with the Big Bang. NONE ?d
themselves in>contradiction with it. A buncha disgusting
lying pigs tries to worship>a few dozen that are simply hard
to ?ure out.A few dozen anomalies whose distances were
measured by the redshift. Um...-- Let us learn to dream,
gentlemen, then perhaps we shall ?d the truth... But let us
beware of publishing our dreams before they have been put to
the proof by the waking understanding. -- Friedrich August
Kekul.8e
===
... stuff deleted ...>> If you (or Mr. Arp
or>>anybody) want to claim that any two galaxies are in
interaction with>>each other then the burden of proof is on
you (or Mr. Arp or whoever>>makes the claim).> > Precisely.
And Dr. Arp was well on the way to providing just that.
Which> is --> of course -- why Dr. Arp was kicked off
telescopes. True science in the> modern sense. The burden
of proof is on Arp -- so we'll simply remove from> him the
ability to collect proof (data). Problem solved.> Do you have
any evidence that (1) Arp is not allowed to use a telescope
today. (2) [if 1 is in fact correct] the reason is as cynical
as you are making it out to be. ... the rest deleted ...I
looked brie? found no indication that (1) is correct. I
alsodid not ?d any indication that (1) is incorrect.You
apparently have some ?st-hand information, and it would
beworthwhile (to this reader) to see it presented.> -->
greywolf42> ubi dubium ibi libertas> Dale.
===
> > For
example the tens of millions of measured redshifts that
agree> > perfectly with the Big Bang model. That's DATA
for you, in capital> > letters.> >> > ROTFLMAO! How
do they agree?>> They ?d that objects which are more
redshifted tend to be farther> away.>> Thats how.> 1)
Assume objects with larger redshift are farther away.> 2)
Claim that observing larger redshift proves it is farther
away.> That's one of the shortest circular arguments 
I've
seen in awhile.Why is this so dif?ult for you?1) Measure
redshift of object.2) Measure distance to object.3) Notice
that larger distances correlate linearly withlarger
redshifts.Where does Assume objects with larger redshift are
fartheraway come into what you read above? Where do you think
hesaid that?You measure the redshift. You measure the
distance. Theseare two measurements. There are no
assumptions. You noticethat large values of x go with large
values of y. Where'sthe circle?I know you're not an 
idiot. So
where do you see a circularargument in a description of a
correlation process? - Randy
<3c65f87.0311181923.13b10b42@posting.google.com>
<3c65f87.0311200627.6edb705a@posting.google.com>
<df160b8f.0311211700.3336813@posting.google.com>
<vrvhrq1u8o3l73@corp.supernews.com>
<df160b8f.0311241043.1bc30cd4@posting.google.com>
<vs7bc6rl043447@corp.supernews.com>
<df76407e.0311251339.6f1665a3@posting.google.com>
===
> Why is
this so dif?ult for you?>> 1) Measure redshift of object.> 2)
Measure distance to object.> 3) Notice that larger distances
correlate linearly with> larger redshifts.>> Where does
Assume objects with larger redshift are farther> away come
into what you read above? Where do you think he> said that?In
all fairness, surely it's true that measuring distance only
worksfor relatively near objects. I assume (but don't know)
that we canindependently measure the distance of a minority
of objects whoseredshifts we can measure. For the remainder
of the objects, I supposewe have no recourse but to infer
their distance from their redshifts.It is this collection of
facts that leads to the wrong conclusion thatredshift is the
only measure of distance we have, and the correlationbetween
redshift and distance is circular. It's similar to thealleged
circularity one sees in some arguments regarding the
fossilrecord: The only measure of the age of the rocks are
the fossils foundthere and the age of the rocks give the age
of the fossils.-- Jesse Hughes If you really think there's a
bug you should report a bug. Maybeyou're not using it
properly... It turns out Luddites don't know howto use
software properly, so you should look into that. -- Bill
Gates
===
> Why is this so dif?ult for you?>> 1) Measure
redshift of object.>> 2) Measure distance to object.>> 3)
Notice that larger distances correlate linearly with>> larger
redshifts.>> Where does Assume objects with larger redshift
are farther>> away come into what you read above? Where do
you think he>> said that?>>In all fairness, surely it's true
that measuring distance only works>for relatively near
objects. I assume (but don't know) that we can>independently
measure the distance of a minority of objects whose>redshifts
we can measure. For the remainder of the objects, I suppose>we
have no recourse but to infer their distance from their
redshifts.I'm no expert, but this seems to understate the
case from what I read.I posted a bunch of links about
measuring distance elsewhere in thisthread. Parallax can be
used close by. Cepheid variables can be usedout to extreme
distances, as far as we can see. In between there are anumber
of options, including stellar brightness, that are
apparentlywell calibrated out to many thousands of parsecs.So
I don't think it's the case that the number of points 
in a
HubbleLaw validation study is a few dozen or small. The one
study I foundonline mentioned 800 Cepheid variables, and that
was just one studyfocusing on extreme ranges. - Randy
===
>
Why is this so dif?ult for you?>> 1) Measure redshift of
object.>> 2) Measure distance to object.>> 3) Notice that
larger distances correlate linearly with>> larger
redshifts.>> Where does Assume objects with larger
redshift are farther>> away come into what you read above?
Where do you think he>> said that?>>In all fairness, surely
it's true that measuring distance only works>for relatively
near objects. I assume (but don't know) that we
can>independently measure the distance of a minority of
objects whose>redshifts we can measure. For the remainder of
the objects, I suppose>we have no recourse but to infer their
distance from their redshifts.Sort of, buy there's a little
more to it, because of a lot of different ways to measure
distance. I imagine everyone agrees thatparallax (noting the
difference in apparent position of a star at different places
in the Earth's orbit around the sun and deducing)the distance
from that) gives a reliable measure of distance. Butit only
works nearby. But then there are those Cepheid variable
things. You look at nearbyCepheid variables, where you can
measure absolute brightness bymeasuring the distance using
parallax, and you note I think it'sa relation between period
and absolute brightness. You notice thatthis relation holds
for nearby nearby Cepheids and also for farawaynearby ones,
so you assume that it also holds for faraway ones,and then
you start measuring distance by comparing period
andapparent brightness. That's not guaranteed to work, but
nothingin science is guaranteed - since you can't give any
reasonwhy the relation should be different for faraway stars
it seemsreasonable to assume the relation is the same until
there's somereason not to.Then you notice that the redshift
is correlated with the distanceas measured by the period
and apparent brightness, in exactlythe way it would be if in
fact the Cepheid distance measurementswere accurate and
farther objects were moving away faster. At thispoint the
idea that the universe is expanding is just the simplestway
to explain the data - otherwise we need an explanation for
whythe difference in the way faraway Cepheids works is in
sync withthe redshift that way...[Then someone has problems
with noise in a radio telescope (orradar or something) and
someone notices that it's somethingthat had been predicted to
exist as consequence of the Big Bang.People work out how far
from perfectly isotropic the noise shouldbe, and recently
when they do the Fourier analysis on datameasured by some lah
dee dah satellite they ?d it's exactlyas the model predicts
(except for one detail about a missinglow harmonic, which
people are speculating could mean theuniverse is compact.) So
of course at that point it's perfectlynatural to decide that
there's no basis for any of this whatever,it's just 
stuff
people believe because it's fashionable...]>It is this
collection of facts that leads to the wrong conclusion
that>redshift is the only measure of distance we have, and
the correlation>between redshift and distance is circular.
It's similar to the>alleged circularity one sees in some
arguments regarding the fossil>record: The only measure of
the age of the rocks are the fossils found>there and the age
of the rocks give the age of the fossils.>-- >Jesse Hughes
>If you really think there's a bug you should report a bug.
Maybe>you're not using it properly... It turns out Luddites
don't know how>to use software properly, so you should look
into that. -- Bill GatesDavid C. Ullrich
===
>>You know I've
had two extremely contentious discussions with Randy >>Only
two?It just seems like more. As I recall they both concerned
the correctinterpretation of dL/dt and ran on and on and on.
There might havebeen one other but I can't remember the
subject.>and>>I've never seen him lie. The most one might
say is that he - like all>>- makes occasional mistakes or
misjudges the relevance of issues>>What? Why, I oughta <pant>
<snort>...> and>>even gets irritated once in a while. But a
lie is a purposeful>>manipulative misstatement of truth. You
might consider getting your>>barbs and facts at least a
little straighter before you shoot them.>>Well, thanks. I
guess. Since our relationship has been as you
say>contentious, I'll take this as high praise. I do indeed
get irritated.>There are two hot buttons for me: (1)
something I perceive as massive>illogic (we will of course
disagree as to what constitutes illogic,>and (2) being
accused of lying.>You're welcome I guess. My primary hot
button is being accused of badfaith in presenting or
analyzing issues whether right or wrong. Istill haven't
forgotten the arguments you advanced in support of
yourevaluation of the directon of dr which I still need to
understand andexplain to my own satisfaction (and no I'm not
trying to resurrect theissue here). I've never really
understood why people can't justdiscuss even contentious
issues without being personally abusive andderogatory.
===
>
That he may have made errors doesn't change the fact that
Galileo performed> experiments (with balls on inclined
planes). And ?mechanics' is a> theoretical practice -- not an
experimental one. {Galileo didn't drop> weights from the Tower
of Pisa -- that was a different guy.}Galileo performed
experiments indeed. But the oft-forgotten thing isthat so did
everyone else.Copernicus gets scorn in my book. He was right,
but only by beingwhatsoever to recommend it. Kepler's at
least explained theexperimental observations better than
Ptolemy, for the ?st timeever. It was only until Newton that
the principal objections to theheliocentric model were
adequately explained.Thomas
===
>> That he may have made
errors doesn't change the fact that Galileoperformed>
experiments (with balls on inclined planes). And ?mechanics'
is a> theoretical practice -- not an experimental one.
{Galileo didn't drop> weights from the Tower of Pisa --
that was a different guy.}>> Galileo performed experiments
indeed. But the oft-forgotten thing is> that so did everyone
else.Not everyone else. Most of the hierarchy simply read
Aristotle.> Copernicus gets scorn in my book. He was right,
but only by being> whatsoever to recommend it.Perhaps just
better-read. The Greeks knew about the Earth going around
theSun in 400 B.C.> Kepler's at least explained the>
experimental observations better than Ptolemy, for the ?st
time> ever. It was only until Newton that the principal
objections to the> heliocentric model were adequately
explained.Actually, the principle objections were not
resolved until parallax was?st measured. As the lack of
observable parallax (a requirement of anySun-centered system)
was the major objection.--greywolf42ubi dubium ibi
libertas
===
what's irrelevant is the profundity (or not)of
Arp's research (and the anti-BBers) to Jimi Harris' 
math. >
> Where do you think Hubble's Law came from in the ?st
place?> >> > Not perfect. But a damned good ?. >
Motions, not orbits. But yes, I understand the assumptions
behind the> hubble law.> > But check out the second
graph.>>
http://www.physics.utah.edu/~burko/courses/Phys3740/notes/
hubble.html> Fascinating. This appears to be the supernova
data ... chopped off, I do> believe. I believe the data goes
much farther out -- to 2 or 3 billion> parsecs. And shows a
clear deviation from linear -- destroying the linear>
?hubble law'. (i.e. dark energy)> This is especially
egregious because the author claimed that some had> proposed
quadratic redshift-distance laws have been ?disproved' due
to> the supernova data. This is (politely) called data
selection. Less> politely, it's called deliberate distortion
of the evidence. > We have exactly zero direct measurements of
quasar distances. Regardless of> what numbers we have of
anything else.> > See this page, in the section titled
Determining> H0.>>
http://homepages.tcp.co.uk/~carling/astrocos.html>> Out
to 50 kpc we can use stars. We know how bright they> should
be from stellar evolution theory.> We can assume this. But
it's not a direct measurement. And the brightness> of these
simulations was originally calibrated by assuming the Hubble>
cepheids, high mass and low metallicity stars.> > We use the
brightness> of the star as a distance measure. How many
stars have we> observed within 50000 parsecs? Isn't that
most of the> Milky Way galaxy?--ils duces d'Enron!
===
> My
point is that my discovery is PURE MATH, and PURE KNOWLEDGE,>
representing a previously unknown formula for both ?ding and>
counting primes.Why should I believe you? Here's a previously
unknown formula:283490823481290348 + 1987234817239472345 =
2270725640720762693Is it very interesting? No.Mine is
certainly previously unknown: I'll bet anyone $100 that
nobodyhas ever before published that addition. But it's not
veryinteresting, for two reasons:1) Knowing that addition
isn't very important, and2) Anyone could reproduce the
formula very easily if they needed it.Similarly, methods of
creating ?ite difference equations are verywell known. What
is missing in your previously unknown formula is:1) An
explanation of why it is important--for example, does it
enable faster computation of primes? (no); does it make for
faster factorizations? (no). Did you know there is a
diophantine equation whose roots are exactly the set of
primes? I could tweak that equation in a million ways and
produce a previously unknown formula for primes, but that's
hardly very interesting.2) Your methods: where did your
equation come from? How can we apply the techniques you used
in other cases for other problems? Thomas
<51566c78.0311242033.6cbb94e0@posting.google.com>
<bpuogv$d0r$1@lust.ihug.co.nz>
<PuOwb.325$b77.281@dfw-service2.ext.raytheon.com>
===
>
Reminds me of Lenny Bruce's bit about if, in a movie, a man
approaches a > woman with a pillow, he would rather see him
put it under her rear than > over her face.Or George Carlin's
bit in which he imagines replacing the world killwith the
word fuck in classic movies.-- Jesse F. HughesReally, I'm
not out to destroy Microsoft. That will just be acompletely
unintentional side effect. -- Linus Torvalds
===
These
programs use simulated evolution to create arti?ial
neuralnetworks to solve speci? problems. They are well
documented and arein standard C with all source included.
They will compile and run withall the common compilers and
OS's. They can be downloaded at no
chargefrom:http://annevolve.sourceforge.netMitchell Timin--
Many are stubborn in pursuit of the path they have chosen,
few inpursuit of the goal. - Friedrich
Nietzschehttp://annevolve.sourceforge.net is what I'm into
nowadays.Humans may write to me at this address: zenguy at
shaw dot ca <bohtjh$1em590$1@ID-206850.news.uni-berlin.de>
<3c65f87.0311080513.159f0484@posting.google.com>
<Z4KdnefKqaUznDCiRVn-sA@comcast.com>
<c37480a7.0311250244.a756f96@posting.google.com>
<r7KdnaCbKpkA116iRVn-hA@comcast.com>
<c37480a7.0311251240.16f90f37@posting.google.com>
===
>> Let's
be careful with our selection of terms. JSH didn't offer a
proof of>> FLT, he offered an incorrect argument in an
attempt to prove FLT.>>> Justin Van Winkle>>> > Somehow
I doubt it. Even if your proof were to be found to be
entirely>> > correct, nobody would care. Why? Because the
thing you assert is>> > completely irrelevent. Why don't
you know this? Because you don't know>> any>> >
mathematics. You don't know any mathematics.>> > What you
say is true, for a reason that reveals the *ugliness*>> of
mathematics--and the *ugliness* of mathies. Mathies>> would
turn up their nose at an elementary proof of FLT>> by JSH
but not (say) an elementary proof of FLT by Harvey>>
Friedman, because JSH is an Afro-American without formal>>
training in mathematics--unlike Harvey Friedman, who is
white,>> has a Ph.D, and plays the game as mathies require
it to be>> played. THAT is why a proof of FLT by JSH would
be irrelevant.>> And THAT is why everything JSH says about
professional>> mathematicians is true.>> > Seriously,
think about this.>> > --John> Mathies WOULD turn up
their nose at an elementary proof of FLT> by JSH but not
(say) an elementary proof of FLT by Harvey> Friedman>>
and,>> THAT is why a proof of FLT by JSH WOULD be
irrelevant.>> in response to:>> Even if your proof WERE to
be found to be entirely> correct, nobody WOULD care>>
Refrain from attributing to me words not mine, or reading
what is not there> into things that I say.Yes, you said
those things. Speci?ally, you said that the relevantfacts
are that Friedman is white and has a PhD and plays the
gameand that JSH is black and has no formal training and
presumablydoesn't play the correct game. Unfortunately you
have no evidencethat this ugliness actually exists. I would
be very shocked if Friedman came up with a proof for
FLT,since that isn't his bag as far as I know, but 
that's not
reallyrelevant. Everyone here agrees that if a reputable
mathematician cameup with a proof of FLT, it would be
recognized -- at least reasonablysoon. The dispute is
whether, if someone like JSH did the same, itwould be
recognized.Sadly, no one like JSH has come up with such a
proof so the hypothesisis untested. I know that James thinks
otherwise, but *no one else*has ever said otherwise as far as
I know, yourself included. Maybeyou ought to follow James's
lead and switch to his prime countingthing. Everyone agrees
that it works, at least. Now we can debate(well, not *we*,
but you and others) whether it is ignored because (1)James is
an iconoclast or (b) it is old or a slight variation
onsomething old.I know you think James is a genius, but so
far your only evidence thathe is a genius is that people that
you don't like reject his work.Oh, and also you don't
understand his work. I was never clear: isthe fact that you
don't get it part of your evidence he's a genius?Or is 
that
just a convenient reason to claim he's a genius withouthaving
to explain why?To be fair, maybe you got the impression he's a
genius simply from thenumber of times that claim has been
repeated here. There is a smallbody of posters (previously
one, now two) that seem fond of thatfact.-- Conservative,
n: A statesman who is enamored of existing evils, as
distinguished from the Liberal who wishes to replace them
with others. -- Ambrose Bierce
===
grade of a
mathematics-oriented high-school in Croatia and would like
you tohelp me with an assignment conjured up by my teacher
and put into our schooltest: Prove or disprove the
irrationality of 2^pi.
===
> grade of a mathematics-oriented
high-school in Croatia and would like you to> help me with
an assignment conjured up by my teacher and put into our
school> test:> Prove or disprove the irrationality of
2^pi.> There's a result called the Gelfond-Schneider theorem
that says if ais algebraic and not equal to 0 or 1, and b is
irrational then everyvalue of a^b is transcendental.
Therefore 2^pi is transcendental. Thistheorem is proved in
Chap. 8 of A. H. Rose, A Course in NumberTheory.
===
>grade
of a mathematics-oriented high-school in Croatia and would
like you to>help me with an assignment conjured up by my
teacher and put into our school>test:> Prove or disprove the
irrationality of 2^pi.This is de?itely not a suitable problem
for high-school students. I think it's unsolved. On the other
hand, it follows from a result of Lang that there are at most
two primes p such that p^pi is algebraic (see Baker,
Transcendental Number Theory, Theorem 12.3). And from
thesame result, at least one of 2^pi, 2^(pi^2), 2^(pi^3) is
transcendental.Robert Israel israel@math.ubc.caDepartment of
Mathematics http://www.math.ubc.ca/~israel University of
British Columbia Vancouver, BC, Canada V6T 1Z2
===
> grade of
a mathematics-oriented high-school in Croatia and would like
you to> help me with an assignment conjured up by my teacher
and put into our school> test:> Prove or disprove the
irrationality of 2^pi.> Let me see if I understand this. You
want a reader of sci.math to do the problem for you, so you
can hand it in as your own work, and get credit for it. Have
I got that right? Is there nothing about that that strikes
you as just a teeny-weeny bit, what's the word I'm 
looking
for, dishonest? Or am I missing something? For what it's
worth, this looks like a very hard problem to me, perhaps
even an unsolved problem.--
===
> Let me see if I understand
this.Permission granted.Right.> You want a reader of sci.math
to do the problem for you,> so you can hand it in as your own
work, and get credit for it.Wrong.> Have I got that
right?No.> Is there nothing about that that strikes you as
just a teeny-weeny bit,> what's the word I'm looking 
for,
dishonest?I won't even dignify this weak sarcastic remark
with an answer.> Or am I missing something?Try everything.>
For what it's worth, this looks like a very hard problem to
me,> perhaps even an unsolved problem.Exactly why I posted
it. The test is over, I got an A in it, but I stillwonder
whether it can be solved as I cannot solve it myself (and my
teacherwon't help me in attaining the solution).
===
> they
radiate. The faster they go the greater the radiated energy
per> unit time. String a group of them along a line and you
now have a> current, and a current produces a magnetic ?ld
in surrounding space.> Energy is stored in this ?ld. Thus
there are two reasons that charged> only two reasons. At c
(wrt lab frame) the electron cannot be further> accelerated,
not because the associated magnetic ?ld contains in?ite>
disappears at this speed, i.e. all of the virtual photons
inducing the> this speed, simply because the source of those
photons is the> accelerator itself, which is at rest in lab
frame. OTOH an in?ite> quantity of energy input per unit
time into the accelerator is required> terminal velocity, all
of the input energy thereafter is wasted, not one> analogous
to that of a skydiver. The special relativistic prediction
is> incorrect, and despite the propaganda, that prediction
has never been> veri?d. The collision energy has never been
directly measured, it has> only been force ? to the various
?perceived' byproducts of the> collision, all of which are
based upon ad hoc conjectures which are> the shaman, and this
proves only that the general public is at an> evolutionary
standstill. Morons leading morons, they'll both fall into>
the ditch. Best to suspend judgment, this is the high road,
there are> only ditches on either side.> Richard Perry>
magnetic ?ld and emit photons by accelerating, then how can
we knowwhether their observed behavior is really due to
relativistic mass orjust an electromagnetic ?ld effect?Seems
like the only way to tell is if we could ?ure out a way
toDouble-A
===
> > they radiate. The faster they go the
greater the radiated energy per> unit time. String a group
of them along a line and you now have a> current, and a
current produces a magnetic ?ld in surrounding space.>
Energy is stored in this ?ld. Thus there are two reasons
that charged> only two reasons. At c (wrt lab frame) the
electron cannot be further> accelerated, not because the
associated magnetic ?ld contains in?ite> disappears at
this speed, i.e. all of the virtual photons inducing the>
this speed, simply because the source of those photons is
the> accelerator itself, which is at rest in lab frame.
OTOH an in?ite> quantity of energy input per unit time
into the accelerator is required> terminal velocity, all of
the input energy thereafter is wasted, not one> analogous
to that of a skydiver. The special relativistic prediction
is> incorrect, and despite the propaganda, that prediction
has never been> veri?d. The collision energy has never
been directly measured, it has> only been force ? to the
various ?perceived' byproducts of the> collision, all of
which are based upon ad hoc conjectures which are> the
shaman, and this proves only that the general public is at
an> evolutionary standstill. Morons leading morons, they'll
both fall into> the ditch. Best to suspend judgment, this is
the high road, there are> only ditches on either side.>>
Richard Perry>> magnetic ?ld and emit photons by
accelerating, then how can we know> whether their observed
behavior is really due to relativistic mass or> just an
electromagnetic ?ld effect?> Seems like the only way to
tell is if we could ?ure out a way to> Double-Aspace,
matter, and ?ld are interchangeable terms. Richard
Perry
===
>>>>> Is the force on a moving charge equal to
qE?>> How do you know?>>Experimental evidence.>> That is
true only if you assume that its mass effectively increases by
gamma.Nonsense. It is experimentally veri?d, assuming
nothing.>As you know. You even claim to never have said
otherwise.Because you have claimed to accept that the
electrongains the same amount of energy every time it passes
through>E = Fs = qV = qEs s = distance between
electrodes.Sorry, ambiguous!The ?st E is energy, the second
is electric ?ld.> I have never used those expressions here.>
I did not state that.of the fact that the electron gains the
same amount ofenergy every time it passes through the
RF-cavity.Gained energy is force times distance, Fs.This
energy is independent of the speed, thusmust the force F be
independent of the speed.The energy gained by a charge going
througa potential drop is qV, the electric ?ld E = V/sqV =
qEs = FsThus F = qE independent of the speed,which answers
your question.Got it now?Paul
===
>>> >> Is the
force on a moving charge equal to qE?>>> How do you know?>>
>Experimental evidence.>> That is true only if you
assume that its mass effectively increases by
gamma.>>Nonsense. It is experimentally veri?d, assuming
nothing.It depends what you call ?Kinetic energy'.>As you
know. You even claim to never have said otherwise.>>Because
you have claimed to accept that the electron>gains the same
amount of energy every time it passes throughDon't
misrepresent me Paul.I only agreed that in the steady state
condition, that was true. >E = Fs = qV = qEs s = distance
between electrodes.>>Sorry, ambiguous!>The ?st E is energy,
the second is electric ?ld.Yes I understood that. For a
moving charge, I claim it could be wrong.> I have never
used those expressions here.>> I did not state that.>>of the
fact that the electron gains the same amount of>energy every
time it passes through the RF-cavity.>>Gained energy is force
times distance, Fs.>This energy is independent of the speed,
thus>must the force F be independent of the speed.>The energy
gained by a charge going throug>a potential drop is qV, the
electric ?ld E = V/s>qV = qEs = Fs>Thus F = qE independent
of the speed,>which answers your question.>>Got it now?Well,
that is probably correct - but it isn't really the main
issue. What Iwant to know is ?where that energy goes'. You
say it all goes into KE but onlyif you include the
(effective) mass increase.I say the ?effective mass increase'
is not that at all. It is a reactionagainst the applied ?ld.
It obviously requires a lot of energy to create areverse
?ld ?bubble' inside an applied ?ld.Note: relativity
doeesn't offer any physical explanation for the apparent
massincrease. It simply accepts a seemingly correct equation
derived from boguspostulates.>>Paul>Henri Wilson. See the
Stupidity of
Relativity.www.users.bigpond.com/hewn/index.htm
===
>>>
>>>> >>>Yes.>>>per cycle in the
RF-cavities regardless of the speed.>>>But it will
loose more and more energy per cycle in>>>the bends as
the speed increases.>>>When the two are equal, the
accelerator is in steady state.>>>when it is going at
peak ef?iency.>> >>> gaps. During the rest of their
travels they lose a little speed.>> >>> However the
question is not about the steady state condition - in which
the>>> input energy goes into radiation. It is why
increasingly more energy is>>> balance radiation.>>>
>>The answer is that Nature tells us that the KE is:
m*(gamma-1)*c^2> >> And does a ?change in KE' involve a
term with ?dm/dt' in it?>>In the same way as d/dt(C*x)
where C is constant>involve a term with dC/dt in it.>>>
>>>Thus the gained energy does NOT decrease when the
speed>>>>of the electrons approaches c.>>>>>
>>>>So whatever you think happens to the ?ld in the
RF-cavities>>>>when the speed approaches c, we KNOW
for certain that>>> >>>> But where does that
energy go?>>>>>>Into kinetic energy, of course.>
>>>Einstein says: KE = m*c^2*(((1/sqrt(1 - v^2/c^2)) -
1)>>>Newton says: KE = 0.5*m*v^2>>>>>>Why
do you ?d the second of these equations more natural>>
>>than the ?st?>>>Two different theories, the ?st
is experimentally con?med,>>>the second is
experimentally falsi?d.>>>>You didn't answer this
crucial question, Henry.>>Why are you willing to accept
that KE = 0.5*m*v^2>>as nature tells us is wrong,>>
>and NOT willing to accept KE = m*c^2*(((1/sqrt(1 - v^2/c^2))
- 1)>>as Nature tells us is correct?> >> I do accept
that equation for moving charges Paul. You know I have never
denied>> that is obeys the experimental evidence.>> My
argument is that the SRian idea of a ?mass increase' is not
really a mass>> increase at all but an illusion caused by
a reduction in the force on a moving>> charge due to a
?ld.>>There is no idea of mass increase in SR.>>> I
will go further and speculate that mass is nothing but a
property somehow>> associated with ?lds.>>Henry
Wilson seems to claim that the mass increases, though.>>>
>> Once again it is not the KE itself we have to worry about
but changes in KE.>>> Once again you get a term with
?v.dm/dt'.>>>>No you don't.>>You obviously 
didn't
read what I said!>>The momentum is m*gamma*v where m is
invariant.> >> Once known as ?rest mass'?>>Righ. Or
simply mass.>>>so dp/dt = m*d/dt(gamma*v)> >>
|>You have screwed this up before, and I have shown>> |>you
the correct equation before.>> |>>> |>F = dp/dt = d/dt
(m*gamma*v)>> |>gamma =1/sqrt(1 - v^2/c^2)>> |>>>
|>dv/dt = (1/((v^2/c^2)*gamma^3 + gamma))*F/m>> |>>>
|>((v^2/c^2)*gamma^3 + gamma)*dv = (F/m)*dt>> |>gamma*v =
(F/m)*t + C> >> Since you obviously copied this out of a
book, you didn't see the intermediate>> steps that
included ?v.dm/dt'>>I didn't copy this from anywhere,
Henry.>You are babling.>F = dp/dt = d/dt (m*gamma*v)>F
= m*d/dt(gamma*v)>F = m*((v^2/c^2)*gamma^3 + gamma))*dv/dt>
>dv/dt = (1/((v^2/c^2)*gamma^3 + gamma))*F/m>>Where is the
intermediate step that included v*dm/dt ?>m is a constant
and is treated as such.>> F=dp/dt=d(mv)/dt=m.dv/dt+v.dm/dt...
!!!!!!!!!!!!!!>> Let m=Mo.gamma> So dm/dt=Mo.d(gamma)/dt>>
Then F=Mo.gamma(dv/dt)+v.Mo.d(gamma)/dt> or
F/Mo=gamma(dv/dt)+v.d(gamma)/dt>> Now
d(gamma)/dt=(dv/dt).v.(gamma^3)/(c^2)>> Therefore
dv/dt=(F/Mo)/[(v/c)^2)gamma^3+gamma]>> ....get it?Of course I
get it.This is what I said many postings ago.IF the momentum
is mv, then the mass must increase.BUT MOMENTUM IT IS NOT
mv!According to SR, it is p = m*v*gamma, m = Mo = invariantSo
there isn't any term containing dm/dt in dp/dt.... get it?>>
>The KE is m*(gamma-1)*c^2 where m and c are invariant.>>
>so dE/dt = m*c^2*d/dt(gamma)=m*c^2*gamma^3*v*dv/dt> >>
Yes Paul, you are telling me what is observed.>> I AM
TRYING TO TELL YOU WHERE THE EXTRA ENERGY MIGHT BE STORED.>
>> Do you think the fairies take it?>>Extra energy
relative to what?>Does 0.5*m*v^2 explain where the energy
is stored?>Is there fairies involved here?>SR doesn't
explain where the energy is stored any more than NM does.>
>The theories predicts different energies, neither explains
it.>SR is right, NM is wrong.>That's all.>> NM correctly
says that the KE is (1/2mv^2), where m is constant and v is>
measured in the frame of the ?ld.In the frame of which
?ld?This isn't anything to discuss, Henry.> to get it to
velocity v.Kinetic energy is per de?ition the energy we have
to pump intoAnd you know what that energy is.> stored? In
the ?ld of course.Now you are talking nonsense.Of course NO
kinetic energy is instrinsic.You MUST know that, Henry!>>
>>> That's not what NATURE tells us. That's what SR
says.>>>>That's what Nature through experiments tells
us.>>That SR says the same is why SR isn't falsi?d.>>
>That NM says otherwise is why NM is falsi?d.> >> NM is
perfectly capable of handling this.> >> Simple!>>
>Your only problem is that when we are discussing how much>
>energy is gained in the RF-cavity, then you have to claim>
>that you NEVER stated what you just did. :-)>> I don't see
the connection.Henry, don't tell me you still haven't 
got
this!The gained energy is Fs. It is the same regardlessSo F
can blatantly obvious not decrease when the speed>>>Why
do you have a problem with that?>>>Why not simply accept
it?>> >>> Accepting it means virtually nothing. What's
the point?>>>>This illustrates your problem.>>You
refuse to accept that Nature works as it actually does.>>
>And again I must ask:>>Why don't you say that accepting
KE = 0.5*m*v^2>>means vertually nothing. What's the
point?> >> What is ?v'? That is the point.>>You
didn't answer, Henry.>I ask you why you think KE =
0.5*m*v^2 explains>more than KE = m*(gamma-1)*c^2.>
>What's the big difference?>> I told you. 1/2mv^2 is the
kinetic energy. The additional energy is stored in> the ?ld
?bubble' around the moving charge.You didn't answer, 
Henry.You
just keep asserting that KE is 0.5*m*v^2 and NOT
m*(gamma-1)*c^2.WHY do you claim that KE = 0.5*m*v^2?Do you
know it a priori?Do you know it because it was Newton and not
Einstein who said so?Is it a religious conviction you will
believe blindly despite of anything?>>>>Which proves
you WRONG.>>>>The radiation from an accelerated
charge!>>>>or the ?lds associated with a moving
charge!>>>>or The ?Back EMF' concept.>>>>
>any less when the speed approaches c.>>> >>>>
that does not con?ith what I said.>>>>>>Uh?
:-)>> >>> I repeat, THAT DOES NOT CONFLICT WITH WHAT
I SAID.>>> My argument is about what you claim is an
apparent, ?mass increase'.>>>>You were talking
about:>>radiation from an acceleraed charge!,>>
>which ONLY is relevant when the charge is accelerated,>>
>and: The ?Back EMF' concept. which only is relevant>>
>when there is an EMF (electic ?ld).>>And now you are
insisting that you were NOT talking>>about what happens
in the acceleration part, but how>>accelerated in an
electric ?ld?> >> Paul, if you cannot see what I am
getting at by now I don't think there is any>> hope for
you at all.> >> My argument is about where the additional
energy is stored, when a charged> >> You say it manifests
itself as a mass increase (although you try to disguise>>
the fact)>> I agree that it appears just like a mass
increase. I also claim that it is NOT>> RELATED to a mass
increase but that the energy is stored somehow in the ?lds>
>>Why don't you need enigmatic ?lds to store the KE in
0.5*m*v^2?>> Because ?0.5*m*v^2' is the correct expression
for amount of energy associatedIs it? Why?Both 0.5*m*v^2 and
m*(gamma-1)*c^2 are in accordance withexperimental evidence
when v << c, while only the latter is inaccordance with
experimental evidence when v is big enoughfor the two
equations to give a measurable different prediction.So why do
you claim that the former is the correct expression>>>>
Can you not see that the implications of being able to
eliminate ?mass' from>> the equations are considerable. We
might be able to actually learn something>> about the
nature of mass if we look beyond a simple maths formula.>>
>So you are actually saying that the KE is stored in
enigmatic ?lds>even when c << v? That is the KE =
0.5*m*v^2 is stored in ?lds?>> Not directly. but indirectly
maybe. I reckon all mass is explainable in terms> of ??lds'
whatever they are.May well be. But it is irreleavant to our
issue.I have never seen you have a problem with the KE of low
speedbodies, and insited that the KE must be stored in some
kind of?ld bubbles. Quite the contrary. You have claimed
that it isonly the _differemce_ between m*(gamma-1)*c^2 and
0.5*m*v^2that is stored in these bubbles. So where is the
0.5*m*v^2 stored?>>That doesn't make sense.>>>>
>You are of course free to speculate where the KE>>
>Consider the consequence of that!> >> I am considering
the consequences of far more than that.>> I used to think
that ??lds were made of matter'. Now I think that all
matter>> must be made of ??lds'.>> These ?lds don't
want to move relative to other ?lds, hence the concepts of>
>> ?mass' and ?inertia'.> >> All we need now is an
understanding of what a ??ld' is.>>Good luck.>You
should look at the standard model.> Do you believe in
?solid matter'?>> Anything ?solid' must be made of 
something
smaller...and on and on....>> Where does that end?There are
turtles all the way down, of course.Paul
===
>>>>so
dp/dt = m*d/dt(gamma*v)>>>>> |>You have screwed
this up before, and I have shown>>> |>you the correct
equation before.>>> |>> |>F = dp/dt = d/dt
(m*gamma*v)>>> |>gamma =1/sqrt(1 - v^2/c^2)>>> |>>
|>dv/dt = (1/((v^2/c^2)*gamma^3 + gamma))*F/m>>> |>>
|>((v^2/c^2)*gamma^3 + gamma)*dv = (F/m)*dt>>> |>gamma*v =
(F/m)*t + C>>>> Since you obviously copied this out of
a book, you didn't see the intermediate>>> steps that
included ?v.dm/dt'>> >I didn't copy this from 
anywhere,
Henry.>>You are babling.>>F = dp/dt = d/dt (m*gamma*v)>>
>F = m*d/dt(gamma*v)>>F = m*((v^2/c^2)*gamma^3 +
gamma))*dv/dt>>dv/dt = (1/((v^2/c^2)*gamma^3 +
gamma))*F/m>> >Where is the intermediate step that
included v*dm/dt ?>>m is a constant and is treated as
such.>> F=dp/dt=d(mv)/dt=m.dv/dt+v.dm/dt...
!!!!!!!!!!!!!!>> Let m=Mo.gamma>> So
dm/dt=Mo.d(gamma)/dt>> Then
F=Mo.gamma(dv/dt)+v.Mo.d(gamma)/dt>> or
F/Mo=gamma(dv/dt)+v.d(gamma)/dt>> Now
d(gamma)/dt=(dv/dt).v.(gamma^3)/(c^2)>> Therefore
dv/dt=(F/Mo)/[(v/c)^2)gamma^3+gamma]>> ....get it?>>Of
course I get it.>This is what I said many postings ago.>IF
the momentum is mv, then the mass must increase.>BUT MOMENTUM
IT IS NOT mv!>>According to SR, it is p = m*v*gamma, m = Mo =
invariant>So there isn't any term containing dm/dt in
dp/dtThat's just a smartarse way of expressing the Newtonian
equation for anaccelerating mass that is increasing in
size.Only a real fanatic would fall for it.>>.... get it?>
>>The KE is m*(gamma-1)*c^2 where m and c are invariant.>>
>>so dE/dt = m*c^2*d/dt(gamma)=m*c^2*gamma^3*v*dv/dt>>>
>> Yes Paul, you are telling me what is observed.>>> I AM
TRYING TO TELL YOU WHERE THE EXTRA ENERGY MIGHT BE STORED.>>
>>> Do you think the fairies take it?>> >Extra energy
relative to what?>>Does 0.5*m*v^2 explain where the energy
is stored?>>Is there fairies involved here?>>SR doesn't
explain where the energy is stored any more than NM
does.>>The theories predicts different energies, neither
explains it.>>SR is right, NM is wrong.>>That's
all.>> NM correctly says that the KE is (1/2mv^2), where m
is constant and v is>> measured in the frame of the ?ld.>>In
the frame of which ?ld?>This isn't anything to discuss,
Henry.The Newtonian explanation was dropped after the
Einsteinian red herring wasspotted.If NM is extended to
include the Wilsonian ?reverse ?ld bubble effect', itwill
be quite adequate. > to get it to velocity v.>>Kinetic
energy is per de?ition the energy we have to pump into>And
you know what that energy is.As far as I'm concerned,
KE=0.5.Mo.v^2 at all speeds, (relative to the originalrest
frame). The additional energy used to bring the charge to the
speed shouldnot be included in KE.> stored? In the ?ld of
course.>>Now you are talking nonsense.>Of course NO kinetic
energy is instrinsic.>You MUST know that, Henry!All right,
?intrinsic' might not have been the best word. I think you
know whatI meant though.Please reveal to us your own
physical theory as to where the excess energy isstored
Paul.> >Your only problem is that when we are
discussing how much>>energy is gained in the RF-cavity,
then you have to claim>>that you NEVER stated what you just
did. :-)>> I don't see the connection.>>Henry, don't 
tell me
you still haven't got this!>The gained energy is Fs. It is the
same regardless>So F can blatantly obvious not decrease when
the speedTypical SRian illogic. THE GAINED ENERGY IS THE
SAME IRRESPECTIVE OF SPEED,THEREFORE SPEED INCREASE IS
ALWAYS DIRECTLY PROPORTIONAL TO THE AMOUNT OFENERGY
GAINED.>>>Why do you have a problem with that?>>>
>>Why not simply accept it?>>>>>> Accepting it
means virtually nothing. What's the point?>>> >>This
illustrates your problem.>>>You refuse to accept that
Nature works as it actually does.>>>And again I must
ask:>>>Why don't you say that accepting KE = 0.5*m*v^2>>
>>means vertually nothing. What's the point?>>>> What
is ?v'? That is the point.>> >You didn't answer, 
Henry.>>
>I ask you why you think KE = 0.5*m*v^2 explains>>more
than KE = m*(gamma-1)*c^2.>>What's the big difference?>>
I told you. 1/2mv^2 is the kinetic energy. The additional
energy is stored in>> the ?ld ?bubble' around the moving
charge.>>You didn't answer, Henry.>You just keep asserting
that KE is 0.5*m*v^2 and NOT m*(gamma-1)*c^2.>WHY do you
claim that KE = 0.5*m*v^2?Because that leads to a believeable
physical theory rather than a nebulus mathsequation derived
by pure accident.>Do you know it a priori?>Do you know it
because it was Newton and not Einstein who said so?>Is it a
religious conviction you will believe blindly despite of
anything?Who are YOU to talk about religious convictions.>>
>>> You say it manifests itself as a mass increase
(although you try to disguise>>> the fact)>>> I agree
that it appears just like a mass increase. I also claim that
it is NOT>>> RELATED to a mass increase but that the
energy is stored somehow in the ?lds>> >Why don't you
need enigmatic ?lds to store the KE in 0.5*m*v^2?>>
Because ?0.5*m*v^2' is the correct expression for amount of
energy associated>>Is it? Why?>Both 0.5*m*v^2 and
m*(gamma-1)*c^2 are in accordance with>experimental evidence
when v << c, while only the latter is in>accordance with
experimental evidence when v is big enough>for the two
equations to give a measurable different prediction.>So why
do you claim that the former is the correct expressionI'm
merely de?ing KE as always ?0.5*m*v^2' and distinguishing IT
from theenergy that goes into the ?ld and has all the
appearance of a ?massincrease'..>>> >> Can you not see
that the implications of being able to eliminate 
'mass'
from>>> the equations are considerable. We might be able to
actually learn something>>> about the nature of mass if we
look beyond a simple maths formula.>> >So you are
actually saying that the KE is stored in enigmatic ?lds>>
>even when c << v? That is the KE = 0.5*m*v^2 is stored in
?lds?>> Not directly. but indirectly maybe. I reckon all
mass is explainable in terms>> of ??lds' whatever they
are.>>May well be. But it is irreleavant to our issue.IT IS
NOT>I have never seen you have a problem with the KE of low
speed>bodies, and insited that the KE must be stored in
some kind of>?ld bubbles. Quite the contrary. You have
claimed that it is>only the _differemce_ between
m*(gamma-1)*c^2 and 0.5*m*v^2>that is stored in these
bubbles. So where is the 0.5*m*v^2 stored?'0.5*m*v^2' 
is
hardly ?stored'. It is merely part of the energy released
whenan object is brought to rest in some reference frame. I
claim that the separate energy stored in the ?reverse ?ld
bubble' issimilarly released in this process.You cannot argue
with that Paul. It is simply a matter of de?ition. Mine is
arealistic physical one whereas your is purely a maths one
based on an unprovenpostulate. Whether my equation is
precisely the same as SR's ?Mo.gamma' remainsto be 
seen.>>
>> All we need now is an understanding of what a ??ld' is.>>
>Good luck.>>You should look at the standard
model.>> Do you believe in ?solid matter'?>> Anything
?solid' must be made of something smaller...and on and
on....>> Where does that end?>>There are turtles all the
way down, of course.yes! You found it!>>Paul>Henri Wilson.
See the Stupidity of
Relativity.www.users.bigpond.com/hewn/index.htm
===
>> You
have now changed the question.> You are now specifying that
the electron is moving at high speed when it enters> the
?ld. Do you want me to remind you of your original
question?The original question was how long it takeswhen it
enters a static electric ?ld, regardless ofwas accelerated
from standstill (or a very lowinitial speed). In that case,
you said the forceenters the ?ld.You said however that it
would not do so ifSo my new scenario is:We still have two
electrodes 1 km apart, with a millionvolts potential
difference between them.Behind the cathode, there is an
electron gun shootingelectrons with an initial speed v_o
through a small holein the cathode.The cathode itself isn't
hot, and how the electronis accelerated is of no consequence.
The point isthat it comes out of the hole with the speed
v_o.The question is still:How long time does it take from the
electroncomes out of the hole to a force start acting on it?My
answer is:as it enters a static electric ?ld.What is your
answer?You have given no clear answer.> My previous answer
(repeated twice) stands.I haven't seen it.Would you repeat it
please?Paul
===
>>> You have now changed the question.>> You
are now specifying that the electron is moving at high speed
when it enters>> the ?ld. Do you want me to remind you of
your original question?>>The original question was how long
it takes>when it enters a static electric ?ld, regardless
of>>was accelerated from standstill (or a very low>initial
speed). In that case, you said the force>enters the
?ld.>>You said however that it would not do so if>>So my new
scenario is:>We still have two electrodes 1 km apart, with a
million>volts potential difference between them.>Behind the
cathode, there is an electron gun shooting>electrons with an
initial speed v_o through a small hole>in the cathode.>The
cathode itself isn't hot, and how the electron>is accelerated
is of no consequence. The point is>that it comes out of the
hole with the speed v_o.>>The question is still:>How long
time does it take from the electron>comes out of the hole to
a force start acting on it?>>My answer is:>as it enters a
static electric ?ld.>>What is your answer?>You have given
no clear answer.> My previous answer (repeated twice)
stands.>>I haven't seen it.>Would you repeat it
please?According to my ?reverse ?ld bubble' theory, the
applied ?ld will bereduced in the vicinity of the moving
charge. That will have a similar effectto the applied ?ld
taking time to operate.>>Paul>Henri Wilson. See the Stupidity
of Relativity.www.users.bigpond.com/hewn/index.htm
===
> >>
>> >>Paul>>> >> No silly answers please.>>
>How about the correct answer?>>they radiate. The faster
they go the greater the radiated energy per>unit time.
String a group of them along a line and you now have a>
>current, and a current produces a magnetic ?ld in
surrounding space.>Energy is stored in this ?ld. Thus
there are two reasons that charged>only two reasons. At c
(wrt lab frame) the electron cannot be further>accelerated,
not because the associated magnetic ?ld contains in?ite>
>disappears at this speed, i.e. all of the virtual photons
inducing the>this speed, simply because the source of those
photons is the>accelerator itself, which is at rest in lab
frame. OTOH an in?ite>quantity of energy input per unit
time into the accelerator is required>terminal velocity,
all of the input energy thereafter is wasted, not one>
>analogous to that of a skydiver. The special relativistic
prediction is>incorrect, and despite the propaganda, that
prediction has never been>veri?d. The collision energy has
never been directly measured, it has>only been force ? to
the various ?perceived' byproducts of the>collision, all of
which are based upon ad hoc conjectures which are>the
shaman, and this proves only that the general public is at
an>evolutionary standstill. Morons leading morons, they'll
both fall into>the ditch. Best to suspend judgment, this is
the high road, there are>only ditches on either side.>>
>Richard Perry>> > Relativity is blatant
BULL!!!Relatively moving light clocks are all you need to
disprove SR. If eachis always oriented perpendicular to its
motion relative to the other,then each will expect the other
to tick slower, acceleration doesn'tchange a thing, and
doesn't break any symmetry. The only consistentsolution is
motion wrt a local ?ed medium. Every experiment supportingSR
supports a local aether, in fact demands it, because this
relativemotion crap doesn't ? is contradictory. There
?is' an embankmentagainst which to compare the propagation of
light, namely all of therelative to each other.Richard Perry>
> Henri Wilson.> See the Stupidity of Relativity.>
www.users.bigpond.com/hewn/index.htm
===
>>>> >>
>>Paul>>> >>>> No silly answers please.>>
>How about the correct answer?>> >they radiate. The
faster they go the greater the radiated energy per>>unit
time. String a group of them along a line and you now have
a>>current, and a current produces a magnetic ?ld in
surrounding space.>>Energy is stored in this ?ld. Thus
there are two reasons that charged>>only two reasons. At c
(wrt lab frame) the electron cannot be further>>
>accelerated, not because the associated magnetic ?ld
contains in?ite>>disappears at this speed, i.e. all of
the virtual photons inducing the>>this speed, simply
because the source of those photons is the>>accelerator
itself, which is at rest in lab frame. OTOH an in?ite>>
>quantity of energy input per unit time into the accelerator
is required>>terminal velocity, all of the input energy
thereafter is wasted, not one>>analogous to that of a
skydiver. The special relativistic prediction is>>
>incorrect, and despite the propaganda, that prediction has
never been>>veri?d. The collision energy has never been
directly measured, it has>>only been force ? to the
various ?perceived' byproducts of the>>collision, all of
which are based upon ad hoc conjectures which are>>the
shaman, and this proves only that the general public is at
an>>evolutionary standstill. Morons leading morons, they'll
both fall into>>the ditch. Best to suspend judgment, this is
the high road, there are>>only ditches on either side.>>
>Richard Perry>> >>>> Relativity is blatant
BULL!!!>>Relatively moving light clocks are all you need to
disprove SR. If each>is always oriented perpendicular to its
motion relative to the other,>then each will expect the other
to tick slower, acceleration doesn't>change a thing, and
doesn't break any symmetry. The only consistent>solution is
motion wrt a local ?ed medium. Every experiment
supporting>SR supports a local aether, in fact demands it,
because this relative>motion crap doesn't ? is
contradictory. There ?is' an embankment>against which to
compare the propagation of light, namely all of the>relative
to each other.>>Richard Perry>I agree. SR is just a disguised
aether theory. The LT's cannot be ?real' unlessspace 
and time
are at least locally absolute.>> Henri Wilson.>> See the
Stupidity of Relativity.>>
www.users.bigpond.com/hewn/index.htmHenri Wilson. See the
Stupidity of
Relativity.www.users.bigpond.com/hewn/index.htm
===
How do I
?d that cos(2pi/17)=-1/16+ Sqrt(17)/16 + Sqrt(34 -
2*Sqrt(17))/16 +Sqrt(17 + 3*Sqrt(17) - Sqrt(34 - 2*Sqrt(17))
- 2*Sqrt(34 + 2*Sqrt(17)))/8 ?How do I ?d the right-hand side
given a different angle?
===
> How do I ?d that
cos(2pi/17)=-1/16+ Sqrt(17)/16 + Sqrt(34 - 2*Sqrt(17))/16 +>
Sqrt(17 + 3*Sqrt(17) - Sqrt(34 - 2*Sqrt(17)) - 2*Sqrt(34 +
2*Sqrt(17)))/8 ?>> How do I ?d the right-hand side given a
different
angle?http://www.ace.gatech.edu/~ryanhynd/17gon.pdf-- P.A.C.
SmithThe vast majority of Iraqis want to live in a peaceful,
free world.And we will ?d these people and we will bring them
to justice.
===
> [ snip ] >>> end note:>> Did you ever try
to read Two Dogmas of Empiricism, W.V.O. Quine 1951>> that
I recommended? It's on-line. When I read it for the ?st
time>> View), I thought I understood it, I also found it
boring. Today, I know>> I didn't understand it, I now know
why I didn't understand it, and I>> that's why I 
don't ?d it
boring - putting it mildly, I see it as one>> of the most
important papers written in the 20th century in fact.......>>
>> Give it a shot, and give genuine humility a shot too.
Having a look at>> On What There Is, The Problem of
Meaning in Linguistics, Reference>> and Modality and New
Foundations For Mathematical Logic might help>> ....- hell,
read the whole collection - it'll be a start......>> -- > I
don't know as I'll have the opportunity to do any 
reading
whilst I'm> getting my advanced degree in humility.>
>Actualy, Lester, I suspect you would rather enjoy reading
Two Dogmas ofEmpiricism - Quine's discussion of topics you
seem to ?d engaging is bothseminal and ?It's a short
paper. You can ?d it on-line
at:http://www.ditext.com/quine/quine.htmlI recommend it both
because, as I say, I think you would enjoy reading it,and
because I'm interested to hear what you make of it.The main
thrust of the paper is to undermine two presumptions of
empiricism:| Modern empiricism has been conditioned in large
part by two dogmas. One is| a belief in some fundamental
cleavage between truths which are analytic,| or grounded in
meanings independently of matters of fact and truths which|
are synthetic, or grounded in fact. The other dogma is
reductionism: the| belief that each meaningful statement is
equivalent to some logical| construct upon terms which refer
to immediate experience. Both dogmas, I| shall argue, are ill
founded. One effect of abandoning them is, as we| shall see, a
blurring of the supposed boundary between speculative|
metaphysics and natural science. Another effect is a shift
toward| pragmatism.What practical difference can it make to
anybody's life whether analytic andsynthetic truths differ
in kind or only in degree? The last two sections ofthe
paper, especially the last section, a mostly metaphorical
discussion ofwhat today would be called a theory of belief
revision, belief propagation,or model selection, has direct
bearing on Expert Systems that assist inmaking decisions,
or actualy make decisions on their own, or arrive at
adiagnosis, or a most probable explanation of uncertain
information. Shoulda loan application be approved? Should a
prisoner be paroled? Who shouldget a kidney? What disease or
combination of diseases is behind a set ofsymptoms, and what
lab tests give the most information? My own work istypi?d by
the problem of writing software that, given a digital image of
acheck, reads the dollar amount and decides either to
accept its bestguess as to what the amount is (a most
probable interpretation of allavailable evidence) or route
the image to a workstation for human dataentry. How do we get
from the esoterica of epistomology to algorithms thathand out
kidneys, loans, liberty, treatment, and advice? How do we
evaluatethe performance of such algorithms? Against what
standards can we comparethem?| The totality of our so-called
knowledge or beliefs, from the most casual| matters of
geography and history to the profoundest laws of atomic
physics| or even of pure mathematics and logic, is a
man-made fabric which impinges| on experience only along the
edges. Or, to change the ?ure, total| science is like a ?ld
of force whose boundary conditions are experience.| A
con?ith experience at the periphery occasions
readjustments in the| interior of the ?ld. Truth values
have to be redistributed over some of| our statements.
Re-evaluation of some statements entails re-evaluation of|
others, because of their logical interconnections -- the
logical laws| being in turn simply certain further statements
of the system, certain| further elements of the ?ld. Having
re-evaluated one statement we must| re-evaluate some others,
whether they be statements logically connected| with the ?st
or whether they be the statements of logical connections|
themselves. But the total ?ld is so undetermined by its
boundary| conditions, experience, that there is much latitude
of choice as to what| statements to re-evaluate in the light
of any single contrary experience.| No particular experiences
are linked with any particular statements in the| interior of
the ?ld, except indirectly through considerations of|
equilibrium affecting the ?ld as a whole.| If this view is
right, it is misleading to speak of the empirical content| of
an individual statement -- especially if it be a statement at
all| remote from the experiential periphery of the ?ld.
Furthermore it| becomes folly to seek a boundary between
synthetic statements, which hold| contingently on experience,
and analytic statements which hold come what| may. Any
statement can be held true come what may, if we make drastic|
enough adjustments elsewhere in the system. Even a statement
very close to| the periphery can be held true in the face of
recalcitrant experience by| pleading hallucination or by
amending certain statements of the kind| called logical laws.
Conversely, by the same token, no statement is immune| to
revision. Revision even of the logical law of the excluded
middle has| been proposed as a means of simplifying quantum
mechanics; and what| difference is there in principle between
such a shift and the shift| whereby Kepler superseded Ptolemy,
or Einstein Newton, or Darwin| Aristotle?Are you familiar with
Bayesian Networks, which are also known as BeliefNets? A
Bayesian Network is a directed acyclic graph whose nodes
representrandom variables with arcs representing causal
in?s or class-propertyrelationships (to paraphrase Judea
Pearl). It is a graphical representationof probabilistic
knowledge. The power of such nets is two-fold: they
makemodeling assumptions explicit and clear, and they form
the basis ofef?ient algorithms for calculating joint
probability distributions overthe set of random variables, or
subsets of the variables, or conjunctionsand disjunctions of
sets of variables - you could think of such a network asa
probabilistic database organized to support ef?ient
processing ofprobabilistic queries. Evidence impinges nodes
at the periphery, whichpropagate diagnostic support up the
directed links to parent nodes, andreceive causal support
from parent nodes. A general node, with both parentsand
children, receives both diagnostic support and causal
support. Beliefpropagates through the net until at
equilibrium each node represents aprobability distribution
over its state space. If the graph is connectedthen a change
to any node in the net in?s the state of all nodes inthe
net, but the computation remains tractable because the state
of any nodedepends only on the states of its neighbors
(conditional independance).Just how ef?ient and tractable
such belief propagation algorithms can bedepends on the
topology of the graph - in general the more sparse the
graphthe more ef?ient the algorithm (but potentialy the more
approximate themodel).I ?st learned of Bayesian Networks from
Judea Pearl's 1988 classic:Probablistic Reasoning In
Intelligent Systems: Networks of Plausible This is an
extraordinary book...no philosopher concerned with
probability, decision theory, defeasible logic, or even
epistemology in its formal guises, can afford to be ignorant
of the contents of this book... This is a great book.So far
I've mentioned Bayesian Networks only as static data
structures forthe ef?ient computation of the results of
probabilistic queries, but overthe last ten years there has
been a steady shift in the focus of attentionfrom algorithms
for propagating evidence towards methods for
learningparameters and structure from data (Cowell, Dawid,
Lauritzen, andSpiegelhalter in the introduction to
Probabilistic Networks and ExpertSystems). Adjusting the
parameters of an existing belief net could becalled normal
science, whereas a change in the topology could be called
aparadigm shift.So, what's my point? If nothing else a
look at the history of ExpertSystems provides in microcosm a
chapter in the unfolding book of AppliedEpistemology. Quine's
metaphor has become today's algorithms and datastructures.It
is probably just a coincidence (or is it co-occurrence) that,
whileplugging Quine, Longley did not mention explicitly
Quine's book Word andObject, or his essay Natural Kinds
(in the book OntologicalRelativity), or take the time to
trace the line of thought from thescandal of induction
through innate similarity spaces to, when coupledwith a
heterogeneous world, a mind of adapted computational modules
ratherthan the blank slate, lump of wax, or general-purpose
computer of theStandard Social Science Model (Pinker, The
Language Instinct). But thatis a topic for another
time...Anyhow, Lester. When you ask:>What if the
positivism/empiricism/materialism/behaviorism axis
is>failing? Not just being ignored and passed over as
outmoded scienti?>doctrine but actually being scorned as
ideas which never had any>conceptual validity? What if
quantum mechanics is wrong and there are>special relativity
was never quite right to begin with? If this>happens and
happens in conclusive terms then the scienti? world
and>contemporary visions of reality are going to come apart
at the seams>and I think we'll need all the philosophical
help we can get.the resistance you encounter is probably not
so much a matter of reactionaryscholasticism. Probably it
has more to do with Quine's metaphor: you arenot altering a
single hypothesis or proposition in isolation, you
arealtering the whole interconnecting web of scienti?
belief, which has agreat deal of explanatory power, an
elegant and spare structure - call it alarge epistomological
inertial mass. It will take a LOT of evidence tochange it; or
a HUGE simpli?ation in its structure, or substantialevidence
and substantial simpli?ation whose product is HUGE. If you
wantto move the entire scienti? world you need a lever big
enough and a placeto stand.
===
>> [ snip ]> > end
note:> Did you ever try to read Two Dogmas of Empiricism,
W.V.O. Quine 1951> that I recommended? It's on-line. When I
read it for the ?st time> View), I thought I understood
it, I also found it boring. Today, I know> I didn't
understand it, I now know why I didn't understand it, and
I> that's why I don't ?d it boring - putting it 
mildly, I
see it as one> of the most important papers written in the
20th century in fact.......> Give it a shot, and give
genuine humility a shot too. Having a look at> On What
There Is, The Problem of Meaning in Linguistics,
Reference> and Modality and New Foundations For
Mathematical Logic might help> ....- hell, read the whole
collection - it'll be a start......> -- >> I don't 
know as
I'll have the opportunity to do any reading whilst 
I'm>>
getting my advanced degree in humility.>>>>Actualy,
Lester, I suspect you would rather enjoy reading Two Dogmas
of>Empiricism - Quine's discussion of topics you seem to ?d
engaging is both>seminal and ?It's a short paper. You
can ?d it on-line
at:>>http://www.ditext.com/quine/quine.html>>I recommend it
both because, as I say, I think you would enjoy reading
it,>and because I'm interested to hear what you make of
it.>>The main thrust of the paper is to undermine two
presumptions of empiricism:>>| Modern empiricism has been
conditioned in large part by two dogmas. One is>| a belief
in some fundamental cleavage between truths which are
analytic,>| or grounded in meanings independently of matters
of fact and truths which>| are synthetic, or grounded in
fact. The other dogma is reductionism: the>| belief that each
meaningful statement is equivalent to some logical>| construct
upon terms which refer to immediate experience. Both dogmas,
I>| shall argue, are ill founded. One effect of abandoning
them is, as we>| shall see, a blurring of the supposed
boundary between speculative>| metaphysics and natural
science. Another effect is a shift toward>| pragmatism.>>What
practical difference can it make to anybody's life whether
analytic and>synthetic truths differ in kind or only in
degree? The last two sections of>the paper, especially the
last section, a mostly metaphorical discussion of>what today
would be called a theory of belief revision, belief
propagation,>or model selection, has direct bearing on
Expert Systems that assist in>making decisions, or actualy
make decisions on their own, or arrive at a>diagnosis, or a
most probable explanation of uncertain information.
Should>a loan application be approved? Should a prisoner be
paroled? Who should>get a kidney? What disease or combination
of diseases is behind a set of>symptoms, and what lab tests
give the most information? My own work is>typi?d by the
problem of writing software that, given a digital image of
a>check, reads the dollar amount and decides either to
accept its best>guess as to what the amount is (a most
probable interpretation of all>available evidence) or route
the image to a workstation for human data>entry. How do we
get from the esoterica of epistomology to algorithms
that>hand out kidneys, loans, liberty, treatment, and advice?
How do we evaluate>the performance of such algorithms?
Against what standards can we compare>them?>>| The totality
of our so-called knowledge or beliefs, from the most casual>|
matters of geography and history to the profoundest laws of
atomic physics>| or even of pure mathematics and logic, is a
man-made fabric which impinges>| on experience only along the
edges. Or, to change the ?ure, total>| science is like a ?ld
of force whose boundary conditions are experience.>| A
con?ith experience at the periphery occasions
readjustments in the>| interior of the ?ld. Truth values
have to be redistributed over some of>| our statements.
Re-evaluation of some statements entails re-evaluation of>|
others, because of their logical interconnections -- the
logical laws>| being in turn simply certain further
statements of the system, certain>| further elements of the
?ld. Having re-evaluated one statement we must>| re-evaluate
some others, whether they be statements logically connected>|
with the ?st or whether they be the statements of logical
connections>| themselves. But the total ?ld is so
undetermined by its boundary>| conditions, experience, that
there is much latitude of choice as to what>| statements to
re-evaluate in the light of any single contrary
experience.>| No particular experiences are linked with any
particular statements in the>| interior of the ?ld, except
indirectly through considerations of>| equilibrium affecting
the ?ld as a whole.>>| If this view is right, it is
misleading to speak of the empirical content>| of an
individual statement -- especially if it be a statement at
all>| remote from the experiential periphery of the ?ld.
Furthermore it>| becomes folly to seek a boundary between
synthetic statements, which hold>| contingently on
experience, and analytic statements which hold come what>|
may. Any statement can be held true come what may, if we make
drastic>| enough adjustments elsewhere in the system. Even a
statement very close to>| the periphery can be held true in
the face of recalcitrant experience by>| pleading
hallucination or by amending certain statements of the kind>|
called logical laws. Conversely, by the same token, no
statement is immune>| to revision. Revision even of the
logical law of the excluded middle has>| been proposed as a
means of simplifying quantum mechanics; and what>| difference
is there in principle between such a shift and the shift>|
whereby Kepler superseded Ptolemy, or Einstein Newton, or
Darwin>| Aristotle?>>Are you familiar with Bayesian Networks,
which are also known as Belief>Nets? A Bayesian Network is a
directed acyclic graph whose nodes represent>random variables
with arcs representing causal in?s or
class-property>relationships (to paraphrase Judea Pearl). It
is a graphical representation>of probabilistic knowledge. The
power of such nets is two-fold: they make>modeling assumptions
explicit and clear, and they form the basis of>ef?ient
algorithms for calculating joint probability distributions
over>the set of random variables, or subsets of the
variables, or conjunctions>and disjunctions of sets of
variables - you could think of such a network as>a
probabilistic database organized to support ef?ient
processing of>probabilistic queries. Evidence impinges nodes
at the periphery, which>propagate diagnostic support up the
directed links to parent nodes, and>receive causal support
from parent nodes. A general node, with both parents>and
children, receives both diagnostic support and causal
support. Belief>propagates through the net until at
equilibrium each node represents a>probability distribution
over its state space. If the graph is connected>then a change
to any node in the net in?s the state of all nodes in>the
net, but the computation remains tractable because the state
of any node>depends only on the states of its neighbors
(conditional independance).>Just how ef?ient and tractable
such belief propagation algorithms can be>depends on the
topology of the graph - in general the more sparse the
graph>the more ef?ient the algorithm (but potentialy the
more approximate the>model).>>I ?st learned of Bayesian
Networks from Judea Pearl's 1988 classic:>Probablistic
Reasoning In Intelligent Systems: Networks of Plausible>>
This is an extraordinary book...no philosopher concerned
with> probability, decision theory, defeasible logic, or even
epistemology> in its formal guises, can afford to be ignorant
of the contents of> this book... This is a great book.>>So
far I've mentioned Bayesian Networks only as static data
structures for>the ef?ient computation of the results of
probabilistic queries, but over>the last ten years there
has been a steady shift in the focus of attention>from
algorithms for propagating evidence towards methods for
learning>parameters and structure from data (Cowell, Dawid,
Lauritzen, and>Spiegelhalter in the introduction to
Probabilistic Networks and Expert>Systems). Adjusting the
parameters of an existing belief net could be>called normal
science, whereas a change in the topology could be called
a>paradigm shift.>>So, what's my point? If nothing else a
look at the history of Expert>Systems provides in microcosm a
chapter in the unfolding book of Applied>Epistemology. Quine's
metaphor has become today's algorithms and
data>structures.>>It is probably just a coincidence (or is it
co-occurrence) that, while>plugging Quine, Longley did not
mention explicitly Quine's book Word and>Object, or his
essay Natural Kinds (in the book Ontological>Relativity),
or take the time to trace the line of thought from
the>scandal of induction through innate similarity spaces
to, when coupled>with a heterogeneous world, a mind of
adapted computational modules rather>than the blank slate,
lump of wax, or general-purpose computer of the>Standard
Social Science Model (Pinker, The Language Instinct). But
that>is a topic for another time...>>Anyhow, Lester. When
you ask:>What if the
positivism/empiricism/materialism/behaviorism axis
is>>failing? Not just being ignored and passed over as
outmoded scienti?>>doctrine but actually being scorned as
ideas which never had any>>conceptual validity? What if
quantum mechanics is wrong and there are>>special relativity
was never quite right to begin with? If this>>happens and
happens in conclusive terms then the scienti? world
and>>contemporary visions of reality are going to come apart
at the seams>>and I think we'll need all the philosophical
help we can get.>>the resistance you encounter is probably
not so much a matter of reactionary>scholasticism. Probably
it has more to do with Quine's metaphor: you are>not altering
a single hypothesis or proposition in isolation, you
are>altering the whole interconnecting web of scienti?
belief, which has a>great deal of explanatory power, an
elegant and spare structure - call it a>large
epistomological inertial mass. It will take a LOT of evidence
to>change it; or a HUGE simpli?ation in its structure, or
substantial>evidence and substantial simpli?ation whose
product is HUGE. If you want>to move the entire scienti?
world you need a lever big enough and a place>to
stand.>interconnected. A chain is only as strong as its
weakest link. Whatmakes me think it might actually happen is
that I'm not arguing in avacuum any more. I'm 
actually
beginning to understand what I'm talkingabout. The same
mistakes seem to apply pretty much across the board. By the
way I'll check out the online Quine reference.
===
I'm kind of
confused by a de?ition I've seen of independence oftwo random
events or random variables, which contradicts my notionof
independent random variables/events.What I understand as
independent variables or events is two variables/events that
are the outcome of experiments that have norelationship
whatsoever -- hence the term *INdependent* (as in,they don't
*depend on* each other).One immediate consequence of the
above de?ition is that theprobability of two particular
outcomes for both eventssimultaneously is the product of the
probabilities of each eventalone. Notice that the above
implies this, but this does notimply the above.Now, how can
the above be stated as a de?ition of independence??That
seems to me completely illogical!!In particular, I hear that
for variables with Gaussian distribution,independence and
uncorrelation (uncorrelatedness?) are equivalentthings,
because it turns out that the product of the PDF's of
twouncorrelated Gaussian variables happens to be equal to the
productof the PDF's. I ?d this completely absurd!!!
Independence ofcourse implies uncorrelated; but uncorrelated
does not imply independence!!In cryptograhpy, the data and
the encrypted data are uncorrelated(this is one of many
condition for a good encryption system -- sothat an attacker
can not take advantage of statistical correlationof the
gibberish with the data being hidden). But what a jokewould
it be to say that the data and that data passed through
amathematical function are independent!!!!I can take a
pseudo-random generator algorithm that generatesGaussian
pseudo-random numbers, and construct two random variablesas
follows: draw a random number from some physical (i.e.,
trulyrandom) process; seed the pseudo-random generator with
the randomnumber, and get the ?st variable, then re-seed the
pseudo-randomgenerator with the value of the ?st variable,
and get the secondvariable.The PRNG can be made such that
these two variables will beuncorrelated. But is it not a joke
to claim that the two randomvariables are *independent*?? The
second variable is explicitlya function of the ?st one!!!
Just because of a big coincidenceabout the Gaussian
distribution that makes the joint distributionequal to the
product of the marginals are we going to say thatthey are
independent??Is this a matter of point of view? Or am I being
misled and thosede?itions are indeed
incorrect?Carlos--
===
>I'm kind of confused by a de?ition
I've seen of independence of>two random events or random
variables, which contradicts my notion>of independent random
variables/events.>What I understand as independent variables
or events is two >variables/events that are the outcome of
experiments that have no>relationship whatsoever -- hence the
term *INdependent* (as in,>they don't *depend on* each
other).Well then, your understanding is imperfect. That's an
example, not the de?ition. The de?ition of independent
events is as follows: A and B are independent if Pr(A
intersect B) = Pr(A) Pr(B).>One immediate consequence of the
above de?ition is that the>probability of two particular
outcomes for both events>simultaneously is the product of the
probabilities of each event>alone. Notice that the above
implies this, but this does not>imply the above.>Now, how can
the above be stated as a de?ition of independence??>That
seems to me completely illogical!!Once the de?ition has been
made, and accepted by the mathematicalcommunity, you just have
to accept it. Yell and scream all you want,it's not going to
change.Robert Israel israel@math.ubc.caDepartment of
Mathematics http://www.math.ubc.ca/~israel University of
British Columbia Vancouver, BC, Canada V6T 1Z2
===
I'm trying
to recall an example of a proposition S(n) where we can
showS(k)->S(k+1) but S(1) fails and, in general, S(n) is
false. In such a case,one might falsely conclude S(n) holds
for all natural no's n .L
===
>I'm trying to recall an example
of a proposition S(n) where we can show>S(k)->S(k+1) but S(1)
fails and, in general, S(n) is false. In such a case,>one
might falsely conclude S(n) holds for all natural no's n .The
easiest I can come up with is:S(n) = n is greater than
100(or greater than 10, or greater than anything other than
0).To prove the induction step:Suppose that S(k) holds; then
k>100. Since k+1>k>100, we conclude thatk+1>100; that is, if
S(k) holds, then S(k+1)
holds.
===
==============================

===
=======It's not denial. I'm just very selective
about what I accept as reality. --- Calvin (Calvin and
Hobbes)
===
============================

===
========Arturo
Magidinmagidin@math.berkeley.edu
===
to be true for n>100. I
seem to recall examples which are generally, if notalways,
false, despite the factS(k)->S(k+1).Len>I'm trying to
recall an example of a proposition S(n) where we can show>
>S(k)->S(k+1) but S(1) fails and, in general, S(n) is false.
In such acase,>one might falsely conclude S(n) holds for
all natural no's n .>> The easiest I can come up with is:>>
S(n) = n is greater than 100>> (or greater than 10, or
greater than anything other than 0).>> To prove the induction
step:>> Suppose that S(k) holds; then k>100. Since k+1>k>100,
we conclude that> k+1>100; that is, if S(k) holds, then
S(k+1) holds.>>

===
=================================

===
===> It's not denial. I'm just very selective about>
what I accept as reality.> --- Calvin (Calvin and Hobbes)>

===
=================================

===
===>> Arturo Magidin> magidin@math.berkeley.edu>===How
about something like if n is irrational then n + 1 is is
irrational? This implication is true for all real, or even
complex, n, but the antecedent and consequent are false for
all integers, indeed all rationals.> to be true for n>100. I
seem to recall examples which are generally, if not> always,
false, despite the fact> S(k)->S(k+1).> Len> >I'm trying
to recall an example of a proposition S(n) where we can
show> >S(k)->S(k+1) but S(1) fails and, in general, S(n)
is false. In such a> case,> >one might falsely conclude
S(n) holds for all natural no's n .>> The easiest I can
come up with is:>> S(n) = n is greater than 100>>
(or greater than 10, or greater than anything other than 0).>
>> To prove the induction step:>> Suppose that S(k)
holds; then k>100. Since k+1>k>100, we conclude that>
k+1>100; that is, if S(k) holds, then S(k+1) holds.>>

===
=================================

===
===> It's not denial. I'm just very selective about>
> what I accept as reality.> --- Calvin (Calvin and
Hobbes)>

===
=================================

===
===>> Arturo Magidin> magidin@math.berkeley.edu>
>>
===
>to be true for n>100. I seem to recall examples
which are generally, if not>always, false, despite the
fact>S(k)->S(k+1).S(n) = There are at least n+1 positive
integers strictly smaller than n.S(n) = there are only
?itely many integers smaller than n.S(n) = The set
{1,....,n} can be bijected with a proper subset of
itself.S(n) = There is an integer x such that n<x<n+1.
S(n) = There is an n x n matrix with integer coef?ients
which has determinant 0 and is
invertible.etc.
===
========================

===
============It's not denial. I'm just very
selective about what I accept as reality. --- Calvin
(Calvin and
Hobbes)
===
============================

===
========Arturo
Magidinmagidin@math.berkeley.edu
===
Leonard M. Wapner
<lwapner@adelphia.net> grava .88 la saucisse et au
marteau:> to be true for n>100. I seem to recall examples
which are generally, if not> always, false, despite the
fact> S(k)->S(k+1).The proposition being an irrational
works.-- Nicolas
===
> The proposition being an irrational
works.I was thinking k is not an integer, but this might be
morestraightforward.There is also k = k+1, where the
induction adds 1 to both sides.-- Daniel W.
Johnsonpanoptes@iquest.nethttp://members.iquest.net/~panoptes
/039 53 36 N / 086 11 55 W
===
Perfect!> Leonard M. Wapner
<lwapner@adelphia.net> grava .88 la saucisse et
aumarteau:>turn out> to be true for n>100. I seem to recall
examples which are generally,if not> always, false, despite
the fact> S(k)->S(k+1).>> The proposition being an
irrational works.>> -- > Nicolas
===
im trying to come up
with an algorithm to output an euler tour, given anundirected
eulerian graph G. Can someone give me of a counter example of
agraph for which my algorithm below does not ?d the tour
even though thereis one:1. Do Depth-First Search traversal of
G and number the vertices inDFS-preorder2. Reinitialize all
vertices and edges of G as unused.3. Produce a cycle by:start
from vertex with preorder number 1 computed in step 1, and
repeatedlygo to the vertex with highest preorder number
possible along an unused edge.Stop when all edges incident
to the current vertex are used.X-Cise:
tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate:
admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose:
george_cox@btinternet.comX-Punge: Micro$oft
===
>I know this
is not a very deep question, but as I've come
accross>rigorous texts like Halmos's and Lax's on 
linear
algebra and noticed>their emphasis on ?ite dimensional
vector spaces (certainly to be>expected in Halmos's case), it
makes me wonder if linear algebra is>regarded mainly as the
study of these spaces.No. However, because the ?ite
dimensional case is simpler, it iscommon for itnroductory
texts to only cover it.>Also, are there linear algebra books
(as opposed to functional>analysis texts) that treat in?ite
dimensional vector spaces?-- Shmuel (Seymour J.) Metz,
SysProg and JOATnot reply to spamtrap@library.lspace.org
===
>
I know this is not a very deep question, but as I've come
accross> rigorous texts like Halmos's and Lax's on 
linear
algebra and noticed> their emphasis on ?ite dimensional
vector spaces (certainly to be> expected in Halmos's case),
it makes me wonder if linear algebra is> regarded mainly as
the study of these spaces.If linear algebra is understood
as the study of linear maps,the restriction to ?ite
dimensional vector spaces is not essential.The fundamental
de?ions (e.g. of linear map) make sense formodules over
rings.However, there are results about f.d. vector spaces,
that do notgeneralize to the more general setting, or can be
obtained onlywith more work.In the restricted situation of an
introductory course (or book),it is probably a reasonable
choice, to consider only f.d.v.s.Moreover, in the study of
the in?ite-dimensional case, methodsfrom analysis are
helpful, so the author might want to avoidyet more
methods.When I was a beginning student, the course Linear
Algebraintroduced (in this order)Sets, groups, rings,
modules, ?itely generated vector spacesand restricted to the
more special setting whenever there wouldbe too much of a
technical burden otherwise.Marc
===
I have big questions in my
mind about linear algebra concept. Cananyone help me? (Because
my linear algebra is not good, maybe some ofmy descriptions
are wrong. Please help me to point them out)We can use inner
product to ?d the projection of a vector to thebasis of a
vector space. However, before we de?ing inner product,
wehave already learned the fact that every vector can be
uniquelywritten as linear combination of basis. That means
the concept ofprojection is independent of inner product.
Then why should we de?einner product again? (without using
the concept of projection?)Also, inner product can be de?ed
in different way(say, dot product,Hermitian inner product...
I can't list many examples, because i don'tknow). Do 
all
these inner product give me the same projection? Can Ide?e
my own kind of inner product?Another question is, do in?ite
dimensional vector spaces has theconcept of basis? Also, I
wanna ask the following is correct or not:Given an in?ite
dimensional vector space V, I have already showedthat an
in?ite set of vectors A(subset of V) is an othonormal
set.Can I say that there exists another subset B of V and B
contains A (Ais subset of B also)such that the vectors in B
form the basis of V?(or can I say that every element of V can
be uniquely written aslinear combination of vectors in
B?)
===
>I have big questions in my mind about linear algebra
concept. Can>anyone help me? (Because my linear algebra is
not good, maybe some of>my descriptions are wrong. Please
help me to point them out)>>We can use inner product to ?d
the projection of a vector to the>basis of a vector space.
However, before we de?ing inner product, we>have already
learned the fact that every vector can be uniquely>written
as linear combination of basis. That means the concept
of>projection is independent of inner product. Then why
should we de?e>inner product again? (without using the
concept of projection?)The inner product is independent of
any chosen basis. It addsgeometry (lengths, angles, areas,
etc.) to the vector space. The twokinds of projection you
mention are different if the basis is notorthonormal. For
example, in R^2 with the standard inner product,
theprojection (in the ?st sense) of (1, 1) onto (1, 0) is
(1, 0). Ifyou use the basis {(1, 0), (1, 1)}, then the
projection in the secondsense is the zero vector.>Also, inner
product can be de?ed in different way(say, dot
product,>Hermitian inner product... I can't list many
examples, because i don't>know). Do all these inner product
give me the same projection? Can I>de?e my own kind of inner
product?Hermitian inner products are used in complex vector
spaces and aremore special than an arbitrary inner product on
the underlying realvector space. I suggest ignoring the
complex case until you'recomfortable with the real
case.In?itely many inner products can be de?ed on a vector
space. OnR^n, with respect to the standard basis, an inner
product can berepresented by a positive-de?ite symmetric
n-by-n matrix, and thereare clearly many such matrices.
Different inner products will yielddifferent projections in
general (if by projection you meanorthogonal projection).
One interesting example is a smooth surfacein R^3. At each
point, the tangent space of the surface is a plane,and the
plane inherits an inner product from R^3. The inner product
onthe tangent space de?es the in?itesimal geometry of the
surfaceat the chosen point. A major theme of Differential
Geometry is torelate the in?itesimal geometry to the global
geometry and topologyof the surface (or more generally,
Riemannian manifold).>>Another question is, do in?ite
dimensional vector spaces has the>concept of basis?series).
There's also a general result that any vector space has
abasis, whose proof uses trans?ite induction.John
Mitchell
===
>>Another question is, do in?ite dimensional
vector spaces has the>concept of basis?> series). There's
also a general result that any vector space has a> basis,
whose proof uses trans?ite induction.There's a strikingly
simple and familiar example of an in?ite dimensional vector
space. Consider the set R[x] of all polynomials in one
variable whose coef?ients are real numbers. You can add two
polynomials to get another one, and you can multiply any
polynomial by a real number to get another polynomial. One
can easily verify that R[x] satis?s the axioms of a vector
space, where each polynomial is regarded as a vector, and the
real numbers are the scalars.Now consider the set of
polynomials B = {1, x, x^2, x^3, x^4, . . .}. Every
polynomial in R[x] can be written in a unique way as a ?ite
linear combination of elements of B. Further, it's pretty
clear that no ?ite collection of polynomials can have that
property (because any ?ite collection of polys contains one
of maximum degree, and you can't get any higher-degree polys
by adding or multiplying by scalars). Therefore B is a basis
of R[x], and therefore R[x] has countable dimension
considered as a vector space over the real numbers R.
===
I
was wondering about the partial sum of the sum-over-divisors
of mu =Mobius function, but previously was wondering only
about when the sumwas over those divisors of m which were <=
sqrt(m).Not many of my questions regarding the sqrt(m)
-situation wereanswered, but I might as well post the below
(unamazing) equalitywhich generalizes the sum to being over
those divisors <= n.Let m and n be positive integers.sum{k|m,
1<= k <= n} mu(k)= sum{k=1 to n} mu(k) phi(n/k,m),where
phi(x,m) = number of positive integers <= x and coprime to
m.Also, these sums =sum{k|q, n/p <= k <= n} mu(k),where q is
largest divisor of m which is coprime with p = a prime.If n =
?), I can not see how the above sums (if true) can leadto
a closed-form forsum{k|m, 1<= k <= sqrt(m)} mu(k)Any comments?
thanks,Leroy QuetArchive-Name: misc.metric-system
===
RESULT
unmoderated group misc.metric-system passes 211:25[ Note:
This document is multiposted in 3 copies because too many
large service providers implement crossposting limits and
would otherwise drop it. - n.a.n moderation team ]This vote
was conducted by a neutral third party. Questions about
theproposed group should be directed to the
proponent.Proponent: Markus Kuhn
<Markus.Kuhn@cl.cam.ac.uk>Votetaker: Bill Aten
<bill@netagw.com>There were 236 valid YES/NO votes submitted
during the voting period. Eachproposed newsgroup, in order to
pass, must have at least 2/3 YES votes and atleast 100 more
YES than NO votes. The results are as
follows:misc.metric-system results - 236 valid (YES & NO)
votes Yes No : 2/3? >100? : Pass? : Group---- ---- : ----
----- : ----- : -------------------------------------------
211 25 : Yes Yes : Yes : misc.metric-system 6 abstaining
votes and 3 invalid votesThe proposal passed.There is a ?e
day discussion period after these results are posted. If
noserious and signi?ant allegations of voting
irregularities are raised, themoderator of
news.announce.newgroups will create the newsgroup
shortlythereafter.The remainder of the RESULT contains:
Rationale Charter Final Voting AcknowledgementsNEWSGROUPS
LINE:misc.metric-system The International System of
Units.RATIONALE: misc.metric-systemUnits of measurement and
related standards affect many aspects of ourdaily lives. The
global standardization of a single consistentInternational
System of Units was a major breakthrough for
humancivilization and signi?antly simpli?d communication,
learning,work and trade all over the planet.The introduction
of the metric system still faces delays in someareas. Notable
examples are consumer communication and traf?regulations in
the United States and United Kingdom, as well as partsof the
aeronautical and typographic industry. It is therefore
nosurprise that discussions about the metric system ?p
regularlyin many different newsgroups. In particular the slow
progress withmetrication in the United States promises to fuel
such debates formany years to come.A dedicated newsgroup will
focus expertise and will provide a mediumfor professionals
and hobbyists to ?d advice and suggestions onmetric product
standards and conventions. None of the newsgroups inwhich
metric-system issues ?p frequently is particularly
suitedfor this topic by charter and readership. The
popularity of thethat there is a signi?ant number of people
interested in the topic.Considering the important role that
units of measurement play ineveryone's life, this promises to
become a quite lively newsgroup.The name of the proposed group
has been the subject of some debate.The present proposal is
motivated by these considerations: - Although discussions
about the metric system focus much on its slow progress in a
small number of countries, the topic is inherently
international in nature and discussions tend to bene? very
signi?antly from world-wide participation. Therefore,
placing the group under us.* or uk.* would be inappropriate.
- The metric system affects many ?lds, including consumer
communication and road traf?. Discussions about the metric
system range from basic science and applied engineering
considerations to economic, social, psychological, legal,
public policy and media aspects. This excludes sci.* and
leaves misc.* as the most appropriate hierarchy. - The metric
system is the only system of units used in almost every region
and ?ld of application. Imperial and U.S. Customary units
are usually discussed in relation to the metric system, which
is within the scope of the proposed group. Other unit systems
have very limited applications and are better discussed in
specialized science or history groups. The proposed group is
far more likely to have specialized children rather than
equivalent siblings, which speaks against an entire
misc.measurement.* hierarchy and justi?s a place directly
under misc. - The term metric system remains the most well
known and most easily recognized English language term for
what is more formally called the International System of
Units (SI). This speaks against group names such as *.si or
*.metric.The proposed charter has equally been the subject of
some debate. Thepresent proposal is motivated by these
considerations: - It refers equally to both the of?ial
modern name International System of Units (SI) and the
colloquial English term metric system. This is in the
interest of rapid recognition by both readers and
search-engine users. Any further distinction between these
terms is deliberately left to explanatory periodic postings.
- It is broad enough to cover discussions about different
historic variants of the metric system (e.g., CGS, MKS,
various European customary units) as well as contemporary
units that compete with the SI (e.g., inch, pound,
Fahrenheit, calorie). - It is narrow enough to exclude topics
that are not related to metric units (e.g., the history and
rede?ition of calendars). - It covers product standards and
conventions that are not part of any of?ial de?ition of the
metric system, but that are closely related to metric units
(e.g., metric clothing sizes, paper formats, engineering
components, traf? regulations). These can be considerably
more complex topics than the metric system itself, leading to
discussions of particular interest to consumers and
practitioners. - It leaves room for the possible later
creation of a separate misc.metric-system.advocacy group for
those with a particular interest in political activities
related to metrication. - It was written with the expectation
that the topic is unlikely to attract large non-plain-text
postings, commercial advertising or unsocial behaviour in any
particular way, leaving these issues to common sense and
USENET etiquette. - It is brief.CHARTER:
misc.metric-systemThis newsgroup is for discussion about the
International System ofUnits (SI) or metric system, including
its use in scienti?,technical, and consumer applications, its
history and de?ition, andits adoption in ?lds and regions
where other units of measurementare still prevalent
(metrication). Included within its scope arerelated global
standards and conventions, for example metric
productspeci?ations and consumer-product labelling
practice.END CHARTER.FINAL VOTING
ACKNOWLEDGEMENTS:misc.metric-system - Final Voter Listare
provided only to help verify the validity of the interest
poll.Voted
YES----------------------------------------------------------
--------------------72027.3605 [at] compuserve.com Bob
Bailinalan.jamieson [at] ntlworld.com Alan Jamiesonalanh
[at] unc.edu Alan Hoyleaminggs [at] yahoo.co.uk Andrew
InggsAndries.Brouwer [at] cwi.nl Andries Brouweranthony
[at] atkielski.com Anthony Atkielskib4r4n5k1 [at] yahoo.com
Adam Baranskibaron_carter [at] bmc.com Baron Carterbfrost
[at] nyx.net Bonnell Frostblaise.egan [at] btopenworld.com
Blaise F Eganbob [at] cave.org Robert
Wilkinsboldyrev+nospam [at] cgitftp.uiggm.nsc.ru Ivan
Boldyrevbouvin [at] daimi.au.dk Niels Olof
BouvinBrian.Inglis [at] SystematicSw.ab.ca Brian Inglisbuff
[at] pobox.com William DentonCarletonM [at] aol.com Carleton
MacDonaldcdkaese [at] ntlworld.com Chris Kaesecelia.mesure
[at] spritenote.co.uk Celia Mesureceri [at] submonkey.net
Ceri Davieschrajohn [at] alumni.indiana.edu Chris
Johnsonchris [at] kzim.com Christopher Robin Zimmermanchris
[at] lordsutch.com Chris Lawrencechris [at] metric.org.uk
Chris KEENANCirgreeSys [at] aol.com Timothy Moylancoghlan
[at] sympatico.ca Patrick Coghlancroyle [at] gelemna.org Don
Croyledalcorn [at] hoosierlink.net David K Alcorndavep [at]
davep.org Dave Pearsondavid [at] king-usa.com David
Kingdavid [at] rossde.com David E. Rossdbpg [at] telus.net
David Gibsondbreslau [at] ContinuumPhotonics.com David
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helsinki.?Esa Peuha?al [at] vbi.vt.edu Fidel Salas?]
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inforama.co.uk Jeff Grossjeffrey [at] carlyle.org Jeffrey
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[at] erols.com James Silvertonjimmc [at] nova.org Jim
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ntlworld.com Philip S HallPiotr.Zielinski [at] cl.cam.ac.uk
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[at] connected-systems.com Terry Simpsonthefatphil [at]
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[at] lexmediadigital.ph Tom W. Uhingtwid [at] bibulus.org
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Philip NewtonWbauer03 [at] aol.com William Bauerwessels7
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Pottswillner [at] cfa.harvard.edu Steve Willnerworstall
[at] btopenworld.com Michael V Worstallzimnyzenon [at]
interia.pl Zenobiusz Zimnyzvr [at] pobox.com Alexios
ZavrasVoted
NO-----------------------------------------------------------
-------------------aaron [at] aaroncity.com Aaron
O'Donnellbillv1939 [at] netscape.net Bill Vajkbrougham5
[at] yahoo.com E. Broughamclzeni [at] mindspring.com
clzenidc [at] panix.com David W. Crawforddevin [at]
thecabal.org Devin L. Gangerdougbob [at] charter.net Bob
Douglaseaking [at] ulape.org.uk Edward Kingfwbrown [at]
bellsouth.net Wayne Browngaillard [at] panix.com Ed
Gaillardgprrspw [at] mindspring.com G.P. Ryangraham.drabble
[at] lineone.net Graham Drabblejohnl [at] iecc.com John
Levinejosephb [at] panix.com Joe Bernsteinmiliff [at]
qnet.com Mary Shafermsb [at] vex.net Mark Bradernaddy [at]
mips.inka.de Christian Weisgerberpatrick [at] texier.info
Patrick Texierrick [at] bcm.tmc.edu Richard
Millerru.igarashi [at] usask.ca ru igarashistainles [at]
realtime.net Dwight Brownwiz [at] verinet.com WizVoted
ABSTAIN------------------------------------------------------
------------------------aahz [at] pobox.com Aahzhkt79 [at]
earthlink.net Henrietta K Thomasmmontcha [at] OregonVOS.net
Matthew Montchalintempdog [at] erols.com A [Temporary]
Dogyan [at] storm.ca Yves BellefeuilleInvalid
Votes--------------------------------------------------------
---------------------- ! Ack bounced: No such userm.collado
[at] aaron.ls.?upm.es Manuel Collado ! Ack bounced: No such
usermedawar [at] panix.com ! No name given-- Bill Aten, UVV
<bill@netagw.com>
===
Let us say we have a 4-by-4 same-sum
grid (ie. magic square), whichconsists of 1 to 16 placed so
that every row/column/main-diagonal sumto 30.We could then
subdivide each of the 16 grid-squares each into smaller4-by-4
grids. And then we could place the same numbers we had in
the?st-order grid, then multiply each of these by the
integer that wasplaced into this larger square (1/16 of
?st-order grid) at thestart.And we could continue this
inde?itely...But to avoid in?ities, we could divide each
integer at eachgeneration by 30.We would then be be left, at
the nth generation, with a grid of 16^nsquares containing
rationals, where each row and column andmain-diagonal sums to
1.And I wonder then, what can be said about the
in?ite-generationgrid?What about if the grid is subdivided
at each generation, not into4-by-4, but by some other
subdivision(s)??There must be something interesting to say
and ask about thesefractal(-like) magic squares.thanks,Leroy
Quet
===
>Let us say we have a 4-by-4 same-sum grid (ie. magic
square), which>consists of 1 to 16 placed so that every
row/column/main-diagonal sum>to 30.<snip>That might be
dif?ult to accomplish -- 34 is easier.-- Wally Farley
(WWFiv)
===
Let b(1) = 1;for 2 <= m,b(m) = (1/ln(m)) sum{p|m}
sum{k=1 to a(p,m)} b(m/p^k) /k,where the p-sum is over the
distinct primes p which divide,and p^a(p,m) = highest power
of p which divides m.then:I believe that:sum{m=1 to oo} b(m)/
m^y =exp(integral{y to oo} ln(zeta(x)) dx)which islimit {n
->oo} product{k >= yn} zeta(k/n)^(1/n)(perhaps).Right?My
attempt at trolling...And, by the way, if we altered the sum
for complex-integration, andtook its analytical continuation
if necessary (to get B(z) de?ed onwhole complex plane), we
*might* be able to use B(z) to helpinvestigate the Riemann
hypothesis, for if zeta = 0 anywhere along thecontour of
integration, then B(z) = 0.(Not as easy as it seems, sadly,
for B might still be 0 without anyzeta(z) = 0 in its
integrated domain, and because B seems to becomeindeterminate
anyway as the upper-limit of integration approachescomplex
in?ity.);((I know, I know, I am full o'
bull-spit...);)thanks,Leroy Quet
===
> Anselm's argument is an
excercise in begging the question.An excellent example of
question-begging yourself!> It is -worthless-. Nothing exists
by de?ition. Existence has to be> proven or observed.An
interesting thesis. I'm not sure what it does to the
foundationsof math. I'd simply ask how you would go about
proving this thesis.Thomas
===
Do you profess a religion or at
least believe in God?If you don't, why bother with God?If you
do, does your religion lack a theology? I really wonder why
onewould want to prove or even argue on a mathematical plane,
theexistence of God. I suggest you approach God by faith and
you will?d God
===
> >> This is an interpretation of
Christopher Langan's CTMU, www.ctmu.org>> , and Saint
Anslem's ontological argument.> > Anselm's argument is 
an
excercise in begging the question.> It is -worthless-.
Nothing exists by de?ition. Existence has to be > proven or
observed.> I beg to differ. If I de?e God as the best
football player of all times, then he by de?ition exists.
Of course the de?ition is nonsense, and I'm unlikely to get
anyone else to follow my arguments, unless I start my speach
In this three hour lecture, I will use the word ?God' to
denote the best football player of all times. Forget about
who created the world. Forget about the meaning of life. This
lecture will talk only about football and hot dogs.Russell's
de?ition is very much of the football player kind.Acke
===
>
>>>>> This is an interpretation of Christopher Langan's
CTMU, www.ctmu.org>> , and Saint Anslem's ontological
argument.>>> Anselm's argument is an excercise in
begging the question.>> It is -worthless-. Nothing exists
by de?ition. Existence has to be> proven or observed.>
I beg to differ. If I de?e God as the best football player
of all> times, then he by de?ition exists.Assuming your
de?ition de?es anything. There is no reason whythere should
be any such football player.Compare and contrast your
de?ition with:Ide?e Dog as the largest negative number
greater than 7,does this exist just by this de?ition?> Of
course the de?ition is> nonsense, and I'm unlikely to get
anyone else to follow my arguments,> unless I start my speach
In this three hour lecture, I will use the> word ?God' to
denote the best football player of all times. Forget about>
who created the world. Forget about the meaning of life. This
lecture> will talk only about football and hot dogs.>>
Russell's de?ition is very much of the football player
kind.>> Acke>
===
>>The idea is not worthless, but doesn't
have a lot to go on...Its like de?ing, giving a limit to
in?ity, its not going to comedown to an equation but an
idea...in?ite knowledge...hmmm....yea
===
> This is an
interpretation of Christopher Langan's CTMU, www.ctmu.org> ,
and Saint Anslem's ontological argument.> 1.] If it is
possible for a mind to perfectly understand[model] every>
aspect and detail of reality, then the mind that perfectly
models> reality is a super-intelligence, for all intents and
purposes, the> super-intelligence is God.Since reality is
ever-changing it's not possible to comprehendall of totality
at once. Perhaps the better de?ition ofGod-like intelligence
is having the knowledge and powerof creation.>> 2.]If the
perfect correspondence can be approached via a convergent>
analytic-synthetic propositional limit, then the limit
exists, even> though a sentient mind within reality can only
approach the limit.The knowledge of creation, God-like
knowledge, can be modeledmathematically as a limit. The
mathematics of evolutionary emergentproperties serves as a
basis for such a conjecture.An Introduction to Complex
SystemsTorsten Reil, Department of Zoology, University of
Oxfordhttp://users.ox.ac.uk/~quee0818/complexity/
complexity.htmlUnderstanding that evolution is a property of
randomness, inboth non-living and living structures, provides
a relationshipbetween time and emergent steps for the
following reasons.Random interactions are the driving force
for evolution.Evolution produces emergent properties not
de?ablefrom analysis of the parts.Emergent properties happen
when an evolutionarysystem reaches self-organized
criticality.Evolutionary systems attain self-organized
criticalityas fast as practical.These properties of evolution
are not constrainedto living systems, but also apply to
non-living andintellectual systems as well.Emergent
properties occur in power law bursts, largeevolutionary steps
are normal.Self-Organizing Systems (SOS)
FAQhttp://www.calresco.org/sos/sosfaq.htmINVESTIGATIONSSTUART
A.
KAUFFMANhttp://www.santafe.edu/s?People/kauffman/
Investigations.htmlOnce one understands the mathematics of
evolution it isfairly easy to show that our level of
intelligence doesnot de?e an upper limit.Taken to it's
limit, these properties make clear that theemergence of a
level of intelligence as far a above usas we are above
animals...is a mathematical certainty.The universe, taken as
a single evolving system, is certainto evolve what it needs
to reproduce itself. And thatconclusion provides a basis for
the conjecture thatthis universe was created in a similar
way. Bynaturally evolved intelligent design.>> 3.] If the
limit exists, the exact mental correspondence exists in the>
mind of a super-intelligence.>> 4.] That is to say, if the
limit exists then a description exists.>> 5.] If the
description exists then the describer exists, since the>
description is isomorphic.>> 6.]The describer is a
super-intelligence.>> 7.] By de?ition, the
super-intelligence is God.>> The burden of proof becomes the
burden of proving the convergence,> to an exact
correspondence, between the mental construct[in?ite> number
of axioms] and realityIn?ite number of axioms? There is only
one.All creation and structure in the universe takes the
formof an iterated loop into itself of two variables. The
?stbeing classical motion, the second quantum motion.>> At
the limit>> [MIND]<--->[REALITY]The ...convergence of quantum
and classical motion over timeto evolutionary processes is the
limit.>> M = R>> [axiomatic method]--->[exact
correspondence]<---[scienti? method]
[light]--->[self-organization]<---[motion]As time goes
towards in?ity, so does order and
intelligence.Jonathans
===
> Do these sequence, especially the
Beth sequence have ?ed points? Yes, they must, by an
elementary theorem. The alephs are wellordered, and all the
alephs together are a proper class. (Likewisefor the beths.)
Any such class is isomorphic to the class of allordinals, so
there must be a bijection between the class of alephs andthe
class of ordinals. (That bijection is, in fact, the
functionaleph.)And you know this!> Does the Oswald Veblen's
theorem, that normal ordinal functions have an > unbounded
trans?te series of ?ed points, apply to Aleph and Beth >
sequences?Why wouldn't it?Thomas
===
> All this looks ok now.
For your next mind-boggling trip, may i suggest you> have a
look at inaccessible cardinals?>Finally! Ok, I've pulled
down some web stuff about them, so at least Iknow the
de?ition. So yes, the aleph and beth sequences we
consideredare limit cardinals and strong limit cardinals, but
not very big limitsfor being singular.In addition, from
previous reading I recall that tho it's possible toprove the
consistency of rejecting inaccessible cardinals, it's
impossibleto prove acceptance of inaccessibles is consistent
(relative toconsistency of set theory).I'm much humored by
those results, in effect saying tho it's quitereasonable to
reject the unimaginably big, there's no assurance
whatsoeverthat accepting the unimaginably big is reasonable;
posing pesky overtonesfor those who entertain similar
overwhelming concepts such as almighties.I'm also aware these
are the smallest of a likely trans?ite series ofever bigger,
ever more ghost like cardinals, of which only a ?ite
starthas been perceived thru the fog of whimsy existence:
Mahlo, compact,measurable, huge, extendible, remarkable; just
to mention a notable few.There yet remains the expressions
?unusual', ?notable', ?too 
much','weird', 
?incomprehensible',
?transcendent', ?unmentionable'. ;-)It'll 
take me awhile to
ponder the de?itions and perhaps reviewco?ality before
attempting an intelligent question about them.Have you
anything you'd like to point out about them?
===
> All this
looks ok now. For your next mind-boggling trip, may i>>
suggest you have a look at inaccessible cardinals?>
Finally! Ok, I've pulled down some web stuff about them, so
at least> I know the de?ition. So yes, the aleph and beth
sequences we> considered are limit cardinals and strong limit
cardinals, but not> very big limits for being singular.>> In
addition, from previous reading I recall that tho it's
possible to> prove the consistency of rejecting inaccessible
cardinals, it's> impossible to prove acceptance of
inaccessibles is consistent> (relative to consistency of set
theory).>> I'm much humored by those results, in effect
saying tho it's quite> reasonable to reject the unimaginably
big, there's no assurance> whatsoever that accepting the
unimaginably big is reasonable; posing> pesky overtones for
those who entertain similar overwhelming concepts> such as
almighties.>> I'm also aware these are the smallest of a
likely trans?ite series> of ever bigger, ever more ghost
like cardinals, of which only a> ?ite start has been
perceived thru the fog of whimsy existence:> Mahlo, compact,
measurable, huge, extendible, remarkable; just to> mention a
notable few.>> There yet remains the expressions ?unusual',
?notable', ?too much',> ?weird', 
?incomprehensible',
?transcendent', ?unmentionable'. ;-)>> 
It'll take me awhile
to ponder the de?itions and perhaps review> co?ality before
attempting an intelligent question about them.> Have you
anything you'd like to point out about them?Actually, before
going there, you might ?st be entertained by this verynice
essay (alas, in french) about large objects and in?ite ones
:http://www.eleves.ens.fr:8080/home/madore/math/
in?ity.pdfBut already, measurable cardinals are really
interesting, as it is not clearat all at ?st sight that
they must be huge, then not clear at all how muchhuge (with
respect to inaccessible) that is. In fact, a good start is
toponder why inaccessible cardinals are so much bigger that,
say, theBeth^(omega0)(0)-th ?ed point of the Beth
sequence...
===
I'm college freshmen currently taking
multivariable calculus andconsidering double majoring in math
and computer science. One thingthat's been dif?ult for me
lately is that my teacher grades not justbased on the
correctness of our work, but also the readability. Sheseems
to grade on the organization of our work (if she can't follow
itwhen grading, she takes off points), as well as how well we
explainwhat we have done in words, with graphs, and worked
solutions. I've been programming for quite a while, and I
know that in thatcontext it is very important to write code
that is readable tosimplify debugging and modi?ation.
However, I have only beenstudying mathematics seriously for a
short while and I have yet toovercome many of the bad habits I
developed in high school. If anyone has any tips or knows of
any online/of?esources formath style, it would be much
appreciated. On a side note, I am also not as used to
communicating withmathematicians verbally or in writing as I
am with programmers orengineers in general. When I run into
terminology I am unfamiliarwith, I turn to
www.mathworld.wolfram.com; however, if anyone knows ofany
other and/or better resources I could certainly use
them.-Brendan
===
I want to prove the following:Let f: R --> R
be a continuous function such that(1) f((a + b)/2) >= (f(a) +
f(b))/2, where 0 > a > b,(2) f(x) + f(y) >= f(x') + 
f(y')
implies f(r*x) + f(r*y) >= f(r*x') +f(r*y'), for any 
x, y,
x', y', and r (with r > 0).Then(3) there exists some 
negative
x such thatf(x) - f(2x) >= f(0) - f(x).Now, I believe I can
prove that (1) and (2) imply:(4) for arbitrarily small
negative e, there exists some x < e suchthatf(x) - f(2x - e)
>= f(e) - f(x).And, although I can think of functions that
satisfy (4) but not (3),none of these is continuous. (For
example: let f be such that f(0) =0, and f(x) = x - 1 for all
negative x; then f satis?s (4) but not(3), and f is not
continuous at 0.)So here's my hunch: (4) plus the continuity
of f implies (3). But Ican't work out how to prove it.Any
help would be gratefully received.
===
Thought some of you
might like to know about this
site.http://www-neos.mcs.anl.gov/neos/Enjoy, Flip