mm-104
im 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 rst
derivatives withn = oor(alpha) 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 rst derivatives with> n =
oor(alpha) equal to zero.Careful, careful; he didnt 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 MITs website to nd theindenite
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 Mathematicacant do it its probably not
integrable in terms of elementaryfunctions. Well, after about
two months of toiling over this problem,I think I might
nally have a solution. However, theres 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 theres any aws.So, the problem is to nd in
closed formIntegral[f(x) dx], where f(x) =
x/Sqrt[x^4+4x^3-6x^2+4x+1]So, rst 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, rst 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 xs with us
in the rstintegral 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 theresa (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
website to nd the> indenite 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> cant do it
its probably not integrable in terms of elementary>
functions. Well, after about two months of toiling over this
problem,> I think I might nally have a solution. However,
theres 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 theres any
aws.>> So, the problem is to nd in closed form>>
Integral[f(x) dx], where f(x) = x/Sqrt[x^4+4x^3-6x^2+4x+1]>>
So, rst 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, rst 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 xs with us in the rst> 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 theres> 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 xs with us in
the rst> integral and get>> 2I = Integral[(1 - 1/u^2)f(u)
du]It would be true if the integrals would be denites. But
not if they areindenites. The rst 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 rst
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
anyway, rst 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 xs with
us in the rst> integral and get>> 2I = Integral[(1 -
placeholder, because you aredoing an indenite 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 nd in
closed formIntegral[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 $deg D(x)=2g+2$.
Specically, $f(x)=p(x)/q(x)$ (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 elds 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 elds.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 MITs website to
nd the> indenite 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> cant do it its probably
not integrable in terms of elementary> functions. Well, after
about two months of toiling over this problem,> I think I
might nally have a solution. However, theres 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 theres any aws.So, the problem is to
nd in closed formIntegral[f(x) dx], where f(x) =
x/Sqrt[x^4+4x^3-6x^2+4x+1]So, rst 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 a> substitution
of the form t = x+1/x. So I use this to my advantage>
later.So anyway, rst 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 xs with us
in the rst> integral 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 a>
poloynomial 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 theres> 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 givesI = 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 denitely made one mistake in that
substituting v should notgive you what I put. I think the
nal 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 MITs website to nd the> indenite 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> cant do it its probably not integrable in
terms of elementary> functions. Well, after about two months
of toiling over this problem,> I think I might nally have a
solution. However, theres 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
theres any aws.So, the problem is to nd in closed
formIntegral[f(x) dx], where f(x) =
x/Sqrt[x^4+4x^3-6x^2+4x+1]So, rst 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 a> substitution
of the form t = x+1/x. So I use this to my advantage>
later.So anyway, rst 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 xs with us
in the rst> integral 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 a>
poloynomial 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 theres> 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 givesI = 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 denitely made one mistake in that
substituting v should not> give you what I put. I think the
nal 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 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)*(
fractions in x, thenintegrate.> About 2 months ago I ran
across a problem on MITs website to nd the> indenite
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> cant do it its probably not
integrable in terms of elementary> functions. Well, after
about two months of toiling over this problem,> I think I
might nally have a solution. However, theres 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 theres any aws.>> So, the problem is
to nd in closed form>> Integral[f(x) dx], where f(x) =
x/Sqrt[x^4+4x^3-6x^2+4x+1]>> So, rst 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, rst 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 xs with us in the rst> 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 theres> 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
its 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 MITs website to nd the> indenite
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> cant do it its 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 nally have a solution.
However, theres 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 theres
any aws.>> So, the problem is to nd in closed form>>
Integral[f(x) dx], where f(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 MITs website to nd
the> indenite 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> cant do it its 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 nally have a solution.
However, theres 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 theres
any aws.>> So, the problem is to nd in closed form>>
Integral[f(x) dx], where f(x) = x/Sqrt[x^4+4x^3-6x^2+4x+1]>
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 dont have Mathematica, and its a
painful function to differentiateand simplify. Of course I
could have done it but theres alot of roomfor error, and if
I make a mistake differentiating Im likely to thinkthat my
answer is wrong, when in fact its not. However, you said
itis wrong, so I assume you differentiated the result using
Mathematicaand didnt 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 dont have
Mathematica, and its a painful function to differentiate>
and simplify. Of course I could have done it but theres alot
of room> for error, and if I make a mistake differentiating
Im likely to think> that my answer is wrong, when in fact
its not. However, you said it> is wrong, so I assume you
differentiated the result using Mathematica> and didnt 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 sufcient precision on the calculatorand use
common sense in selecting delta-X.
=theorems. Not being
able to gure out that proof was really buggingme, Ill sleep
better now.>>If F is given consider the sequence of
functions (dened 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 dened>on any circle, not just the unit
circle.]Well you can relax, its true for any closed disk.
(In fact its true> in much greater generality, although its
not so easy to prove; > Runges theorem, included in most
complex books, gives a> generalization, and Mergelyans
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 Cauchys criterion for uniform
convergence, just>Cauchys criterion for convergence - too
obvious an extension to state>explicitly no doubt. But I did
nd 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 cant seem to nd a proof that>>seems needlessly
complex or just plain doesnt work... can someone see>>a
simple argument? I would be much appreciated - I dont even
need>>this result for anything but somehow I just cant 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 cant
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 thats not what the hint suggests - read the hint
again...>>If the polynomial sequence is given then its
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?>> Dont 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 efcient. 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
efciency 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 efciency is maximized at e.This
argument is my own, but Im sure it has beendone better
elsewhere, because everything I thinkI have thats 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 Eulers formula e^(ix) =cos(x) + i sin(x)),
but this is a bit of an over-simplication.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
knowthats a bad description, but its 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 Hals ideas on metric engineering.Excerpts from
Puthoff & IbisonsH.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. Specically, quantum eld theory tells us
that, even in empty space, elds (e.g., the electromagnetic
eld) continuously uctuate about their zero baseline
values.So far, so good. However IMHO it is the QED PV virtual
electron-positron zero point vacuum uctuations that dominate
the low energy effective macro-quantum eld theory out of
which Einsteins gravity together with exotic vacuum unied
dark energy/matter elds co-emerge like love and marriage all
together as phase and amplitude modulation patterns,
respectively, of the vacuum coherence eld 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 uctuations is called
zero point energy (ZPE), reecting 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 ...Thats a strange way of putting it. The local
net random micro-quantu zero point stress-energy density
tensor eld isTuv(Exotic Vacuum) = (String
Tension)/ZPEguvusing Ed Wittens natural units h = c =
k(Boltzmann) = 1Tuv(Exotic Vacuum) = (String
Tension)^2[(String Tension)^3/2|Vacuum Coherence|^2
-1]guvWhat is wrong with Puthoffs 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
elds 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
eld 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 rst 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 Collectors 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.
Thats 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!
;-)Thats the Brute Force approach.That is notThe Tao Chi of
ET!STRIKE THREE!Youre 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 rst 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-uctuation 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 le)to be
continued.Engineering the Zero-Point Field and Polarizable
Vacuum for Interstellar Flight1. IntroductionThe concept of
engineering the vacuum found its rst expression in the
mainstream physics literature when it was introduced by T. D.
Lee in There he stated:The 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.This
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 eld uctuations, 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 yleaf of a
book of essays by Einstein and others on the properties of the
vacuum we nd the statement The 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-ight
applications. To begin, in the case of quantum uctuations
there are uncertainties that remain to be claried regarding
global thermodynamic and energy constraints. Furthermore, the
energetic components of potential utility involve very
small-wavelength, high-frequency elds and thus resist facile
engineering solutions.This 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 eld 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 Einsteins
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 M Theory that yields
something like:G*-String Tension = e^-(r*/Lp)^?(G-String
Tension)Where r* = compactication 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:With 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 empty 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 eld
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.2. Propellantless Propulsion2.1 Global
ConstraintRegardless of the mechanisms that might be
entertained with regard to propellantless or eld propulsion
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 eld 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 FlightIt has been known
since the late 1950s how to circumvent this problem provided
one has exotic stuff with a strong enough negative pressure
to provide what Herman Bondi called negative matter in the
late 1950s and Kip Thorne called exotic matter in the late
1980s in his traversable wormhole time travel papers with
additional work by Igor Novikov and others. Robert Forward
summarized Bondis early work in a paper written in the early
1990s. 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 dark 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 modied 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
dark energy exotic vacuum that is ~ 3/4 of large-scale stuff
of the Universe./zpf < 0 is positive pressure dark matter
exotic vacuum that is ~ 1/4 of the large scale stuff of the
Universe.Gravity effect of exotic vacuum stuff ~ 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
Newtons.The key idea not found in any of Puthoffs papers on
the subject of metric engineering is the local Einstein
stress-energy tensor current densitys covariant divergence
Tuv^;v where ;v is the Diff(4) covariant derivative relative
to the metric torsion free connection eld for parallel
transport of tensors along tangent vector elds 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
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 soft 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 Ofcer 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 rst 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 ying saucers.
They, with Jacques Vallee, have been active in the NASA
Breakthrough Propulsion Program and in UFO groups some
nanced 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
ying 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 y. The 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 NOVAs Elegant Universe with Brian Greene, in Stephen
Hawkings The Universe in a Nutshell, Michio Kakus
Hyperspace and Igor Novikovs 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
dened as the practical control of space and time warps. I
have discussed this in Paramount Pictures DVD Special
Collectors Edition of Star Trek IV: The Voyage Home in Time
Travel: The Art of the Possible and in Learning Channels
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 ction 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 Disneys Fantasia we could, in our incompetence, destroy
the entire Universe just as we are now surely destroying
Earths 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-fullling prophecy in a precognitive remote
viewing of our future by Saint John in The Bibles
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 Einsteins
general theory of relativity and micro-quantum mechanics
including the More is Different emergence of macro-quantum
superuid order in ground state and vacuum instabilities
developed by condensed matter Princeton physicist P.W. in the
inationary chaotic cosmology of A. Linde consistent with the
issue of Scientic American.Lets start with Einsteins
gravity eld equation of ~ 1915. Einstein viewed pure
space-time geometry like the marble in a statue by
Michelangelo with gross matter and radiation as wood.
Einsteins 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 conrms that this way of
writing Einsteins 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 unies the ve 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 Einsteins
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 Einsteins 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)Einsteins 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
magnication is so large that quantum gravity zero point
energy density uctuations in the space-time geometry itself
get large. There may also be torsion elds not included in
Einsteins 1915 geometrodynamics that may require a
modication 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. Hawkings imaginary time and its connection to
inverse temperature is part of that same story.The key
concept for metric engineering is the Einstein current!
Einsteins eld 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 eld, 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 eld is added, like a torsion eld,
there is an additional piece added to this connection eld.
Einsteins 1915 theory is a degenerate case of the bigger
unied eld 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.
Einsteins 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
eld 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 Einsteins
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 Einsteins 1915
theory is the nal 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
=Im 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.Ive 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 at oval).
Ive 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 denite
line, aninward slope as the two squares/patches meet. As far I
can tell thetangents line up along the join, I just dont have
a clue why the joinisnt 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
=Im 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.Ive 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 at oval). Ive 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 denite line, aninward slope as the two
squares/patches meet. As far I can tell thetangents line up
along the join, I just dont have a clue why the joinisnt
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 ns 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 nite 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 ns??I assume that
each n must approach innity 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
Because
Khinchtines constant exists and is nite, we know that the>
distribution of the integers among the terms of the simple
continued> fraction of a sufcently 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 xs> continued
fraction terms approaches Khinchtines 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
rst m CF terms of mostpositive reals would exceed that, on
average, among the rst mpositive integers in general, on
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 grids
squares in> any order the players wish.(We do not actually
need the counters, for players can simply write> the numbers
in the grids 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-players
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*4Columns
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 games strategy.)Any interesting
=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=0Thats true. Now youve got many methods to
solve 3rd degreesequations, 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
hexagons 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 reected>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 paths nal>direction is heavily dependent upon
the accuracy in which the path is>drawn and the accuracy of
the angles reected.>>But I will give the order of the
hexagons 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 awed.)>>(I have no idea.
You best use exact-rational arithetic to get this,if>it is
possible to gure 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
angle chase, heres one description of it.> Ive tried it
three times now, so either its right or Im 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*thetaBut 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 modied version of the problem
might have the pattern as A, E, B, D, 2, F, E, 2, E, 2,
startGeoff.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 innitely towards point V.But a
number of other things could perhaps happen, for my
diagram(hand-drawn) is ambiguous.(Around points V and Z,
Quet
=sum[k^2/k!] from k=1 to k=infSurely there are some
cute algebraic tricks to get 2e out of this, Ijust dont 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 dont 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=infThat equals
sum_(k=1,oo) k/(k-1)! = sum_(k=0,oo) (k+1)/k! = d(xe^x)/dx at
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 dont 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 dont 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/heropponents 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 players 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 foradvanced play). If we have n colors,
we would have (2^n -1) piecesto get all combinations of colors
=I was looking through
my les the other day and found a simple Fermatme, but thats
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 :-)Fermats 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).Fermats 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 lled by James Harris and has beenfor a decade or more.
Youll 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 les the other day and
found a simple Fermat>me, but thats 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
:-)>>Fermats 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).>>Fermats
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 cant 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)^nYou 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 cant 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 orsatised 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 youre crazy. Statistically, itspretty
much true, and hardly any seasoned mathiewill take the time
to nd the aw in your proof.If you want to learn math, and
interact withmore experienced mathies, I suggest nding
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 tond the mistakes in your proof for
yourself.Believe me, unless you really suck at math(or are a
crank),you truly are capable of nding your
ownmistakes.Finally, and a bit hypocritically, I tried tond
the aw in your proof but right away Icant 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 xedpositive integer
triple x,y,z and xed 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 les the other day and
found a simple Fermat> me, but thats 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
:-)Fermats 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)^nNotice 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).Fermats 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+1Sequential 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 qualies as a
proof,we must rst agree on a denition 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
conrm theaxiomatic nature of this statement, perhaps someone
else can conrmor 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 denition 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 condent that we caneither conrm 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 youre crazy. Statistically,
its> pretty much true, and hardly any seasoned mathie> will
take the time to nd the aw in your proof.If you want to
learn math, and interact with> more experienced mathies, I
suggest nding 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> nd the mistakes in your proof for
yourself.> Believe me, unless you really suck at math> (or
are a crank),> you truly are capable of nding your own>
mistakes.Finally, and a bit hypocritically, I tried to> nd
the aw in your proof but right away I> cant 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 xed> positive
integer triple x,y,z and xed 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.TylerI was looking through my les the
other day and found a simple Fermat> me, but thats 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 :-)Fermats 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)^nNotice 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).Fermats 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+1Sequential 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 qualies as a proof,> we must
rst agree on a denition of the word.No, we already have a
denition. 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 Peanos axioms, which are the
verydenition 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 dont 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)^nYouve 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 conrm the> axiomatic
nature of this statement, perhaps someone else can conrm> or
deny it.The equation as stated above certainly does not
require any particularcommon DIVISOR for (n-1) and (n). And
Im 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
dont 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 conrm the>axiomatic
nature of this statement, perhaps someone else can conrm>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 Fermats Last Theorem. Dont 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 Fermats Last Theorem
has ton and n-1 being relatively prime, but that assertion is
notjustied. 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 denition 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.
about what I accept as reality. --- Calvin (Calvin and
Hobbes)
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 dont enter into it.- -
-snip-> Your understanding of what an axiom is is certainly
incorrect.I appreciate you attempting to polish up my
denition, 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 denition: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> nd the aw
in your proof but right away I> cant 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 youre 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, its> pretty
much true, and hardly any seasoned mathie> will take the time
to nd the aw in your proof.Thats 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 les the other day and found a simple Fermat> me, but
thats 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
shy, 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 doesnt fail for n = 2 then it proves
something wrong, so even if you dont 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 shy, 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
circlescircumference to its diameter in a rather subtle,
not-obvious way.
=fuffy> I was looking through my les the
other day and found a simple Fermat> me, but thats 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 :-)>> Fermats 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).>>
Fermats 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 rened 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 difcult to nd
recent comments>> in more than one.> What is to be done
about Fuffy?> Duh-h-h. Dont ask me.> Well, I suspect
virtually every Google user but you>> already knew about sort
by date and how to nd >> recent posters.> Im 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 wont bother to read the last
postingThen youll have no idea whether or not 15 people have
posted sincethe last time you read the thread.
>
fuffy> fuffy> Fuffy has rened 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
difcult to nd recent comments>> in more than one.> What
is to be done about Fuffy?> Duh-h-h. Dont ask me.>
Well, I suspect virtually every Google user but you>> already
knew about sort by date and how to nd >> recent posters.>
Im 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 wont bother to read the
last postingThen youll 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, Ill bet
there are Google Users other than myself who dontyou seem to
want to believe, and for people who dont know how itworks,
McCallum s 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
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 dont have to be
identical, though. Lindebergs 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 -> innity 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 -> innity.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 dont 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 < innity,with probability 1 all but
nitely 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 dont have to be
identical, though. > Lindebergs 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 -> innity 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). Ill 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 Lindebergs 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 dened
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
categry!----------------------------------------------------
--------------------------
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: Whats 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 nd 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>> Ive got a list of
points (in x,y form) that are the corners of a> polygon, in
order. Its 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
-Adrian> This is probably in the faq at
comp.programming.games.algorithms, or> something like that.
Google will probably also help.>> I know its 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 Euclids proof of innitude 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 wont y, butIthere> is no largest
prime (and therefore an innite number of primes) is as>
follows:>> If there is a largest prime, P, then the number of
primes is nite. 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. Youve simply proved that
the nite list of> known primes is not the list of all
primes.>> If you use Euclids 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 nd the new primes
{547,607,1033,31051}, all of which> are less than the
currently largest known prime 3263443.>> Dont forget the
second case! :-)>> Dont forget that if youre pretending
that you dont know what> the primes are, you cant suddenly
pull otherwise well-known> features of the prime numbers out
of a hat.>> Phil> --> Unpatched IE vulnerability:
ADODB.Stream local le writing> Description: Planting
arbitrary les on the local le 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. > Its 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 theres 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 its 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
shy!)-------------------------------------------------------
----------------------- Bill Taylor
W.Taylor@math.canterbury.ac.nz-------------------------------
----------------------------------------------- Camouage
condoms: So they wont see you
coming.------------------------------------------------------
------------------------
I should note that the above
carries no attribution. It is>>copyrighted material
originally published in The Onion,>>but I couldnt locate any
of the original Onion material>>online anywhere. >>Ive
already posted the Onion material (actually, Ive posted the
link>to the material fron the Onions own website) elsewhere
in this thread.Huh. I didnt realize their archive went back
that far. I once triedand eventually gave up. Its 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) dened 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 thats 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) dened 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 theres 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:
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 doesnt 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 eld 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 denition
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
r,s to (r^2,r) and (s^2,s), resp., then r^2.s= s^2.r iff r =
s...>
what I accept as reality.> --- Calvin (Calvin and Hobbes)>
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 (justied 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 nal 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 (justied 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
=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
odentity that allows you to write the answer down
immediately,similar to the binomial theorem?
Set R_1 =
x+y, R_2 = xyIs 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}^innity (x^n + y^n)t^n.ThenF(t) = 1/(1-xt) + (1-yt)
= (2 - ut)/(1 - ut + vt^2) = (2 - ut)sum_{m=0}^innity (u +
vt)^m t^m = (2 - ut)sum_{n=0}^innity t^n sum_m {m choose
n-m}u^{2m-n} v^{n-m}etc. Pulling out the coefcient of t^n
gives D_n(u,v)the Dickson polynomial of the rst 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?Heres 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 coefcients A and B may
be found by considering the initial
conditions.-Michael.
=Set R_1 = x+y, R_2 = xy> Heres 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 Newtons 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.Dont you think that the
solutions to this equation would be x and y.... > The
coefcients A and B may be found by considering the initial
conditions.... with A=1 and B=1? So this approach wont 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>> Heres 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
Newtons 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.>> Dont you think that the solutions to this
equation would be x and y....Indeed....>> The coefcients A
and B may be found by considering the initialconditions.>>
... with A=1 and B=1? So this approach wont 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 =
xyIs 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 Newtons identities (a useful recurrence
relation) andWarings formula. They are also helpful for a
larger number of variables(x,y,z,etc.). I dont know what
would be the simplest formula for twovariables. Wouldnt any
formula for Dicksons 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 dont personally meet.>
Its just that extraordinary claims require extraordinary
evidence. What you > present on your web site cannot even be
considered evidence, let alone> extraordinary evidence. Im
supposed to take your word that if presented> correctly, the
lines would be parallel? Thats not how science is done.I
agree with that. As I said, the main investigation was done
byCrater&McDaniel. I didnt 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 tted 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 t).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-rectied 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 lets 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.Dont you have your priorities wrong? Shouldnt 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 orthorectied, 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, wouldnt 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 wont 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 dened one, testable
with analitical geometry.But based on a diagram, so... could
I have missed some conguration?I recommend to read most of
the material I present, at leastsupercially, before making
comments, and let the childish I am moreintelligent than you
issues out of the conversation.
Message-id:
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> dont 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. Thats 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 didnt 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:
>> You just cant _prove_ its 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 conguration>> has this property.> Duh.>>
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? Dont you mean a
formation?>>Instead of wasting your time uploading gifs and
accusing people, why>dont you Why doesnt 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. Thats all there
is to it.Youre 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 cant bet>on it. Have you thought about _why_ they wont
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 didnt see this
example, so Ill 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 shouldnt,
roullette payouts are 35 to 1. Of course, you would know
thisif youve ever been to a casino.>Just take your money
back and thats>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:
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
cant explain it. Put a coin in a box and shake it. Without
looking, what is the probability ofshowing heads? This isnt
quantum mechanics. It does not depend on whether youlook or
not. It is either heads or it isnt. 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 dont 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 isnt quantum mechanics. It does not
depend on whether you> look or not. It is either heads or it
isnt. 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 dont 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 isnt quantum mechanics. It does not
depend on whether you> look or not. It is either heads or it
isnt. 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 dont 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 wouldnt 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 lets say
you want to use it anyway. What isthe probability that you
will get heads on two consecutive coin ips?Before we start,
we know that the probability of both is the probabilitiesof
the individual ips multiplied together: 1/2 * 1/2 = 1/4.You
ip the coin and it comes up tails. NOW what is the
probability thatboth ips will be heads? We know the answer
is 0, we know the probabilityof the second ip is 1/2. The
probability of the past event being headsmust therefore be
0.Similarly, had the rst ip been heads, we succeed if and
only if thesecond ip is heads. Thus, the chance of success
(at this stage) is 1/2.But thats the probability of the
second individual ip. Therefore, 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 (lets not
re-argue the geometry issue)2: they DID form that way and
your estimate of likelyhood is wrong3: coincidenceThats
where statistics comes into play, to help distinguish case 2
from cant prove anything.
=Hey, Mensanator, that post is
from other poster.
Message-id:
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 gment 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. Ill 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
denition 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 isnt 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. Ill 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 phrasea miss is as good
as a mileLet 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 denition 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 isnt going to get you
anywhere.
=The paper I refer was published in Journal of
Scientic 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 isnt 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 nd 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
scientic test, except a lot of helpfulcommentaries with
Crater & McDaniel, and some personal replications ofpart of
their work.I personally veried my model of parallel lines
(which differs fromtheirs though the ideal polygon is
completely identical), getting agure 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 t-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 gure deviates from an ideal
model. I didnt 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 scientically 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 wouldnt 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 scientic
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 its on. Why is D six pixels below and 3
pixels > to the right of the mound you claim its on?That is
greater deviation from an adjusted t by computer.
Anyway,there is a deviation for an ideal gure. 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 dont
know if the t is better (or worst) when measuringground
distances, instead of aerial.3) Orthorectication 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
t, 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 doesnt
need the likes of me sticking up for his > character, but the
evidence is that you are spouting gibberish > and displaying
agrant disregard for principles of probability,> and
Mensanator is displaying a perfectly sufcient knowledge > of
geometry to interpret and invalidate your illusions, a
likwise> sufcient 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 dont get
money, fame or whatever with thisissue. My quest is
completely a curiosity-based one.I am a physicist, I have
talked with PhDs 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 youve 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 difcult than
what it was.You didnt miss anything. On the contrary, that
got me to the --
KindlyKonrad-------------------------------------------------
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 denition the commutative eld
(R2, +, r.s.m., * ) and * is given in
(a,b)*(u,v)=(au-bv,av+bu) ?The pleasure of nding out (and to
be the rst to publish) i leaveto yourself. My opinion ill
tell You in about a month time, in themeantime You might nd
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-elds have to be worked on.C was strolling
around with his apologist Riemann sur Face, so calledcause he
could give a broad smile on his face- as the reection 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 ank of 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 nd 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 isnt a word>> in the UK?
Does that mean that trillion isnt 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.> -->
In message
<bp4jq1$91$1@sparta.btinternet.com>, Richard Heatheld> A
quote from:>> The New Fowlers Modern English Usage, Third
Edition, Edited by R. W.>> Burcheld, 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
isnt 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 bres 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 brewise 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 reect ** 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 bres 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 brewise an isometric embedding? If so,
what is a reference>to this result?>>Ignorantly,>Allan
Adler>ara@zurich.ai.mit.edu>> ...Im not sure, but you may be
able to show this for compact M bychoosing a trivial bundle
over T of sufciently 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 Coefcients , and the
leadingcoefcient 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
coefcient of fn must be 1.e.g e to the power x^2 is in V
because it satisesf(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 Coefcients , and the leading> coefcient 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 coefcient of fn must be 1.e.g >
e to the power x^2 is in V because it satises>
f(x)-2xf(x)=0prove 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 coefcient of the ODE MUST BE 1is equivalent to the
more reasonable statementthe leading non-zero coefcient is
constant. (just divide through; its 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. Couldnt 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.? Arent
points and sets of points and the aformentioned ideas
abstractions of spatial concepts?I get your (and Shmuels)
point. I apologize for my ippant, tendenciously presented
relationship between geometry and topology.Philosophically
-everything- is distinct. But then also everything has
similarities with everything else. All Im trying to point
out is that in the nebulous cloud of intuitively described
mathematical areas, some areas are more similar than others.
I think Ive watered this down enough to say absolutely
nothing. Or possibly not. Maybe.Mitch
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>
doesnt say much about it, does it?>> Oh yes, it does. That
is the beauty of Sanskrit. It is not verbose;> it just denes
the heart of things, as briey as 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 doesnt 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 doesnt 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.)
you only have to remember one> number, all along the
process.>> fact.>> But is _that aspect_ of the method
Vedic? The expression crosswise vertical> doesnt say much
about it, does it?>> Oh yes, it does. That is the beauty of
Sanskrit. It is not verbose;> it just denes the heart of
things, as briey as 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 doesnt 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 totallyhorried 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 fromrst 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 doesnt 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 nd out a new and
better multiplicationmethod. Who knows what will happen to the
eld 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.
certainly true that the stupidest mistakes are often> made by
the smartest people.>> No, this is illogical.Its 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 wont. It would feellike explaining a
joke.Herman Jurjus
stupidest mistakes are often> made by the smartest people.>>
No, this is illogical.Its 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 ndout 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 wont. It would feel> like explaining a
joke.Herman Jurjus
true that the stupidest mistakes are often>>made by the
smartest people.>>No, this is illogical.>>Its 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 supercial. 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,denition of
smart is somewhat different. In general, the acceptedidea is
related to innovativeness/creativeness in your eld 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 nd> 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 wont. 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 cant gure out what happens to the series. Also, does
thismatrix have a special name? I think Ive seen it before,
but dont knowwhere.JR
If A can be diagonalized as PAP,
then computing A^n is as simple asI think one of these As 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 cant gure out what happens to the series. Also, does
this>matrix have a special name? I think Ive seen it before,
but dont 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 1s on thejth superdiagonal and 0s elsewhere. If
f(z) = sum_{j=0}^innity c_j (z-L)^j, then f(J) =
sum_{j=0}^innity 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
cant gure out what happens to the series. Also, does this>
matrix have a special name? I think Ive seen it before, but
dont 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 nd 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 cant gure out what happens to the
series. Also, does this> matrix have a special name? I think
Ive seen it before, but dont 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 nd
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 nd 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 dont
see the point of this added requirment as the nalembedded
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 nite number ofend to end line segments
all contained in a nite 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, 11If I were assuming independence, Id
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 = 1.3But are they independent? Does event A depend
on when B or> C occurred, or how many times A has
occurred?Lets 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 time15. So the mean
inter-event measure is rather out-of-date. It looksto me as
though this event Cs 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, Id 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 cantfall 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 reect the 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 Cs cluster of occurences has probably> ended, and I
have little expectation that the event will occur again> in
the near future.Yes, youre right. Somewhere along the line I
got confused between Bs data and Cs data. The analysisabove
reects 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 dont get
ourselves a legalght, wait for part III of my journal and
youll see what I mean.Happy however to have my own
observations totally conrmed.see you around, Guenther>
However there is something else I want 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. Im going to try and> gure 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. Im 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 ctitious 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.Im
quite convinced that my Action Device works, because,
although it> didnt y off 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!
Im left to conclude that it actually did y off> with
uniform acceleration into space.Good luck,> -Mike
=loathe
as i am to use the jsh thing in the subject, im aware that
itwould only offend jsh if i didnt, and that isnt the point
of this.sorry, james, but i am one of those dreaded
mathematicians. i hope youdont 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 at because weare
mathematicians) and deny anyones claim that disagrees with
ournarrow viewpoint. i wish that i could just write something
down andhave it instantly enter the annals, but it doesnt
work like that.instead, you ,hopefully, nd something
interesting to think about.then when youve gured out why
the rst three attempts went wrong,you talk to some other
people. then when youve possibly got somethingthat isnt
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 youve changed what needs to be
changed youmight get it submitted to some journal and the
referee points out somemore errors, where youve 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
yourereasonable. 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, ive assumed i can
invert 7 here, and it implies1/7 is in the ring of algebraic
integers. of course it isnt (you canprove that i hope, its
a simple exercise most undergrads should beable to do), but
instead of checking your assumtion youre off on acrusade. i
hope youve read enough of the commentaries to understandwhy
you cant divide by seven. it would then seem reasonable
tocorrect these aspects. of course as has been pointed out
you cant.and i mean that as an impersonal you, rather than a
personalcriciticism.however, heres 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 claritys sake
wouldnt 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 youve uncovered
ahundred year old problem, but dont state what it is, just
launch intosome unmotivated reasoning and say, voila, thats
wrong.your prime counting algorithm is at best a party trick
im afraid. ifyou could nd some new primes for us, thatd be
interesting.if you wish to demonstrate your skills, then how
about nding 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 dont 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, im aware that it> would only offend
jsh if i didnt, and that isnt 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,
im aware that it> would only offend jsh if i didnt, and that
isnt the point of this.JSH in the subject line isnt due to
Jamess vanity. It is includedso that other readers can
killle 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 *dened* to be specic
solutions to specic linear differentialequations. The fact
that an ODE is linear basically implies that we onlyhave to
solve a small number of ODEs (read: dene a small number
ofelementary functions), and this allows us to - essentially
- write down asolution to *all* linear ODEs in terms of
these functions.One could try the same with nonlinear ODEs,
but here a solution to a someODE can not be generalized to
other non-linear ODEs. 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 difcult to solve?>> Because they are
not easy to solve. There are very simple-looking> rst-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 rst-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 eqns
have a related property, found in all thetextbooks.LH
=Id
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.>>Reect on 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>
>conict and war for power and riches?????>Tom
Potter>I dont 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 conict and>war for power and riches...... or
what other reasons are>there for killing each
other?......and if the conditions were>right, Tom, youd
be in the forefront doing the same, given>the personality
you exhibit on the Usenet......ahahahaha...>Gee, I
dont get this. How do I get into this power elite?
Imusually>>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>>havent 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 inuence 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 dont get it.
Its very simple.>It is not so-called riches and power
that gets Jews into deep >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>
ing,>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
wont 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> conict
and war for power and riches.......ahahahha.....ahahaha>
>>Are you asserting that fkasner>>is against the instigation
of conict and war>>for power and riches, by non-Jews like
Bush, Blair and Rumfeld,>>and that fkasner thinks that the
root causes of how conict>>comes about should be
discussed, researched, and something>>done to eliminate the
root causes?>> Suit yourself Tom, but let me repeat for
your benet, 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
conict and 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
arguments sake.> Let me repeat for your benet, 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. Hes 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 wont 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 1900s,> and are instigating the
religious wars of the 2000s,> that fkasner implies that all
instigators of conict and 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
conict and war for prot,> 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 prot are not Jews.>> It is
interesting to see that fkasner uses the same> boilerplate
tactic that the war-for-prot 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 conict and 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
(eld 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 (eld 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 (eld 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 theres 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 (eld 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 theres 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.?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, Im 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 eld for the
characteristic polynomial of A.Or if you want to stay within
the original eld, 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.>Youre saying theres a non-measurable set E with
m(E) > 0?Depends somewhat on your denitions. If m(E) is the
outer measure,then yes. If it is only dened 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.>Youre saying theres a
non-measurable set E with m(E) > 0?Depends somewhat on your
denitions. If m(E) is the outer measure,> then yes. If it is
only dened 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. Theres 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 nd a single example where outermeasure is CALLED
Lebesgue measure.--Ron Bruck
=[...]>Youre saying theres a
non-measurable set E with m(E) > 0?Depends somewhat on your
denitions. If m(E) is the outer measure,> then yes. If it is
only dened 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. Theres 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 nd a single example where outer> measure is CALLED
Lebesgue measure.> Heres one...H. Federer, GEOMETRIC MEASURE
THEORY (1969).2.1.2 (page 53), denition 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
Russells 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 xs
and the ys are not of the same type is not
relevant.Russells 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
Rs.Witt
Theres 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 re bricks.>Whats 6T mean?> The puddle covers
about 5 square feet and didnt get smaller last> night, but
no larger either. I dont know the relative humidity or> how
well water will perk, and the water meter isnt 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 ow test results pouring water into a quart
measuring> cup while timing it:>> A) ow rates that were not
sufcient to move the water meter> 1) *1.24 gph> 30 gpd> 2)
1.77 gph> 42.5 gpd>> B) ow rates that make the water meter
move very slowly> 1) 2.63 gph> 63 gpd> 2) 1.11 gph> 27 gpd>As
A1&2 conict with B2, data is suspect.Typo? Measurement or
recording mistake?Inaccuraces of meter?Perhaps mininum useful
data is 5 gph.
Theres 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 re bricks.>> Whats 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 spilled 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.--
P.A.C. SmithThe vast majority of Iraqis want to live in a
peaceful, free world.And we will nd 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 isnt
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 severalows measured. I did not
think there was sufcient data, but (assumingreading skills)
I thought someone here was at least bright enough to know
ifthere was an obtainable answer...If you didnt glean this
from the OP, Im the plumber and I didnt want to digup the
pipes for nothing while Im dealing with the u - obviously
now, Ihave to, but -Is there an answer to the original
post?>> This aint math, its plumbing psychology detective
work.> At bed time, securely tape all the toilet and faucet
handles with duck> tape. Take ashlight, read water meter and
note time. Go to bed and> dont 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 xed the
privious> leak. Otherwise its water companys 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,> orine or whatever other
locally popular water pollutants the water> company is found
of making you drink and bath in.>>Theres 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 re bricks.> 6 what? Inches,
feet, meters?>>The puddle covers about 5 square feet and
didnt get smaller last>night, but no larger either. I
dont know the relative humidity or>how well water will
perk, and the water meter isnt 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
^spilled^ 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.> 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 isnt moving during the leak, if it is
a> leak.>Didnt seem you got any results. Rate of ow not
important, whatsimportant is what volumn is needed for meter
to register noticeablechange. Take a shower and see how much
meter moves. If much, then ushtoliet 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 sufcient 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? Thatd make for
easy spot digs tond what direction leak is moving.Did
watching puddle ll give any hint entry point into puddle of
water?When my sewer was xed, contractor dug 3 ft thru dry
hard packed dirt tohave miss his target. Eventually he gured
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> ows
measured. I did not think there was sufcient data,>> If you
didnt glean this from the OP, Im the plumber and I didnt
want to dig> up the pipes for nothing while Im dealing with
the u - obviously 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?>Ive made some comment thereto in other
post.
=One thing that simply cant be denied is that
Legendres Method isugly.Besides being ugly its clunky.Its
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 oor(10/2) -
1 = 4, to get the composites with 2 as afactor, and those
composites are 4, 6, 8, and 10. Next you takeoor(10/3) - 1
to get the composites with 3 as a factor, and thoseare 6 and
9.OOPS! What the !!! Youve DOUBLE COUNTED as you have 6
in bothfreaking lists!!!So now what to do, what to do, oh,
you subtract oor(10/6) = 1, sothat you get the correct count
which is 5 for the 5 composites4,6,8,9, and 10.Thats the
single big idea that mathematicians managed to come up
withover a hundred years for counting primes.Then they
tweaked it, ddled 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 ing stupid. I
hate mathematicians.Its 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
ing twit mathematicians.James HarrisMy math discoveries,
found for prothttp://mathforprot.blogspot.com/
One
thing that simply cant be denied is that Legendres Method
is> ugly.>> Besides being ugly its clunky.>> Its 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
oor(10/2) - 1 = 4, to get the composites with 2 as a>
factor, and those composites are 4, 6, 8, and 10. Next you
take> oor(10/3) - 1 to get the composites with 3 as a
factor, and those> are 6 and 9.>> OOPS! What the !!!
Youve DOUBLE COUNTED as you have 6 in both> freaking
lists!!!>> So now what to do, what to do, oh, you subtract
oor(10/6) = 1, so> that you get the correct count which is 5
for the 5 composites> 4,6,8,9, and 10.>> Thats the single big
idea that mathematicians managed to come up with> over a
hundred years for counting primes.>> Then they tweaked it,
ddled 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
ing stupid. I hate mathematicians.>> Its 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 ing twit
mathematicians.>If I knew someone was evil, out to get me,
stupid, etc., I wouldntcontinuecorrespondence with them. Why
havent you gured this out yet? Isthere some sort of problem
you have that you need to have acceptance? Maybesome father
issues ??(Im assuming of course that you know who your
fatheris.)Well see...MB
If I knew someone was evil, out
to get me, stupid, etc., I wouldnt> continuecorrespondence
with them. Why havent you gured this out yet? Is> there
some sort of problem you have that you need to have
acceptance?Maybe> some father issues ??(Im 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 ts ofgrandeur)!Here is a clinical
denition:<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 ve (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***
=thats a
nice explanation, if its true of Legendres crappola, butwhat
does oor mean -- 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 oor(10/2) - 1 = 4, to get the
composites with 2 as a> factor, and those composites are 4,
6, 8, and 10. Next you take> oor(10/3) - 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 oor(10/6) = 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://mathforprot.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 wouldnt expect to see it in the books
for another year or two.Why dont you go away in the
meantime.V.-- homepage: cs utk edu tilde lastname
One
thing that simply cant be denied is that Legendres Method
is> ugly.>> Besides being ugly its clunky.> You stupid
ing twit mathematicians.> James HarrisJimmy Baby!You
are friggin beautiful! I am rolling on the oor laughing my
ass off!
You stupid ing 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, haventthey? Do you believe your method is
superior, then prove it (and we allknow the answer dont 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 Xand Y to Y?Now, I dont think its true in
general, so, what if Y=Y=Z ? What ifX is known to be
free/projective ? Id 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 dont think
its true in general, so, what if Y=Y=Z ?For ->nitely
generated<- Z-modules, it is true that if X+Y = X+Y,then
Y=Y. For non-nitely 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
nite or innitecountable 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.
what I accept as reality. --- Calvin (Calvin and
Hobbes)
have one question regarding Goldbachs Conjecture.It states
that every even integer greater than 2 issum of two prime
numbers.Im 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 specied above?--Krzysiek
I
have one question regarding Goldbachs Conjecture.> It states
that every even integer greater than 2 is> sum of two prime
numbers.> Im 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.Thats 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 specied 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 doesnt 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 specied 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
=the Stanford low-orbit thing
sounds interesting, buthow could it be denitive, 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 dEnron!
<kZhwb.6937$9O5.66@fed1read06>
tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate:
admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Pose:
george_cox@btinternet.comX-Punge: Micro$oft
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 youre thinking ofseems to be back on track.-- Shmuel
(Seymour J.) Metz, SysProg and JOATnot reply to
This is the rst 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 gure.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 cant move in downward
move freely and independently of each otherthey can.> I am
going to recover myself psychologically and physically.Great
> 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_.
Theresnothing like having a system crasher inuence
developersdecisions about whether theyll write new code or
x 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
arent successful if the attacker retreats as soon asfatigue
sets in.People here havent 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 isnt 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, Im sure.A tip: Stop posting to this kind of
newsgroup if you cant handle rationalarguments against your
point of view, without perceiving yourself as beingunder
attack.
thing coming are probably the >>same ones who say I could
care less.>>Gib> I could care less is just sarcasm isnt
it? Makes perfect sense then.I dont know. What do you mean
by sarcasm in this context?GibI mean, I could care less said
sarcastically means the same thingas I couldnt care less
meant literally. Whenever Ive heard peoplesay the former
(which is not too often) theyve been careful to say itin a
_very_ sarcastic tone - but maybe where you are its 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
isnt it? Makes perfect sense then.>>I dont 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
couldnt care less meant literally. Whenever Ive heard
people> say the former (which is not too often) theyve been
careful to say it> in a _very_ sarcastic tone - but maybe
where you are its caught on to> such an extent that people
no longer do this.Where I am (New Zealand) nobody says could
when they mean couldnt - it is a usage that Ive 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 dont think this is way Ive usually heard
it, rather I get the impression that the speaker just doesnt
realize what hes saying.To bring us back to mathematics, Im
reminded of the mad hatters tea-party, where Alice is
offered more tea, when she hasnt 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.Ive had nothing yet, Alice replied in an offended
tone: so I cant take more.You mean you cant take less, said
the Hatter: its very easy to take more than nothing.Nobody
asked your opinion, said Alice.Whos making personal remarks
now? the Hatter asked triumphantly.Its amazing how much
entertainment JSH provides, isnt 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
isnt it? Makes perfect sense then.>>I dont 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
couldnt care less meant literally. Whenever Ive heard
people> say the former (which is not too often) theyve been
careful to say it> in a _very_ sarcastic tone - but maybe
where you are its 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 couldnt care less). Whether
theyreirony-decient or simply dont 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 couldnt 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, youre going
to nd out that the truth is Y, another thing.Well, he said it
much better:> When one says, If you think X, youve 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 rst thing is not the thought about X, but X
itself (very> loosely). If you expect X, you have another
thing coming. If you> think Ill 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 Id just have to ignore
you. please,dyspose of yourself at once -- and dont start a
forest re! >I could care less is just sarcasm isnt it?
Makes perfect sense then.>>I dont know. What do you mean
by sarcasm in this context?--ils duces dEnron!
> Right.
Its 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 Im 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 youll know what Im talking about with
Texas language. Ididnt 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?Im a damn librl
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>
<87smkdvi03.fsf@phiwumbda.org>
<adm5svc6vcokof2i0n0frpckcrd0ifp9r1@4ax.com>
Right.
Its 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 Im 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 youll know what Im talking about
with Texas language. I> didnt really mean proverb. Probably
simile. Or is it metaphor.>> That boy is slower than ....>
This newsgroup is slicker n.... [owl on a brass>
doorknob]At least thats how I nish #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 its a bit
affected. I think its 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 likeIll
be on you like ugly on an ape, and if it was a snake,
itwouldve bit you, and That was biggern all get out.>> Im
a damn librl 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 cant
be sure they are any good until youveused them --- and then
its too late. John Steinbeck
Im a damn librl
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, Ill 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>
<87smkdvi03.fsf@phiwumbda.org>
<adm5svc6vcokof2i0n0frpckcrd0ifp9r1@4ax.com>
<87y8u4gb9c.fsf@phiwumbda.org>
> Im a
damn librl 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, Ill bite. What eight years? Is this a
reference to> some presidential administration?No, its the
eight years I spent in Pittsburgh doing my damnedest
toinuence speech patterns. My damnedest was essentially
limited toteaching a German or two a few key phrases and the
ner points of theone-syllable vs. the two-syllable versions
of .My main German student was willing to learn and got
the usage patternsdown pretty well, but Hoo-doggies! That
puppy was slickern owlon a brass doorknob! just doesnt
ow when your accent is similar toSchwarzeneggers[1].It
probably doesnt 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, Im sure that my buddys 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
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
Apples 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 cant 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.| | | | | | | | | Its
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 dened
(the z: x*z = y), then 0/0 = 0, and 1/0 = 0.> Frege surely
wouldnt 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 Ivenever 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 Ive>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 theres 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
wasntcertain 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 rst.>> I agree with this.>> For
example the tens of millions of measured redshifts that
agree> perfectly with the Big Bang model. Thats DATA for
you, in capital> letters.>> ROTFLMAO! How do they agree?>>
They nd 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.Thats one of the shortest
circular arguments Ive seen in awhile.>> Because we simply
*assume* that redshift> gives us a distance.>> You would
assume such a thing, yes.>> A scientist wouldnt.>> Thats
why youll never be a scientist.Too bad so many have. That is
the basis for the Big Bang. But we agreethat Big Bang
cosmologists arent 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
condently state that you are lying.Fascinating. Citation,
please. (Ill 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 ne agreement with Big Bang cosmology.Yawn. Since
you get to adjust parameters both on distance and on the
BBside, Im 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 doesnt
really matter, just pick> some object you can recognize) and
you measure their brightness and> some are bright and some
are dim and theres some distribution of> brightness between
the brightest and the dimmest and when you look at> objects
that are farther away then youll see the exact same>
distribution of brightness except that all objects will
appear dimmer> because theyre farther away.Sure, with the
adjustible parameter of extinction to help you out. And
youcant 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 nd 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 ow.And 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 nd an
additional redshift the farther things are away -- by>
whatever means the distance is determined.But thats 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 nds a singleconnected group of objects with signicantly
different redshift.> And it doesnt 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
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 well simply remove fromhim the ability to collect proof
(data). Problem solved.> But JUST to make sure were not
missing something somewhere, we> actually look at the stuff
Mr. Arp hands us and we nd 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
Arps objects is the wordcoincidence. (So your claiming fact
is a bold-faced lie.)Lets take an example. An (apparently)
physically connected group of threegalaxies. Two which appear
to be ejected from the main one. The outer twohave
signicantly 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 conrm
AGAIN that he has no point.> Then we do the same thing AGAIN
and AGAIN and after the tenth or> hundredths time we can
condently 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 cant gure out
(difcult observations, all of them) and it is> simply a
*lie* to proclaim that they pose some kind of contradiction>
to the Big Bang. They dont. No matter how long we stare into
the> heavens, there will always be some observation at the
sheer limits of> observability that is difcult to interpret
-- but thats all they> are: difcult 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
Arps is wrong). You slime over to claiming they aredifcult
to interpret, and only a few dozen cant be gured 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 Arps reinstatement to viewing time. Then you
could evaluateArps 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 youd 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,
specically, are you referring? With ination, 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 youre talking about.I gured
youd 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 conrms and renes the
previous one.ROTFLMAO!!! Those are pretty major renings!>
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 youre
willing to consider ination> to be a part of the Big Bang
Model now in which case noting has been> added to the Big
Bang since ination. Thats 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 rst
order (and to second one) the CMB should be isotropic and>
thus we build a satellite to measure that and it conrms 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 nd 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, thats
theticket!> Then someone says> well, if were really nitpicky
about the equations then there should> actually be this tiny,
most minute little anisotropy in there, but way> beyond
COBEs capability. So we scratch our heads, build a
satellite> more than an order of magnitude more sensitive
(WMAP) and sure enough> it nds 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.Its 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 theyre at it, they squeeze these few oddball cases
into> roles they never had: taking an object whos redshift
seems to place> it outside the hubble ow to pretend that
there IS NO hubble ow and> the entire theory explaining it
must be wrong.>> Yes. That is the inescapable logic of
science.>> No, it isnt. 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 youll nd 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 thats
exactly what is> observed.>> The radial velocity of any one
single object doesnt 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 ow.Ive always
understood your theory. And Ive not in any fashion
disagreedthat your above statement describes the theory and
assumptions behind thebig bang. The problem is that you dont
appear to understand the scienticmethod, or even minimal
history of your own theory and itsconstantly-changing
content and predictions.--greywolf42ubi dubium ibi
libertas
Real science is data rst.>> I
agree with this.>> For example the tens of millions of
measured redshifts that agree> perfectly with the Big Bang
model. Thats DATA for you, in capital> letters.>>
ROTFLMAO! How do they agree?>> They nd 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
wouldnt.Thats why youll never be a scientist.> Thats one
of the shortest circular arguments Ive seen in awhile.Of
course it is circular: it is YOUR argument, not anybody
elses.>> Because we simply *assume* that redshift> gives
us a distance.>> You would assume such a thing, yes.>> A
scientist wouldnt.>> Thats why youll 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 arent scientists.More that
youll ever be.> There are *tens* *of* *millions* of
measurements of distance versus> redshift in perfectly ne
agreement with Big Bang cosmology.Yawn. Since you get to
adjust parameters both on distance and on the BB> side, Im
not impressed.Yes, you would do such a thing.A scientist
wouldnt.Thats why youll 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 doesnt really matter, just pick> some
object you can recognize) and you measure their brightness
and> some are bright and some are dim and theres some
distribution of> brightness between the brightest and the
dimmest and when you look at> objects that are farther away
then youll see the exact same> distribution of brightness
except that all objects will appear dimmer> because theyre
farther away.Sure, with the adjustible parameter of
extinction to help you out. And you> cant 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 nd that
by and large redshift increases as> brightness decreases.Ah!
But the by and large is the assumption. No, it is what
everybody nds whos done the observation. Theresnothing
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 ow IN 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 ow.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 isnt 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 nd an additional redshift the farther
things are away -- by> whatever means the distance is
determined.But thats the whole point! The whatever means is
always the redshift.It isnt. 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 dont. You
are a disgusting lying pig.> Beyond that is> theory --
adjustible to meet the BB postulate. UNLESS one nds a
single> connected group of objects with signicantly
different redshift.1) Any small group of objects can have any
local relative velocitiesand the Big Bang wouldnt care -- it
is a *global* theory of theuniverse as a whole.2) No single
connected group of objects with signicantly
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 doesnt 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
agree there are uncertainties. So why are you so certain
you are> correct?I wasnt saying anything whatsoever about
uncertainties. You areTens of millions of astronomical
observations nd themselves inperfectly ne agreement with
the Big Bang. NONE nd themselves incontradiction with it. A
buncha disgusting lying pigs tries to worshipa few dozen that
are simply hard to gure 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 wasnt.You are a disgusting
lying pig.> But JUST to make sure were not missing something
somewhere, we> actually look at the stuff Mr. Arp hands us and
we nd 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 havent 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*hes 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
liketheyre close together then youve already lost right
here.> Two which appear to be ejected from the main one. The
outer two> have signicantly 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 youre 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 dont.> 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,
wouldnt.Which is why youre never going to be a scientist.>
which is why you weaseled and never mention *HOW* each>
observation of Arps is wrongTheres hardly weaseling
anywhere involved in not answering aquestion that nobody
asked.But heres a chance for you to learn what the
youre talkingabout for the rst time in your worthless life:
why dont 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 youd never actually poison your retarded
brain withactual DATA.> You slime over to claiming they are>
difcult to interpret, They are.It is difcult to interpret
observations whenever objects happen tolie close to the same
line of sight. Thats it. Theres 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 dont.You are a
disgusting lying pig.> If you had an ounce of honesty or
professional integrity,Id advise that you look less
ridiculous when you dont use such wordsright after lying
publicly.> There are a dozen versions of the BB, over the
past 50 years.One.> To which> version, specically, are you
referring? With ination, 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 doesnt 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 youre talking about.I gured youd
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 youre 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> cosmologyLOL!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 rst
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 conrms and renes
the previous one.ROTFLMAO!!! Those are pretty major
renings!Yes. Renements. Your pretending that WMAP didnt
rene the COBEresults doesnt 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
conrmed and rened the COBE results. Thats it. Theres
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 youre
willing to consider ination> to be a part of the Big Bang
Model now in which case noting has been> added to the Big
Bang since ination. Thats 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. > Its
just that the BB predictions change (as always) as soon as
another> contradiction is found.If you had the slightest clue
about science, youd know that aprediction cannot change after
the fact. What is predicted ispredicted. And then we go and
see if we can conrm the predictionobservationally. And
sometimes we can and sometimes we cant. Andsometimes we were
too optimistic in what might be detectable andsometimes were
too pessimistic.Nobody has a monopoly on the magnitude of
*any* observed effect UNTILit has been observed. At that
point, theres 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 ction.>> And
while theyre at it, they squeeze these few oddball cases
into> roles they never had: taking an object whos redshift
seems to place> it outside the hubble ow to pretend that
there IS NO hubble ow and> the entire theory explaining it
must be wrong.>> Yes. That is the inescapable logic of
science.>> No, it isnt. 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 youll nd 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 thats
exactly what is> observed.>> The radial velocity of any one
single object doesnt 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 ow.Ive always
understood your theory. You didnt in the past and you dont
now. Otherwise you wouldntcontinue to assert that the redial
velocity of a single observedobject can somehow disprove the
Big Bang.You are merely a disgusting lying pig.But Im
repeating myself.
> Whatever population of things you
measure the redshift distribution>> of, you nd an additional
redshift the farther things are away -- by>> whatever means
the distance is determined.> But thats the whole point!
The whatever means is always the redshift.[...]>Tens of
millions of astronomical observations nd themselves
in>perfectly ne agreement with the Big Bang. NONE nd
themselves in>contradiction with it. A buncha disgusting
lying pigs tries to worship>a few dozen that are simply hard
to gure out.A few dozen anomalies whose distances were
measured by the redshift. Um...-- Let us learn to dream,
gentlemen, then perhaps we shall nd 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 well 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 briey and found
no indication that (1) is correct. I alsodid not nd any
indication that (1) is incorrect.You apparently have some
rst-hand information, and it would beworthwhile (to this
reader) to see it presented.> --> greywolf42> ubi dubium ibi
libertasDale.
For example the tens of millions of
measured redshifts that agree> perfectly with the Big Bang
model. Thats DATA for you, in capital> letters.>>
ROTFLMAO! How do they agree?>> They nd 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.Thats one of the shortest circular arguments Ive seen
in awhile.Why is this so difcult 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. Wheresthe circle?I
know youre not an idiot. So where do you see a
circularargument in a description of a correlation process? -
<vrvhrq1u8o3l73@corp.supernews.com>
<vs7bc6rl043447@corp.supernews.com>
Why is
this so difcult 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 its true that measuring distance only
worksfor relatively near objects. I assume (but dont 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. Its 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 theres a
bug you should report a bug. Maybeyoure not using it
properly... It turns out Luddites dont know howto use
software properly, so you should look into that. -- Bill
Gates
Why is this so difcult 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 its true
that measuring distance only works>for relatively near
objects. I assume (but dont 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.Im 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 dont think its 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 difcult 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 its true that
measuring distance only works>for relatively near objects. I
assume (but dont 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 theres 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
Earths 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 itsa 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. Thats not guaranteed to work, but nothingin
science is guaranteed - since you cant give any reasonwhy
the relation should be different for faraway stars it
seemsreasonable to assume the relation is the same until
theres 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 its 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 nd its 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 its perfectlynatural to decide that
theres no basis for any of this whatever,its just stuff
people believe because its 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.
Its 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 theres a bug you should report a bug.
Maybe>youre not using it properly... It turns out Luddites
dont know how>to use software properly, so you should look
into that. -- Bill GatesDavid C. Ullrich
>You know Ive
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 cant remember the
subject.>and>>Ive 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
guess. Since our relationship has been as you
say>contentious, Ill 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.>Youre welcome I guess. My primary hot
button is being accused of badfaith in presenting or
analyzing issues whether right or wrong. Istill havent
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 Im not
trying to resurrect theissue here). Ive never really
understood why people cant justdiscuss even contentious
issues without being personally abusive andderogatory.
That he may have made errors doesnt change the fact that
Galileo performed> experiments (with balls on inclined
planes). And mechanics is a> theoretical practice -- not an
experimental one. {Galileo didnt 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. Keplers at
least explained theexperimental observations better than
Ptolemy, for the rst timeever. It was only until Newton that
the principal objections to theheliocentric model were
adequately explained.Thomas
> That he may have made
errors doesnt change the fact that Galileoperformed>
experiments (with balls on inclined planes). And mechanics
is a> theoretical practice -- not an experimental one.
{Galileo didnt 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.> Keplers at least explained the> experimental
observations better than Ptolemy, for the rst 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 wasrst
measured. As the lack of observable parallax (a requirement of
anySun-centered system) was the major
objection.--greywolf42ubi dubium ibi libertas
=whats
irrelevant is the profundity (or not)of Arps research (and
the anti-BBers) to Jimi Harris math. > Where do you think
Hubbles Law came from in the rst place?>> Not perfect.
But a damned good t. > 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.htmlFascinating. 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, its 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 its 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? Isnt that most of the> Milky
Way galaxy?--ils duces dEnron!
My point is that my
discovery is PURE MATH, and PURE KNOWLEDGE,> representing a
previously unknown formula for both nding and> counting
primes.Why should I believe you? Heres a previously unknown
formula:283490823481290348 + 1987234817239472345 =
2270725640720762693Is it very interesting? No.Mine is
certainly previously unknown: Ill bet anyone $100 that
nobodyhas ever before published that addition. But its not
veryinteresting, for two reasons:1) Knowing that addition
isnt very important, and2) Anyone could reproduce the
formula very easily if they needed it.Similarly, methods of
creating nite difference equations are verywell known. What
is missing in your previously unknown formula is:1) An
explanation of why it is important--for example, does it
enable faster computation of primes? (no); does it make for
faster factorizations? (no). Did you know there is a
diophantine equation whose roots are exactly the set of
primes? I could tweak that equation in a million ways and
produce a previously unknown formula for primes, but thats
hardly very interesting.2) Your methods: where did your
equation come from? How can we apply the techniques you used
in other cases for other problems? Thomas
<bpuogv$d0r$1@lust.ihug.co.nz>
<PuOwb.325$b77.281@dfw-service2.ext.raytheon.com>
Reminds me of Lenny Bruces 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 Carlins
bit in which he imagines replacing the world killwith the word
in classic movies.-- Jesse F. HughesReally, Im not out
to destroy Microsoft. That will just be acompletely
unintentional side effect. -- Linus Torvalds
=These
programs use simulated evolution to create articial
neuralnetworks to solve specic problems. They are well
documented and arein standard C with all source included.
They will compile and run withall the common compilers and
OSs. 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 Im into
nowadays.Humans may write to me at this address: zenguy at
shaw dot ca <bohtjh$1em590$1@ID-206850.news.uni-berlin.de>
<Z4KdnefKqaUznDCiRVn-sA@comcast.com>
<r7KdnaCbKpkA116iRVn-hA@comcast.com>
> Lets
be careful with our selection of terms. JSH didnt 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 dont you know this?
Because you dont know>> any>> mathematics. You dont 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. Specically, 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 presumablydoesnt 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 isnt his bag as far as I know, but
thats 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 Jamess
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 dont like reject his work.Oh, and also you dont
understand his work. I was never clear: isthe fact that you
dont get it part of your evidence hes a genius?Or is that
just a convenient reason to claim hes a genius withouthaving
to explain why?To be fair, maybe you got the impression hes 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.> Theres 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 denitely not a suitable
problem for high-school students. I think its 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, whats the word Im looking
for, dishonest? Or am I missing something? For what its
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,> whats the word Im looking for,
dishonest?I wont even dignify this weak sarcastic remark
with an answer.> Or am I missing something?Try everything.>
For what its 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
teacherwont 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 eld
in surrounding space.> Energy is stored in this eld. 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 eld contains innite>
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 innite> 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> veried. The collision energy has never been
directly measured, it has> only been force t 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, theyll both fall into>
the ditch. Best to suspend judgment, this is the high road,
there are> only ditches on either side.Richard Perry>
magnetic eld and emit photons by accelerating, then how can
we knowwhether their observed behavior is really due to
relativistic mass orjust an electromagnetic eld effect?Seems
like the only way to tell is if we could gure 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 eld in surrounding space.> Energy is
stored in this eld. 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
eld contains innite> 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 innite> 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> veried. The
collision energy has never been directly measured, it has>
only been force t 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, theyll both fall into> the ditch. Best to
suspend judgment, this is the high road, there are> only
ditches on either side.>> Richard Perry>magnetic eld and
emit photons by accelerating, then how can we know> whether
their observed behavior is really due to relativistic mass
or> just an electromagnetic eld effect?Seems like the only
way to tell is if we could gure out a way toDouble-Aspace,
matter, and eld 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 veried, 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 rst E is energy, the second
is electric eld.> 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 eld 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 veried, 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 throughDont 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
rst E is energy, the second is electric eld.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 eld 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
isnt 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 eld. It obviously requires a lot of energy to
create areverse eld bubble inside an applied eld.Note:
relativity doeesnt 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 efciency.>> 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 eld 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 nd the
second of these equations more natural>than the rst?>Two
different theories, the rst is experimentally conrmed,>the
second is experimentally falsied.>>You didnt 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 eld.>>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 elds.>>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
dont.>>You obviously didnt 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 didnt see the intermediate>> steps that included
v.dm/dt>>I didnt 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 isnt 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
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 doesnt 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.>Thats all.>> NM correctly says that the KE is
(1/2mv^2), where m is constant and v is> measured in the
frame of the eld.In the frame of which eld?This isnt
anything to discuss, Henry.> to get it to velocity v.Kinetic
energy is per denition the energy we have to pump intoAnd
you know what that energy is.> stored? In the eld of
course.Now you are talking nonsense.Of course NO kinetic
energy is instrinsic.You MUST know that, Henry!>> Thats
not what NATURE tells us. Thats what SR says.>>Thats what
Nature through experiments tells us.>>That SR says the same is
why SR isnt falsied.>>That NM says otherwise is why NM is
falsied.>> 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 dont
see the connection.Henry, dont tell me you still havent 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. Whats the
point?>>This illustrates your problem.>>You refuse to
accept that Nature works as it actually does.>>And again I
must ask:>>Why dont you say that accepting KE =
0.5*m*v^2>>means vertually nothing. Whats the point?>>
What is v? That is the point.>>You didnt answer, Henry.>I
ask you why you think KE = 0.5*m*v^2 explains>more than KE =
m*(gamma-1)*c^2.>Whats the big difference?>> I told you.
1/2mv^2 is the kinetic energy. The additional energy is
stored in> the eld bubble around the moving charge.You
didnt 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 elds associated with a moving
charge!>>or The Back EMF concept.>>any less when the
speed approaches c.>> that does not conict with 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
eld).>>And now you are insisting that you were NOT
talking>>about what happens in the acceleration part, but
how>>accelerated in an electric eld?>> Paul, if you cannot
see what I am getting at by now I dont 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
elds>>Why dont you need enigmatic elds 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 elds>even
when c << v? That is the KE = 0.5*m*v^2 is stored in elds?>>
Not directly. but indirectly maybe. I reckon all mass is
explainable in terms> of elds 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 ofeld 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 doesnt 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 elds were made of matter. Now I think that all
matter>> must be made of elds.>> These elds dont want to
move relative to other elds, hence the concepts of>> mass
and inertia.>> All we need now is an understanding of
what a eld 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 didnt see the intermediate> steps that included
v.dm/dt>>I didnt 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 isnt any term containing dm/dt in
dp/dtThats 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
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 doesnt 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.>>Thats all.>> NM correctly says that the KE is
(1/2mv^2), where m is constant and v is>> measured in the
frame of the eld.>>In the frame of which eld?>This isnt
anything to discuss, Henry.The Newtonian explanation was
dropped after the Einsteinian red herring wasspotted.If NM is
extended to include the Wilsonian reverse eld bubble
effect, itwill be quite adequate. > to get it to velocity
v.>>Kinetic energy is per denition the energy we have to
pump into>And you know what that energy is.As far as Im
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 eld 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 dont see the connection.>>Henry, dont tell me you
still havent got this!>The gained energy is Fs. It is the
same regardless>So F can blatantly obvious not decrease when
IRRESPECTIVE OF SPEED,THEREFORE SPEED INCREASE IS ALWAYS
do you have a problem with that?>>Why not simply accept
it?>> Accepting it means virtually nothing. Whats the
point?>>This illustrates your problem.>You refuse to
accept that Nature works as it actually does.>And again I
must ask:>Why dont you say that accepting KE =
0.5*m*v^2>means vertually nothing. Whats the point?>>
What is v? That is the point.>>You didnt answer,
Henry.>>I ask you why you think KE = 0.5*m*v^2 explains>>more
than KE = m*(gamma-1)*c^2.>>Whats the big difference?>> I
told you. 1/2mv^2 is the kinetic energy. The additional
energy is stored in>> the eld bubble around the moving
charge.>>You didnt 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
elds>>Why dont you need enigmatic elds 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 expressionIm merely dening KE as always 0.5*m*v^2
and distinguishing IT from theenergy that goes into the eld
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
elds>>even when c << v? That is the KE = 0.5*m*v^2 is stored
in elds?>> Not directly. but indirectly maybe. I reckon all
mass is explainable in terms>> of elds 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>eld 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 eld bubble issimilarly released in
this process.You cannot argue with that Paul. It is simply a
matter of denition. Mine is arealistic physical one whereas
your is purely a maths one based on an unprovenpostulate.
Whether my equation is precisely the same as SRs Mo.gamma
remainsto be seen.> All we need now is an understanding of
what a eld 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
eld. Do you want me to remind you of your original
question?The original question was how long it takeswhen it
enters a static electric eld, regardless ofwas accelerated
from standstill (or a very lowinitial speed). In that case,
you said the forceenters the eld.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 isnt
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 eld.What is your
answer?You have given no clear answer.> My previous answer
(repeated twice) stands.I havent 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 eld. 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 eld, regardless
of>>was accelerated from standstill (or a very low>initial
speed). In that case, you said the force>enters the
eld.>>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 isnt 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 eld.>>What is your answer?>You have given no
clear answer.> My previous answer (repeated twice)
stands.>>I havent seen it.>Would you repeat it
please?According to my reverse eld bubble theory, the
applied eld will bereduced in the vicinity of the moving
charge. That will have a similar effectto the applied eld
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 eld in surrounding space.>Energy is
stored in this eld. 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 eld contains innite>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
innite>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>veried. The collision energy has never been directly
measured, it has>only been force t 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, theyll 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
doesntchange a thing, and doesnt break any symmetry. The
only consistentsolution is motion wrt a local xed medium.
Every experiment supportingSR supports a local aether, in
fact demands it, because this relativemotion crap doesnt y,
it is contradictory. There is an embankmentagainst which to
compare the propagation of light, namely all of therelative
to each other.Richard PerryHenri 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 eld in surrounding space.>>Energy
is stored in this eld. 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 eld contains innite>>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 innite>>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>>veried. The collision energy has never been directly
measured, it has>>only been force t 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, theyll 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 doesnt>change a thing, and doesnt break any
symmetry. The only consistent>solution is motion wrt a local
xed medium. Every experiment supporting>SR supports a local
aether, in fact demands it, because this relative>motion crap
doesnt y, it 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 LTs 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
nd 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 nd the right-hand
side given a different angle?
How do I nd 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 nd 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 nd 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? Its on-line. When I read it for the rst
time>> View), I thought I understood it, I also found it
boring. Today, I know>> I didnt understand it, I now know
why I didnt understand it, and I>> thats why I dont nd 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 - itll be a start......>> -- > I dont know
as Ill have the opportunity to do any reading whilst Im>
getting my advanced degree in humility.>Actualy, Lester, I
suspect you would rather enjoy reading Two Dogmas
ofEmpiricism - Quines discussion of topics you seem to nd
engaging is bothseminal and uid. Its a short paper. You can
nd 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 Im 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 anybodys 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 istypied 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 gure, total| science is like a eld of force whose
boundary conditions are experience.| A conict with
experience at the periphery occasions readjustments in the|
interior of the eld. 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 eld. Having re-evaluated one
statement we must| re-evaluate some others, whether they be
statements logically connected| with the rst or whether they
be the statements of logical connections| themselves. But the
total eld 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 eld, except
indirectly through considerations of| equilibrium affecting
the eld 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 eld. 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
inuences 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 ofefcient 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 efcient
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 inuences 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 efcient and tractable
such belief propagation algorithms can bedepends on the
topology of the graph - in general the more sparse the
graphthe more efcient the algorithm (but potentialy the more
approximate themodel).I rst learned of Bayesian Networks from
Judea Pearls 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
Ive mentioned Bayesian Networks only as static data
structures forthe efcient 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, whats my point? If nothing else a look
at the history of ExpertSystems provides in microcosm a
chapter in the unfolding book of AppliedEpistemology. Quines
metaphor has become todays algorithms and datastructures.It
is probably just a coincidence (or is it co-occurrence) that,
whileplugging Quine, Longley did not mention explicitly
Quines 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 scientic>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 scientic world
and>contemporary visions of reality are going to come apart
at the seams>and I think well 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 Quines metaphor: you arenot altering a single
hypothesis or proposition in isolation, you arealtering the
whole interconnecting web of scientic 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
simplication in its structure, or substantialevidence and
substantial simplication whose product is HUGE. If you
wantto move the entire scientic 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? Its on-line. When I
read it for the rst time> View), I thought I understood it,
I also found it boring. Today, I know> I didnt understand
it, I now know why I didnt understand it, and I> thats
why I dont nd 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 - itll be a
start......> -- >> I dont know as Ill have the
opportunity to do any reading whilst Im>> getting my
advanced degree in humility.>>Actualy, Lester, I
suspect you would rather enjoy reading Two Dogmas
of>Empiricism - Quines discussion of topics you seem to nd
engaging is both>seminal and uid. Its a short paper. You
can nd 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 Im 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 anybodys 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>typied 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 gure, total>| science is like a eld of force whose
boundary conditions are experience.>| A conict with
experience at the periphery occasions readjustments in the>|
interior of the eld. 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 eld. Having re-evaluated
one statement we must>| re-evaluate some others, whether they
be statements logically connected>| with the rst or whether
they be the statements of logical connections>| themselves.
But the total eld 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 eld, except indirectly through considerations of>|
equilibrium affecting the eld 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 eld.
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 inuences 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>efcient
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 efcient
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 inuences 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 efcient and tractable
such belief propagation algorithms can be>depends on the
topology of the graph - in general the more sparse the
graph>the more efcient the algorithm (but potentialy the
more approximate the>model).>>I rst learned of Bayesian
Networks from Judea Pearls 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 Ive mentioned Bayesian Networks only as static data
structures for>the efcient 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, whats 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. Quines
metaphor has become todays algorithms and
data>structures.>>It is probably just a coincidence (or is it
co-occurrence) that, while>plugging Quine, Longley did not
mention explicitly Quines 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 scientic>>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 scientic world
and>>contemporary visions of reality are going to come apart
at the seams>>and I think well 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 Quines metaphor: you are>not altering
a single hypothesis or proposition in isolation, you
are>altering the whole interconnecting web of scientic
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 simplication in its structure, or substantial>evidence
and substantial simplication whose product is HUGE. If you
want>to move the entire scientic 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 Im not arguing in avacuum any more.
Im actually beginning to understand what Im talkingabout.
The same mistakes seem to apply pretty much across the board.
By the way Ill check out the online Quine reference.
=Im
kind of confused by a denition Ive 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 dont
*depend on* each other).One immediate consequence of the
above denition 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 denition 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 PDFs of
twouncorrelated Gaussian variables happens to be equal to the
productof the PDFs. I nd 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 rst variable, then re-seed the
pseudo-randomgenerator with the value of the rst 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 rst 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 thosedenitions are indeed
incorrect?Carlos--
Im kind of confused by a denition
Ive 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 dont *depend on* each
other).Well then, your understanding is imperfect. Thats an
example, not the denition. The denition 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 denition 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 denition of independence??>That
seems to me completely illogical!!Once the denition has been
made, and accepted by the mathematicalcommunity, you just have
to accept it. Yell and scream all you want,its 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
=Im 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 nos n .L
Im 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 nos 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.
what I accept as reality. --- Calvin (Calvin and
Hobbes)
be true for n>100. I seem to recall examples which are
generally, if notalways, false, despite the
factS(k)->S(k+1).Len>Im 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 nos 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.>>
what I accept as reality.> --- Calvin (Calvin and Hobbes)>
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>Im 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 nos 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.>>
about>what I accept as reality.> --- Calvin (Calvin and
Hobbes)>
>
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
nitely 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 coefcients which has
determinant 0 and is
invertible.etc.
selective about what I accept as reality. --- Calvin (Calvin
and
Hobbes)
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 nd 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 Ive come
accross>rigorous texts like Halmoss and Laxs on linear
algebra and noticed>their emphasis on nite dimensional
vector spaces (certainly to be>expected in Halmoss case), it
makes me wonder if linear algebra is>regarded mainly as the
study of these spaces.No. However, because the nite
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 innite
dimensional vector spaces?-- Shmuel (Seymour J.) Metz,
I know this is not a very deep question, but as Ive come
accross> rigorous texts like Halmoss and Laxs on linear
algebra and noticed> their emphasis on nite dimensional
vector spaces (certainly to be> expected in Halmoss 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 nite dimensional
vector spaces is not essential.The fundamental denions (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 innite-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, nitely 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 nd the projection of a vector to thebasis of a
vector space. However, before we dening 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 deneinner product again? (without using
the concept of projection?)Also, inner product can be dened
in different way(say, dot product,Hermitian inner product...
I cant list many examples, because i dontknow). Do all
these inner product give me the same projection? Can Idene
my own kind of inner product?Another question is, do innite
dimensional vector spaces has theconcept of basis? Also, I
wanna ask the following is correct or not:Given an innite
dimensional vector space V, I have already showedthat an
innite 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 nd
the projection of a vector to the>basis of a vector space.
However, before we dening 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 dene>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 rst 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 dened in different way(say, dot
product,>Hermitian inner product... I cant list many
examples, because i dont>know). Do all these inner product
give me the same projection? Can I>dene 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 yourecomfortable with the real
case.Innitely many inner products can be dened on a vector
space. OnR^n, with respect to the standard basis, an inner
product can berepresented by a positive-denite 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 denes the innitesimal geometry of the
surfaceat the chosen point. A major theme of Differential
Geometry is torelate the innitesimal geometry to the global
geometry and topologyof the surface (or more generally,
Riemannian manifold).>>Another question is, do innite
dimensional vector spaces has the>concept of basis?series).
Theres also a general result that any vector space has
abasis, whose proof uses transnite induction.John
Mitchell
>Another question is, do innite dimensional
vector spaces has the>concept of basis?series). Theres also
a general result that any vector space has a> basis, whose
proof uses transnite induction.Theres a strikingly simple
and familiar example of an innite dimensional vector space.
Consider the set R[x] of all polynomials in one variable
whose coefcients 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] satises 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 nite
linear combination of elements of B. Further, its pretty
clear that no nite collection of polynomials can have that
property (because any nite collection of polys contains one
of maximum degree, and you cant 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 =
oor(m), 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?
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 therst-order grid, then multiply each of
these by the integer that wasplaced into this larger square
(1/16 of rst-order grid) at thestart.And we could continue
this indenitely...But to avoid innities, 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 innite-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
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 difcult 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) dened 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
innity.);((I know, I know, I am full o
Anselms argument is an
excercise in begging the question.An excellent example of
question-begging yourself!> It is -worthless-. Nothing exists
by denition. Existence has to be> proven or observed.An
interesting thesis. Im not sure what it does to the
foundationsof math. Id simply ask how you would go about
proving this thesis.Thomas
=Do you profess a religion or at
least believe in God?If you dont, 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 willnd God
> This is an interpretation of
Christopher Langans CTMU, www.ctmu.org>> , and Saint
Anslems ontological argument.> Anselms argument is an
excercise in begging the question.It is -worthless-. Nothing
exists by denition. Existence has to be > proven or
observed.> I beg to differ. If I dene God as the best
football player of all times, then he by denition exists. Of
course the denition is nonsense, and Im 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.Russells
denition is very much of the football player
kind.Acke
This is an interpretation of Christopher
Langans CTMU, www.ctmu.org>> , and Saint Anslems
ontological argument.> Anselms argument is an excercise in
begging the question.>> It is -worthless-. Nothing exists by
denition. Existence has to be> proven or observed.> I beg
to differ. If I dene God as the best football player of all>
times, then he by denition exists.Assuming your denition
denes anything. There is no reason whythere should be any
such football player.Compare and contrast your denition
with:Idene Dog as the largest negative number greater than
7,does this exist just by this denition?> Of course the
denition is> nonsense, and Im 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.>> Russells
denition is very much of the football player kind.>>
Acke>
>The idea is not worthless, but doesnt have a
lot to go on...Its like dening, giving a limit to innity,
its not going to comedown to an equation but an
idea...innite knowledge...hmmm....yea
This is an
interpretation of Christopher Langans CTMU, www.ctmu.org> ,
and Saint Anslems 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 its not possible to comprehendall of totality
at once. Perhaps the better denition 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
denablefrom 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 dene an upper limit.Taken to its
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 denition, the
super-intelligence is God.>> The burden of proof becomes the
burden of proving the convergence,> to an exact
correspondence, between the mental construct[innite> number
of axioms] and realityInnite 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
rstbeing 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]<---[scientic method]
[light]--->[self-organization]<---[motion]As time goes
towards innity, so does order and
intelligence.Jonathans
Do these sequence, especially the
Beth sequence have xed 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 Veblens
theorem, that normal ordinal functions have an > unbounded
transnte series of xed points, apply to Aleph and Beth >
sequences?Why wouldnt 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, Ive pulled down
some web stuff about them, so at least Iknow the denition. 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 its possible toprove the consistency of
rejecting inaccessible cardinals, its impossibleto prove
acceptance of inaccessibles is consistent (relative
toconsistency of set theory).Im much humored by those
results, in effect saying tho its quitereasonable to reject
the unimaginably big, theres no assurance whatsoeverthat
accepting the unimaginably big is reasonable; posing pesky
overtonesfor those who entertain similar overwhelming
concepts such as almighties.Im also aware these are the
smallest of a likely transnite series ofever bigger, ever
more ghost like cardinals, of which only a nite 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. ;-)Itll take me awhile to
ponder the denitions and perhaps reviewconality before
attempting an intelligent question about them.Have you
anything youd 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, Ive pulled down some web stuff about them, so
at least> I know the denition. 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 its
possible to> prove the consistency of rejecting inaccessible
cardinals, its> impossible to prove acceptance of
inaccessibles is consistent> (relative to consistency of set
theory).>> Im much humored by those results, in effect
saying tho its quite> reasonable to reject the unimaginably
big, theres no assurance> whatsoever that accepting the
unimaginably big is reasonable; posing> pesky overtones for
those who entertain similar overwhelming concepts> such as
almighties.>> Im also aware these are the smallest of a
likely transnite series> of ever bigger, ever more ghost
like cardinals, of which only a> nite 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. ;-)>> Itll take me awhile
to ponder the denitions and perhaps review> conality before
attempting an intelligent question about them.> Have you
anything youd like to point out about them?Actually, before
going there, you might rst be entertained by this verynice
essay (alas, in french) about large objects and innite ones
:http://www.eleves.ens.fr:8080/home/madore/math/
innity.pdfBut already, measurable cardinals are really
interesting, as it is not clearat all at rst 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 xed point of the Beth
sequence...
=Im college freshmen currently taking
multivariable calculus andconsidering double majoring in math
and computer science. One thingthats been difcult 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 cant follow
itwhen grading, she takes off points), as well as how well we
explainwhat we have done in words, with graphs, and worked
solutions. Ive 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 modication.
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/offline resources for math 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 satises (4) but not(3), and f is not
continuous at 0.)So heres my hunch: (4) plus the continuity
of f implies (3). But Icant 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