mm-297

===

Subject: Re: Antidiagonal, Infinity
>What I propose is that given any
>rational that the value greater than it and less than any
other
>greater is irrational,
There is no such number, as several different people have
shown you.
> In non-standard analysis, there might be, however.
> See Alain Robert's book about NSA. Rather than being
> irrational, it would be non-standard, though.
I have yet to see any standard or non-standard model of the
reals in
which there is a smallest positive number.
In the various non-standard versions, there tend to be rather
more
numbers between any positive number x and zero, there are all
those
y such that y/x are ifinitesimal but positive, then all those
z such
that z/y is infinitesimal but positive, and so on ad infinitum.

===

Subject: Re: Antidiagonal, Infinity
>What I propose is that given any
>rational that the value greater than it and less than any
other
>greater is irrational,
There is no such number, as several different people have
shown you.
In non-standard analysis, there might be, however.
> See Alain Robert's book about NSA. Rather than being
> irrational, it would be non-standard, though.
> I have yet to see any standard or non-standard model of the
reals in
> which there is a smallest positive number.
Who says that that is meant?
In nsa, there can be a nonstandard number that can be said to
be
'greater than (a given standard rational) and less than any
other (standard)
greater'
That it is not unique, who cares?
In nsa, though, there are both rational and irrational
non-standard numbers
with this property. So, the statement that such values are
necessarily
irrational
is not true, there.
But who knows what this poster's intuition may eventually lead
to?
Herman Jurjus

===

Subject: Re: Antidiagonal, Infinity
>What I propose is that given any
>rational that the value greater than it and less than any
other
>greater is irrational,
There is no such number, as several different people have shown
you.
In non-standard analysis, there might be, however.
> See Alain Robert's book about NSA. Rather than being
> irrational, it would be non-standard, though.
I have yet to see any standard or non-standard model of the
reals in
> which there is a smallest positive number.
> Who says that that is meant?
> In nsa, there can be a nonstandard number that can be said
to be
> 'greater than (a given standard rational) and less than any
other
(standard)
> greater'
> That it is not unique, who cares?
Ross wants to use non-standard numbers to confirm his
hypothesis
that there is a next real after any real, and that the
irrationals
and rationals alternate on the real line or on some
non-standard
real line. So he cares. On the other hand, his hypothesis is
way out
in left field, where he has been stuck for months, if not
years.

===

Subject: Re: Antidiagonal, Infinity
(On RAF)
> But who knows what this poster's intuition may eventually
lead to?
We should all worry :-(
--

===

Subject: Re: Antidiagonal, Infinity
>What I propose is that given any
>rational that the value greater than it and less than any
other
>greater is irrational,
There is no such number, as several different people have
shown you.
> In non-standard analysis, there might be, however.
> See Alain Robert's book about NSA. Rather than being
> irrational, it would be non-standard, though.
> I have yet to see any standard or non-standard model of the
reals in
> which there is a smallest positive number.
> In the various non-standard versions, there tend to be
rather more
> numbers between any positive number x and zero, there are
all those
> y such that y/x are ifinitesimal but positive, then all
those z such
> that z/y is infinitesimal but positive, and so on ad
infinitum.
Between any two odd integers is an even integer, between any
two even
integers is an odd integer. The density in their union of
either is
one half.
Here I equate density with measure in the unit interval.
I don't care if you ignore gravity, it won't do you much good,
I'm
here only concerned with considering a model where the
rationals and
irrationals alternate in the reals.
If there are more irrationals than rationals and rationals and
irrationals are disjoint and distinct, then, where they are
each
totally ordered, then there necessarily would be irrationals
with no
rationals between them. Yet, there are not.
I'm trying to think of a function between the unit interval's
reals
and irrationals. The claim is that one exists because the
rationals
map onto the integers and the integers don't map to the reals,
thus
that the irrationals map onto the reals else the reals would
be a
union of two sets that don't map onto the reals. Yet, a
construction
explicitly mapping each element of the irrationals to each
element of
the reals is not given. I'm also still looking for a mapping
between
R[0,1)^N and R[0,1).
I like to think that the rationals and irrationals alternate
and that
the function f(x)=x+iota maps Q[0,1) onto P(0,1), and
f(x)=x-iota maps
Q(0,1] to P(0,1).
Then again I think the impulse function evaluates to half
infinity
at zero, and consider the Gamma function on negative integers
to have
values of various finite multiples of a scalar infinity.
Now I'm looking at the post about mapping R <-> P. I don't
immediately grasp vector space over a field and linearly
independent. You have the sequence b being a sequence of reals
each
linearly independent over Q, and a set C of reals of {b_0,
b_1, ...}
linearly independent over Q, with the initial sequence element
b_0
being a rational. RQ=P, you claim that R injects into P by
f(b_n)=b_{n+1} and f(c)=c. Why do you have braces around n+1
instead
of parentheses? Then you have F(c)=c, for c in C. I think you
mean
that c in C is not an element of the sequence b. Then you say
to
extend that to all of R by linearity over Q. So you claim a
function
f(r)=p for r in R and p in P to be defined for all reals.
What's r
for f(r)=pi? What's p for f(2)? Why f and F, presumably a
shift-key
error?
http://mathworld.wolfram.com/LinearlyIndependent.html
http://mathworld.wolfram.com/VectorSpace.html
http://www.wikipedia.org/wiki/Vector_space:
A set V is a vector space over a field F, if given an operation
vector addition defined in V, deno v+w for all v, w in v, and
an
operation scalar multiplication in V, deno a*v for all v in V
and a
in F, the following 10 properties hold for all a, b, in F and
u, v,
and w in V:
1. v+w E V
2. u+(v+w) = (u+v)+w
3. v+0 = v
4. v-v = 0
5. v+w = w+v
6. a*v E V
7. a*(b*v) = (a*b)*v
8. 1*v=v
9. a*(v+w) = a*v + a*w
10. (a+b)*v = a*v + b*v
Those each hold for V = R and F = Q.
Properties 1 through 5 indicate that V is an abelian group
under
vector addition.
The intersection of all subspaces containing a given set of
vectors
is called their span; if no vector can be removed without
diminishing
the span, the set is called linearly independent.
So you say each element of the sequence represents a set of
vectors or
a set of a vector, I'm not sure which, and that it is linearly
independent over Q because removing that vector from the set of
vectors would diminish the span of the intersection of the
subspaces
of the vector space.
Please neaten that up provide a more self-contained
explanation. Also
explain. While you're at it show a bijection between R^N and R.
Some talk here is about the nosntandard treatment of the
reals: the
hyperreals. One thing to note is that *R, the hyperreals, as a
set
contains the same elements as R, the reals. It's just a
different way
to consider them.
About the uniform probability distributions over intervals of
reals,
that's not about making some new definition of what a
probability
distribution is. It's about applying the characteristics of a
probability distribution to an infinite population. We were
talking
about the probability of an infinite binary seqence having one
element
being on, the rest off. That probability is expressed as
n/2^n, as n
diverges to infinity. The probability of any possible sequence
is
equal to 2^n/2^n, in the limit: one. So anyways out of those n
possible sequences with one on bit and the rest off bits, each
is
equally probable. The probability of each among all possible
infinite
binary sequences is being1/2^n, the probability of each among
all
infinite binary sequences with one on bit is 1/n. So a
theoretical
(read: thought experiment) method to generate an element of N
is to
once again flip infinitely many coins. At this point it's a
crazy, or
rather, unconventional thought experiment in that the first
coin toss
says whether it is oo/2 or greater or less than oo/2. Assume
it's a
long sequence of zeros, then it would be saying about whether
the
result is greater than or equal to oo/4, oo/8, oo/16, etcetera.
Without a method to generate a sample from a uniform
distribution over
the natural numbers, it's still that the probability of
selecting any
is 1/|N|.
Of course that's ludicrous but at the same time it allows us to
consider the realm of thought in concern of this issue and to
then
talk about the probability of selecting a given element of the
natural
integers assuming a uniform probability distribution over the
integers. At least we seem to have some agreement that a
uniform
probability distribution over an interval of the reals exists,
and a
simple method to sample an element of an interval of the reals
exists.
infinitesimals, it talks about 1-infinitesimals,
2-infinitesimals,
etcetera, n-infinitesimals, with the oo-infinitesimal being
zero.
Ross

===

Subject: Re: Antidiagonal, Infinity
> Between any two odd integers is an even integer, between any
two even
> integers is an odd integer. The density in their union of
either is
> one half.
What does that last sentence mean?
> Here I equate density with measure in the unit interval.
Then the integers have density zero.
> I don't care if you ignore gravity, it won't do you much
good, I'm
> here only concerned with considering a model where the
rationals and
> irrationals alternate in the reals.
Which is still as stupid as trying to ignore the law of
gravity.
> If there are more irrationals than rationals and rationals
and
> irrationals are disjoint and distinct, then, where they are
each
> totally ordered, then there necessarily would be irrationals
with no
> rationals between them. Yet, there are not.
In the set of rationals, there are no irrationals.
In the set of irrationals, there are no rationals.
In the set of reals, there are countably many rationasl ans
uncontably many reals arranged so that between any two distinct
reals there are countably many rationals and uncountably many
reals.
In fact, there is a order preserving bijection from any open
interval, (a,b), with a < b, to the set of all reals, namely
f:(a,b) -> R: x |-> (a-b)*x/[(x-a)*(x-b)], and if a and b are
rational, the mapping f carries rationals to rationals and
irrationals to irrationals.
> I'm trying to think of a function between the unit
interval's reals
> and irrationals. The claim is that one exists because the
rationals
> map onto the integers and the integers don't map to the
reals, thus
> that the irrationals map onto the reals else the reals would
be a
> union of two sets that don't map onto the reals. Yet, a
construction
> explicitly mapping each element of the irrationals to each
element of
> the reals is not given. I'm also still looking for a mapping
between
> R[0,1)^N and R[0,1).
A reverse bijection, from the reals to the irrationals, was
given in
at least 2 versions in a prior posting in this thread, so just
take
the inverse bijection of either of them.
> I like to think that the rationals and irrationals alternate
and that
> the function f(x)=x+iota maps Q[0,1) onto P(0,1), and
f(x)=x-iota maps
> Q(0,1] to P(0,1).
You may like to think a lot of things, but that does not make
them
true.
> Then again I think the impulse function evaluates to half
infinity
> at zero, and consider the Gamma function on negative
integers to have
> values of various finite multiples of a scalar infinity.
Again you reveal that your wiring is short circui.
> Now I'm looking at the post about mapping R <-> P. I don't
> immediately grasp vector space over a field and linearly
> independent. You have the sequence b being a sequence of
reals each
> linearly independent over Q, and a set C of reals of {b_0,
b_1, ...}
> linearly independent over Q, with the initial sequence
element b_0
> being a rational. RQ=P, you claim that R injects into P by
> f(b_n)=b_{n+1} and f(c)=c. Why do you have braces around n+1
instead
> of parentheses?
Standard newsnet notation for a compound subscript.
Then you have F(c)=c, for c in C. I think you mean
> that c in C is not an element of the sequence b. Then you
say to
> extend that to all of R by linearity over Q. So you claim a
function
> f(r)=p for r in R and p in P to be defined for all reals.
What's r
> for f(r)=pi? What's p for f(2)? Why f and F, presumably a
shift-key
> error?
> http://mathworld.wolfram.com/LinearlyIndependent.html
> http://mathworld.wolfram.com/VectorSpace.html
> http://www.wikipedia.org/wiki/Vector_space:
> A set V is a vector space over a field F, if given an
operation
> vector addition defined in V, deno v+w for all v, w in v,
and an
> operation scalar multiplication in V, deno a*v for all v in
V and a
> in F, the following 10 properties hold for all a, b, in F
and u, v,
> and w in V:
> 1. v+w E V
> 2. u+(v+w) = (u+v)+w
> 3. v+0 = v
> 4. v-v = 0
> 5. v+w = w+v
> 6. a*v E V
> 7. a*(b*v) = (a*b)*v
> 8. 1*v=v
> 9. a*(v+w) = a*v + a*w
> 10. (a+b)*v = a*v + b*v
> Those each hold for V = R and F = Q.
> Properties 1 through 5 indicate that V is an abelian group
under
> vector addition.
> The intersection of all subspaces containing a given set of
vectors
> is called their span; if no vector can be removed without
diminishing
> the span, the set is called linearly independent.
> So you say each element of the sequence represents a set of
vectors or
> a set of a vector, I'm not sure which, and that it is
linearly
> independent over Q because removing that vector from the set
of
> vectors would diminish the span of the intersection of the
subspaces
> of the vector space.
I have this set of reals B = {b_0,b_1,b_2, ....} whose members
are
are, as vectors over Q, linearly independent and so that b_0
is a
non-zero rational. Every real in span(B) is a linear
combination of
finitely many of the members of B. There are other reals which
are
linearly independent of the span of B, the set of which I
called C,
and which is a (vector) subspace of R. Given any real r, then
there
are unique rationals p,q, and a unique real b in span(B) and a
unique real c in C such that r = p*b+q*c. Thus R is the direct
sum
of subspaces span(B) and C.
It may be proven that there is only one Q-linear function, say
f,
from R to RQ and such that f(b_0) = b_1, f(b_1) = b_2 etc., and
f(c) = c for every c in C. This function is a bijection.
> Please neaten that up provide a more self-contained
explanation. Also
> explain. While you're at it show a bijection between R^N and
R.
The explanation is sufficient for those who know a little math.
Learn a bit about vector spaces and it may become clear to
you, or
don't and it won't.
> Some talk here is about the nosntandard treatment of the
reals: the
> hyperreals. One thing to note is that *R, the hyperreals, as
a set
> contains the same elements as R, the reals. It's just a
different way
> to consider them.
The Robinson formulation of *R has more than a single member
corresponding to a member of R, it has uncountably many
coresponding
to eanc member of R, which are infinitesimally close to each
other,
as well as some which do not correspond to any member of R.
> About the uniform probability distributions over intervals
of reals,
> that's not about making some new definition of what a
probability
> distribution is.
It is well known, to those who understand what pdf's
(probability
density functions) are that there cannot be a uniform pdf on R.
Uniform pdf's require finite real intervals or finite sets to
operate on. You cannot make one on a countably infinite set
nor on
an unbounded real interval.
> It's about applying the characteristics of a
> probability distribution to an infinite population.
Since a uniform pdf must have certain properties to be a pdf,
there
are limits on what sets they can exist on, and N, Q and R are
outside those limits.
[garbage dele]
> Of course that's ludicrous but at the same time it allows us
to
> consider the realm of thought in concern of this issue and
to then
> talk about the probability of selecting a given element of
the natural
> integers assuming a uniform probability distribution over the
> integers. At least we seem to have some agreement that a
uniform
> probability distribution over an interval of the reals
exists, and a
> simple method to sample an element of an interval of the
reals exists.
We have no agreement that such a distribution exists, because
it
cannot.
> infinitesimals, it talks about 1-infinitesimals,
2-infinitesimals,
> etcetera, n-infinitesimals, with the oo-infinitesimal being
zero.
Your knowledge of infinitesimals is nil. Until you get a
reasonable
understanding of the standard reals, your hopes of
understanding
anything about non-standard reals is unreal.

===

Subject: Re: Naive Q: Set theory, logic - which comes first?
<vcb8yogpvht.fsf@beta13.sm.luth.se>
<vcbk780kpyu.fsf@beta13.sm.luth.se>
<1xUcb.14254$O85.6040@pd7tw1no>
<3f79e264$7$fuzhry+tra$mr2ice@news.patriot.net>
at 01:50 PM, mattias_wikst71@hotmail.com (Mattias Wikstr?m)
said:
>G.9adel did not show this. He showed that for a given sound
formal
>system in which the statements about natural numbers can be
>formula, there are statements that can neither be proven nor
>disproven.
Google for G.9adel number. He represen statements as natural
numbers. The theorems he proved about statements were also
theorems
about the integers representing them.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
Unsolici bulk E-mail will be subject to legal action. I reserve
the right to publicly post or ridicule any abusive E-mail.
Reply to domain Patriot dot net user shmuel+news to contact
me. Do

===

Subject: Re: Grid: Integers = To Sum Of Some Divisors
> Take an n-by-n-grid, n>= 3.
Place the integers 2 to (n^2 +1) into the grid,
> one DISTINCT integer per grid-square, so that:
If s(k,j) = a grid-square (ie. an element of an
> n-by-n matrix), then
> (for all k and j where n >= k >= 3 and n >= j >= 1)
> s(k,j) = (any divisor >= 2 of s(k-1,j)) +
> (any divisor >= 2 of s(k-2,j)),
and> (for all k and j where n >= k >= 1 and n >= j >= 3)
> s(k,j) =(any divisor >= 2 of s(k,j-1)) +
> (any divisor >= 2 of s(k,j-2))
>[...] an n=3 example is:
> 5 3 8
> 2 6 4
> 7 9 10
> Is there an n=4 example??
There appear to be 2 basic solutions and of course their
transposes:
5 2 7 9
3 12 6 15
8 4 10 14
11 16 13 17
5 3 8 11
2 12 4 16
7 6 10 13
9 15 14 17
11 2 13 15
3 12 6 9
14 4 16 8
5 10 7 17
11 3 14 5
2 12 4 10
13 6 16 7
15 9 8 17
which a bit iously could be extended to 5x5 or 6x6
but not much further because it uses nes for loops,
one level per cell, rather than a recursive approach.
The program takes about 1 second to exhaust the 4x4 case.
-jiw

===

Subject: Re: Grid: Integers = To Sum Of Some Divisors
> Take an n-by-n-grid, n>= 3.
Place the integers 2 to (n^2 +1) into the grid,
> one DISTINCT integer per grid-square, so that:
If s(k,j) = a grid-square (ie. an element of an
> n-by-n matrix), then
> (for all k and j where n >= k >= 3 and n >= j >= 1)
> s(k,j) = (any divisor >= 2 of s(k-1,j)) +
> (any divisor >= 2 of s(k-2,j)),
> and> (for all k and j where n >= k >= 1 and n >= j >= 3)
> s(k,j) =(any divisor >= 2 of s(k,j-1)) +
> (any divisor >= 2 of s(k,j-2))
>[...] an n=3 example is:
> 5 3 8
> 2 6 4
> 7 9 10
 Is there an n=4 example??
> There appear to be 2 basic solutions and of course their
transposes:
> 5 2 7 9
> 3 12 6 15
> 8 4 10 14
> 11 16 13 17
> 5 3 8 11
> 2 12 4 16
> 7 6 10 13
> 9 15 14 17
> 11 2 13 15
> 3 12 6 9
> 14 4 16 8
> 5 10 7 17
> 11 3 14 5
> 2 12 4 10
> 13 6 16 7
> 15 9 8 17
> which a bit iously could be extended to 5x5 or 6x6
> but not much further because it uses nes for loops,
> one level per cell, rather than a recursive approach.
> The program takes about 1 second to exhaust the 4x4 case.
> -jiw
Ahh...So there ARE solutions after all!
I was neglecting the likelyhood of bigger integers, such as
the 11 and
12, being in the upper-left, perhaps.
Leroy Quet

===

Subject: Re: Deep Thoughts # 1: A new limitation to the human
mind
> 1. Mathematics is the science in which we make something out
of
> nothing.
> Wrong. Mathematics is built on the 13 Axioms, which are not
nothing.
And where did those 13 axioms come from? What is needed to
develop
them?
> 2. All of man-made Mathematics consists of an abstraction
from
> physical processes.
> Wrong again. Mathematics consists of theorems deriva by
following a
> set of formal rules.
And when is this not just an abstraction from physical
processes
(example, please)?
> 3. Since Mathematics in general needs nothing to be crea,
then the
> human mind seems to be limi in that it can only consider
> possibilities that are analogous to physical processes.
> Well, I think first of all mathematics exist all by
themselves, as a
> logical system.
And so needs nothing to be crea.
> However one may argue that for humans this exists in
> its precise own way, because the human mind works the way it
works.
That's right. Exactly. And in particular, the human mind seems
to
only discover the mathematics that is an abstraction from
physical
processes.
> Thus mathematics actually may help delineate the limits of
the human
> mind.
That is exactly what I am postulating.
Cambridge, MA

===

Subject: Re: Deep Thoughts # 1: A new limitation to the human
mind
7e3Z2dkQEBC5ubm60r3HAAACeUlEQVR4nIXTQW+
bMBQAYChLfcWzUq5pipUr7lPptUNGXBMC8zUZ
qnt1aNz392c7pGXSpvkSyV/ee37POFr/Y0X/g56rxd+
gb2IQC8X79brdLtovaBgACIsRMVTr4hN4
Ah54xy0rtdbjFRCAupizIhUDB69X2IHAGKCrxyrxcJyA76CQKwZn2ckEXK6h/
YwoOld/zJUMEYds
ghMIYhj0kjehxqGYFacgalk1PpM+jBOkvg0othFR4bjDVHyjQx+FZ+
b29fMEe114SIC6UB/RXoCX
enAhGQORAdxp/TaN5N63lNqGJZACO2n9/
QL81qfVQjEXAkBKPyoPm7BfPqFLJ0qBt1fwmfTAFiYW
MbAC1WmCnyEitcYImtI07fY3FwglBuwMUiqoiLv98gL+
4PrNSmXdWGJK7Q4D8JBpWeXGxi6CpqOJ
QoOrkOksKzMaRinN7LsI8OChsGOl+ogiE8SeaIAP9/
9BRIs1xzgzESWteQ7gDgWQpmRdmcgtNFYJ
/
5W4CWpRRsTyHE2UBgifjzvUQMGMOfYYO1CI7wFWfhwsWXSNq0qQcMQkgBvhADvI
rOrduRArpW7D
Rflb0gjHvO5kpUhX93h3DJ3v9VB+S8pMulW1kis73aAqh/
LpBYalbA2qiLZKP16G+KHLowIYojSm
UTys+/
LrogaFVFTKpN2P45qT6UX58qrbEFcil81h9tQSfVSyIZ0v3zzNYEMPVtbR2e3XL
8v547w/
uDymk7wfXx7n8PBscrm1fW/G1a8/gKiz3GJeYbt5ncMJ+
Y1ckVxi19zMwaIcXXUreb6yc1gQ9ypr
tLKSCueQjQ4qxFzKXo0zwNDbxv/U2ynkN+Dj/u9oA6vhAAAAAElFTkSuQmCC
>> 1. Mathematics is the science in which we make something
out of
>> nothing.
> Wrong. Mathematics is built on the 13 Axioms, which are not
nothing.
Which 13?
--
And yes, for those who think that just maybe I did find a
short proof
of Fermat's Last Theorem, and THE prime counting function, if I
succeed at what I'm working on now world economy as you know
it will
be gone. -- branches out.

===

Subject: Re: Deep Thoughts # 1: A new limitation to the human
mind
>> 1. Mathematics is the science in which we make something
out of
>> nothing.
Wrong. Mathematics is built on the 13 Axioms, which are not
nothing.
> Which 13?
I am sorry, I let myself get carried away by indignation. I
would have
to look them up. My math has become extremely rusty since I
studied,
but I can recall that our first semester analysis class consis
in
laying out the foundation of mathematics by learning 23 axioms
and how
they sufficed to build up the rest. I believe we star with the
peano Axioms.

===

Subject: Re: Deep Thoughts # 1: A new limitation to the human
mind
>> 1. Mathematics is the science in which we make something
out of
>> nothing.
Wrong. Mathematics is built on the 13 Axioms, which are not
nothing.
> Which 13?
> --
> And yes, for those who think that just maybe I did find a
short proof
> of Fermat's Last Theorem, and THE prime counting function,
if I
> succeed at what I'm working on now world economy as you know
it will
> be gone. -- branches out.
There is a guy who is really a fundamentalist religious
preacher calling
himself Paniagua (Bread 'n water)
He got nicknamed something like Taco Bill.
I hope this is not the same guy!!
Bob Pease

===

Subject: Re: Deep Thoughts # 1: A new limitation to the human
mind
> There is a guy who is really a fundamentalist religious
preacher calling
> himself Paniagua (Bread 'n water)
> He got nicknamed something like Taco Bill.
> I hope this is not the same guy!!
> Bob Pease
No, Im not, that is my name. I am religious, but neither am I a
preacher nor am I a fundamentalist.
Just wan to mention that Springer-Verlag's annual mathematics
book
sale, the Yellow Sale, is finally back.
Here's the URL:
http://www.springer-ny.com/yellowsale/
The book sale is available in bookstores across North America,
as well
as online at the URL above to customers in the Americas.
Springer is
the world's leading publisher in the field of mathematics.
Jason Roth
Springer-Verlag NY

===

Subject: irreducible polynomial ?
Is the polynomial 2*cos(2^k*arccos(x/2)) irreducible for each
positive integer k? If so, how is it proved?
--Jim Buddenhagen
------------
To reply copy jbuddenh@REMOVEtexas.net to address bar and edit
out REMOVE

===

Subject: Re: irreducible polynomial ?
I asked:
> Is the polynomial 2*cos(2^k*arccos(x/2)) irreducible for each
> positive integer k? If so, how is it proved?
I received email from Joe Silverman answering this question
and giving a good deal of additional information on the
factorization of Chebeshev polynomials.
---------------------------------------------------------------
Comments from Joe Silverman:
The n'th Chebeshev polynomial (up to factors of 2 used by
different
authors)
is the polynomial F_n(x) with the property that
F_n(2*cos(t)) = 2*cos(n*t).
So if we put t = arccos(x/2), then we get
F_n(x) = 2*cos(n*arccos(x/2)).
This is your formula.
If we write cos(t) = (e^{it} + e^{-it})/2, and for ease of
notation,
let z = e^{it}, then we get
F_n(z+z^{-1}) = z^n + z^{-n}.
This is another characterization of F_n.
The roots of F_n(x), from your formula or from this
alternative, are
x = 2*cos(pi*(j+1/2)/n) for j = 0,1,2,...,n.
Notice this can also be written as
x = e^{it} + e^{-it} with t = (pi*(j+1/2))/n.
it's easy to see that the splitting field of F_n(x) is the
real subfield
of the 4n'th cyclotomic field. In other words, the roots of
F_n(x)
generate the field
K_{4n} = Q(e^{pi*i/2*n}+e^{-pi*i/2*n}).
The degree of this field is [K_n:Q] = phi(4*n)/2, where phi(n)
is Euler's
phi function. On the other hand, the degree of F_n is n. So
the conclusion
is the following:
Proposition: F_n(x) is irreducible over Q if and only if
phi(4*n) = 2*n.
The case you're asking about is n = 2^k, and indeed
phi(4*2^k) = phi(2^{k+2}) = 2^{k+1} = 2*2^k,
so your polynomials are irreducible. Further, I think this is
probably the
only case that phi(4*n) = 2*n, so the only case that F_n is
irreducible.
Hope this is of some help. Feel free to post this, if you want.
------------------------------------------------------
--Jim Buddenhagen
To reply copy jbuddenh@REMOVEtexas.net to address bar and edit
out REMOVE

===

Subject: Re: irreducible polynomial ?
> I asked:
> Is the polynomial 2*cos(2^k*arccos(x/2)) irreducible for each
> positive integer k? If so, how is it proved?
> I received email from Joe Silverman answering this question
> and giving a good deal of additional information on the
> factorization of Chebeshev polynomials.
>
---------------------------------------------------------------
> Comments from Joe Silverman:
> The n'th Chebeshev polynomial (up to factors of 2 used by
different
authors)
> is the polynomial F_n(x) with the property that
> F_n(2*cos(t)) = 2*cos(n*t).
> So if we put t = arccos(x/2), then we get
> F_n(x) = 2*cos(n*arccos(x/2)).
> This is your formula.
> If we write cos(t) = (e^{it} + e^{-it})/2, and for ease of
notation,
> let z = e^{it}, then we get
> F_n(z+z^{-1}) = z^n + z^{-n}.
> This is another characterization of F_n.
> The roots of F_n(x), from your formula or from this
alternative, are
> x = 2*cos(pi*(j+1/2)/n) for j = 0,1,2,...,n.
> Notice this can also be written as
> x = e^{it} + e^{-it} with t = (pi*(j+1/2))/n.
> it's easy to see that the splitting field of F_n(x) is the
real subfield
> of the 4n'th cyclotomic field. In other words, the roots of
F_n(x)
> generate the field
> K_{4n} = Q(e^{pi*i/2*n}+e^{-pi*i/2*n}).
> The degree of this field is [K_n:Q] = phi(4*n)/2, where
phi(n) is Euler's
> phi function. On the other hand, the degree of F_n is n. So
the
conclusion
> is the following:
> Proposition: F_n(x) is irreducible over Q if and only if
phi(4*n) = 2*n.
> The case you're asking about is n = 2^k, and indeed
> phi(4*2^k) = phi(2^{k+2}) = 2^{k+1} = 2*2^k,
> so your polynomials are irreducible. Further, I think this
is probably
the
> only case that phi(4*n) = 2*n, so the only case that F_n is
irreducible.
> Hope this is of some help. Feel free to post this, if you
want.
by essentially the same argument you see that the odd
chebyshev polynomials
of
prime order are irreducible if you eliminate the trivial
factor x (zero
x=0)
sincerely
Klaus
> ------------------------------------------------------
> --Jim Buddenhagen
> To reply copy jbuddenh@REMOVEtexas.net to address bar and
edit out REMOVE

===

Subject: Re: irreducible polynomial ?
> Is the polynomial 2*cos(2^k*arccos(x/2)) irreducible for each
> positive integer k? If so, how is it proved?
and the proof is essentially trivial:
Each polynomial is the square of its predecessor less 2.
This follows immediately from the identity
(cos z)^2 = (1 + cos(2*z))/2. Now for k=0,1 the
polynomials are x and x^2-2 so it is clear that for
each k the leading coeff is 1, the constant is 2 and
all other coeffs are even, so Eisenstein applies as
Arturo suggests.
--Jim Buddenhagen

===

Subject: Re: irreducible polynomial ?
http://icm.mcs.kent.edu/reports/1998/ICM-199802-0001.pdf

===

Subject: Re: irreducible polynomial ?
Isn't it something like Chebychev polynomial?

===

Subject: Re: irreducible polynomial ?
Visiting Assistant Professor at the University of Montana.
>Is the polynomial 2*cos(2^k*arccos(x/2)) irreducible for each
>positive integer k? If so, how is it proved?
Is it a polynomial? In what? Irreducible over what?

===

Subject: Re: irreducible polynomial ?
>Is the polynomial 2*cos(2^k*arccos(x/2)) irreducible for each
>positive integer k? If so, how is it proved?
this is up to normalization the n-th (=2^k) chebyshev
polynomial (for
|x|<=2)
http://mathworld.wolfram.com/
ChebyshevPolynomialoftheFirstKind.html
its roots are rela to the n-th roots of unity, and the
irreducibility to
the
special value of phi(n)
hth (and hope it is not nonsense)
klaus
> Is it a polynomial? In what? Irreducible over what?
 It's not denial. I'm just very selective about
> what I accept as reality.
> --- Calvin (Calvin and Hobbes)
>
magidin@math.berkeley.edu

===

Subject: Re: irreducible polynomial ?
>Is the polynomial 2*cos(2^k*arccos(x/2)) irreducible for each
>positive integer k? If so, how is it proved?
> Is it a polynomial? In what? Irreducible over what?
in x, irreducible over Q
For example,
2*cos(2^4*arccos(x/2)) =
16 14 12 10 8 6 4 2
x - 16 x + 104 x - 352 x + 660 x - 672 x + 336 x - 64 x + 2
is irredicible.

===

Subject: Re: irreducible polynomial ?
Visiting Assistant Professor at the University of Montana.
>>Is the polynomial 2*cos(2^k*arccos(x/2)) irreducible for each
>>positive integer k? If so, how is it proved?
>> Is it a polynomial? In what? Irreducible over what?
>in x, irreducible over Q
>For example,
>2*cos(2^4*arccos(x/2)) =
> 16 14 12 10 8 6 4 2
>x - 16 x + 104 x - 352 x + 660 x - 672 x + 336 x - 64 x + 2
>is irredicible.
Never seen that before; but would it not be possible to prove
it
irreducible by using Eisenstein's Criterion? Looks like the
leading
coefficient is 1, the constant coefficient is 2, and all the
other
terms have even coefficient. If the pattern holds for
arbitrary k,
then there you are.

===

Subject: Re: irreducible polynomial ?
>Is the polynomial 2*cos(2^k*arccos(x/2)) irreducible for each
>>positive integer k? If so, how is it proved?
>Is it a polynomial? In what? Irreducible over what?
>>in x, irreducible over Q
>>For example,
>>2*cos(2^4*arccos(x/2)) =
>>16 14 12 10 8 6 4 2
>>x - 16 x + 104 x - 352 x + 660 x - 672 x + 336 x - 64 x + 2
>>is irredicible.
> Never seen that before; but would it not be possible to
prove it
> irreducible by using Eisenstein's Criterion? Looks like the
leading
> coefficient is 1, the constant coefficient is 2, and all the
other
> terms have even coefficient. If the pattern holds for
arbitrary k,
> then there you are.
Right you are. Let p_k(x) = cos(2^{k} * arccos(x/2)), then
p_1(x) = x^{2} / 2 - 1
and
p_{k+1}(x) = 2 * (p_k(x))^2 - 1
in x^2, has constant term equal to 2 or -2 and that 2p_k(1) =
-1 and
2p_k(2) = 2. As you say, there you are. You don't even need to
invoke
Eisenstein.
Rick

===

Subject: Re: irreducible polynomial ?
>Is the polynomial 2*cos(2^k*arccos(x/2)) irreducible for each
>positive integer k? If so, how is it proved?
> Is it a polynomial? In what? Irreducible over what?
Yes. In x. Over Z or Q.
--JB

===

Subject: Re: MTL and MatLab
iQBVAwUAP4MJpEHMCo9UcraBAQEGZgH/WYWzR/
00hb6naqYilcbwMgCo3o9rDreJ
mEK3cJTjmuxifF+wuAqYurB6n2KECVpTlR7nLnRfEDHEXqXCt6tWuA==
=Gqcd
> Hello!
> I am trying to make Matrix Template Library
> (http://www.osl.iu.edu/research/mtl/) interact with MatLab
6.5. If
> anybody has done it, please give me your comments on what
you think
> about it. I would greatly appreciate it!
> Iryna
I have used both. What do you mean by having MTL 'interact'
with
Matlab? Do you want to call matlab functions via the mex
interface? Or
do want to generate a C++ program that is callable from matlab?
I recently did a project where results had to be read in
Matlab, but the
simulation which generates them is far too complex to run
efficiently as
a Matlab program. I crea a class for results that allowed easy
measurments from a simulation, and automates the writing of a
corresponding mat file. If this is what you are trying to do,
let me
know and I can send you the code (I have a version on the web,
but it
needs to be upda).
Regarding MTL... It's been a long time since any updates came
from MTL?
In fact he object orien numerics websit (oon.org) is looking a
bit
out of date. Is there a better reference? Has anything to
replace
blitz or MTL come out ?
G.S.
[ See http://www.gotw.ca/resources/clcm.htm for info about ]
[ comp.lang.c++.modera. First time posters: Do this! ]

===

Subject: Re: Brownian motion approximation
Can you spell out how to do it? I can see that the general
theorem
follows easily if you can prove it where f(t) is a linear
function(by
scaling, Markov property, etc.) but how do you show easily
that f(t)
is approxima when f is linear? As I mentioned, I can handle the
case f(t)=0 for all t, because then you can use the reflection
principal. Is B_t - f(t) with f linear a BM with drift(I have
heard
the term before but don't really know what it is)?
>> A while back I pos a question about whether or not
> P[sup_{0<=t<=1}|B_t - f(t)| < d] > 0 for all d > 0 and f(t)
continuous
>> on [0,1] with f(0) = 0.
> In other words, does Brownian motion uniformly approximate
any
>> continuous function(with f(0)=0) with positive probability?
Someone
>> replied that it does, and this follows from first proving
it for f(t)
>> = 0 for all t and then applying the Cameron-Martin Theorem.
I can do
>> it for f(t)=0, but I don't seem to be able to find a
reference for the
>> Cameron-Martin Theorem, though it seems to be rela to
Girsanov's
>> Theorem, and maybe even follows from it. Can someone give
me some help
>> or lead me to a reference?
I don't believe there is any significant difference between
the
>Cameron-Martin-Girsanov theorem, the Girsanov theorem and the
Cameron-Martin
>theorem. As I understand it the same theorem was discovered
independently
>and is now attribu to all three.
Rather like the Green-Gauss-Ostrogradsky theorem.
> This does not need quoting any complica theorems. Construct
> a polygonal function h such that |f - h| < d/3 on the
interval,
> bound (from below) the probability that |B_t - h(t)| < d/3 at
> all vertices, and bound the probability that B differs by d/3
> from its polygon at those vertices. The Markov nature of B
and
> the boundedness and continuity of f enable all of this to be
> carried out easily.

===

Subject: Vedctor Calculus Question
Could someone help me to understand how to find the minimum
distance
between a surface (say f1(x,y,z)=c1) and a line (f2(x,y,z)=c2.
i
believe i should be using gradients.
thank you very much!

===

Subject: Re: Vedctor Calculus Question
> Could someone help me to understand how to find the minimum
distance
> between a surface (say f1(x,y,z)=c1) and a line
(f2(x,y,z)=c2. i
> believe i should be using gradients.
> thank you very much!
A single equation, such as f2(x,y,z)=c2, can describe in a 3d
space
a surface, possibly a plane, but not a line.
The vector parametric form for a line is
g(t) = (u1 + u2*t, v1+v2*t,w1+w2*t),
where (ui,v1,w1) is a point on the line and (u2,v2,w2) is a
vector
parallel to the line.
If the surface and line intersect, i.e., the distance between
the
surface and the line is zero, then f1(u1+u2*t,v1+v2*t.w1+w2*t)
= c1
is true for some real value of t.
If the surface and line do not intersect and the surface has
continuous gradients and no boundary curves, then the gradient
of f2
at any extremal point (closest to or furthest from the line)
must be
perpendicular to (u2,v2,w2).

===

Subject: Re: Vedctor Calculus Question
>A single equation, such as f2(x,y,z)=c2, can describe in a 3d
space
>a surface, possibly a plane, but not a line.
I could *swear* that {(x,y,z) in R^3 : x^2+y^2=0} was a line,
last time I looked.
Lee Rudolph

===

Subject: Re: Vedctor Calculus Question
>A single equation, such as f2(x,y,z)=c2, can describe in a 3d
space
>a surface, possibly a plane, but not a line.
> I could *swear* that {(x,y,z) in R^3 : x^2+y^2=0} was a line,
> last time I looked.
>
I stand correc, but it is not a very efficient way of doing
lines, and certainly not the standard way.

===

Subject: Re: Vedctor Calculus Question
>A single equation, such as f2(x,y,z)=c2, can describe in a 3d
space
>a surface, possibly a plane, but not a line.
> I could *swear* that {(x,y,z) in R^3 : x^2+y^2=0} was a line,
> last time I looked.
>
Note that z is unrestric. It might not hurt to look again.

===

Subject: Re: Vedctor Calculus Question
>>A single equation, such as f2(x,y,z)=c2, can describe in a
3d space
>>a surface, possibly a plane, but not a line.
>> I could *swear* that {(x,y,z) in R^3 : x^2+y^2=0} was a
line,
>> last time I looked.
>Note that z is unrestric.
Oh, I do note that, I do, I do.
>It might not hurt to look again.
You go first!
Lee Rudolph

===

Subject: Re: Vedctor Calculus Question
>A single equation, such as f2(x,y,z)=c2, can describe in a 3d
space
>a surface, possibly a plane, but not a line.
> I could *swear* that {(x,y,z) in R^3 : x^2+y^2=0} was a line,
> last time I looked.
>
This actually happened.
Many years ago a problem similar to this came up in a CalcIII
class I was
teaching.
I mentioned that situations like this are called Degenerate
case
A girl raised her hand and said
But that's what my Father says *I* am!!
Bob Pease

===

Subject: Vigier V Conference Topics for Debate
Comments by Jack Sarfatti on excerpts from:
attention.)
ABSTRACT
Each approach to the quantum-gravity problem originates from
expertise
in one or another area
Hi Jack,
I can feel your intense interest to find the mechanism of
gravity and
objects but are instead wave structures in a quantum space. Our
perception of their properties was 'schaumkommen' of the wave
structures. (appearances.)
I disagree. I agree with the deBroglie-Bohm-Vigier pilot
theory that
from information waves.
IT FROM BIT
matter cores /zpf < 0 that balance the centrifugal repulsion
from
quantized rotation about their centers of mass and from the
repulsive
self electric charge.
the pilot wave information BIT landscape it is rolling on in a
generalized gradient flow including the fiber space
connections or
gauge potentials as in the Bohm-Aharonov effect that is the
Josephson
effect in the macro-quantum case. That is action without
reaction for
the micro-quantum approximation with signal locality that
applies to
not apply to complex macro-quantum systems.
Einstein agreed, but nobody worked it out.
False. It's all worked out in Bohm and Hiley's The Undivided
Universe.
Now, it has been worked out. see QuantumMatter.com and
SpaceandMotion.com
The results are amazing. 1) All the natural laws are found as
properties
of the wave structure of the electron.
2) Everything grows out of only two principles which are
properties of
one thing - space.
Awesome. Gravity is the simplest piece of cake. Take a look. I
would
love to have your thoughts.
I do not know what you mean. Have you derived the equations
for general
relativity from the information wave? That is precisely what I
have
done for the giant vacuum pilot wave along with the unified
dark
energy/matter local field.
Any new proposal must be couched in mathematical language and
must in
suitable limiting cases yield the battle tes equation of
theoretical
physics such as
Guv = (8piG/c^4)Tuv
Maxwell's equations
etc.
Otherwise it is not legitimate physics IMHO.
Also there must be contact with experimental observations both
in terms
of prediction and explanation as nicely presen in David
Deutsch's
book The Fabric of Reality for example in the chapters on
proper
methodology in theoretical physics.
Ditto for the excess verbal baggage of less by David C.
Williams below
on the nature of c in E = mc^2. The hard core of what is
behind this
can be found in the book by Wheeler and Taylor's introductory
text on
Einstein's relativity. The basic idea of geometrodynamics is
the block
universe of 4 dimensions with time and space mixing together
in changes
of perspective of uniformly moving observers in the globally
flat case
without gravity to begin with.
The issue is the invariance of the 4D line element ds under
the relevant
groups of local frame transformations at a fixed spacetime
event P.
Given a local frame of reference with coordinates x,y,z, t
ds^2 = dx^2 + dy^2 + dz^2 - c^2dt^2 = dx'^2 + dy'^2 + dz'^2 -
c^2dt'^2
Here c must be invariant under the group as in the Lorentz
transformations
dx' = (1 - (v/c)^2)^-1/2[dx - vdt]
dt' = (1 - (v/c)^2)^-1/2[t - vx/c^2]
dy' = dy
dz' = dz
for a nonaccelerating frame shift at constant velocity v along
the
common direction x parallel to x' of the two global inertial
frames S
and S'.
One cannot describe gravity this way if one insists on retarded
causality of no teleological future causes of past effects
associa
with the ideas of destiny, fate and purpose.
could use Newton's gravity with global special relativity to
produce the
three classic tests of GR provided one introduced the
Wheeler-Feynman-Dirac-Hoyle-Narklikar trick of advanced
potentials in
addition to retarded potentials. The fact that Puthoff gets
those tests
as well with his variable dielectric vacuum model is no great
achievement either because Einstein's classical
geometrodynamics goes
beyond those tests, e.g. gravimagnetism and gravity waves and
black holes.
With gravity one must use LOCALLY curved spacetime in which at
spacetime
point event P
ds^2(P) = guv(P)dx^udx^v
with summation convention u,v, = 0,1,2,3
There are two LOCAL symmetry groups here.
The Poincare group is no good anymore. The translation
subgroup symmetry
of the Poincare group is broken by the locally variable Diff(4)
curvature tensor that is the essence of gravity. The local
Lorentz group
of invariant tipped light cone structure is obeyed in the
tangent
fiber space attached to P.
The base space of the tangent bundle obeys the Diff(4) group
of LOCAL
general coordinate tensor transformations xu' = x^u'(x^u) that
replaces
the translation subgroup of the globally flat Poincare group
of special
relativity.
This is all for zero torsion of course.
All LOCAL observables must be tensors or spinors under both
groups. A
spinor is a square root of a tensor.
Einstein's equivalence principle is mathematically represen by
the
tetrad map eu^a(P) from locally flat tangent space inertial
coordinates
a to locally curved base space non-inertial coordinates u. The
a -> a'
transformation is via the 6-parameter Lorentz group of special
relativity. The u -> u' transformation is via the 4-parameter
Diff(4)
group of general relativity.
dx^u = eu^a(P)dx^a
nab = ea^u(P)eb^v(P)guv(P)
Local elimination of the non-tensor connection field for
parallel
transport of tensors along world line paths normally associa
with
Newtonian gravity acceleration g for example. These are
g-forces or
inertial forces from accelerating non-inertial frames like the
surface
of the rotating Earth for example. They are not inhomogeneous
tidal
variations in the g-force from the curvature tensor, which are
never
locally elimina although they are here on Earth very small of
order
(scale of measurement) 10^-13 in centimeters.
Where nab is the flat space-time constant metric tensor of
special
relativity and guv(P) is the locally variable metric which
represents a
real gravity field only when the 4th rank curvature tensor of
tidal
forces does not vanish. You can have a variable guv(P) without
a real
gravity field from using a non-inertial local frame that is
accelerating
without tidal forces. This is not physically of great interest
however.
A LOCAL tensor that vanishes in one LOCAL frame vanishes in
ALL LOCAL
frames at fixed point event P for ALL relevant symmetry groups.
If tuv = 0 then tab = 0 and vice versa.
There is no such thing as a gravity force anymore in this
non-Newtonian
paradigm of geometrodynamics. Newton's gravity equations are
regained
in the limit of weak curvature and slow speeds compared to c.
If there
that minimize their dynamical action in the given metric field
guv(P)
Light rays move on null geodesics ds^2 = 0.
The invariance of the speed of light c in global special
relativity is
replaced by the above remark!
In general there are gravimagnetic cross terms in the case of
nonstationary metrics and this complicates what is meant by
the speed of
light.
For example, if there are no gravimagnetic cross terms in a
simple case
with spherical polar coordinates for a non-geodesic observer
subject to
spin 1 gauge forces like
ds^2 = grr(P)dr^2 - gthetatheta(P)dtheta^2 + gphiphi(P)dphi^2 -
c^2gtt(P)dt^2
For a light ray we have
0 = grr(P)dr^2 - gthetatheta(P)r^2dtheta^2 +
gphiphi(P)(rsin(theta))^2dphi^2 - c^2gtt(P)dt^2
with the usual convention that the LOCAL metric field guv(P)
is a pure
dimensionless number and r(P) is the Schwarzschild curvature
radial
coordinate defined such that the surface area surrounding the
center in
the static spherically symmetric spacetime geometry has the
Euclidean
area 4pir(P)^2 .
Consider the null radial geodesic, dtheta = dphi = 0
0 = grr(P)dr^2 - c^2gtt(P)dt^2
0 = dR^2 - c^2dT^2
where
dR = grr(P)^1/2dr
dT = gtt(P)^1/2dt
dR and dT are physically measured space and time intervals for
the
light ray using meter sticks and clocks or radars by the
non-geodesic
observer for small separations between two lightlike separa
events P
and P'. Small means compared to the local radii of spacetime
curvature.
For example in the static spherically symmetric Schwarzschild
vacuum
metric solution of
Ruv = 0
for r > 2Gm/c^2
gthetatheta = 1
gphiphi = 1
grr(P) = (1 - 2Gm/c^2r)^-1
gtt(P) = (1 - 2Gm/c^2r)
The black hole event horizon is at
gtt(P) = 0
dR = (1 - 2Gm/c^2r)^-1/2dr
But the circumference C = 2pir
The change in C for dr is
dC = 2pidr
Therefore
dC/dR = 2pi(1 - 2Gm/c^2r)^1/2
--> 0 at the event horizon.
The physical radius R = integral dR is much larger compared to
physical
circumference C as it would be in flat 3D space. If one keeps
R fixed ~
h/mc ~ G*m/c^2, the micro-geon of Wheeler's Mass without mass
Charge without charge Spin without spin shrinks to a point
in high resolution Heisenberg microscope scattering probes
with r ~ h/p
for momentum transfer p like SLAC deep inelastic electron
scattering off
protons.
This is a semi-classical theory without quantum electrodynamic
vacuum
polarization zero point energy density effects, however the
latter were
shown by Feynman and Schwinger to obey the Poincare group in
the absence
of gravity. The price for this is the obscure renormalization,
which may
not be internally mathematically consistent although its
predictions are
in remarkable agreement with experiments. Indeed, the
requirement of
renormalization with a finite number of fudge factors or
epicycles
:-) has been a very useful rule of thumb. Directly
micro-quantizing
Einstein's general relativity is not renormalizable, i.e. one
needs an
infinity of epicycle fudge factors. That tells us we have
asked the
wrong question. As John A. Wheeler says:
The Question is: What is The Question?
Resemblances of Wheeler's remark to Cantor's diagonal argument
and
Godel's incompleteness theorem of self-referential spontaneous
self-organization are not accidental and random. Wheeler
thinks of
the universe as a self-exci circuit of observer-participators.
The velocity of light c in ordinary non-gravitating vacuum
with /zpf ~
0 is directly measured by a variety of techniques. It is an
observable
measurable property. The velocity of light in a medium is c/n
where n is
the index of refraction. The physical vacuum also has an index
of
refraction n(vac) that is very close to 1 in most situations.
This
small variation comes primarily from vacuum polarization zero
point
fluctuations of the off mass shell or virtual
electron-positron plasma
electrically neutral ionized plasma inside the vacuum. This is
vacuum.
One must be careful in how to use both special and general
relativity
when polarization effects in real on mass shell media are
considered.
In the case of special relativity one should, for example,
consult the
text books by Arnold Sommerfeld, Panofsky and lips, Landau and
Lifz. A real on mass shell medium in a sense spontaneously
breaks
translational symmetry on scales larger than the unit cell of
the
lattice as described in more detail by P. W. Anderson in his
More is
different series of papers collec in A Career in Theoretical
Physics (World Scientific) When one includes all dynamical
degrees of
freedom including those inside the unit cell of the lattice,
global
special relativity is restored to the propagation of light in
a real
medium in the usual situation where gravity tidal forces are
negligible.
appropriate
parties (bcc's to you and Toby) to stimulate wider discussion
and
understanding of the important points you have raised towards
re-evaluation
and correction of fundamental errors in scientific
conceptualizations since
the key departure of Newton's work neutering science by his
using his
mathematicalizations which did not incorporate properly his
own determined
religious faith in the absolute nature of truth as per his
belief in God.
I
edi you posts lightly for clearer reading per common American
English
punctuation etc., and changed one word from proof to idea
relating
to
the Theosophical science quote so that your point was not
mistaken as that
quote you ci being some kind of empirical experimental proof
of the
your choice of proof over idea please expand on this in your
next
post
to
correct my misunderstanding and I will forward appropriately
with
apologies.
Your discussion properly considered, by those to whom it was
sent, should
also go a long way towards restoring, or integrating, ethics
into the
scientific discipline of thought since the only way out of
present
conundrums in science is to replace uncertainty about
uncertainty with
certainty about the absolute nature of truth itself and let
the chips
fall where they may in terms of how this change in principle
modifies
scientific perception of the nature of reality, the various
theory
equations etc.
I noticed in your writeups that you reference C as the
constant value of
the
speed of light in several places and then in one place you
seem to
reference it as the velocity of light. I understand that in his
earlier
works, especially in German, Einstein used the term velocity
and then
over years he too began routinely using speed apparently due
to the
mathematical convention dictating that by definition the value
of
the
square of the direction component of the square of a vector
property is
defined identically equal to one.
Thus, in trying to understand, for example, what is the true
value of, say,
(10mph-North)^2, the square of the velocity of ten miles per
hour in a
northerly direction, the simple (2+2=4) logic of ordinary math
is thrown
out the window by this mathematical convention per this dictum
and
the
normal logical value of the square of this vector quantity
(100m^2/h^2-North^2) is defined for all intents and purposes
as equal
to
(100m^2/h^2), ie, like saying because we cannot mentally grasp
or
understand what it means (North)^2, we define it to be
identical to
one,
ie, no meaning at all.
While this may seem a trivial point for all purposes within
the realm of
considering physical systems dynamics from the point of view
of current
science since apparently Galileo's time, ie, the exclusively
objective
nature of reality, ie, that observer and observed are separate
and all
physical systems that are real exist independenttly of and
identically
observable by all observers, or they are not considered real
by science
when they are not reproducible by all observers at all times,
in the
case
of C^2 as the proportionality constant between the value of
the Energy of
any given system of reality under observation and the value of
the mass of
that system, and since C itself appears in so many equations of
electrodynamics etc., herein apparently lies another
overlooked fundamental
misconception of scientific thought compromising the absolute
nature of
truth since C itself is not pegged to the notion of an external
objective reality reference frame but is pegged to the
identity of
the
observer. That is to say, since C is non-additive and its
square (C^2)
is
the proportionality constant between the values of energy and
mass of
real
physical systems then this mathematical convention seems
evidently
the
main limit of current mathematics to overcome for a full
understanding
of
the relative nature of reality, ie, the relationship depic in
Buddhism
(and in Hinduism I believe) that each person's mind is the
creator of
the universe relative to the identity of that person, ie, the
ego-centric
nature of the universe.
While this view seems rightly to folks like Dr. Sarfatti as
psycho-babble
it nevertheless is the view that offers a mathematical handle
for a
starting point to conceptualize and integrate into modern
physical theories
this precept that there is a consciousness factor at work
between the
nature of the observer and the nature of the system of reality
under
observation, ie, a relationship of mind between observer and
observed
as well acknowledged by quantum physics experiments over last
decades
where he describes this principle as observership.
Many scientists are wrestling with this notion of the
relationship of mind
between observer and observed, eg, deno in O'Leary's books as
the consciousness factor and in Dr. Sarfatti's post quantum
physics
of consciousness theory equations by their complexities that
include
the Uncertainty Principle mathematics and depict the operation
of
intent on the mental field of matter causing a 'back-action in
time'
reflecting back via that mental field of matter in a
'cybernetic feedback
loop of consciousness' of predictible 'moment of
consciousness' duration
which corellates well with latest experimental results in
neuroscience
etc
even though in Sarfatti's view, in my words, the observer is
transparent
ie, there is no accounting for variations between observer
identities
and corresponding variations in observable systems dynamics
from
observer to observed (non-reproducibility by all observers at
all
times of all physical systems dynamics, eg, psi-phenomena),
ie, the
differences in each observer's 'mind' impacting the physical
systems
dynamics of the 'reality' which each observer observes.
Back-action is not temporal. It is the direct reaction if IT
back on
its BIT field. IT's BIT field quantum potential Q now has
sources and
is not fragile, but has macro-quantum phase rigidity of which
Andrei
Sakharov's metric elasticity is an example. IT is no longer a
test
from in spontaneous self-organization - the participatory
universe as a
self-exci circuit. It is not to be confused with the
Wheeler-Feynman
advanced potential from future to past. It is true that when
the limits
of micro-quantum theory without back-action hence with signal
locality
are transcended, then one gets presponse signal nonlocality
which
mimics an advanced potential effect. Back-action in my sense
as quantum
action + post-quantum reaction, and advanced potentials are,
of course,
not incompatible in my theory in which EPR micro-quantum
nonlocality
with signal locality are as in John Cramer's transactional
generalization of the Wheeler-Feynman classical advanced
potential as
first no by Costa de Beauregard back in the 1970's. Antony
Valentini
shows that post-quantum back-action that pushes the system
away from
sub-quantal heat death causes signal nonlocality like what is
seen by
Dick Bierman in his mind-matter presponse experiments.
Macro-quantum
BEC systems in particular have post-quantum back-action IMHO.
The above is my latest attempt to explain in words what I have
rendered
since 1974 into low level mathematical equations of a unified
field
theory (referencing field of awareness and the human mind's
consciousness orientation function of light as correction
factor to
add back this omit true extra value of the square of the
direction
component of C^2 which is more than (north)^2 because of this
fact
that C is non-additive and therefore as such is fundamental,
being
the only physical not 'pegged' to the notion of the
exclusively objective
nature of reality but rather 'pegged' to the conceptualization
of the
'relative' (or 'ego-centric') nature of reality a
conceptualization which
is obvious in human daily life via common sense in all other
realms and
is throughout the history of human thought a fundamental
principle as
discussed in new age psycho-babble in such terms as we are the
center of our own universe and by application of the human mind
we can influence and change the nature of our universe.
The above paragraph by David Williams is New Age cargo cult
pseudoscience IMHO on a par with crystal power, orgone, spiral
dynamics, monoatomic highly deformed nuclei superconducting
Ormus
powder et-al. It is not even wrong in Wolfgang Pauli's sense.
Bruce DePalma is the only person who has looked at the four
equations
of my Tetron Natural Unified Field Theory and the minimal
discussion
of their meaning presen to him soon after I met him in May
1979,
and his reaction in subsequent soon talk with me was, David,
you
know, I understand your theory. 'Seeing is believing, right?'
was his
response with a glimmer of humor behind his eyes that made me
know that he had hit the nail on the head with this response
which
zero's in on the paradoxical relationship of mind between
seeing
and believing as it relates to all levels of human thought
including
science, religion, spirituality, politics, etc.
The above paragraph by David Williams is New Age cargo cult
pseudoscience IMHO on a par with crystal power, orgone, spiral
dynamics, monoatomic highly deformed nuclei superconducting
Ormus
powder et-al. It is not even wrong in Wolfgang Pauli's sense.
Exclusive of Dr. Sarfatti's vigorous ... criticisms,
everyone else since has either said so what? in response, or
like
O'Leary have just ignored and refused to respond to this theory
of mind, with the single exception of Dr. Fred Wood who
listened
to my first-ever public lecture on my theory, on September 10,
2001,
at my alma mater California State University at Northridge,
and after
made a point of telling me that he would think seriously about
what
I had said (a prepared written formal read aloud lecture
archived at
http://groups.yahoo.com/group/gcsc-csun/message/6 )
Subsequently, I emailed Dr. Wood and sugges that the next step
in the application of my tetron thesis is beyond the
mathematics
of my education as a Bachelor of Science in Chemistry, ie, how
to understand the (tensor?) mathematical relationship between
this correction factor Tetron -- the mathematical function
applied
to the square of the speed of light to correct it to its true
value
of
the square of the velocity of light orien relative to the
identity
of the observer, ie, adding back the square of the vector
component of C to overcome above described mathematical
convention, ie, compromise about the nature of light and the
absolute nature of truth itself in all places where C^2
appears in
physics equations -- as Tetron applies to C^2, and since many
equations particularly in electromagnetism contain C, there
must
be a similar correction factor to apply in such equations
which is
also the missing link in understanding the consciousness factor
as it relates to these new space energy technologies and the
expec reconciliation of their presently conflicting theories of
operation.
The above paragraph by David Williams is New Age cargo cult
pseudoscience IMHO on a par with crystal power, orgone, spiral
dynamics, monoatomic highly deformed nuclei superconducting
Ormus
powder et-al. It is not even wrong in Wolfgang Pauli's sense.
Since your theoretical approach well includes deep background
understanding on this Eastern osophy concept of the mind
as the creator of the universe per Vedic knowledge you write
and feedback on my above views will help stimulate productive
thinking towards a deeper understanding of this consciousness
factor as it also needs to be formalized in order to correct
the
mind of science and fully understand what is going on with all
these various new space energy technology experiments and
the theories posi to try and understand these results as well
as how some of the theories (like yours and Joseph Newman's)
have predic and perhaps even empowered the successful
experimental results that you have each obtained by different
configurations of rotating magnetic systems with entirely
different
theoretical underpinnings yet with similar overunity results.
Consciousness IMHO is when one has post-quantum back-action
which
excites the MACRO-QUANTUM BIT BEC field into self-awareness.
Our minds are macroscopic non-classical information fields of
both
active and inactive information in the Bohm-Hiley sense.
Flashes of
memory are the activation of unoccupied basins of attraction
of the
MACRO-QUANTUM BIT LANDSCAPE when the IT system point of
sub-microtubular
hydrophobically caged electric dipole Frohlich collective
modes roll
into that basin.
In the torsion field theory developed in recent decades from
the
Russian language-mind views expressed in mathematical
language and with deeper consideration of many documen
psi-phenomena experiments in former Soviet Union, going
back to the 1950's-60's apparently, there may also be this idea
of a relationship of mind between observer and observed but
the details of how this is represen in this theory are unclear
to me at my level of math education and literature
availability.
But it is clear from my personal conversation with Dr. Shipov
courtesy of Dr. Sarfatti's kind invitation for my informal
participation with his group in San Francisco one day a few
years ago, that torsion field theory also is perplexed by the
results of DePalma showing variation in gravitational behavior
between spinning and non-spinning ball bearing drop results
discussed in earlier post.
I do not believe DePalma's claims. Gennady's beliefs in psi
are not
directly connec with his torsion math. Bill Page talked about
the
latter at Vigier IV.
Creon Levit investiga such claims at ISSO 1999-2000 that
Williams
refers to here. Creon was not impressed with any of the New
Age Free
Energy claims promo by David. But Creon can speak for himself.
My hunch here on this is that rotating objects too have a
vector property which analogous to C discussed above, is
being overlooked in its importance as it relates to the
observer
because every rotating system also may, similar to C, be seen
in the view of its orientation and rotational properties as
relates
to the relative identity of the observer. I think that this
other
application of what I talk about above will not come here until
this issue about C is resolved, but that there is an important
connection between the correc true value of C^2 as per
above and a deeper understanding and correction also with
torsion field theory, although the math involved is way beyond
my education level to deal with as a language to express the
principles I have tried my best above to explain in my style of
California Chemical English
The above paragraph by David Williams is New Age cargo cult
pseudoscience IMHO on a par with crystal power, orgone, spiral
dynamics, monoatomic highly deformed nuclei superconducting
Ormus
powder et-al. It is not even wrong in Wolfgang Pauli's sense.
David Crockett Williams 661-822-3309
Chartered Life Underwriter
Bachelor Science Chemistry
www.angelfire.com/on/GEAR2000/vision.html
http://groups.yahoo.com/group/drums-of-peace
http://groups.yahoo.com/group/new-energy-solutions
http://www.josephnewman.com
Red Silk Road Peace March Project
USA, California, Japan, Korea, China, Nepal,
India, Pakistan, Afghanistan, Iran, Iraq, Jerusalem,
http://groups.yahoo.com/group/silk-road-to-peace
http://groups.yahoo.com/group/dharma-walks
One Person Can Make a Difference
www.kucinich.us
Rep. Kucinich
http://groups.yahoo.com/group/Kucinich-for-President
Spiritualism, The Highest Form of Politics
For a Culture of Peace & Community
Our Spiritual Unity Re-Awakening
www.leonardpeltier.org
www.horseforgovernor.com
www.prophecykeepers.com
http://web.mahatma.org.in
www.peaceinspace.com
www.cesarechavez.org
www.brianwillson.com
www.dharmawalk.org
www.sathyasai.org
www.tewari.org
www.prop1.org

===

Subject: Re: Is this newsgroup useless?
> Moderating this group is not an option. The USENET Powers
that Be
> (otherwise knows as the moderators of
news.announce.newgroups) have
> decreed that proposals for changing the moderation status of
existing
> groups will be not be accep. If you want a modera version of
> sci.math then you'll have to propose a new newsgroup.
But doesn't sci.math.research exist? And isn't it modera? So
what is
all the fuss about?
Joachim
--
Trau niemals einem Stollentroll!

===

Subject: Re: Is this newsgroup useless?
> But doesn't sci.math.research exist? And isn't it modera? So
what is
> all the fuss about?
sci.math.research doesn't allow discussions of elementary
topics
(homework problems), and other math rela things like Latex,
whether Maple is better than Mathematica, how easy it is to
lose
a job by going to the Joint Meetings, whether electronic
journals
are as good as paper ones, and so on.
Bart

===

Subject: Re: Is this newsgroup useless?
> But doesn't sci.math.research exist? And isn't it modera? So
what is
> all the fuss about?
sci.math.research is orien towards mathematical research at a
fairly
high level. sci.math isn't (although such topics arise here);
it's scope is
much broader.

===

Subject: Re: Is this newsgroup useless?
> The real
> problem is that this is an *unmodera* newsgroup. All posters
are
> permit.
> I have two other main newsgroup interests. One of them is
> christianity. In that case, I avoid the unmodera groups
> at like a Pharisee avoids a leper. They make sci.math look
like
> a candle next to the space shuttle taking off. It takes a
hardworking,
> patient and wise moderator to keeps christians behaving
like...uh..
> christians. So in that sense, your half right.
> But my other interest is homebrewing,
What's your favorite style? I like to brew IPA's. But, so far,
my best
brew is a lagered Bohemian Pilsner. Smooooth. I'm having a
little
problem with mashing ... poor yield. Math + Homebrew = FUN.
> and that group is UNmodera,
> but the most pleasant and helpful and unflaming group on all
of USENET.
> Every newbie who comes along asking the same dumb question
that's
> been asked 17 trillion times is welcomed with open arms,
given
> kind advice and we cheer that another lost sheep has been
conver
> from the evils of Budmilloors and demon megaswill. So your
half
> wrong.
> Half the problem here is that it's unmodera. The other half
is
> that it's popula by overly-anal-retentive jerks who think
it's
> going to be important to point out that I spelled you're your
> twice in the preceding paragraphs.
> There's something the same about fanatic legalists in the
religious
> groups and anal mathematicians, in that they think they've
scored
> something if they catch you in an error. AHA! You've typed
> slander when you should have typed libel! So what? Therefore
> I'm an idiot and he's a savant and now I have to erase his
> chalkboards for him? Hardly. I'm going to have a homebrew and
> hang out with the cute chicks while he re-catalogues his
PowerRanger
> Collector's Cards.
> I _thought_ we had beaten these guys up sufficiently in the
high
> school locker room that they'd be quiet by now. All I did was
> explain that if we'd all place our duffel bags
perpendicularly
> on the benches, there would be room for all of us to get
dressed
> at once. I really don't think that was sufficient reason for
> them to put Nair in my bottle of Prell.
> Maybe the real problem is that a guy can't e-mail a matburn.
> Bart

===

Subject: Re: Is this newsgroup useless?
>> But my other interest is homebrewing,
> What's your favorite style? I like to brew IPA's. But, so
far, my best
> brew is a lagered Bohemian Pilsner. Smooooth. I'm having a
little
> problem with mashing ... poor yield. Math + Homebrew = FUN.
I probably to an IPA every other batch, because they're ready
early and I seem to run out of beer so fast. I often make
the Sister Star variety because I like heavily hopped versions.
But my favorites are double bocks, even though they have to
age forever.
The thing that increased my mash efficiency the most was doing
a mash-out. (In a 10-gallon batch) just before sparging I drain
off 3 gallons of wort, bring to almost boil, and then add it
back
to the mash. This brings the temp up into the 165 range and the
sparging rinses out much more fermentables. (But you gotta' be
careful not to go higher than 170.) But maybe you do that
already.
The other thing that helped me a lot was paying attention to pH
and adding the right amount of calcium carbonate to the mash
water.
Gotta keep those enzymes happy.
rec.crafts.brewing is a great group. I've nicknamed my liver
Kenny,
by the way...
Bart

===

Subject: Re: Is this newsgroup useless?
at 02:32 PM, No One <NoAdress@nowhere.net> said:
> I guess that the time has come for me to ditch this group
>altogether. It seems to have been taken over by deranged
individuals
>exposing their misguided ideas over and over again (Mr.
Harris being
>the best representative of this increasingly vocal group,)
impervious
>to any kind of logic reasoning; by people trying to get the
group to
>do their homework for them, and by plain idiots who post the
most
>irrelevant babbling (like this moron who posts numerological
stuff,
>while changing his profile frequently.)
Kooks are endemic to Usenet. Your filter is your friend.
Actually,
sci.math is in good shape compared to some, even in the sci.*
hierarchy.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT
Unsolici bulk E-mail will be subject to legal action. I reserve
the right to publicly post or ridicule any abusive E-mail.
Reply to domain Patriot dot net user shmuel+news to contact
me. Do

===

Subject: Re: Is this newsgroup useless?
>> The O.P. may well be a troll.
> Believe me, I am not. I am just expressing despondency about
the status
> quo nowadays. I have been reading sci.math, on and off, for
some 14 years
> now, and it has gotten steadily worse. With all the trolls,
morons,
> lunatics and otherwise deranged contributors to this group
so prevalent
> today, going through new headers every day has become a
painful exercise.
Why? Do you talk to the nntp-server via the raw protocol?
Just use a decent news-reader and ignore certain Subject-lines
and authors.
Marc

===

Subject: Re: Is this newsgroup useless?
> Just use a decent news-reader and ignore certain
Subject-lines and
> authors.
He does use a decent newsreader, and he could ignore certain
Subject-lines
and authors with it.
Joachim
--
Trau niemals einem Stollentroll!

===

Subject: Re: Is this newsgroup useless?
>> I'd certainly read sci.math.modera but who would moderate
it?
>> It would be a lot of hard work.
>It wouldn't be so hard to read a message and decide whether to
>post it. And having decided not to post it, to return the
message
>to the author with the (first) offending sentence highligh and
>a pointer to the website with the posting guidelines.
>The REAL hard work would be in the fielding of the the
challenges
>to the moderator's decisions. If the moderator was strict and
did
>his work well, the posting-decisions work would be minor (but
a
>daily chore, no doubt) since posters would quickly learn what
was
>acceptable and abide by the rules. The moderator, by being
consistant
>and fair, could keep his own work minimal.
>It's when some dingbat engages him in an argument about
whether
>his post should have gone through and starts whining about
free
>speech and starts posting libelous crap about the moderator
and
>his fascist policies on sci.math and (as in one case I know)
actually
>takes the moderator to court, that things get out of hand.
(The
>idiot was laughed out of court, of course, but what a pain
for the
>moderator.) One buffoon on the christianity group got mad at
the
ads
>with active embedded scripts, which, upon opening, of course,
To bring the thread back on-topic, this reminds me of a
mathematical
problem recently raised by a member of my department, who
shall remain
anonymous. Suppose you (applies to male readers only!) had
taken advantage
of every offer of the above type received, and all had worked
as claimed.
Estimate how long it would be.
Derek Holt.

===

Subject: Re: Is this newsgroup useless?
>>[...] One buffoon on the christianity group got mad at the
ads
>>with active embedded scripts, which, upon opening, of course,
>To bring the thread back on-topic, this reminds me of a
mathematical
>problem recently raised by a member of my department, who
shall remain
>anonymous. Suppose you (applies to male readers only!) had
taken advantage
>of every offer of the above type received, and all had worked
as claimed.
You're assuming we haven't, and that if we had they wouldn't
work.
>Estimate how long it would be.
One learns not to step on anything pretty quickly...
>Derek Holt.
************************

===

Subject: Re: Is this newsgroup useless?
> The number of replies is the thread's rating. Apparently,
post's quality
has
> nothing to do with the rating. Now, face it: the JSH show
has the
highest
> rating in sci.math.
> That's what makes it so sad - that the rantings of that
pathetic
loony
> seem to be stuff that so many contributors to sci.math are
so interes
> in.
I'm puzzled why this bothers you so much. The huge majority of
posts on
sci.math are about mathematics. It's easy to skip the other
threads,
assuming a moderately decent newsreader.

===

Subject: Re: Is this newsgroup useless?
>> The number of replies is the thread's rating. Apparently,
post's
quality has
>> nothing to do with the rating. Now, face it: the JSH show
has the
highest
>> rating in sci.math.
>> That's what makes it so sad - that the rantings of that
pathetic
loony
>> seem to be stuff that so many contributors to sci.math are
so interes
>> in.
>I'm puzzled why this bothers you so much. The huge majority
of posts on
>sci.math are about mathematics.
I'm glad you said that, because I wan to but hesita to do so
because after all I'm one of the guys who's part of the problem
according to various posts here.
Seems to me that an even huger majority of the _threads_ are
about mathematics, making it even easier to filter out the crap
if you want.
>It's easy to skip the other threads,
>assuming a moderately decent newsreader.
************************

===

Subject: Re: Is this newsgroup useless?
> The number of replies is the thread's rating. Apparently,
post's
quality has
>> nothing to do with the rating. Now, face it: the JSH show
has the
highest
>> rating in sci.math.
That's what makes it so sad - that the rantings of that
pathetic
loony
>> seem to be stuff that so many contributors to sci.math are
so
interes
>> in.
I'm puzzled why this bothers you so much. The huge majority
of posts on
>sci.math are about mathematics.
> I'm glad you said that, because I wan to but hesita to do so
> because after all I'm one of the guys who's part of the
problem
> according to various posts here.
Why not tell it like it is, Ullrich? You're the epitome of a
mathie
sadist!
--John
> Seems to me that an even huger majority of the _threads_ are
> about mathematics, making it even easier to filter out the
crap
> if you want.
>It's easy to skip the other threads,
>assuming a moderately decent newsreader.
> ************************
>

===

Subject: Re: Is this newsgroup useless?
>>I'm puzzled why this bothers you so much. The huge majority
of posts
>>on sci.math are about mathematics.
> I'm glad you said that, because I wan to but hesita to do so
> because after all I'm one of the guys who's part of the
problem
> according to various posts here.
Well, as I think about it more, the social aspect of
the newsgroup is part of its appeal also. My original outline
for the strict moderation would leave the atmosphere pretty
antiseptic, after all.
We have to feel like we're sitting around the math lounge
chatting and if we want to go off on tangents then we can if
we feel like it. As someone said, a modera group was tried
before, and almost no one participa. Nice argument Are
you going to Phoenix this year? I never go to those silly
things. Waste of time. Waste of time! I'm giving a short
course!
So you're one of those idiots padding his vita by pretending to
be an expert on a non-topic? What a bozo! 'Bozo'? I hope that
clown sues you for copyright infringement! It's not 'copyright'
it's 'trademark', Krusty.......
After all, most of us are nerds, and this may be our only
social life....;-)
So perhaps the _real_ problem is that the math lounge has
2000 people in it, and the cross conversation requires a bit
more subtle etiquette than a normal math lounge with only 5 or
10 people.
Several people have commen that the filtering systems should
work fine. Sure they do, if you continuously update them.
Maybe some semi-official header guidelines (that is some sort
of netiquette for sci.math) that aids in efficient filtering
would be the thing to persue. Violators could be politely
informed
of the conventions, and if they choose to ignore them, then
there's
nothing we can do about it, except ignore them, but that
should be
sufficient.
I guess having JSH in certain headers was a step in that
direction,
but I don't really have any good ideas as to what header
guidelines
should be. I'm just thinking outloud.
Bart

===

Subject: Re: Is this newsgroup useless?
>[...]
>Several people have commen that the filtering systems should
>work fine. Sure they do, if you continuously update them.
If you continuously update them, right. That's really a _lot_
of work with a decent newsreader.
Like take Agent, for example. I download the new headers.
Years ago I'd just sit there for a few minutes with a finger
on the N key, moving to the next new post and hitting
Enter if I wan to read it. These days I sit there with one
finger on the N key and one on the I key - when a wacky
thread appears (in some other group where I don't want
to see the wacky threads, not here) I hit the I key instead
of the N key, and I'm never bothered by that thread again.
Which is to say that simultaneously scanning for things
you want to read and killfiling things you don't want to
read in the future is _no_ harder than just browsing
through the headers, if you're using a decent newsreader.
You must have internet access - Agent has a free version,
so I don't see what the problem is.
Well, you do need to be able to find the N key and the
I key. That can be tricky at first, but you only have to
do it once per session; just leave your fingers there.
>Maybe some semi-official header guidelines (that is some sort
>of netiquette for sci.math) that aids in efficient filtering
>would be the thing to persue. Violators could be politely
informed
>of the conventions, and if they choose to ignore them, then
there's
>nothing we can do about it, except ignore them, but that
should be
>sufficient.
>I guess having JSH in certain headers was a step in that
direction,
>but I don't really have any good ideas as to what header
guidelines
>should be. I'm just thinking outloud.
You're certainly doing _something_ out loud - hard to say
whether
thinking is the right word, since the problems you're
complaining
about are so easy to fix. Luckily this is not a modera group,
so you're free to bounce ideas off us just like JSH is...
>Bart
************************

===

Subject: Re: Is this newsgroup useless?
>so I don't see what the problem is.
>Well, you do need to be able to find the N key and the
>I key. That can be tricky at first, but you only have to
>do it once per session; just leave your fingers there.
Actually that could be a hard task, to locate those two keys.
R'ght 'ow, for ''sta'ce, ' ca''t seem to do 't.
dave (short for dav'd rus'')

===

Subject: Re: Is this newsgroup useless?
> You're certainly doing _something_ out loud - hard to say
whether
> thinking is the right word, since the problems you're
complaining
> about are so easy to fix. Luckily this is not a modera group,
> so you're free to bounce ideas off us just like JSH is...
I haven't been complaining at all. Just chatting with the
original complainer. Everyone knows what the really, really,
absolutely true problem is, however, and it is the insufferable
rudeness of several posters. But like you say, these things
are easy to fix.
<plonk>

===

Subject: Re: Is this newsgroup useless?
>[...]
>Several people have commen that the filtering systems should
>work fine. Sure they do, if you continuously update them.
>> You're certainly doing _something_ out loud - hard to say
whether
>> thinking is the right word, since the problems you're
complaining
>> about are so easy to fix. Luckily this is not a modera
group,
>> so you're free to bounce ideas off us just like JSH is...
>I haven't been complaining at all. Just chatting with the
>original complainer. Everyone knows what the really, really,
>absolutely true problem is, however, and it is the
insufferable
>rudeness of several posters. But like you say, these things
>are easy to fix.
><plonk>
See, that wasn't all that hard.
************************

===

Subject: Re: Is this newsgroup useless?
>So perhaps the _real_ problem is that the math lounge has
>2000 people in it, and the cross conversation requires a bit
>more subtle etiquette than a normal math lounge with only 5 or
>10 people.
Nah. The _real_ problem is that the math lounge has 2000
people in it, of whom a number (probably not out of proportion
to that in the general population) have the bad qualities of
mind
that made life hell for John Nash and those around him, without
having the good qualities of mind that made Nash a
mathematician
of note.
That is: it's not the nerds, it's the nuts.
Lee Rudolph

===

Subject: hw help -- continuity
Folks,
I have a couple questions. This is homework, so please post a
nudge,
not a solution.
1)prove that if f,g continuous, then so are max(f,g) and
min(f,g)
After drawing some graphs, this seems pretty obvious for the
single
point a0 -- max(f,g) has 2 cases: it equals to f or g. Either
is
continuous. However, this question implies continuous on R,
not just
at a single point. Any ideas how to approach this?
2)Let f be a function with the property that every point of
discontinuity (ie the lim (x->a) f(x) exists, but is not equal
to
f(x)) is a removeable discontinuity. This implies lim
(y->x)f(y)
exists for all x, but f may be discontinuous at some (even
infinitely
many) numbers x.
Define g(x) = lim (y->x) f(x). Prove g is continuous.
--I don't even know where to start with this one.
-earl-

===

Subject: Re: hw help -- continuity
> Folks,
> I have a couple questions. This is homework, so please post
a nudge,
> not a solution.
> 1)prove that if f,g continuous, then so are max(f,g) and
min(f,g)
> After drawing some graphs, this seems pretty obvious for the
single
> point a0 -- max(f,g) has 2 cases: it equals to f or g.
Either is
> continuous. However, this question implies continuous on R,
not just
> at a single point. Any ideas how to approach this?
I'm confused. If you can show that max(f,g) is continuous at
each
point, then you have shown that max(f,g) is continuous.
However, you still have to prove that max(f,g) is continuous at
a point given that f and g are both continutous at that point.
Remember, the ideal of continuity is that one is making a
statement
about how control over the independent variable allows control
over the dependent variable. For example, if h(t) represen the
height of a balloon (h) as a function of time (t), then to
say that h is continous is to say that if at a given time t0,
if I confine my attention to times sufficiently close to t0,
then the height of the ballon is guaranteed to be sufficiently
close to h(t0). Formally, the two occurances of sufficiently
are replaced by delta and epsilon, respectively. Thus,
if I want h is a continutous function of t, suppose
I want a guarantee that the balloon will now have change
height by more than 1000ft. You might respond that this
will be true if I confine my attention to a time period
of 60seconds. So you have told me that if |t-t0|<60,
then |h(t)-h(t0)|<1000. Suppose I want better--I want a
guarantee
that the balloon has not changed altitude by more than 100 ft.
Well, you might tell me that now I must confine my attention
to 20 seconds. You have told me that if |t-t0|<20, then
|h(t)-h(t0)| < 100. Now, the above is the basic idea
of continuity. In the first case, I demanded what to
do of an epsilon=1000, and you told me that a delta=60 would
suffice. Then I asked what I would need to get a guarantee
for epsilon=100, and you told me delta=20. Note, but the
way, that if delta=20 works for epsilon=100, then delta=10
also works for epsilon=100. That is, for each epsilon you
have to produce a delta small enough so that changes in the
independent variable of less than delta result in a change
of less than epsilon in the dependent variable. For checking
continuity, one need not provide an optimal or largest
delta--this
is part of the abstraction. Of course, if you told me that
for epsilon=100ft I would need a delta of 0.0000000001sec,
then this might not be useful for practical purposes; a
continuous
function can still jump around a lot. But for the abstract
idea of continuity, you must merely argue that for at an
arbitrary point, for a given epsilon>0, there exists a delta
(perhaps a very small delta, of no real use) such that
a change in the independent variable of less than delta
results in a change of the dependent variable that is
*logically guaranteed* to be less than epsilon.
Now back to the problem at at. We want to guarantee that
max(f,g) doesn't change too quickly. We know that f
and g don't change too quikly, so this is looking reasonable.
Now we follow our nose and try the standard openning
Let a0 be a point, and epsilon>0.
[Ok, what now. Well we know that f and g are continuous. Let's
write down what that means]
So, for the particular epsilon lis about, there exists delta_1
such that |x-a0|<delta_1 implies |f(x)-f(a0)|<epsilon
[Note, we call the delta that comes from f delta1 because
we know were going need another delta fog g, as we do now
Again, for the epsilon chosen above, there exists delta_2
such that |x-a0|<delta_2 implies |f(x)-f(a0)|< epsilon.
[Ok, so this is looking good--we've used continuit of f and
g, and it's making sense--we are saying that we can use
bounds on the independent variable to constrain the
dependent varialbe. Now we need to choose a delta for
the function max(f,g).
So we write somethiing like delta = {some function of delta_1
and
delta_2}
then show that that delta works for max(f,g)]
> 2)Let f be a function with the property that every point of
> discontinuity (ie the lim (x->a) f(x) exists, but is not
equal to
> f(x)) is a removeable discontinuity. This implies lim
(y->x)f(y)
> exists for all x, but f may be discontinuous at some (even
infinitely
> many) numbers x.
> Define g(x) = lim (y->x) f(x). Prove g is continuous.
Try the one above again--if you still have problems, write
back.
-MIke
> --I don't even know where to start with this one.
> -earl-

===

Subject: Re: hw help -- continuity

===

Subject: hw help -- continuity
>1)prove that if f,g continuous, then so are max(f,g) and
min(f,g)
>After drawing some graphs, this seems pretty obvious for the
single
>point a0 -- max(f,g) has 2 cases: it equals to f or g. Either
is
>continuous. However, this question implies continuous on R,
not
>just at a single point. Any ideas how to approach this?
Show min(x,y), max(x,y), RxR -> R are continuous.
Then as f,g are continuous, so are compositions min(f,g),
max(f,g)
----

===

Subject: Re: hw help -- continuity
> 1)prove that if f,g continuous, then so are max(f,g) and
min(f,g)
> After drawing some graphs, this seems pretty obvious for the
single
> point a0 -- max(f,g) has 2 cases: it equals to f or g.
Either is
> continuous. However, this question implies continuous on R,
not just
> at a single point. Any ideas how to approach this?
There is a slight complexity. The point may lie on both f and
g.
You should try working directly from the definition for
continuity and
playing around with the inequalities associa with the min and
max
functions.

===

Subject: Re: hw help -- continuity
> Folks,
> I have a couple questions. This is homework, so please post
a nudge,
> not a solution.
> 1)prove that if f,g continuous, then so are max(f,g) and
min(f,g)
> After drawing some graphs, this seems pretty obvious for the
single
> point a0 -- max(f,g) has 2 cases: it equals to f or g.
Either is
> continuous. However, this question implies continuous on R,
not just
> at a single point. Any ideas how to approach this?
Proving a function continuous at any point proves it
continuous at
every point.
> 2)Let f be a function with the property that every point of
> discontinuity (ie the lim (x->a) f(x) exists, but is not
equal to
> f(x)) is a removeable discontinuity. This implies lim
(y->x)f(y)
> exists for all x, but f may be discontinuous at some (even
infinitely
> many) numbers x.
> Define g(x) = lim (y->x) f(x). Prove g is continuous.
> --I don't even know where to start with this one.
Does you definition of g(x) actually read g(x) = lim (y->x)
f(x)
or should it be g(x) = lim (y->x) f(y)?
In the first case, g(x) = f(x) at all x, so is not continuous
at
discontinuities of f.
In the latter case, does g(x) = lim (y -> x) g(y), for all x?

===

Subject: Re: hw help -- continuity

===

Subject: Re: hw help -- continuity
>> 1)prove that if f,g continuous, then so are max(f,g) and
min(f,g)
>> Any ideas how to approach this?
>Proving a function continuous at any point
>proves it continuous at every point.
f(x) = 0 when x <= 0
= 1 when 0 < x
is continuous at 1. Thus it's continuous at 0?
----

===

Subject: Re: hw help -- continuity

===

> Subject: Re: hw help -- continuity
>> 1)prove that if f,g continuous, then so are max(f,g) and
min(f,g)
>> Any ideas how to approach this?
>Proving a function continuous at any point
>proves it continuous at every point.
> f(x) = 0 when x <= 0
> = 1 when 0 < x
> is continuous at 1. Thus it's continuous at 0?
But 1 is not any point, it is a specific point.
To restate less ambiguously:
If you prove f continuous at an arbitrary point, then it is
continuous at every point
And to prove it at an arbitrary point, you only need show that
for
an arbitrary x, g(x) = lim_{y -> x} g(y)

===

Subject: Re: After Coxeter
> At one point or another I read parts of these books:
> * Manfredo P. do Carmo: Riemannian Geometry
> a very nice reading. good exercises
> * R.W. Sharpe : Differential Geometry. Cartan's
Generalization of
> Klein's Erlangen Program.
> very broad minded. not allways accurate.
> * Kobayashi and Numizu
> enciclopedic.
> * J. Jost
> the analytic view. very dense.
> * Spivak's pentalogy
> surpsingly, some parts of it are actually nice.
Yes, now it's manufactured as a proper book, I'm saving up my
pennies to
buy it.
> HTH
> D.L.
--
G.C.

===

Subject: Exponentative closure
Can the reals be defined using repea exponentative closure?
By the exponentative closure F, I define F/x as the set of all
the
zeroes of all the polynomial functions with coeffeicients AND
exponents in F. For example, the the algebraics are the
exponentative
closure of the integers. Thus, it can be written A=Z/x. Does
C=A/x? If not what does A/x equal? Can C be genera by
repealy exponentatively closing the integers a finite number of
times? If so, how many? A countable number of times? An
uncountable
number of times?

===

Subject: Re: Exponentative closure
>Can the reals be defined using repea exponentative closure?
>By the exponentative closure F, I define F/x as the set of
all the
>zeroes of all the polynomial functions with coeffeicients AND
>exponents in F. For example, the the algebraics are the
exponentative
>closure of the integers. Thus, it can be written A=Z/x. Does
>C=A/x? If not what does A/x equal? Can C be genera by
>repealy exponentatively closing the integers a finite number
of
>times? If so, how many? A countable number of times? An
uncountable
>number of times?
The set of all that is obtained is countable. This is because
each polynomial has only a finite number of coefficients and
exponents, and returns only a finite number of results. This
is assuming that a clear definition exists of x^y if x is
negative and takes the value |x|^y; more can be done.
Doing this more than omega (the order type of the integers)
times yields nothing new, because there are only a finite
number
of arguments for each extension.
--
This address is for information only. I do not claim that
these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558

===

Subject: Re: Exponentative closure
>Can the reals be defined using repea exponentative closure?
>By the exponentative closure F, I define F/x as the set of
all the
>zeroes of all the polynomial functions with coeffeicients AND
>exponents in F. For example, the the algebraics are the
exponentative
>closure of the integers. Thus, it can be written A=Z/x. Does
>C=A/x? If not what does A/x equal? Can C be genera by
>repealy exponentatively closing the integers a finite number
of
>times? If so, how many? A countable number of times? An
uncountable
>number of times?
Unless I misunderstand you, F/x is countable if F is
countable. So no,
a finite or even a countable number of exponentative closures
won't
do it: the union of countably many countable sets is
countable. I don't
know about an uncountable number of times.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2

===

Subject: Re: Exponentative closure
>Can the reals be defined using repea exponentative closure?
>By the exponentative closure F, I define F/x as the set of
all the
>zeroes of all the polynomial functions with coeffeicients AND
>exponents in F. For example, the the algebraics are the
exponentative
>closure of the integers. Thus, it can be written A=Z/x. Does
>C=A/x? If not what does A/x equal? Can C be genera by
>repealy exponentatively closing the integers a finite number
of
>times? If so, how many? A countable number of times? An
uncountable
>number of times?
> Unless I misunderstand you, F/x is countable if F is
countable. So no,
> a finite or even a countable number of exponentative
closures won't
> do it: the union of countably many countable sets is
countable. I don't
> know about an uncountable number of times.
> Robert Israel israel@math.ubc.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia
> Vancouver, BC, Canada V6T 1Z2
A further question along those lines is what is is the set, E,
of
numbers genera by repea sums of products of rational powers of
rationals? It is a subset of the algebraics, but does it form a
field? What can be said about the set of sums and products of
elements of E to the power of elements of E?

===

Subject: Re: Exponentative closure
>A further question along those lines is what is is the set,
E, of
>numbers genera by repea sums of products of rational powers of
>rationals? It is a subset of the algebraics, but does it form
a
>field?
I believe so: if x_1 is a member of E, then so are its
conjugates x_2,
x_3, ..., x_n.
Note that x_1 x_2 ... x_n is rational (being the constant term
of a
monic polynomial over the rationals whose roots are x_1, x_2,
..., x_n).
And then 1/x_1 = x_2 ... x_n / (x_1 x_2 ... x_n) is in E.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2

===

Subject: Re: Exponentative closure
>A further question along those lines is what is is the set,
E, of
>numbers genera by repea sums of products of rational powers of
>rationals? It is a subset of the algebraics, but does it form
a
>field?
> I believe so: if x_1 is a member of E, then so are its
conjugates x_2,
> x_3, ..., x_n.
> Note that x_1 x_2 ... x_n is rational (being the constant
term of a
> monic polynomial over the rationals whose roots are x_1,
x_2, ..., x_n).
> And then 1/x_1 = x_2 ... x_n / (x_1 x_2 ... x_n) is in E.
> Robert Israel israel@math.ubc.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia
> Vancouver, BC, Canada V6T 1Z2
If we say that a set, P, is exponentially closed if for all
p,q in P,
p^q is P, then can the complex numbers be defined as the field
that is
exponentially closed? Or even more simply, the set that is
exponentially closed and contains -2 and 0? If not, what are
the
properties of the smallest field, S, that is exponentially
closed
(cardinality, algebraic completeness, convergence of
Cauchy-Sequences)?
Note: that I am only considering fields with characteristic 0.

===

Subject: Re: Exponentative closure
Visiting Assistant Professor at the University of Montana.
>>A further question along those lines is what is is the set,
E, of
>>numbers genera by repea sums of products of rational powers
of
>>rationals? It is a subset of the algebraics, but does it
form a
>>field?
>> I believe so: if x_1 is a member of E, then so are its
conjugates x_2,
>> x_3, ..., x_n.
>> Note that x_1 x_2 ... x_n is rational (being the constant
term of a
>> monic polynomial over the rationals whose roots are x_1,
x_2, ..., x_n).
>> And then 1/x_1 = x_2 ... x_n / (x_1 x_2 ... x_n) is in E.
>> Robert Israel israel@math.ubc.ca
>> Department of Mathematics http://www.math.ubc.ca/~israel
>> University of British Columbia
>> Vancouver, BC, Canada V6T 1Z2
>If we say that a set, P, is exponentially closed if for all
p,q in P,
>p^q is P, then can the complex numbers be defined as the
field that is
>exponentially closed? Or even more simply, the set that is
>exponentially closed and contains -2 and 0?
>Note: that I am only considering fields with characteristic 0.
No; either set is too small.
I assume that your operation ^:PxP -> P is just a partial
operation,
not defined at (0,0); so just redefine it to give ^(0,0) = 0.
The smallest field of characteristic zero that is
exponentially closed
would be the closure of Q under the operation ^. Since ^ is a
finitary
operation (takes only a finite number of arguments), the
closure of Q
is obtained as follows:
S_0 = Q;
T_{n} = S_n cup ^(S_n,S_n)
where ^(S_n,S_n) is the image of (S_n,S_n) in C; that is, all
numbers
of the form p^q with p,q in S_n, under that definition;
S_{n+1} = T_{n} cup +(T_{n},T_{n}) cup -(T_n) cup *(T_n,T_n)
cup
^{-1}(T_n-{0})
where +(T_{n},T_{n}) is the sum of any two elements of T_n,
-(T_n) is
the additive inverse of any element of T_n, *(T_n,T_n) is the
product
of any two elements of T_n, and ^{-1}(T_n-{0}) is the
multiplicative
inverse of any nonzero element of T_n
Then the closure of Q is
S_{omega} = Union_{n<w} S_n.
Going beyond n=omega does not give you anything new, since the
operations are all of arity strictly less than w. Or, it is
easy to
verify that S_{omega} will be closed under all the operations,
and so
is a field closed under your exponentiation.
Now, S_0 is countable; clearly, if S_n is countable, then so
are
^(S_n,S_n) and therefore T_n, and if T_n is countable, then so
is
S_{n+1}. So S_{omega} is the countable union of countable
sets, hence
is itself countable. Thus, it cannot equal the complex numbers.
>If not, what are the
>properties of the smallest field, S, that is exponentially
closed
>(cardinality, algebraic completeness, convergence of
>Cauchy-Sequences)?
So S=S_{omega}.
Cardinality is clearly countable, by the argument above (a
standard
argument of General Algebra). You would not get convergence of
Cauchy
sequences: since this is countable, there is a real number
which is
not in the set, and of course that real number is the limit of
a
cauchy sequence of rationals, which are all in S.
I suspect algebraic completeness will also fail: if r is an
algebraic
number such that Q(r) is not solvable by radicals, how would
you get r
in S? Of course, since you also have non-algebraic numbers in
S, this
is hardly a 'proof', more of a 'I would look at these kinds of
numbers
to try to settle the question in the negative'.

===

Subject: Given the radius of convergence for one series, I'm
trying to find
the radius for 2 simliar series
Let f(z) = sum_k a_k*z^k be a formal power series with radius
of
convergence 1. Put s_n = a_0 + ... + a_n and t_n = (s_0 + ... +
s_n)/(n+1).
Let g(z) = sum_k s_k*z^k and h(z) = sum_k t_k*z^k.
How would I show that the radius of convergence of h and g are
both 1
as well? (The lim sup method doesnt seem to work).
Greg

===

Subject: Re: Given the radius of convergence for one series,
I'm trying to
find the radius for 2 simliar series
>Let f(z) = sum_k a_k*z^k be a formal power series with radius
of
>convergence 1. Put s_n = a_0 + ... + a_n and t_n = (s_0 + ...
+
>s_n)/(n+1).
>Let g(z) = sum_k s_k*z^k and h(z) = sum_k t_k*z^k.
>How would I show that the radius of convergence of h and g
are both 1
>as well? (The lim sup method doesnt seem to work).
Some hints:
Note that for any epsilon > 0, there is C such that
|a_k| < C (1+epsilon)^k for all k. What does that say about
|s_k|?
For the other direction, note that a_n = s_n - s_{n-1}.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2

===

Subject: What is C[[x]] (generating functions) : algebra
structure
I have a problem in combinatorics (chapter is generating
functions)
where f(x) = sum_{x>=0} a_n x^n is in C[[x]].
What is C[[x]]?
I have to show that composition of two f,g in C[[x]] is again
in C[[x]]
if the constant term of g is 0.
Our professor defined this structure in class but I can't find
the
definition.
Any help on the definition of this structure is greatly
apprecia.
Also, what are WEIGH generating functions?

===

Subject: Re: What is C[[x]] (generating functions) : algebra
structure
NNTP-Posting-User: [DNl+kfRPf6mkEI2haabWc7pkPCvSlQMj]
> I have a problem in combinatorics (chapter is generating
functions)
> where f(x) = sum_{x>=0} a_n x^n is in C[[x]].
> What is C[[x]]?
Ring of formal power series. Elements are power series in x
with
coefficients in C, not necessarily convergent.
> I have to show that composition of two f,g in C[[x]] is
again in C[[x]]
> if the constant term of g is 0.
> Our professor defined this structure in class but I can't
find the
> definition.
> Any help on the definition of this structure is greatly
apprecia.
> Also, what are WEIGH generating functions?
Michael A. Van Opstall
Padelford C-113
opstall@math.washington.edu
http://www.math.washington.edu/~opstall/

===

Subject: Re: What is C[[x]] (generating functions) : algebra
structure
Visiting Assistant Professor at the University of Montana.
>I have a problem in combinatorics (chapter is generating
functions)
>where f(x) = sum_{x>=0} a_n x^n is in C[[x]].
>What is C[[x]]?
Usually, power series on nonnegative powers of x. Like
polynomials with
coefficients in C, except they are allowed to have infinitely
many
nonzero monomials.

===

Subject: Re: What is C[[x]] (generating functions) : algebra
structure
> I have a problem in combinatorics (chapter is generating
functions)
> where f(x) = sum_{x>=0} a_n x^n is in C[[x]].
> What is C[[x]]?
> I have to show that composition of two f,g in C[[x]] is
again in C[[x]]
> if the constant term of g is 0.
> Our professor defined this structure in class but I can't
find the
> definition.
> Any help on the definition of this structure is greatly
apprecia.
> Also, what are WEIGH generating functions?
C[[x]] are simply formal power series, i e you do not care
for convergence. Weigh? Just ask him (as you could do
for the first Q), may be he says: 'put some factors at the
coefficients'.
Hm ... there is no shame to ask questions in lectures ...

===

Subject: looking for smooth function
Hello sci.math,
I've got an array size 100 of integer values (range: 0-5).
Each plot in
the array represents a 1ms time window. I currently have these
graphed
as stairs in MatLab (which looks like a bar graph). I want to
make
this function smooooooth, but i'm not sure of a method to
apply to it.
What steps should I take to complete this?
Dustin

===

Subject: Express As Single Fraction
How do I do this?
Express the following as a single fraction:
4/3ab - 5/6bc
and
(m^2 + 2)/(m^2 + m) - (m - 2)/m

===

Subject: Re: Express As Single Fraction
>How do I do this?
>Express the following as a single fraction:
>4/3ab - 5/6bc
Multiply the first term by 2c/2c and the second by a/a.
>and
>(m^2 + 2)/(m^2 + m) - (m - 2)/m
Note that the denominator of the first term is m*(m+1). Use
the same
approach described above.
<<Remove the del for email>>

===

Subject: Re: Express As Single Fraction
>How do I do this?
Express the following as a single fraction:
4/3ab - 5/6bc
> Multiply the first term by 2c/2c and the second by a/a.
I'm obviously doing it wrong but that appears to leave two
different
denominators. I thought we were trying to get a common one?

===

Subject: Re: Express As Single Fraction
> How do I do this?
> Express the following as a single fraction:
> 4/3ab - 5/6bc
> (m^2 + 2)/(m^2 + m) - (m - 2)/m
You do it the same way you do it for fractions in arithematic.
The general formula is derived thus
a/b + r/s = as/bs + br/bs = (as + br)/bs

===

Subject: Re: Express As Single Fraction
> How do I do this?
> Express the following as a single fraction:
4/3ab - 5/6bc
> (m^2 + 2)/(m^2 + m) - (m - 2)/m
You do it the same way you do it for fractions in arithematic.
> The general formula is derived thus
> a/b + r/s = as/bs + br/bs = (as + br)/bs
Yep, I understand the basic priniciple, but I just don't know
how to put it
in practise with these types of fractions.

===

Subject: Re: Express As Single Fraction
> How do I do this?
> Express the following as a single fraction:
> 4/3ab - 5/6bc
> and
> (m^2 + 2)/(m^2 + m) - (m - 2)/m
(1) find a common denominator. The least common denominator is
a
good one to use.
(2) convert each fraction to an equivalent fraction, all
having the
same (common) denominator found in step 1.
(3) Add (or subtract) numerators and put the result over the
common
denominator from step 1.
(4) Reduce the result to lowest terms. Note that if you have
used
the least common denominator, the fraction may already be in
lowest
terms.

===

Subject: Re: Express As Single Fraction
> How do I do this?
Express the following as a single fraction:
4/3ab - 5/6bc
 (1) find a common denominator. The least common denominator
is a
> good one to use.
What, for example in the first one, is the lowest common
denominator? Is it
(3ab)(6bc)? If it is, do I then multiply that out? If not,
what is it?

===

Subject: Re: Express As Single Fraction
How do I do this?
Express the following as a single fraction:
4/3ab - 5/6bc
(1) find a common denominator. The least common denominator is
a
> good one to use.
> What, for example in the first one, is the lowest common
denominator? Is
it
> (3ab)(6bc)? If it is, do I then multiply that out? If not,
what is it?
6abc = (3ab)(2c) = (6bc)(a), so 6abc is a common multiple of
3ab and
6bc.
Since (2c) and (a) have no common factor greater than 1,
nothing can
be left out of 6abc and still have a common multiple of 3ab
and 6bc,
so 6abc is the least common multiple.

===

Subject: Re: Express As Single Fraction
>How do I do this?
Express the following as a single fraction:
4/3ab - 5/6bc
>
(1) find a common denominator. The least common denominator is
a
>> good one to use.
> What, for example in the first one, is the lowest common
denominator?
> Is it (3ab)(6bc)? If it is, do I then multiply that out? If
not, what
> is it?
Factorize all denominatars as much as possible, including
numbers:
3ab = 3*a*b
6bc = 2*3*b*c
The lowest common denominator is the lowest common multiple of
the
denominators. To get it, take all factors raised to the
greatest exponent
that appears. In this case,
lcm(3ab, 6bc) = 2*3*a*b*c = 6abc
Then multiply and divide each fraction by the quotient between
the common
denominator and its denominadtor:
4/(3ab) - 5/(6bc) = 4*(2c)/((3ab)*(2c) - 5*a/((6bc)*a) = (8c -
5a)/(6abc)
--
Ignacio Larrosa Ca.96estro
A Coru.96a (Espa.96a)
ilarrosaQUITARMAYUSCULAS@mundo-r.com

===

Subject: Re: Express As Single Fraction
>How do I do this?
Express the following as a single fraction:
4/3ab - 5/6bc
>
(1) find a common denominator. The least common denominator is
a
>> good one to use.
What, for example in the first one, is the lowest common
denominator?
> Is it (3ab)(6bc)? If it is, do I then multiply that out? If
not, what
> is it?
> Factorize all denominatars as much as possible, including
numbers:
> 3ab = 3*a*b
> 6bc = 2*3*b*c
> The lowest common denominator is the lowest common multiple
of the
> denominators. To get it, take all factors raised to the
greatest exponent
> that appears. In this case,
> lcm(3ab, 6bc) = 2*3*a*b*c = 6abc
> Then multiply and divide each fraction by the quotient
between the common
> denominator and its denominadtor:
> 4/(3ab) - 5/(6bc) = 4*(2c)/((3ab)*(2c) - 5*a/((6bc)*a) = (8c
- 5a)/(6abc)

===

Subject: consecutive composite integers
Examining a table of factors and primes, I found that for any
sequence
of consecutive composite numbers there is always one integer
that has
a prime factor larger than any other prime factor of any of
the other
integers. Further, this prime is not raised to any power. My
question
is: Is this true for all consecutive composite sequences and
if so, is
there a proof?
(Here's an example: 1500, 1501, ... 1509. The last integer has
the
prime factor 503.)

===

Subject: Re: consecutive composite integers
> Examining a table of factors and primes, I found that for
any sequence
> of consecutive composite numbers there is always one integer
that has
> a prime factor larger than any other prime factor of any of
the others
Nice conjecture!
It is equivalent to saying (more neatly perhaps) that in any
adjacent
sequence of numbers, the largest prime factor occurs only once.
It seems obviously true, though I can't see a slick proof
off-hand.
I'm sure there must be one. It is possible to produce a
heuristic
demo that can doubtless be turned into a formal proof with
effort.
Suppose your two largest equal prime factors are P, and we may
as well
put them at the ends of the interval WLOG. Then we have an
interval of
length P which must be filled up by factors less than P. This
would mean
the P interior numbers have to be removed one-by-one by
ditching the evens,
then the 3-multiples, then the 5-multiples, and so on up to P.
But we can remove at most (1/2)*(2/3)*(4/5)*(6/7)*...*(1 -
1/P) this way;
and a quick heuristic integral integral shows this to be
1/(log P).
So the most numbers we can reduce to by removing those with
(even at least one!) small prime factor is one log-Pth of the
lot,
and the lot is of size P. That's P/logP. Nowhere near zero or
one.
So it can't be done... not by a huge amount!
Nice question though.
--------------------------------------------------------------
--------------
--
Bill Taylor W.Taylor@math.canterbury.ac.nz
--------------------------------------------------------------
--------------
--
Chebychev said it -- I'll say it again;
There is always a prime between n and 2n.
--------------------------------------------------------------
--------------
--

===

Subject: Re: consecutive composite integers
>Examining a table of factors and primes, I found that for any
sequence
>of consecutive composite numbers there is always one integer
that has
>a prime factor larger than any other prime factor of any of
the other
>integers. Further, this prime is not raised to any power. My
question
>is: Is this true for all consecutive composite sequences and
if so, is
>there a proof?
>(Here's an example: 1500, 1501, ... 1509. The last integer
has the
>prime factor 503.)
How does the Further,... apply to this example: 8, 9?
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2

===

Subject: Re: consecutive composite integers
<blvhk9$3fe$1@nntp.itservices.ubc.caExamining a table of factors and primes, 
I found that for any
sequence
>of consecutive composite numbers there is always one integer
that has
>a prime factor larger than any other prime factor of any of
the other
>integers. Further, this prime is not raised to any power.
> How does the Further,... apply to this example: 8, 9?
Perhaps he intended maximal sequence, 8,9,10 has 5 as max.
However the maximal sequences 4 and 12 have 2 squared and
the maximal sequence 18 has 3 squared.
Thus the conjecture needs be limi to
maximal sequences of composite numbers longer than 1.

===

Subject: Re: consecutive composite integers
>>Examining a table of factors and primes, I found that for
any sequence
>>of consecutive composite numbers there is always one integer
that has
>>a prime factor larger than any other prime factor of any of
the other
>>integers. Further, this prime is not raised to any power.
>> How does the Further,... apply to this example: 8, 9?
>Perhaps he intended maximal sequence, 8,9,10 has 5 as max.
But the example he gave was 1500, 1501, ... 1509 which is not
maximal.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2

===

Subject: Re: consecutive composite integers
>Examining a table of factors and primes, I found that for any
sequence
>of consecutive composite numbers there is always one integer
that has
>a prime factor larger than any other prime factor of any of
the other
>integers. Further, this prime is not raised to any power.
> How does the Further,... apply to this example: 8, 9?
>>Perhaps he intended maximal sequence, 8,9,10 has 5 as max.
> But the example he gave was 1500, 1501, ... 1509 which is
not maximal.
Also, the largest prime factor of a number in that range is
751, not
503. The conjecture still holds, but not for the reason given.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&
bookid=228>

===

Subject: Re: Did mathematicians know?
well, isn't this teensy duality the same,
for any proof that one hasn't actually read,
all the way through (or to the point where
one can see it through, already) ??
> you're saying that the L-wing thing is just an argument
> *about* an actual (R-wing) proof?
--les ducs de Buffet et Schulz (R&D Chair Assoc.Intl.);
vote NONE OF THE BELOW
on Trickier Dick Cheney's California Recall/e-Dereg!
http://larouchepub.com
http://members.tripod.com/~american_almanac/

===

Subject: Re: The Bible Code
it's not even wrong. it's just simple numbertheory --
skipcodes are that, and they were apparently used
by (some?) Torah writers/copyists to ensure accuracy,
as with the old CRC in 8-bit communications programs.
I read taht they summed the letters on every 70th,
or skipped to every 70th, or some thing.
the computer can be set to find any message
in any ring of an alphabet, and Drosnin et al know this ... or
maybe they can't learn it, not because they're dumb.
there was ambiguity in _The Bible Code_ taht he ignored,
like with the variant translations and the fact that
Old Hebrew has no vowels.
repeat, _War and Peace_ or just the 26 letters in any order
can be used with the infinite set of co-prime skips,
with teh resulting hits being further massaged
into some m by n array (or what ever).
>FAILED.
Is this accurate? And does this say something about the New
Testament
>and the belief in a Christ figure?
--les ducs de Buffet et Schulz (R&D Chair Assoc.Intl.);
vote NONE OF THE BELOW
on Trickier Dick Cheney's California Recall/e-Dereg!
http://larouchepub.com
http://members.tripod.com/~american_almanac/

===

Subject: Re: The Bible Code
> it's not even wrong. it's just simple numbertheory --
> skipcodes are that, and they were apparently used
> by (some?) Torah writers/copyists to ensure accuracy,
> as with the old CRC in 8-bit communications programs.
> I read taht they summed the letters on every 70th,
> or skipped to every 70th, or some thing.
> the computer can be set to find any message
> in any ring of an alphabet, and Drosnin et al know this ...
or
> maybe they can't learn it, not because they're dumb.
> there was ambiguity in _The Bible Code_ taht he ignored,
> like with the variant translations and the fact that
> Old Hebrew has no vowels.
> repeat, _War and Peace_ or just the 26 letters in any order
> can be used with the infinite set of co-prime skips,
> with teh resulting hits being further massaged
> into some m by n array (or what ever).
>FAILED.
Is this accurate? And does this say something about the New
Testament
>and the belief in a Christ figure?
> --les ducs de Buffet et Schulz (R&D Chair Assoc.Intl.);
> vote NONE OF THE BELOW
> on Trickier Dick Cheney's California Recall/e-Dereg!
> http://larouchepub.com
> http://members.tripod.com/~american_almanac/
A defective theory when applied does neither prove or disprove
a
hypothesis.
The Code stuff has been applied successfully to Moby Dick to
predict
moderns events with Astounding accuracy. (NOT!)
rj Pease

===

Subject: Re: Re fermat by Tomas
what is found at your site is known as Sophistry. the fact that
Fernat's Last Theorem is negative is not problem, just as
proofs (of neg or pos statements) by contradiction are good.
of course, three per cent is rational by definition.
Negative statements about numbers are unverifiable. Take the
definition of an irrational as a number that is not rational,
where
being rational means having periodic decimal expansion. Suppose
someone wants to verify that %3 is irrational. Since the
indirect
proof is out he computes its decimal expansion to the
trillionth digit
and, satisfied that no evidence of periodicity looms on the
horizon,
declares that %3 is irrational. A mischievous fellow quickly
forms a
periodic decimal whose first period is that first trillion
> http://www.users.bigpond.com/pidro/home.htm
--les ducs de Buffet et Schulz (R&D Chair Assoc.Intl.);
vote NONE OF THE BELOW
on Trickier Dick Cheney's California Recall/e-Dereg!
http://larouchepub.com
http://members.tripod.com/~american_almanac/

===

Subject: A polynomial problem
Hi all,
I'm trying to prove the following theorem:
Let P be a polynomial with real coefficients such that P(x)
>=0 for
every real x. Then, there are polynomials R and S such that
P(x) =
R^2(x) + S^2(x) for every complex x.
It's easy to see that the degree of P must be even. If r is a
real
root of P, then the restriction of P to the reals has an
absolute
minimum at r and, from the differentiability of P, it follows
there's
an even number k such that the k-1 first derivatives of P
vanish at r
and the k_th is positive. Therefore, the k-1 derivatives
admits the
root r with multiplicity 1, which implies P admits r as a
rooth with
multiplicity k. So, we see every real root of P, when they
exist, must
have an even multiplicity (I think we could come to this same
conclusion a bit faster, considering only the continuity of
polynomial
functions).
Corollary - If all of the roots of P are real, then, P = Q^2
for some
polynomial Q. So, for this particular case the theorem has
just been
proved.
To prove the theorem, for the general case, I tried to use
mathematical induction. It didn't work, that's why I'm asking
for
help. What I did is as folows:
the previous paragraph, it's also enough to cover the case of
even-degree polynomials with real coefficients and no real
roots. It's
well known that every monic trinomial T of the 2nd degree that
satisfies such conditions can be written as T(x) = (x-a)^2 +
C^2,
where a and C<>0 are real. Therefore, for such trinomials the
theorem
holds trivially. Now, suppose there's a natural k such that the
theorem holds for i=1,...k-1 for every 2i-degree polynomial
with real
coefficients and no real roots. If P is a 2k-degree polynomial
with
these same properties, then at some (or several) real r's the
restriction of P to the reals attains an absolute minimum m>0.
This
implies that, for every real x, the polynomial P-m is
non-negative,
has one (or several) real root(s) and (i) P(x) - m = (x-r_1)^p1
*...(x-r^_n)^p_n * Q(x), where r_1, ..r_n are the real roots
and the
numbers p_1,...p_n are even. In addition, Q is monic, has even
degree
< 2k, has no real root and is strictly positive on the real
line. By
the induction assumption, the theorem holds for Q and, in
virtue of
(i), also holds for P-m. But now, to complete the induction, it
remains to prove the theorem is good for P, in other words, it
remains
to prove that if the theorem holds for some polynomial P then
it holds
for P+m for every m>0. That's where I got stuck.
Actually, I think I chose a very cumbersome way to prove the
theorem,
there certainly is a neater proof.
Any suggestions are welcome.
Amanda.

===

Subject: Re: A polynomial problem
> I'm trying to prove the following theorem:
> Let P be a polynomial with real coefficients such that P(x)
>=0 for
> every real x. Then, there are polynomials R and S such that
P(x) =
> R^2(x) + S^2(x) for every complex x.
Show P factors over the reals as a product of irreducible
quadratics
times a product of squares of linear polynomials.
Show that an irreducible quadratic is a sum of 2 squares.
Show that a product of two sums of two squares is a sum of two
squares.
--

===

Subject: Re: A polynomial problem
>Hi all,
>I'm trying to prove the following theorem:
>Let P be a polynomial with real coefficients such that P(x)
>=0 for
>every real x. Then, there are polynomials R and S such that
P(x) =
>R^2(x) + S^2(x) for every complex x.
>It's easy to see that the degree of P must be even. If r is a
real
>root of P, then the restriction of P to the reals has an
absolute
>minimum at r and, from the differentiability of P, it follows
there's
>an even number k such that the k-1 first derivatives of P
vanish at r
>and the k_th is positive. Therefore, the k-1 derivatives
admits the
>root r with multiplicity 1, which implies P admits r as a
rooth with
>multiplicity k. So, we see every real root of P, when they
exist, must
>have an even multiplicity (I think we could come to this same
>conclusion a bit faster, considering only the continuity of
polynomial
>functions).
>Corollary - If all of the roots of P are real, then, P = Q^2
for some
>polynomial Q. So, for this particular case the theorem has
just been
>proved.
>To prove the theorem, for the general case, I tried to use
>mathematical induction. It didn't work, that's why I'm asking
for
>help. What I did is as folows:
>the previous paragraph, it's also enough to cover the case of
>even-degree polynomials with real coefficients and no real
roots. It's
>well known that every monic trinomial T of the 2nd degree that
>satisfies such conditions can be written as T(x) = (x-a)^2 +
C^2,
>where a and C<>0 are real. Therefore, for such trinomials the
theorem
>holds trivially. Now, suppose there's a natural k such that
the
>theorem holds for i=1,...k-1 for every 2i-degree polynomial
with real
>coefficients and no real roots. If P is a 2k-degree
polynomial with
>these same properties, then at some (or several) real r's the
>restriction of P to the reals attains an absolute minimum
m>0. This
>implies that, for every real x, the polynomial P-m is
non-negative,
>has one (or several) real root(s) and (i) P(x) - m =
(x-r_1)^p1
>*...(x-r^_n)^p_n * Q(x), where r_1, ..r_n are the real roots
and the
>numbers p_1,...p_n are even. In addition, Q is monic, has
even degree
>< 2k, has no real root and is strictly positive on the real
line. By
>the induction assumption, the theorem holds for Q and, in
virtue of
>(i), also holds for P-m. But now, to complete the induction,
it
>remains to prove the theorem is good for P, in other words,
it remains
>to prove that if the theorem holds for some polynomial P then
it holds
>for P+m for every m>0. That's where I got stuck.
>Actually, I think I chose a very cumbersome way to prove the
theorem,
>there certainly is a neater proof.
I believe there is. The real roots all have even order and the
complex
roots come in conjugate pairs - this means there exists a
polynomial
F (with complex coefficients) such that P is the product of F
and the
complex conjugate of F...
>Any suggestions are welcome.
>Amanda.
************************

===

Subject: non-Hausdorff homeomorphism to R^n
is there a non-Hausdorff space locally homeomorphic to R^n?

===

Subject: Re: non-Hausdorff homeomorphism to R^n

===

Subject: non-Hausdorff homeomorphism to R^n
>is there a non-Hausdorff space locally homeomorphic to R^n?
No. I use theorem
locally closed Hausdorff S ==> S Hausdorff
S locally closed Hausdorff when for all p in S,
some open U nhood p for which cl U is a Hausdorff subspace
Let the non-Hausdorff space be S. As I don't know definition
of locally
homeomorphic I'll do my best to intuit what it means.
If p in S, then there's some open U nhood p homeomorphic
to an a nhood V of R^n about q, the image of p
As R^n is normal, we can find a closed nhood K of q contained
in V.
Now the homeomorphic image of K is a closed Hausdorff nhood of
p.
Thus I hope I've shown S is locally closed Hausdoff, hence
Hausdorff.
----

===

Subject: Re: non-Hausdorff homeomorphism to R^n

===

> Subject: non-Hausdorff homeomorphism to R^n
>is there a non-Hausdorff space locally homeomorphic to R^n?
> No. I use theorem
> locally closed Hausdorff S ==> S Hausdorff
So what do you do about the real line with double origin?
(Quotient of R x {0,1} by the smallest equivalence ~
with (x,0) ~ (x,1) for x =/= 0 ).
--

===

Subject: Re: non-Hausdorff homeomorphism to R^n
>is there a non-Hausdorff space locally homeomorphic to R^n?
Sure thing. Look for a current thread which mentions
doubling points in one of the posts (I think by Simeon
Stefanov).
Not if n=0, though.
Lee Rudolph

===

Subject: Open form for the integral of x^x dx
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h973a2W15776;
I've been trying to find a generalized (open) formula for the
integral of
x^x dx.....does anyone know if it's been derived at all?
Curious

===

Subject: real analysis: construct this set ...
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h974dnb19740;
Can you construct a set E in [0, 1] s.t. for every open
interval I in [0,1]
m(I intersect E) > 0 & m(I intersect E^c) > 0
m is lebesgue measure
E^c is the complement of E
This is so tricky! I was thinking something with the
generalized Cantor set
but everything I'm trying isn't working. Any suggestions?
Ideas?

===

Subject: Re: real analysis: construct this set ...
>Can you construct a set E in [0, 1] s.t. for every open
interval I in [0,1]
m(I intersect E) > 0 & m(I intersect E^c) > 0
>m is lebesgue measure
>E^c is the complement of E
>This is so tricky! I was thinking something with the
generalized Cantor
set but everything I'm trying isn't working. Any suggestions?
Ideas?
I have previously pos an example of such a set, which is
Borel measurable. Use the Cantor set idea, but with different
sized parts removed from the intervals still in, and added to
the intervals already removed. It can be done in terms of the
digits to any base.
--
This address is for information only. I do not claim that
these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558

===

Subject: Re: real analysis: construct this set ...
> Can you construct a set E in [0, 1] s.t. for every open
interval I in
[0,1] m(I intersect E) > 0 & m(I intersect E^c) > 0
> m is lebesgue measure
> E^c is the complement of E
A doubt. You say the Lebesgue measure. Are you implying that
the
construc set must be Lebesgue measurable? Or the problem
somehow
gives us a notion that, even if the wording Lebesgue measure is
replaced by Lebesgue outer measure,it is impossible to find a
Lebesgue immeasurable set satisfying the condition as the
problem
indicates?
> This is so tricky! I was thinking something with the
generalized Cantor
set but everything I'm trying isn't working. Any suggestions?
Ideas?

===

Subject: Re: real analysis: construct this set ...
>> Can you construct a set E in [0, 1] s.t. for every open
interval I in
>[0,1] m(I intersect E) > 0 & m(I intersect E^c) > 0
>> m is lebesgue measure
>> E^c is the complement of E
>A doubt. You say the Lebesgue measure. Are you implying that
the
>construc set must be Lebesgue measurable? Or the problem
somehow
>gives us a notion that, even if the wording Lebesgue measure
is
>replaced by Lebesgue outer measure,it is impossible to find a
>Lebesgue immeasurable set satisfying the condition as the
problem
>indicates?
By saying m(...) where m is Lebesgue measure, it's clearly
asking for
measurable sets.
As far as outer measure m^* is concerned, you can do a lot
better:
there is a (nonmeasurable) set E such that for every open
interval I,
m^*(E intersect I) = m(I) and m^*(E^c intersect I) = m(I).
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2

===

Subject: Re: real analysis: construct this set ...
> Can you construct a set E in [0, 1] s.t. for every open
interval I in
[0,1]
> m(I intersect E) > 0 & m(I intersect E^c) > 0
> m is lebesgue measure
> E^c is the complement of E
take away fat disjoint Cantor sets K1 and J1. Now from B2 (K1
U J1) take
away fat disjoint Cantor sets K2 and J2. Continue, and set E =
... (If this
is a homework problem and you use this hint, be sure to give
credit to
sci.math.)

===

Subject: dissolving Russel's Paradox
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h97D3uP19205;
I studied an alternative set theory which dissolves Russel's
Paradox.
In this theory, it is possible to get good theorems of ordinal
number.
More description about the set theory is available on my home
page.
http://boat.zero.ad.jp/~zbi74583/another02.htm
I appreciate any comment about it.
--------------------
@ 1.Axiom of Free Class.
@
A. ma [A]A[A]B[A]x[A]y A ma x &@B ma y & A=B -> x=y
This axiom means that any Atsumari makes only one class.
A. el x[A]A[A]B ([A]a a el A <-> a el B) -> A=B
This axiom means that an Atsumari is decided by its members.
A.F [E]x{E]B ([A}a a el B <-> F(a)) & B ma x
This is the Schema of comprehention axioms.
Where A,B,C,...are variables for Atsumari, which means
collection or set
or pile in Japanese.
a,b,c,...are variables for Class,
[A] means all, [E] means exist, [E]! means exist only one, ma
means
makes,
el means element, -> means then, <-> means equivalent,
V means or, & means and, Atsu is an abbreviation of Atsumari
~ means negation, {a,b,c} means the Atsu which has members
a,b,c
2.What does Free Class mean?
@
[E]B a el B & B ma b means a el b in ZF (TR)
For example if {a,c} ma b & {x,y} ma b are true,
then a el {a,c} & {a,c} ma b
so [E]B a el B & B ma b
it means a el b in ZF.
FC ZF
{a,c} ma b corresponds a el b, c el b
{x,y} ma b x el b, y el b
3. Russel's class
[A]a a el R <-> ~([E]B a el B & B ma a)
The right side of formula means ~(a el a) in ZF.
This Atsu R makes Russel's Class r, so R ma r.
Let a=r then,
r el R <-> ~([E]B r el B & B ma r)
If r el R is true, considering that R ma r is also true,then
[E]B r el B & B ma r is true, then right side of formula is
false. This
is a contradiction.
So, the following formulas are gotten.
~(r el R), [E]B r el B & B ma r.
This conclusion means that the diagonal logic was dissolved.
So, it is possible to think that 2^aleph(0)=aleph(0).@

===

Subject: Re: dissolving Russel's Paradox
> I studied an alternative set theory which dissolves Russel's
Paradox.
> In this theory, it is possible to get good theorems of
ordinal
number.
> More description about the set theory is available on my
home page
> http://boat.zero.ad.jp/~zbi74583/another02.htm
> I appreciate any comment about it.
> .81@ 1.Axiom of Free Class.
> .81@
Not using plain text in newsgroup makes reading difficult for
some
brousers. For example what does ^A@ mean?
> A. ma [A]A[A]B[A]x[A]y A ma x &.81@B ma y & A=B -> x=y
What does that mean?
A makes for all A, for all B, for all x, for all y
A makes x and ?? B makes y and A = B implies x = y
Don't understand. What do you mean by 'A makes' ?
> This axiom means that any Atsumari makes only one class.
> A. el x[A]A[A]B ([A]a a el A <-> a el B) -> A=B
> This axiom means that an Atsumari is decided by its members.
> A.F [E]x{E]B ([A}a a el B <-> F(a)) & B ma x
> This is the Schema of comprehention axioms.
> Where A,B,C,...are variables for Atsumari, which means
collection or
set or pile in Japanese.
> a,b,c,...are variables for Class,
> [A] means all, [E] means exist, [E]! means exist only one,
ma means
makes,
> el means element, -> means then, <-> means equivalent,
> V means or, & means and, Atsu is an abbreviation of Atsumari
> ~ means negation, {a,b,c} means the Atsu which has members
a,b,c

===

Subject: a coloring problem
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h97D48o19230;
I studied a generalized coloring problem.
The ordinary coloring problem is defined as follows.
@
@@Place different colors on two vertices which are next each
other
in a plane graph.
@@How many colors are necessary and enough?
@
@@And new coloring problem is the following.
@
@@A definition
@@D.1.
@@Place different colors on two vertices which are near each
other in a plane graph.
@@How many colors are necessary and enough?
@
@@The term near is defined as follows.
@@For different two vertices a and b,@either condition 1. or 2.
is filled.
@@c.1. a is next of b
@@c.2. There are three paths of length 2 between a and b.
@
@@c.1 and c.2 is represen as (1,1) and (2,3) respectively.
@
@@If G is a plane graph and
@@if edges are added between all pairs of vertices which
satisfy
(2,3)@in G,
@@then this graph is written as Near_2,3(G)
@
@@New problem is also defined as follows.
@
@@Another definition
@@D.2.
@@What is chr(Near_2,3(G))?
@@@@ where chr(x) means the chromatic number of x.
@
I proved a theorem as follows.
[ The@@7 colors theorem. ]
@@T.7C.@@7 colors are necessary and enough.
More description about this problem is available on my HP.
@
http://boat.zero.ad.jp/~zbi74583/another02.htm

===

Subject: Re: The Octic x^8-x^7+29x^2+29 Revisi
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h97D4OF19281;
,
By a stroke of luck, I managed to find the missing piece to the
puzzle of how to solve the resolvent septic. So here is the
complete solution:
Given: x^8-x^7+29x^2+29 = 0
Then,
x1=(1+(a-b-c-d+e-f-g))/8
x2=(1-(a-b-c-d-e+f+g))/8
x3=(1-(a+b-c+d+e-f+g))/8
x4=(1+(a+b-c+d-e+f-g))/8
x5=(1-(a+b+c-d+e+f-g))/8
x6=(1+(a+b+c-d-e-f+g))/8
x7=(1-(a-b+c+d-e-f-g))/8
x8=(1+(a-b+c+d+e+f+g))/8
where the 7 variables a,b,c,d,e,f,g are the SQUARE ROOTS
(positive
case) of 4v+1 and the v's are the roots of the septic:
8903+47647v+39672v^2+7192v^3-522v^4-174v^5+v^7 = 0 (eq.3)
with the solution:
v=2(w^11+w^13+w^16+w^18)-2(w+w^12+w^17+w^28)-(w^2+w^5+w^24+w^
27)+
(w^3+w^7+w^22+w^26)+(w^4+w^10+w^19+w^25)-(w^8+w^9+w^20+w^21)
where w is ANY complex root of unity <>1 such that w^29=1.
Note: Though there would be 28 such roots, the properties of
these
roots ensure that v will ONLY have 7 distinct values.
I found the solution of v in Dave Rusin's website, and it's by
P.Montgomery, though the solution wasn't used in the same way
I used
it. Eq.3 wasn't explicitly mentioned there but when I used the
Integer Relations applet for a particular v, eq.3 popped out.
It
looked familiar and I realized i saw it before while trying to
solve the resolvent (eq.2) of the prior post, namely:
z^7-7z^6-2763z^5-19523z^4+1946979z^3+34928043z^2+119557031z-
3247^2=0
(eq.2)
and by letting z=4v+1, we get the new septic:
8903+47647v+39672v^2+7192v^3-522v^4-174v^5+v^7 = 0 (eq.3)
So there it is. It's so nice to have completion. :)
By the way, do SOME solvable 12-deg polys need an 11-deg
resolvent?
--Tito

===

Subject: Re: Quaternion Extensions
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h97D4L919274;
>> I have recently extended the Quaternions to larger sets by
requiring
>> some (new) group elements to commute. In doing so, I found
this
process
>> and its results to be very asthetic. For one, the law of
association
is
>> regained. However, the algebra involved is no longer a
division
algebra,
>> i.e. we may not always follow x = 0 or y = 0 from xy=0
(when x and y
are
>> certain elements taken from a linear combination of group
vectors).
Has this type of thing been done before and are its
conclusions of
>> interest?
It's obvious that there exist extensions of the quaternions H,
>eg H + H (direct sum),
>or algebras of matrices with quaternion elements.
You'd have to say what properties your extension has
>before anyone could say if it is of interest.
>I am neither refering to the ring of quaternion matrices nor
to the
>group
>product... have sent you a copy of this work.
>Apologies for not replying to your email ... lectures have
just begun.
>My point was that you seemed surprised to find that
>there were algebras extending the quaternions,
>and I no that this wasn't too surprising.
>So the fact that you have construc such an algebra
>could not be considered interesting in itself --
>any interest would have to lie in the special properties of
the algebra.
>--
>Timothy Murphy
>e-mail: tim /at/ birdsnest.maths.tcd.ie
>(all email over 80k dispatched to /dev/null)
>tel: +353-86-2336090, +353-1-2842366
>s-mail: School of Mathematics, Trinity College, Dublin 2,
Ireland
Absolutely no need to apologize for not having replied yet; it
is good to
know you recieved it, as I now asume to be the case. The group
diagram
picture I made on page 4 or 5 (I forgot) and its explanation
in the text
should tell a lot about the group's properties very quickly.
By the way, the
same process of extension (I call it reflection) can be used
again and
again to create more group elements, presumably proceeding to
higher
dimensions in the process- but Im not quite sure if this is
somehow
equivilant or to the procedure for extending Clifford Algebras
(or, indeed,
if my group is perhaps a Clifford algebra in disguise. Admit, I
don't know enough about Clifford algebras at the momement and
am currently
checking this possibility myself). Note also that many types
of groups can be
reflec, but this does not always give rise to a new group. For
example,
the trivial group {1,-1} does not change after reflection (the
complex
trivial group {1,-1, i, -i} does).
Creighton Dement

===

Subject: Chessboard knight metric?
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h97E9gZ23741;
>Take a chessboard (with or without infinetely many squares)
let the
>distance d((x_1,x_2),(y_1,y_2)) between two squares x and y
of the
>chessboard be defined as the minimum number of moves a knight
takes
>to reach y from x.
>Is d a metric? Not trying to suggest that this is some new
>question that hasn't been asked/answered before. Is there a
general
>formula for calculating d? More generally, the same question
may be
>asked for the other pieces (queen, king, knight, biship)?
Actually, I
>asked myself this question a few years ago. If I remember
back to the
>notes I took, I had something like (x_1-y_1, x_2-y_2)= (even
number,
>even number), then d(x,y) = even number. If (x_1-y_1,
x_2-y_2)= (even
>number, odd number), then d(x,y) = odd number. Finally, if
(x_1-y_1,
>x_2-y_2)= (odd number, odd number), then d(x,y) = even
number. In other
words, the same rules for adding natural numbers...

===

Subject: Re: Chessboard knight metric?
>Take a chessboard (with or without infinetely many squares)
let the
>distance d((x_1,x_2),(y_1,y_2)) between two squares x and y
of the
>chessboard be defined as the minimum number of moves a knight
takes
>to reach y from x.
>Is d a metric?
With this distance, the triangular equation is obviously true,
and that
makes it a metric.
> Not trying to suggest that this is some new
>question that hasn't been asked/answered before. Is there a
general
>formula for calculating d? More generally, the same question
may be
>asked for the other pieces (queen, king, knight, biship)?
Actually, I
>asked myself this question a few years ago. If I remember
back to the
>notes I took, I had something like (x_1-y_1, x_2-y_2)= (even
number,
>even number), then d(x,y) = even number. If (x_1-y_1,
x_2-y_2)= (even
>number, odd number), then d(x,y) = odd number. Finally, if
(x_1-y_1,
>x_2-y_2)= (odd number, odd number), then d(x,y) = even
number. In other
>words, the same rules for adding natural numbers...
>

===

Subject: Re: Pointless
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h97E9jb23746;
>mathedman <mathedman@hotmail.cut.com> scribbled the following
>on comp.lang.c:
>> Why is this being discussed in comp.lang.c???
>Obviously because IT IS FULL OF JEWS!!! Jews are taking over
>comp.lang.c because their greedy grubby need to take over
everything is
>finally seeping into the C language. Next thing you know they
will be
>trying to rewrite the standard. The entire reason for my low
IQ and
>inability to succeed in life can be attribu to jews. If it
wasn't
>for all the damn JEWS in science I wouldn't have to study!
They keep
>taking all the women too, being all nice and treating them
with
>respect and making me look like a complete ass. They took all
the
>jobs, now there is no point even looking for one. All I can
do is sit
>around all day filled with self pity and loathing for the
damn JEWS who
>did this to me. I hate my life and its all the fault of the
Jew!
>> YOU are a total idiot.
>I thought he was being sarcastic.
>--
>/-- Joona Palaste (palaste@cc.helsinki.fi)
---------------------------|
Kingpriest of The Flying Lemon Tree G++ FR FW+ M- #108 D+ ADA
N+++|
>| <a
href=http://www.helsinki.fi/~palaste>http://www.helsinki.fi/~
palaste</a>
W++ B OP+ |
>----------------------------------------- Finland rules!
------------/
>It sure is cool having money and chicks.
> - Beavis and Butt-head
If he was, then he is very bad at it.

===

Subject: Re: Quaternion Extensions
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h97FaaV30213;
> I have recently extended the Quaternions to larger sets by
requiring
> some (new) group elements to commute. In doing so, I found
this
process
> and its results to be very asthetic. For one, the law of
association
is
> regained. However, the algebra involved is no longer a
division
algebra,
> i.e. we may not always follow x = 0 or y = 0 from xy=0
(when x and y
are
> certain elements taken from a linear combination of group
vectors).
 Has this type of thing been done before and are its
conclusions of
> interest?
>>It's obvious that there exist extensions of the quaternions
H,
>>eg H + H (direct sum),
>>or algebras of matrices with quaternion elements.
>>You'd have to say what properties your extension has
>>before anyone could say if it is of interest.
>>I am neither refering to the ring of quaternion matrices nor
to the
>group
>>product... have sent you a copy of this work.
>>Apologies for not replying to your email ... lectures have
just begun.
>>My point was that you seemed surprised to find that
>>there were algebras extending the quaternions,
>>and I no that this wasn't too surprising.
>>So the fact that you have construc such an algebra
>>could not be considered interesting in itself --
>>any interest would have to lie in the special properties of
the algebra.
>>--
>>Timothy Murphy
>>e-mail: tim /at/ birdsnest.maths.tcd.ie
>>(all email over 80k dispatched to /dev/null)
>>tel: +353-86-2336090, +353-1-2842366
>>s-mail: School of Mathematics, Trinity College, Dublin 2,
Ireland
>Absolutely no need to apologize for not having replied yet;
it is
>good to know you recieved it, as I now asume to be the case.
The
>group diagram picture I made on page 4 or 5 (I forgot) and its
>explanation in the text should tell a lot about the group's
>properties very quickly. By the way, the same process of
extension
> (I call it reflection) can be used again and again to create
more
> group elements, presumably proceeding to higher dimensions
in the
>process- but Im not quite sure if this is somehow equivilant
or to
>the procedure for extending Clifford Algebras (or, indeed, if
my
>group is perhaps a Clifford algebra in disguise. Admit, I
>don't know enough about Clifford algebras at the momement and
am
>currently checking this possibility myself). Note also that
many
>types of groups can be reflec, but this does not always give
rise
>to a new group. For example, the trivial group {1,-1} does
not change
> after reflection (the complex trivial group {1,-1, i, -i}
does).
>Creighton Dement
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) with ESMTP id h97G9do32618
id 1A6uP0-0003DA-6d
include it with
any abuse report
12]
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===

Subject: Re: The ... spacetime; answer to critic.
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h97JmJl15824;
I know it is a bit stupid, but
1> how do you prove that a discrete topology is metrizable?
2> X is an set of all positive integers and
T={{},{1,2,3,4...},{2,3,4...},{3,4...},{4...},...} Why is
(X,T) not
metrizable?
><pre>
<av95@comtv.ru> As a topological space spacetime is metrizable, but one 
does
not,
> in general, look at any particular metric (in the
topological
sense)
> on it.
Why Severian thinks that this Hausdorf topology on Space-Time
> is adequate to its physical sence and geometrical structure?
I care nothing for physical sence (sic). All I was doing was
pointing
> out that the term metric was being used in two different ways
> (like many words in mathematics).
Do you mean as two
ways:::::::::::::::::::::::::::::::::::::::::::::::::::::::
> ### (pseudo-)Riemann metric -- quadratic form
> and
> ### metric (distance)
> ?
Yes, they're different, but the first is applicable only to
manifolds.
Yes, I know that, but the original poster was confused by
these two
> distinct concepts having the same word.
>Of course the two concepts are rela this way. So? The
> original poster was talking about one version of metric
> in terms of the definition of the other version - that's
going
> to cause confusion, regardless of the fact that the two
> are rela, so pointing out that they are two different
> things seems like a good idea.
>The original poster. In the original question. Where he said
something
>about metrics that applied to one version, then asked about
what
>he said in re an instance of the other version.
>And note how these concepts are rela and how it can be
applied to
S.T.
>>seems also a good idea...
>More precisely:
The standard topology of Euclidean space is induced by metric.
You are failing to specify which usage of the term metric you
are using
> here.
>>Directly, I used the metric (distance).
>>But the Euclidean distance is a partial case of Riemann
distance,
>>so not important which I use distance or quadratic form.
>The standard topology of Minkowski space isn't.
Minkowski space is metrizable.
>>Himmel! ### Not by Minkowski metric ! ! !
>He didn't say it _was_ metrizable by the Minkowski metric.
>He said it was metrizable. It is.
>You said The standard topology of Euclidean space is induced
>by metric. The standard topology of Minkowski space isn't.
>We assumed that induced by metric meant induced by
>some metric.
>What _did_ you mean by induced by metric??? If
>induced by metric means induced by Minkowski
>metric then the standard topology on euclidean
>space is _not_ induced by metric...
>>You implicitly defined on it
>>the standart (Hausdorf) topology of finite-dimension linear
space
>>and said that it's metrizable, as all finite-dimension
linear spaces!
>>Original question was NOT ABOUT SUCH TOPOLOGY,
>>but IS Minkowski A METRIC on the Space-Time? .
>>The answer is:
>>____________________________________________________________
_______
>>[ Minkowski metric on the Space Time ( dx0^2 -dx1^2 -dx2^2
-dx3^2 )
>>[ not leads to any metric (distance function) in classical
sence.
>That's correct. Several people have explained this already.
Nobody has
>disagreed with this.
>>~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
>The standard topology of Euclidean space is
> the only (non-trivial) topology invariant to motions
(rotation,
mirror, shift).
Is it? Proof?
>>Sorry, in general it isn't. I mistaken.
>>It's true that any non-one-point and non-discrete invariant
topology
>>is identical to standard IN ANY BOUNDED DOMAIN,
>Say _exactly_ what this means (and give a hint of the proof.)
>Seriously, I cant figure out what it means. Because I can't
>see what it means for a topology on a BOUNDED DOMAIN
>to be invariant under shifts, for one thing. Nor under
rotations,
>unless it happens that the domain is invariant under
rotations.
>>but at infinity exist some different cases...
>>Except standard topology, it may be some COMPACTIFIED
topologies.
>>One of them is like standard but accepts only bounded closed
subsets
>>(or, the same, only open subsets which are neighbourhoods of
infinity).
>>Is there another cases, I don't know yet.
>>I would ignore this fact a long time...
>The standard topology of Minkowski space is invariant but not
unique.
In Euclidean case, all smooth maps from R to the Space
are CONTINIOUS
> and may represent a point trajectory in Newtonian mechanics.
> In the Space-Time, all smooth maps R->M are continious
(in std. top.),
> but not all are physically allowed.
physically allowed!
>>This mean that the 4-speed is not space-like:
>>( d x / d tau )^2 >= 0
>>and that we have a correct time sign:
>>( d x0 / d tau ) >= 0
>What bull.
>>*** Anecdote:
>>enter expression: cos ( pi / 2 )
>>Syntax error!
>>enter expression: 1 * 0
>>Syntax error!
>>enter expression: 2 + 2
>>Syntax error!
>>enter expression: .8f.9b.90 .99.8c.89.8c
>>.8f.9b.90 is not an argument!
>>--
>>qq~~~~>/ / > /_/ /
>> ____/
>>

===

Subject: Re: The ... spacetime; answer to critic.
> I know it is a bit stupid, but
> 1> how do you prove that a discrete topology is metrizable?
Write down a metric for it
The simplest possible metric should give the discrete topology
...
> 2> X is an set of all positive integers and
> T={{},{1,2,3,4...},{2,3,4...},{3,4...},{4...},...} Why is
(X,T) not
> metrizable?
Is it Hausdorff?
--

===

Subject: Re: Chessboard knight metric?
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h97JmW515882;
>>Take a chessboard (with or without infinetely many squares)
let the
>>distance d((x_1,x_2),(y_1,y_2)) between two squares x and y
of the
>>chessboard be defined as the minimum number of moves a
knight takes
>>to reach y from x.
>>Is d a metric? Not trying to suggest that this is some new
>>question that hasn't been asked/answered before. Is there a
general
>>formula for calculating d? More generally, the same question
may be
>>asked for the other pieces (queen, king, knight, biship)?
Actually, I
>>asked myself this question a few years ago. If I remember
back to the
>>notes I took, I had something like (x_1-y_1, x_2-y_2)= (even
number,
>>even number), then d(x,y) = even number. If (x_1-y_1,
x_2-y_2)= (even
>>number, odd number), then d(x,y) = odd number. Finally, if
(x_1-y_1,
>>x_2-y_2)= (odd number, odd number), then d(x,y) = even
number. In other
words, the same rules for adding natural numbers...
Sorry for not quite completing the question above (even
though,
>perhaps, obvious):
>Is d a metric over the product space of whole/natural numbers
>(corresponding to two different cases of an infinite chess
board)
>or the restriction of the product space of natural numbers to
64 elements?
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h97JmTA15867;
A grammar would be
S -> BaB
B -> BB
B -> aBb
B -> bBa
B -> lambda
I doubt there is a way to prove it, or find a base case
>A grammar would be
>S -> BaB
>B -> BB
>B -> aBb
>B -> bBa
>B -> lambda
>I doubt there is a way to prove it, or find a base case
It's easy to see that B gives any string with the same number
of a's and b's, so we just need to show that if a string has
one more a's than b's it can be written as BaB.
Say s is such a string, and say s[n] is the initial substring
of s of length n. Now s = s[length(s)] has more a's than
b's, so there exists an n such that s[n] has this property.
Let N be the _smallest_ n such that s[N] has more a's
than b's. Then s[N-1] must have the same number of a's
and b's and the N-th character must be a, QED.
************************

===

Subject: Does Goldbach imply Reimann
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h97KV2u19406;
It is possible prove the Ternary Goldbach Conjecture (TGC) and
the Twin
Prime Conjecture (TPC) are true, if the Generalized Riemann
Hypothesis (GRH)
is true.
See
http://www.ams.org/era/1997-03-15/S1079-6762-97-00031-0/S1079-
6762-97-00031-
0.pdf
for the proof of GRH-->TGC.
Is there a similar paper for the converse? If the TGC or TPC
is true then,
the GRH or (RH) is true.
My question is can either of the following be proved?
TGC-->GRH
or
TPC-->GRH
John Washburn

===

Subject: Re: Does Goldbach imply Reimann
> It is possible prove the Ternary Goldbach Conjecture (TGC)
and the Twin
Prime
> Conjecture (TPC) are true, if the Generalized Riemann
Hypothesis (GRH) is
> true.
GRH implies twin primes? News to me.
> Is there a similar paper for the converse? If the TGC or TPC
is true
then,
> the GRH or (RH) is true.
Not to my knowledge.
--

===

Subject: Re: Class of computable functions
>Suppose I want a 'large' computably enumerable collection of
functions
>f_i : N --> N with the following properties:
>1. Each primitive recursive function is included.
>2. Each f_i is total by construction.
>3. Given i and n in N, there is a computable function time: N
x N --N which tells me that the value of f_i(n) will take at most
time(i,n)
>to compute by a turing machine or equivalent.
>['Take as long as you want, but PLEASE tell me when you will
be
>done!']
>4. The function time is computable in 'Ackermann + constant'
time,
>or at least it's behavior is boundedly nasty in some sense. :)
There was a paper 40 years ago somewhat along these lines. If
you want to be able to predict how long a computation will
take,
then you have severely restric the class of functions. Here
is the reference:
Robert W. Ritchie
Classes of predictably computable functions
Transactions of the American Mathematical Society,
vol. 106 (1963), pp. 139-173
It might help.
--Herb Enderton

===

Subject: Re: Class of computable functions
> I took a class in basic foundations and computability last
year, but
> there is a question that has been bothering me.
> Suppose I want a 'large' computably enumerable collection of
functions
> f_i : N --> N with the following properties:
> 1. Each primitive recursive function is included.
> 2. Each f_i is total by construction.
> 3. Given i and n in N, there is a computable function time:
N x N --N which tells me that the value of f_i(n) will take at
most time(i,n)
> to compute by a turing machine or equivalent.
> ['Take as long as you want, but PLEASE tell me when you will
be
> done!']
> 4. The function time is computable in 'Ackermann + constant'
time,
> or at least it's behavior is boundedly nasty in some sense.
:)
> I feel rather queasy about the prospects for this, but it
has been
> bothering me for too long. The basic theme is this: I want
to know
> worst case scenario computation time. Will I not get much
besides
> primitive recursive functions?
> Rex Butler
> PS I've heard the term 'strongly computable.' Is this rela?
take a look at http://members.ozemail.com.au/~chess3/tm1.html
Its 10 lines of code (actually a 5 state Turing Machine) and
there is no
proof if it terminates. Only 10 lines and no value for f_i.
In theory your class could be all well defined and well
behaved functions,
but in practice they will only be trivial.
Say you are nesting loops, then the invariants at each level
would have to
be combined into parallel equations, you have to solve all
enumerations
of possible equations, and show they are satisfied for each
loop to
terminate.
Even then the class of functions has no scope, here is a tiny
3 state TM
http://members.ozemail.com.au/~chess3/tm2.html you can
construct it easy
enough but how do you categorise what function it is? ~ binary
counter.
Herc