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Vallenstream science forum beginner
Joined: 17 Nov 2005
Posts: 2

Posted: Tue Nov 22, 2005 10:30 am Post subject:
Re: Conjecture Related to Goldbach's Conjecture



Thanks, Gerry. I see there is an interesting graph of numbers for Levy's Conjecturedoesn't actually look much like Goldbach's Comet.
Levy's Conjecture isn't exactly the same as the one I described, since he apparently doesn't require that the "twice a prime" number be twice an odd prime. I also guess that his conjecture allows the "twice a prime" to be twice the prime that is added to it, to get the odd integer. I don't use that, because it is, of course, simply a multiple of three, which just repeats the odd number rather than being a total with new primes.
My idea was to formulate a conjecture that used only the odd primes as factors of the "twice a prime" term, since the same primes would then be available for that conjecture as for Goldbach'sof course, it's impossible to use integer two as one of the primes in Goldbach's strong conjecture, but many odd integers are four more than a prime.
Looks like quite a bit of research has been done on Levy's Conjecture, so maybe there is, in the research, some analytical method that I can use.
Another interesting "sums of primes" investigation, is the problem of using only primes that are twin primeswhich have some unusual propertiesas pairs of primes that equal even integers. The two primes, of course, don't have to be the two successive twin primes of the pairs they belong to.
I guess the proposition would be, that, at some magnitude, most even numbers above that magnitude are not the sums of two primes that are both twin primes. It's rather interesting that, among small even integers, only three even numbers between 90 and 100 are not the sum of two twin primes. Since twin primes thin out faster at large magnitudes than primes in general do, I have to think that, above a certain number, very few integers are the sum of two twin primes. Would be interesting to see how the diminishing of sums works out. 

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John Baez science forum Guru Wannabe
Joined: 01 May 2005
Posts: 220

Posted: Fri Nov 18, 2005 12:45 pm Post subject:
Re: Presheaf as functor



In article <djj3uo$1ss$1@news.ks.uiuc.edu>,
joe.shmoe.joe.shmoe <joe.shmoe.joe.shmoe@gmail.com> wrote:
Quote:  One could add slightly modify the definition of a presheaf, starting
with a fixed "base" functor R: Opens(X) > B, and a "projection"
functor P:C > B, and then saying a presheaf into C ("relative to R
and P") is a functor F:Opens(X)> C such that P o F=R. However, this
does make things a lot more cumbersome, and I have not seen this sort
of thing done anywhere.
Any thoughts on this?

What you're suggesting here is okay  and it's just another way
of doing what Roig was suggesting.
Roig suggests that we consider the category of "rings with modules",
whose objects are a ring and a module of that ring. Call it RingMod.
There's an obvious forgetful functor
U: RingMod > Ring
that throws out the module and keeps the ring.
A "presheaf of rings with modules" is a contravariant functor
F: Opens(X) > RingMod
and composing with U we get a presheaf of rings
U o F: Opens(X) > Ring
You're just talking about the reverse process! You're starting
with a presheaf of rings
P: Opens(X) > Ring
and then you're looking for some presheaf of rings with modules
F: Opens(X) > RingMod
that gives P when we forget the module:
P = U o F
That's fine too. There's kind of a choice here as to whether you
want to take the presheaf of rings as "fixed", or part of what you
want to vary. The same issue shows up when we consider "bundles over
a fixed manifold" versus "bundles over manifolds". 

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Robert Clark science forum Guru Wannabe
Joined: 30 Apr 2005
Posts: 129

Posted: Tue Nov 15, 2005 7:22 pm Post subject:
Re: Math question about trajectory for beamed propulsion.



I'm concerned with means to minimize the distance from
the launch point to a beamed propulsion craft to so
that the beamed energy reaching the craft is not
diminished too greatly.
Here's the scenario. The craft has to reach a certain
speed in order
to reach orbital velocity. It's on the order of 7800
m/s. I'll just call it
10,000 m/s for simplicity. You also would like the
acceleration not to be too
large The reason is that for beamed propulsion, for a
given amount of beamed energy
transmitted, a fixed thrust is going to be provided.
Quote:  From Newton's equation F=ma, so
if you make the acceleration large, the mass, or 
payload will be reduced. So
let's say the acceleration is kept below 40 m/s^2, or
about 4 g's.
But then it takes time to build up the large velocity
required. And
traveling at the limited acceleration allowed would
force a long distance to
be used to build up the necessary speed. Indeed for
the space shuttle
for example the distance from the launch pad is on the
order of 2000 km for
powered thrust. This large distance is what I'm trying
to avoid. A few hundred km is
what I'm looking for.
So here's the suggestion. What's really needed for
the beamed
propulsion is keeping the *straightline* distance to
the craft low. Then what you
could do would have the craft travel in say a helical
path. Then it could be
traveling a long distance, the length of the helical
path, while the
straightline distance from start to finish might be
low.
Yet this introduces more problems. For when the craft
is turning in
that helical path, it is undergoing acceleration. And
if the turning radius
is small, the acceleration could be high, and your low
acceleration
condition might be contradicted anyway.
So as a math problem, what I'm looking for is a
trajectory in 3dim'l space s(t) such that the
magnitude of the acceleration is low, s"(t) <= 40
m/s^2, the final speed s'(t) is 10,000m/s, but the
curve is contained in a sphere of minimal diameter.
This is a calculus of variations problem, like the
brachistochrone problem, though of a different kind
since it is not an integral that has to be minimized.
Note that without the condition that the acceleration
is limited we could make the straightline distance
arbitrarily small by making the curve make many twists
and turns while enclosed in a small box.
With the acceleration condition, I'm inclined to
believe you can't do any better than a straightline
but I don't see how to prove that.
 Bob Clark
[ Moderator's note:
Clearly he also wants the initial condition s'(0) = 0.
 ri ] 

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Ali Taghavi science forum addict
Joined: 14 May 2005
Posts: 73

Posted: Wed Nov 02, 2005 3:34 pm Post subject:
Re: IS quantum mechanics a limit cycle theory??



"Due to the main subject of this topic I present the following new question(related to the title of this Topic)":
Assume X is a non vanishing Vector field on a manifold M,it generates an operator(say X) on functional space
C^inf(M),with the standard formule f>X.f
(the derivative of f along the trajectories of X)
Does there exist a linear operator S on C^inf (M) such that XSSX=1 (1=identity)
Thank you Ali Taghavi
alitghv_at_yahoo.com 

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Ali Taghavi science forum addict
Joined: 14 May 2005
Posts: 73

Posted: Mon Oct 31, 2005 4:00 pm Post subject:
Re: IS quantum mechanics a limit cycle theory??



The projection of all closed orbits of Hamiltonian is mentioned by
Robert Israel, is a single circle in 3 dimensional position space.
But the total number of closed orbits in 6 dimension is infinite. Thus
this is not the answer of my Question:"I search for a Hamiltonian
vector Field with a finite (but nonzeros) number of closed orbits) As
an example: What is the number of closed orbits of the following
system (hamiltonian vector field) in R^4: H=(y(x^3x))zxw. The
Hamiltonian vector field is a vector field in R^4 which projection in
2 Dimensional space is the standard vander Pol equation Thank you
Ali Taghavi
Quote: 
Consider a particle moving in 3 dimensions in the
potential
V(x,y,z) = (1  x^2  y^2)^2 z + z^3 + x^2 + y^2
Note that dV/dz > 0 except on the circle x^2 + y^2
=
1, z = 0, so
the only possible closed orbits are on that circle.
On that circle
there are closed orbits in both directions.
Robert Israel
israel@math.ubc.ca
Department of Mathematics
http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver,
BC, Canada



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Ali Taghavi science forum addict
Joined: 14 May 2005
Posts: 73

Posted: Fri Oct 28, 2005 1:30 pm Post subject:
Re: IS quantum mechanics a limit cycle theory??



The Hamiltonian vector field corresponding to your H has infinite number
of closed orbits! But my question searched for a hamiltonian with a
finite but non zero number of nontrivial periodic solutions(of
course, it is impossible in 2 dim.) For example how many closed orbits
exist for the following Hamiltonian in R^4 H=(y(x^3x))zxw
Note that the projection in 2 first component is the standarrd vander pol equation.
Finaly : a question about your hamiltonian: as you said there is only
one periodic orbit in (3 dimensional) position space. What is the
Quantum interpretation for this unique orbit (and for this
uniquness)??I mean look at the variable as Hermitian operator....
Quote:  The complete
Hamiltonian is u^2/2 + v^2/2 + w^2/2 + V(x,y,z) where
u, v, w are
conjugate to x, y, z respectively.
Robert Israel
israel@math.ubc.ca
Department of Mathematics
http://www.math.ubc.ca/~israel
University of British Columbia Vancouver,
BC, Canada



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Robert B. Israel science forum Guru
Joined: 24 Mar 2005
Posts: 2151

Posted: Thu Oct 27, 2005 2:30 pm Post subject:
Re: IS quantum mechanics a limit cycle theory??



Ali Taghavi wrote:
Quote:  the domain of hamiltonian is even dimensional space!
Consider a particle moving in 3 dimensions in the
potential
V(x,y,z) = (1  x^2  y^2)^2 z + z^3 + x^2 + y^2
Note that dV/dz > 0 except on the circle x^2 + y^2 =
1, z = 0, so
the only possible closed orbits are on that circle.
On that circle
there are closed orbits in both directions.

I think you may have misunderstood my example. The Hamiltonian system
is 6dimensional (3 components for velocity and 3 for position),
representing a particle moving in the potential V. The complete
Hamiltonian is u^2/2 + v^2/2 + w^2/2 + V(x,y,z) where u, v, w are
conjugate to x, y, z respectively.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada 

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Ali Taghavi science forum addict
Joined: 14 May 2005
Posts: 73

Posted: Wed Oct 26, 2005 2:30 pm Post subject:
Re: IS quantum mechanics a limit cycle theory??



May you more Explain on "Physical relevance"? Further in this example
the hamiltonian vector field in R^6 has an invariant torus with
infinite number of closed orbits. I search for an example of an
analytic Hamiltonian in R^2n with a finit number of closed orbits....
Is there is any example of such Hamiltonian, with k number of closed
orbits, what is the quantum interpretation for this "k",after an
appropriate quantization of H! Let's try this question for the
following this question for the following hamiltonian:
H=(y(x^3x))zzw (in R^4)
Quote:  israel@math.ubc.ca wrote:
Consider a particle moving in 3 dimensions in the
potential
V(x,y,z) = (1  x^2  y^2)^2 z + z^3 + x^2 + y^2
Note that dV/dz > 0 except on the circle x^2 + y^2
= 1, z = 0, so
the only possible closed orbits are on that circle.
On that circle
there are closed orbits in both directions.
IMHO this cannot have any physical relevance, because
these orbits are
highly _unstable_. Am I wrong?
Han de Bruijn
.



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Ali Taghavi science forum addict
Joined: 14 May 2005
Posts: 73

Posted: Tue Oct 25, 2005 9:00 pm Post subject:
Re: IS quantum mechanics a limit cycle theory??



Thank you for your example:
Let's try my question about the following example in R^4:
H=(y(x^3x))zxw
consider the corresponding hamiltonian vector field X_H in R^4 with
respect to standard symplectic structure of R^4 (how many periodic
solutions do exist)?? Now try to construct a Hermitian operator with
H(as usual methods), my question is that what is the operator theoretic
interpretation for the number of closed orbits of X_H...
Quote:  the domain of hamiltonian is even dimensional space!
But can Morse
theory help to my question, namely is dynamic fix if
we do not pass a
critical value (However I prefer to not change the
main subject of my
question, a possible relation between limit cycle
theory and
quantization)
Consider a particle moving in 3 dimensions in the
potential
V(x,y,z) = (1  x^2  y^2)^2 z + z^3 + x^2 + y^2
Note that dV/dz > 0 except on the circle x^2 + y^2
=
1, z = 0, so
the only possible closed orbits are on that circle.
On that circle
there are closed orbits in both directions.
Robert Israel
israel@math.ubc.ca
Department of Mathematics
http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver,
BC, Canada



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Ali Taghavi science forum addict
Joined: 14 May 2005
Posts: 73

Posted: Tue Oct 25, 2005 2:30 pm Post subject:
Re: IS quantum mechanics a limit cycle theory??



the domain of hamiltonian is even dimensional space! But can Morse
theory help to my question, namely is dynamic fix if we do not pass a
critical value (However I prefer to not change the main subject of my
question, a possible relation between limit cycle theory and
quantization)
Quote:  Consider a particle moving in 3 dimensions in the
potential
V(x,y,z) = (1  x^2  y^2)^2 z + z^3 + x^2 + y^2
Note that dV/dz > 0 except on the circle x^2 + y^2 =
1, z = 0, so
the only possible closed orbits are on that circle.
On that circle
there are closed orbits in both directions.
Robert Israel
israel@math.ubc.ca
Department of Mathematics
http://www.math.ubc.ca/~israel
University of British Columbia Vancouver,
BC, Canada



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Han de Bruijn science forum Guru
Joined: 18 May 2005
Posts: 1285

Posted: Tue Oct 25, 2005 2:30 pm Post subject:
Re: IS quantum mechanics a limit cycle theory??



israel@math.ubc.ca wrote:
Quote:  Consider a particle moving in 3 dimensions in the potential
V(x,y,z) = (1  x^2  y^2)^2 z + z^3 + x^2 + y^2
Note that dV/dz > 0 except on the circle x^2 + y^2 = 1, z = 0, so
the only possible closed orbits are on that circle. On that circle
there are closed orbits in both directions.

IMHO this cannot have any physical relevance, because these orbits are
highly _unstable_. Am I wrong?
Han de Bruijn
.. 

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Robert B. Israel science forum Guru
Joined: 24 Mar 2005
Posts: 2151

Posted: Mon Oct 24, 2005 5:30 pm Post subject:
Re: IS quantum mechanics a limit cycle theory??



Ali Taghavi wrote:
Quote:  Due to material of my question in "Is quantum mechanics a limit cycle
theory" , it seems that I should ask, in priori, for an example of a
classical hamiltonian H with a finite (but non zero) number of closed
orbits (of course it is impossible in 2 dim.) If it is impossible to
present an example of hamiltonian with a finite number of closed
orbits, so the quantum interpretation for this "number" would be
meaningless!

Consider a particle moving in 3 dimensions in the potential
V(x,y,z) = (1  x^2  y^2)^2 z + z^3 + x^2 + y^2
Note that dV/dz > 0 except on the circle x^2 + y^2 = 1, z = 0, so
the only possible closed orbits are on that circle. On that circle
there are closed orbits in both directions.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada 

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joe.shmoe.joe.shmoe science forum beginner
Joined: 19 Oct 2005
Posts: 7

Posted: Mon Oct 24, 2005 5:00 pm Post subject:
Re: Presheaf as functor



Thanks for the replies. This sounds like a good way to do it, but it's
not exactly what I had in mind. I started thinking about this after
reading the definition of a presheaf in many textbooks, along the lines
of:
"On a topological space X, a presheaf F of sets (or abelian groups,
rings, etc) is an assignment for each open subset..." etc.
Clearly, it seems beneficial to prove statements about presheaves in
the general context of "contravariant functors from opens(X) to C" for
some C. Then looking at stalks, for example, works the same for all
categories in which direct limits exist. Or another example 
constructing the sheaf associated to a presheaf, which in the case of
sets, abelian groups etc., gives you a (pre)sheaf with values in the
same category you started with. This is the sort of thing I'm trying to
achieve with some degree of generality.
One could add slightly modify the definition of a presheaf, starting
with a fixed "base" functor R: Opens(X) > B, and a "projection"
functor P:C > B, and then saying a presheaf into C ("relative to R
and P") is a functor F:Opens(X)> C such that P·F=R. However, this
does make things a lot more cumbersome, and I have not seen this sort
of thing done anywhere.
Any thoughts on this? 

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Han de Bruijn science forum Guru
Joined: 18 May 2005
Posts: 1285

Posted: Mon Oct 24, 2005 2:30 pm Post subject:
Re: minimum volume ellipse



John wrote:
Quote:  I am looking for a solution that works in high dimensions.
Most of these solutions are exponential in the dimension.

How about trying to generalize the theory at:
http://huizen.dto.tudelft.nl/deBruijn/grondig/crossing.htm#BE
to multiple dimensions? Involving a quadratic form with the mean values
of the points as a midpoint, and the inverse of a matrix with variances
giving the coefficients.
Must be not too bad. Just my 5 cents worth ...
Han de Bruijn
.. 

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Ali Taghavi science forum addict
Joined: 14 May 2005
Posts: 73

Posted: Mon Oct 24, 2005 12:30 pm Post subject:
Re: IS quantum mechanics a limit cycle theory??



Due to material of my question in "Is quantum mechanics a limit cycle
theory" , it seems that I should ask, in priori, for an example of a
classical hamiltonian H with a finite (but non zero) number of closed
orbits (of course it is impossible in 2 dim.) If it is impossible to
present an example of hamiltonian with a finite number of closed
orbits, so the quantum interpretation for this "number" would be
meaningless!
Quote:  hello
let we have a classical hamiltonian H,and we quantize
it.
I want to learn some relations between the behavior
of solutions of X_H(the classic hamiltonian vector
field) and some operetor theoretic invariants of the
quantum operatores.In particular a nice
interprewtation of the number of closed orbits of X_H
in quantum language!
Thank you for your suggestions for some deep
references
Ali Taghavi
Hi
I Am Intersted in the Hilbert 16th problem which
main
object is "Limit
Cycle"!
Last Year I Found in the web a paper by Cetto and
De
La Penna :with
The Title "Is Quantum Mechanics A Limit Cycle
Theory?"
this paper is available in Mathscinet.
I Invite you to review this paper and discuss on a
possible relation
between Hilbert 16th Problem And Mathematical
Aspect
Of QM,
for begining:Let We Have A Planner Vector field
L(Lienard Equation )
as follow:
x'=yF(x) y'=x where F is A non even_polynomial
,we
are intersted in
the number of Limit Cycles Of L,consider the
following two
questions:(Assume F'(0) is not zero)
1)Does there exist a correspondence between {closed
orbites} of L and
{closed orbits} of 4 dimensional (classical)
Hamiltonian (yF(x))zxw?
please see also the similar question in :
http://www.arxiv.org/abs/math.CA/0409594
2)Let's Quantize the above 4 dimensional
hamiltonian:x,y stand for
operators multiplication by x and y,while z,w stand
for partial
derivative with respect to x ,y resp.
What Is The Quantum interpretations for (the number
of) closed orbits
of classical Hamiltonian
(yF(x))zxw?
Thank you
Ali Taghavi
Iran



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