New journal article from Brilliant Light Power
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teh example works by M such that of O is such that O z points into the downward sphere it maps O onto Flip O, where Flip just result in the same great circle
as the original but the resulting normal points to the upper sphere. You don't change the density on the sphere by this, but the end result is that there are no
normal pointing to the lower half sphere. and because Flip O is again an orthogonal matrix the induced from M(O) z has the desired properties. you cannot take
-z becaue z is fixed as of the definition. But surely there is an O2 such that O2 z = O- z in the original metric, but we filter those out where O - z points to the lower
half sphere.
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You don't change the density on the sphere by this
Your formulas have no density term, and you have not given your great circles any (infinitesimal) width. Something feels ill-defined here.
Give me an arbitrarily large set of great circles that comprise a sphere, and I'll take your set and give you back a larger set with great circles inserted between each pair.
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but we filter those out where O - z points to the lower half sphere.
Where in your problem definition do you restrict the unit normals? I think you neglect to do this.
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Your formulas have no density term, and you have not given your great circles any (infinitesimal) width. Something feels ill-defined here.
Give me an arbitrarily large set of great circles that comprise a sphere, and I'll take your set and give you back a larger set with great circles inserted between each pair.
So what you are saying is equivalent to if I give you the set of real numbers you can find new numbers between them which is not real number. This is of cause wrong.
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Where in your problem definition do you restrict the unit normals? I think you neglect to do this.
A measur can put the weight zero on a specific great circel with a specific normal. My mapping (my map M) the set of great circles maps to the upper half sphere the induced meassure on the image
will essentially put zero weight on the normals pointing to the lower half sphere without changing the density.
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Stefan: I'm unwilling to spend much time on your problem until I understand your use of terminology.
Please explain why you use O(3) - and a necessarily surjective map onto closed geodesics, rather than S2, and the natural bijective map with oriented closed geodesics? O(3) introduces a whole extra unnecessary degree of freedom as well as spurious reflections. Since you are using measures this seems an unfortunate and perhaps unwise complication and makes visualising what is going on a lot harder.
It is as though you are using maths to complicate the problem instead of simplify it! I'm sure not deliberately: still it can pay you dividends to think a little about what you are doing before leaping in with a mathematical structure.
But perhaps I'm missing something significant about the use of O(3) here. Could you explain why you need to use the group of 3D rotations and reflections to define uniquely a single unit vector and its unique associated oriented closed geodesic? Why have a map at all? I can't see its utility when you can do surface integrals on S2 and uniform unit vectors on S2 will be an equivalent condition to uniform density of closed geodesics on S2 because of the natural diffeomorphism.
Tom
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Stefan: I'm unwilling to spend much time on your problem until I understand your use of terminology.
Please explain why you use O(3) - and a necessarily surjective map onto closed geodesics, rather than S2, and the natural bijective map with oriented closed geodesics? O(3) introduces a whole extra unnecessary degree of freedom as well as spurious reflections. Since you are using measures this seems an unfortunate and perhaps unwise complication and makes visualising what is going on a lot harder.
It is as though you are using maths to complicate the problem instead of simplify it! I'm sure not deliberately: still it can pay you dividends to think a little about what you are doing before leaping in with a mathematical structure.
But perhaps I'm missing something significant about the use of O(3) here. Could you explain why you need to use the group of 3D rotations and reflections to define uniquely a single unit vector and its unique associated oriented closed geodesic? Why have a map at all? I can't see its utility when you can do surface integrals on S2 and uniform unit vectors on S2 will be an equivalent condition to uniform density of closed geodesics on S2 because of the natural diffeomorphism.
Tom
Maybe you are right it complicates things, I would just as well define a measure on S2. But in order to define the density of the covering I need to, for each point in the sphere, match that to a point in a specific great circle. I therefore reached to
orthoganal matrices to to that association. So the total(G) fould be defined by \int r d\mu where \mu is a direct measure in the S2. That's a correct observation. Then I could define for each point p, a set A on S2, we have a set B_p of all great circles so that s_1 in B_p means that it covers p, then we simply say that the measure \nu defined by \nu(A) = \mu(U_{p \in A}B_p}) is the uniform measure on S^2 s.t. \nu(S^2) = 1. Would you think that that is better?. However I like my formulation because you can consider a non uniform measures on the actual loop, and consider what kind of density you get by using a covering measure $\mu$ of type G. Then you get problems with the definition above because we are using the uniformity of the great circle to simplify the fromulation. A possible generalisation to the conjecture would be to ask if instead of a uniform measure on the loop the same holds for a probability measure on the loop.
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I would just as well define a measure on S2.
No need, you have the measure induced by the standard embedding of S2 in E3.
But in order to define the density of the covering I need to, for each point in the sphere, match that to a point in a specific great circle.
Why is that? I thought you wanted the great circles to be uniformly dense. That is ensured through the natural diffeomorphism if the unit vectors are the same.
if you want to consider some more complex class of variable-thickness great circles we are in a land so unconstrained I can't see the point, nor any reasonable physical analogue.
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I would just as well define a measure on S2.
No need, you have the measure induced by the standard embedding of S2 in E3.
But in order to define the density of the covering I need to, for each point in the sphere, match that to a point in a specific great circle.
Why is that? I thought you wanted the great circles to be uniformly dense. That is ensured through the natural diffeomorphism if the unit vectors are the same.
if you want to consider some more complex class of variable-thickness great circles we are in a land so unconstrained I can't see the point, nor any reasonable physical analogue.
Somtimes a more general problem is easier to prove so I will add that as a comment. I'll add your suggestion as an reformulation tag. Also note that if you have amoving measure of charge on a great circle that does not radiate the indiced density
on the sphere will also not radiate. It can be of interests to try understand what that resulting charge density can be.
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So what you are saying is equivalent to if I give you the set of real numbers you can find new numbers between them which is not real number. This is of cause wrong.
No. It is like saying that if you give me an arbitrary set of real numbers, I can give you back a larger set of real numbers, with additional ones between each consecutive in the original set pair. Which is of course correct.
(1) You must show that your set of great circles is analogous to the real numbers to invoke your counterargument. (2) It is not trivially clear that the real numbers come with a notion of "density" in the manner that you'd like to use it.
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My mapping (my map M) the set of great circles maps to the upper half sphere the induced meassure on the image
will essentially put zero weight on the normals pointing to the lower half sphere without changing the density.
I think you need to show that this is the case in your problem setup rather than just implicitly assuming it.
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closed geodesic?
I see that you can have a "closed geodesic" on a Riemann manifold. I infer that S2 is a Riemann manifold. Note that these are a form of "geodesic loop".
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Somtimes a more general problem is easier to prove so I will add that as a comment. I'll add your suggestion as an reformulation tag. Also note that if you have amoving measure of charge on a great circle that does not radiate the indiced density
on the sphere will also not radiate. It can be of interests to try understand what that resulting charge density can be.
Stepan,
My suggestion is not more general. It is actually more specific, and more precisely tuned to the problem. Charge moving in a great circle can only possibly be modelled in this style as an oriented great circle.
In this case the specific formulation is much simpler than the more general (and less clearly defined, because you posit the existence of a measure subject to various conditions) one.
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I see that you can have a "closed geodesic" on a Riemann manifold. I infer that S2 is a Riemann manifold. Note that these are a form of "geodesic loop".
Hi Eric, I defined (use my definitions, they work) S2 as the surface of the unit sphere in euclidean 3-space. It is indeed a Riemannian manifold but you can just use common or garden vector calculus to do computations - the extra generality of a manifold in this case is not required to be made explicit.
I think closed geodesic is a fair and precise definition of such a great circle, accessible to all. But I'm happy also to call it a geodesic loop. Stefan does not as far as I can see need any mathematical structure beyond vector calculus on a 2D surface embedded on 3D, considerably less foreign to many than manifolds - even though it is an example of one.
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But I'm happy also to call it a geodesic loop.
No, please continue to use "closed geodesic," which seems appropriate. The line above was just an allusion to your complaint about Stefan using "loop" earlier.
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No. It is like saying that if you give me an arbitrary set of real numbers, I can give you back a larger set of numbers, with additional numbers between each consecutive pair (which also happen to be real numbers themselves). Which is of course correct.
(1) You must show that your set of great circles is analogous to the real numbers to invoke your counterargument. (2) It is not trivially clear that the real numbers come with a notion of "density" in the manner that you'd like to use it.
No that's not correct. Take the set [a,b] then for each x,y in [a,b] a point p in between x,y also is located in [a,b] hence adding middle numbers does not incrase the set. However if you take the union of [1,2] and [4,5] you can of cause create a larger set
[1,2]u{3}u[4,5] so you are right sometimes but we consider a measure on all great circles which so this means that all great circkes are included in this set. A measure is define of it's oppinion on measurable sets, what is measurable is defined by the borell
algebra e.g. basically if you take finite unions and intersection and complements between of all open sets a set is open if can be generated by a countable union of the base set of all sets D{p,r} = all x so that distance(x,p) < r on the sphere which is the intersection of all open sets on E^3 and S^2. So technically measures are not defined on all possible sets, that's the technical difficulty in measure theory and makes precise arguments. A set D{p,r} that is not the empty set has the property that for all two
normals in that set all normals between those will be in that set so you can't find a normal between those two that is outside D{p,r} that's how I define between. But for the set of all great circles is those with a normal pointing to the unit sphere S2 so if you find a great circle between two others it too has to have a normal pointing to S^2 which is already included in the set of all great circles. Now with a reasonable notion of between there is sets where indeed you can find a loop between, but then whats the issue with that, it has no essential drawbacks for my conjecture.
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I think you need to show that this is the case in your problem setup rather than just implicitly assuming it.
I will add this information
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I will add this information
If you follow THH's counsel, you will not need to explain non-trivial results such as this one. I would not be surprised if your problem description would become much more manageable and compact if you were to follow his advice.
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Stepan,
My suggestion is not more general. It is actually more specific, and more precisely tuned to the problem. Charge moving in a great circle can only possibly be modelled in this style as an oriented great circle.
In this case the specific formulation is much simpler than the more general (and less clearly defined, because you posit the existence of a measure subject to various conditions) one.
I will add a formulation in the discussion section that I think will be more to the point.
