New journal article from Brilliant Light Power

  • 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.

  • 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.

  • 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.

  • 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.

  • 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.

  • 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.

  • 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.

    Eric I still think I motivated why I would like to include the definition as is, but i also see a value to add what I think THHuxleynew's is indicating. I'm still not convinced that his suggestion can precisely lead to the generalisation I'm looking for to include as well in as suplementary question.

  • 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.


    Unfortunately the underlying structure here is diffeomorphic to E2, 2D real numbers.


    Therefore any Borel algebra derived from great circles will be deficient. You can use a measure, but it needs to be isomorphic to that induced by the E3 metric on the great circle unit normals as they form an E3 embedding. It must therefore be defined on dense subsets of great circles with uncountable numbers of elements.


    Yet another reason why you might want to follow my suggestion.


    Regards, THH

  • Eric I still think I motivated why I would like to include the definition as is, but i also see a value to add what I think THHuxleynew's is indicating. I'm still not convinced that his suggestion can precisely lead to the generalisation I'm looking for to include as well in as suplementary question.


    Just imagine: a Mathematics Stack Exchange question consisting of two short paragraphs and no bold text or headings. Several upvotes. Two well-reasoned responses. This is the future if we try.

  • So what do you like my added retake of the problem ?

  • From the Wikipedia article: "Clearly, the Funk transform annihilates all odd functions, and so it is natural to confine attention to the case when ƒ is even." Can you specify exactly what you believe ƒ(u) to be in this case, for x a unit vector, and u ∈ C(x) a vector from the set of unit vectors perpendicular to x? I can guess what it might be, but I'll probably guess it incorrectly. Is your function ƒ odd or even? We can take a look at it when you write it down.


    In your alternative formulation, by fiat you have restricted the unit normals to the great circles to be ones that point towards the upper hemisphere. That is less interesting, for it makes a result of 1/2 suspiciously unsurprising.

  • From the Wikipedia article: "Clearly, the Funk transform annihilates all odd functions, and so it is natural to confine attention to the case when ƒ is even." Can you specify exactly what you believe ƒ(u) to be in this case, for x a unit vector, and u ∈ C(x) a vector from the set of unit vectors perpendicular to x? I can guess what it might be, but I'll probably guess it incorrectly. Is your function ƒ odd or even? We can take a look at it when you write it down.


    In your alternative formulation, by fiat you have restricted the unit normals to the great circles to be ones that point towards the upper hemisphere. That is less interesting, for it makes a result of 1/2 suspiciously unsurprising.

    f is an assumed density of mu, every function on the sphere can be written as an even blus and odd function f(p) = (f(p) + f(-p))/2 + ((f(p) - f(-p))/2, then a even id a(-p)=p and odd if a(-p)=-a(p). The uniform density is even.

    and the result of the treansform shall be the uniform density e.g. Y_0^0

  • From the Wikipedia article: "Clearly, the Funk transform annihilates all odd functions, and so it is natural to confine attention to the case when ƒ is even." Can you specify exactly what you believe ƒ(u) to be in this case, for x a unit vector, and u ∈ C(x) a vector from the set of unit vectors perpendicular to x? I can guess what it might be, but I'll probably guess it incorrectly. Is your function ƒ odd or even? We can take a look at it when you write it down.


    In your alternative formulation, by fiat you have restricted the unit normals to the great circles to be ones that point towards the upper hemisphere. That is less interesting, for it makes a result of 1/2 suspiciously unsurprising.

    Yes I should nuke the point to the upper part, that's the optimal version

  • Stefan, can you provide a specific ƒ, rather than making a generic argument that could apply to any ƒ? What is your ƒ(u), for u a vector, in concrete terms? Once you write down a specific function we can examine whether it is even or odd (or neither). Your general argument doesn't lead where you want it to lead, for example, if the even part is 0.


    A density function would presumably be everywhere nonnegative, and so not nontrivially odd. But I doubt your ƒ(u) consists of only a density.