New journal article from Brilliant Light Power

  • I assumed that for a point p in the sphere, there is a function F : p -> (f_1(p),...,f_k(p)). Hense the superscript k - that usually is the cardinality.


    I have also have not seen such complicated use of the function arrow notation, e.g., png.latex?%5Cinline&space;%5Clarge&space;F&space;:&space;S%5E2&space;%5Crightarrow&space;%5Cprod%5Ek(I&space;%5Ctimes&space;S%5E1). Usually you have just a specification of the domain and range, and the range is pretty simple, e.g., png.latex?\inline&space;f&space;:&space;\mathbb{R}&space;\rightarrow&space;\mathbb{C} (just as an example). Afterwards one is given the arbitrarily complex function definition. In this instance, if I read your notation closely, it looks like the function is to take a sphere as input to an unspecified product of tuples as the output. Unspecified, in the sense that we do not yet have a product operator gif.latex?\inline&space;g&space;:&space;(I&space;\times&space;S^1)&space;\times&space;(I&space;\times&space;S^1)&space;\rightarrow&space;\mathbb{R}. I doubt that your intention was to say that the range of your function looks like a product of tuples (i.e., a complex structure); probably the range is the non-negative real numbers? Is it possible you're combining the range and domain from the arrow notation with elements of the function itself? Or is this a usage that is common that I just haven't seen before?

  • related to https://math.stackexchange.com/q/2316201/453800?sgp=2


    I redid it again trying to use commentars suggestion here and also the experience I got from trying to prove that Mills orbitspher is of type G.

    I will probably publish a much nicer proof than you find in GUTCP perhaps tonight, well see. But the definitin and question has a much better form now. Hope that you

    enjoy it and agree.


    /Stefan

  • I have also have not seen such complicated use of the function arrow notation, e.g., png.latex?%5Cinline&space;%5Clarge&space;F&space;:&space;S%5E2&space;%5Crightarrow&space;%5Cprod%5Ek(I&space;%5Ctimes&space;S%5E1). Usually you have just a specification of the domain and range, and the range is pretty simple, e.g., png.latex?\inline&space;f&space;:&space;\mathbb{R}&space;\rightarrow&space;\mathbb{C} (just as an example). Afterwards one is given the arbitrarily complex function definition. In this instance, if I read your notation closely, it looks like the function is to take a sphere as input to an unspecified product of tuples as the output. Unspecified, in the sense that we do not yet have a product operator gif.latex?\inline&space;g&space;:&space;(I&space;\times&space;S^1)&space;\times&space;(I&space;\times&space;S^1)&space;\rightarrow&space;\mathbb{R}. I doubt that your intention was to say that the range of your function looks like a product of tuples (i.e., a complex structure); probably the range is the non-negative real numbers? Is it possible you're combining the range and domain from the arrow notation with elements of the function itself? Or is this a usage that is common that I just haven't seen before?


    You can take product of sets and those are again sets. This is a common notation in advanced math litterature.

  • You can take product of sets and those are again sets. This is a common notation in advanced math litterature.


    Bottom line, I doubt you needed that whole complicated apparatus for the range of your function F, given that your conjecture deals with some kind of number, total(G), and one suspects instead that what you wanted was a simple R, or maybe Rk, or maybe (I x S1)k. I see that you've edited your question once more. I will read it again and try to understand it.

  • Bottom line, I doubt you needed that whole complicated apparatus for the range of your function F, given that your conjecture deals with some kind of number, total(G), and one suspects instead that what you wanted was a simple R, or maybe Rk, or maybe (I x S1)k. I see that you've edited your question once more. I will read it again and try to understand it.

    thanks

  • Stefan, from the latest version of your math question:


    Quote

    where μo is as a positive measure on the set of orthogonal 3 dimensional matrises, O(3) s.t. μo(O(3))=1


    By this I understand you to mean that for M ϵ O(3), μo(M) = 1? Note that O(3) is a group.


    Quote

    where s1 is a fixed geodesic to the sphere and μ1 it's the uniform measure on s1 such that μ1(s1)=1


    In this context are you referring to a maximal geodesic which travels from one pole of the sphere to another? (There are shorter geodesics as well.) By "unit normal" for the geodesic, I assume you mean what THH referred to, which is a normal to the plane of the geodesic, presumably at the center of the circle of which the geodesic is a segment? How do we know which direction the normal points? Note that there is no preferred side of a plane.


    Quote

    Now the elemets of O(3) maps a geodesics to a geodesics and we can define a map F:O(3)×s1→S2 with F(O,p)=Op.


    By this I understand you to be referring to transformations of s1 in R3, where M s1 = s’ for M ϵ O(3) and s’ a maximal geodesic on our sphere from S2, which is embedded in R3? Or do you have in mind something like F(s) = { s' : s' = M s for M ϵ S ⊂ O(3) }, where S is related to some axis, and which gives you something like a sphere? Is your thought to construct a sphere by rotating s1 around a preferred axis? I don't recall seeing in the arrow notation a domain specification with a concrete element like s1 in place of the sets that are generally used.


    Quote

    We will assume the normal borell sets for all geometrical objects included in the definition.


    What was your thinking in including this caveat?


    Quote

    there is a supremum for the values that the magnitude of total(G)


    By “supremum,” do you mean “maximum”?


    Does our geodesic s1 have width? With regard to the integrals you give towards the end of the problem statement, which include a vector term n, by symmetry I would expect the result to be 0 when integrated over the domains of the integrals. Is this incorrect? Your formulas take no cognizance of angle and so apply to all angles equally.

  • Guide to terminology used here for those not trained in differential geometry


    S2 . The 2D surface of a unit sphere in E3 (technically other spaces as well, but E3 will do us fine)

    E3. 3D Euclidean space

    (Euclidean) metric. the (normal) measure of distance in an n-dimensional space.

    Maximal geodesic on S2. Great circle on the S2 sphere

    Unit vector. Line from centre of S2 sphere to a point in S2 (the surface of the sphere).

    Normal (to a maximal geodesic). Unit vector perpendicular to the plane of the maximal geodesic.

    Manifold. A set with local euclidean structure which can be used to do calculus. In this case you can happily integrate etc over the surface of a sphere because locally it approximates a plane, even though there is no single global metric that works for this. You do it in school (e.g. using spherical coordinates) implicitly without worrying that it might not work.

    Lebesgue measure. A mathematical structure like a metric but more general (and therefore weaker). A measure is essentially the weakest structure needed to do integration. If a set has a measure, you can integrate over it. Usually, we integrate over something like a plane or line using the euclidean metric. This has stronger properties than a measure, not all of which we use doing the integration. Of course, every metric has a corresponding induced measure, if you want to think of it like that. If armed with a metric (as we usually are) there is however no need to think about measures - just do integration like you were taught at school!

    Diffeomorphism. Isomorphic mapping between two manifolds respecting the local metrical structure.


    I have a more general point here. Stefan is working with a problem that could be solved using the S2 embedding in E3, the natural duality between maximal S2 geodesic normal vectors and unit vectors of S2, and the inherited E3 metric for S2. Essentially everything of interest is either a point in S2, or dual to that, so that the manifold structure is obvious and indeed even accessible to anyone with a bit of school 3D geometry and calculus.


    if you start with the dual space (of unit vectors on S2) you can use the natural mapping onto S2 maximal geodesics. There are, as Eric points out, two normals for each S2 maximal geodesic and without a turn direction for the geodesic no way to distinguish them. I suspect, again, that Stefan actually needs to give his geodesics a direction (clockwise or anticlockwise, etc) in which case the map from S2 unit vectors to S2 maximal geodesics becomes the natural bijection - in fact also obviously a diffeomorphism, and all is sweet. There are even so two natural maps corresponding to which order to you label the 3 axes of E3. You can arbitrarily choose one of them and fix it, since they are isomorphic. (Eric, I feel that letting Stefan continue to justify his stuff without this clarification is likely to use up more time than necessary). The duality here matters because it is much easier to reason about uniform density of unit vectors than uniform density of maximal geodesics.


    This also I believe is the structure that Stefan wants, because he is trying to make some point about classical orbits of charged particles.


    So I think he is giving himself (and us) a hard time by trying to generate something that makes sense and uses a mathematical structure (stand-alone Lebesgue measure) which is both weaker and less natural than that implicit in the problem formulation (E3 metric and induced manifold on S2 from its E3 embedding). I suspect what he is doing is possible, but he'd need much more care in how he used maths for it to work. For example, his measure can easily be too unconstrained and therefore incorporate physically non-real cases. Whereas taking the implicit structure of the problem (making this explicit) the heavy lifting is all done and what he wants to say can be said more easily and also more comprehensibly.


    In Chinese academic discourse it is traditionally seen as a sign of profundity and topic mastery if you say something that is so complex no-one can understand it. This leads, as you can imagine, to a profound lack of academic communication and also a lot of plain bad work. The same tendency can happen in the West but it is not so entrenched. I am firmly myself of the view that if you want to show people you understand something well, you do that by finding the simplest and most appropriate language to explain it, while also respecting rigor. I'm not saying I do this, but it is my ideal. In maths that corresponds to finding the right level of mathematical structure to fit the problem so it becomes simple and (if possible) almost tautologous. One of the marvels of maths is that so often by looking at things a different way something that appeared complex becomes simple.


    I'd recommend Stefan to try that.


    Apologies for not saying this earlier: I am quite slow and when presented with something unclear it takes me some time to work it out.

  • Thank you for the excellent primer, THH.


    Maximal geodesic on S2. Great circle on the S2 sphere


    Can you elaborate on this counterintuitive detail? I would have expected a maximal geodesic to be half a great circle.


    I am firmly myself of the view that if you want to show people you understand something well, you do that by finding the simplest and most appropriate language to explain it, while also respecting rigor.


    I agree entirely. Saying something in terms more complex than needed suggests a weak grasp of the problem. This is not a bad thing, because that's where we necessarily start out, but one should acknowledge this and seek to iterate and put it in simpler and more rigorous terms, and answer questions that are raised. (And, as an observer, become acquainted with the terminology!) If Stefan is unwilling to clarify points as they come up, there is only so much that can be done. His question is posed in a manner that will be ineffective for Mathematics Stack Exchange against good advice, so at this point I am just trying to understand what he's saying out of curiosity.


    Unless Stefan introduces some preferred direction to his setup, I think he is going to have problems with the symmetry argument, as you pointed out above and I also pointed out.

  • Should I use matehmatical nomeclature on a mathematical site the things stays short and tidy. I can add explanation for all the notions in the question but then the question will be very long.

    What I will do add a explanation section and give the needed discussion for normal people to understand what I'm doing

  • Should I use matehmatical nomeclature on a mathematical site the things stays short and tidy. I can add explanation for all the notions in the question but then the question will be very long.


    What you have is not tidy. Somehow you're simultaneously abusing notation, not being clear about what you want to do and leaving things fatally under-specified. If you finally get to a satisfactory problem statement, it will be the opposite: no abuse of the notation, clarity about what is being described, and no obvious questions because something has been left under-specified. The concepts you're dealing with are not all that difficult, but it is hard to pin down what you have in mind.

  • Should I use matehmatical nomeclature on a mathematical site the things stays short and tidy. I can add explanation for all the notions in the question but then the question will be very long.

    What I will do add a explanation section and give the needed discussion for normal people to understand what I'm doing


    I'm all for you using maths. I'd like you to do so precisely, consistently (no loop flipping, indeed no loops) and in the most compact way, which might be as I indicated above.

  • I added a commentary to help you dig out from the Underland dungeon. I added an explanation section at the bottom of the page and changed

    geodesics, which I thought was great circles e.g. straight lines in a spherical geometry. I referred to the Euclidian space.


    I hope things becomes clearer

  • I added a commentary to help you dig out from the Underland dungeon.


    Thank you for helping me dig out of my underground dungeon, and also for not calling a "great circle" a "geodesic." I will take a look at your revised question. It might be most efficient to get THH's best attempt at a clear and rigorous formulation of what it is that you're proposing.


    My suspicion is that symmetry will take the following integral to zero (assuming it's well-defined), and that you're doing something wrong in your example with the polar coordinates:


    gif.latex?total(G)&space;=&space;\int_{O\in&space;O(3)}&space;O&space;\hat{z}&space;d\mu_0

  • Thank you for helping me dig out of my underground dungeon, and also for not calling a "great circle" a "geodesic." I will take a look at your revised question. It might be most efficient to get THH's best attempt at a clear and rigorous formulation of what it is that you're proposing.


    My suspicion is that symmetry will take the following integral to zero (assuming it's well-defined), and that you're doing something wrong in your example with the polar coordinates:


    gif.latex?total(G)&space;=&space;\int_{O\in&space;O(3)}&space;O&space;\hat{z}&space;d\mu_0


    I think the example is correct so each great circle can have two normal if it is pointing towards the lower sphere you can change the normal by mirroring it to the upper sphere keeping the actual great circle the same. So no you have all normals of the great circles pointing to the upper sphere. Note, the mapping I had in the example (wrong by the way, I'll fix it) will change the meassure appropriately and the result is that total(G)=1/2.

  • The problem, it seems to me, is that in order for that integral to be nonzero, you need an asymmetry of some kind when a vector is reflected through the origin. Nothing obvious from your problem setup stands out as providing such an asymmetry. This argument should motivate you to take a closer look at your example.


    This discussion assumes that your problem is well defined, which I do not assume.

  • My suspicion is that symmetry will take the following integral to zero (assuming it's well-defined), and that you're doing something wrong in your example with the polar coordinates:


    That depends on the space used. In polar coordinates all vectors pointing off the surface are positive. Think of the Gauss flux integral law.

  • 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.

  • 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.

  • 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.

  • 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.

  • 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.

  • 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.