New journal article from Brilliant Light Power

Quote from stefan

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., . Usually you have just a specification of the domain and range, and the range is pretty simple, e.g., (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 . 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

Quote from Eric Walker

I have also have not seen such complicated use of the function arrow notation, e.g., . Usually you have just a specification of the domain and range, and the range is pretty simple, e.g., (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 . 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.

Quote from stefan

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 R^k, 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.

Quote from Eric Walker

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 R^k, 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), μ_o(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.

Quote from THHuxleynew

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.

Quote from THHuxleynew

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.

Quote from THHuxleynew

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.

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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 discovered a very simple example that is extreme and is a nontrivial example with nonzero total(G) and where one see that the extreme

value for this special case is the maximum modulo all loop flippings.

Quote from stefan

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.

Eric

Re maximal geodesic you are quite right, I'm using the words rather loosely! So let us call it a locally geodesic closed curve (which happens to be a great circle, and so a maximal circle on S2).

Quote from stefan

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.

Quote from THHuxleynew

Re maximal geodesic you are quite right, I'm using the words rather loosely! So let us call it a maximal locally geodesic curve (which happens to be a great circle).

Ok, so you want great circles. But does Stefan want great circles? (I assume he does, given his use of the word "loop".)

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

Quote from stefan

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:

Quote from Eric Walker

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:

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.

Quote from Eric Walker

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.