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## Spherically symmetric space is conformally flat

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### Spherically symmetric space is conformally flat

An example of a spherically symmetric space can be found in Swarzschild's solution of Einstein's general relativity: a space-like slice where Swarzschild's time coordinate is constant. In study of this and similar models it may be interesting that such spaces are coformally flat, except possibly at the center of the spherical symmetry.

Our assumption is that the space is three dimensional and its geometry is spherically symmetric. The assumption can be epressed with the formula
ds² = A(r)² dr² + B(r)² (dθ² + dφ² sin² θ)
for line element in terms of sherical coordinates, where A and B are two positive-valued functions except that B(0) = 0.

The theorem we shall prove is that where r > 0 the space is conformally flat. This can be expressed with the formula
ds² = C²(r) (dR(r)² + R(r)²(dθ² + dφ² sin² θ))
for line element where C and R are some functions. The theorem is that it is possible to choose these functions so that this formula for line element is the same as the first formula.

In order to prove the theorem we construct C and R in terms of A and B and substitute in the latter formula:
R(r) = exp(∫ₐʳ A(u) / B(u) du) where a > 0 but otherwise the value of a is not important,
and
C(r) = B(r) / R(r).
R(0) (and consequently C(0)) may be ill defined as the expression contains a division by B(u) and B(0) = 0. This is does not matter as we have excluded r = 0 from our theorem.

The substitution is quite simple when we observe that
dR(r) = R(r) A(r) dr / B(r)
and the expression for R need not be used; instead,
C(r) R(r) = B(r)
and
C(r) dR(r) = A(r) dr
can be substitued, giving
ds² = A(r)² dr² + B(r)² (dθ² + dφ² sin² θ)
Q.E.D.
Mikko

Posts: 138
Joined: Mon Feb 17, 2014 1:53 am

### Re: Spherically symmetric space is conformally flat

Mikko wrote:An example of a spherically symmetric space can be found in Swarzschild's solution of Einstein's general relativity: a space-like slice where Swarzschild's time coordinate is constant. In study of this and similar models it may be interesting that such spaces are coformally flat, except possibly at the center of the spherical symmetry.

Our assumption is that the space is three dimensional and its geometry is spherically symmetric. The assumption can be epressed with the formula
ds² = A(r)² dr² + B(r)² (dθ² + dφ² sin² θ)
for line element in terms of sherical coordinates, where A and B are two positive-valued functions except that B(0) = 0.

The theorem we shall prove is that where r > 0 the space is conformally flat. This can be expressed with the formula
ds² = C²(r) (dR(r)² + R(r)²(dθ² + dφ² sin² θ))
for line element where C and R are some functions. The theorem is that it is possible to choose these functions so that this formula for line element is the same as the first formula.

In order to prove the theorem we construct C and R in terms of A and B and substitute in the latter formula:
R(r) = exp(∫ₐʳ A(u) / B(u) du) where a > 0 but otherwise the value of a is not important,
and
C(r) = B(r) / R(r).
R(0) (and consequently C(0)) may be ill defined as the expression contains a division by B(u) and B(0) = 0. This is does not matter as we have excluded r = 0 from our theorem.

The substitution is quite simple when we observe that
dR(r) = R(r) A(r) dr / B(r)
and the expression for R need not be used; instead,
C(r) R(r) = B(r)
and
C(r) dR(r) = A(r) dr
can be substitued, giving
ds² = A(r)² dr² + B(r)² (dθ² + dφ² sin² θ)
Q.E.D.

Nice effort there Mikko, but your Q.E.D is misplaced. That formally correct derivation is I believe the standard way ISM was derived in the first place. Part 3 of article (v3) deals with this by noting true conformal non-flatness has local physical significance - locally non-Euclidean geometry. Standard SM is not conformally flat.
As shown there by considering the differential area/volume ratio, or differential great circle circumference/radial gap ratio. ISM pretends to be at one and the same time conformally flat, yet also representing the same physical metric as SM. A logical impossibility. You either choose ISM as representing a physically different metric - true conformal flatness = locally Euclidean geometry, or that the latter is a misleading artifact of coordinate transformation thus fake. You may say "but you have agreed the derivation is correct!". Formally correct but necessarily having a fatal subtle flaw as just described. What makes sense to me is that the metric given by the transformed coordinate system is 'isomorphic' to the original only in a limited strictly local sense - about some given radial coordinate value. Making it in fact essentially useless. Not something generally recognized obviously but otherwise how to explain the now obvious contradiction?

[when I get time, will do a simple test - compare SM vs ISM for two radius-to-circumference values. That will definitely decide between ISM as actually equivalent to SM or not. Not that it ultimately matters.]
Q-reeus

Posts: 272
Joined: Sat Jun 07, 2014 11:18 pm

### Re: Spherically symmetric space is conformally flat

Q-reeus wrote:
Mikko wrote:An example of a spherically symmetric space can be found in Swarzschild's solution of Einstein's general relativity: a space-like slice where Swarzschild's time coordinate is constant. In study of this and similar models it may be interesting that such spaces are coformally flat, except possibly at the center of the spherical symmetry.

Our assumption is that the space is three dimensional and its geometry is spherically symmetric. The assumption can be epressed with the formula
ds² = A(r)² dr² + B(r)² (dθ² + dφ² sin² θ)
for line element in terms of sherical coordinates, where A and B are two positive-valued functions except that B(0) = 0.

The theorem we shall prove is that where r > 0 the space is conformally flat. This can be expressed with the formula
ds² = C²(r) (dR(r)² + R(r)²(dθ² + dφ² sin² θ))
for line element where C and R are some functions. The theorem is that it is possible to choose these functions so that this formula for line element is the same as the first formula.

In order to prove the theorem we construct C and R in terms of A and B and substitute in the latter formula:
R(r) = exp(∫ₐʳ A(u) / B(u) du) where a > 0 but otherwise the value of a is not important,
and
C(r) = B(r) / R(r).
R(0) (and consequently C(0)) may be ill defined as the expression contains a division by B(u) and B(0) = 0. This is does not matter as we have excluded r = 0 from our theorem.

The substitution is quite simple when we observe that
dR(r) = R(r) A(r) dr / B(r)
and the expression for R need not be used; instead,
C(r) R(r) = B(r)
and
C(r) dR(r) = A(r) dr
can be substitued, giving
ds² = A(r)² dr² + B(r)² (dθ² + dφ² sin² θ)
Q.E.D.

Nice effort there Mikko, but your Q.E.D is misplaced.

Not at all, the Q.E.D. should be at the end and there it is. It's the text before it that should be written differently. At least there should be a bit of prose
after the last formula but a different arrangement of the formulas would have permitted a better presentation. The core of the proof is in the formulas,
and their relationships to each other and to the theorem are expressed clearly, in spite of poor presentation.
That formally correct derivation is I believe the standard way ISM was derived in the first place. Part 3 of article (v3) deals with this by noting true conformal non-flatness has local physical significance - locally non-Euclidean geometry.

What you mean by "locally"? As usually understood, every spacetime is locally Minkowskian except at singularities and therefore spatial slices (whether isochronous or not) are locally Euclidean. If you allow "locally" to include less local measures then you will soon have locally non-Euclidean (or non-Minkowskian) geometries in every curved space (or spacetime).
Standard SM is not conformally flat. As shown there by considering the differential area/volume ratio, or differential great circle circumference/radial gap ratio. ISM pretends to be at one and the same time conformally flat, yet also representing the same physical metric as SM. A logical impossibility. You either choose ISM as representing a physically different metric - true conformal flatness = locally Euclidean geometry, or that the latter is a misleading artifact of coordinate transformation thus fake. You may say "but you have agreed the derivation is correct!". Formally correct but necessarily having a fatal subtle flaw as just described. What makes sense to me is that the metric given by the transformed coordinate system is 'isomorphic' to the original only in a limited strictly local sense - about some given radial coordinate value. Making it in fact essentially useless. Not something generally recognized obviously but otherwise how to explain the now obvious contradiction?

Contradiction is not obvious until you can prove that 1 = 2. You have claimed that that some space or spacetime is or is not conformally flat depending on how you look at it but you have not proved anything (which you could do e.g. by computing the Cotton tensor).
[when I get time, will do a simple test - compare SM vs ISM for two radius-to-circumference values. That will definitely decide between ISM as actually equivalent to SM or not. Not that it ultimately matters.]

Radius is a little bit difficult. The centre of the Swartschild solution is singular. If you replace the interior part with a non-vacuum solution, you must integrate through heterogenous space. It is easier to use two spheres at constant radii so that their areas are A and A+dA so that both are in the vacuum region, and check the distance between these spheres.
Mikko

Posts: 138
Joined: Mon Feb 17, 2014 1:53 am

### Re: Spherically symmetric space is conformally flat

Mikko wrote:Not at all, the Q.E.D. should be at the end and there it is. It's the text before it that should be written differently. At least there should be a bit of prose
after the last formula but a different arrangement of the formulas would have permitted a better presentation. The core of the proof is in the formulas,
and their relationships to each other and to the theorem are expressed clearly, in spite of poor presentation.

And as acknowledged in earlier post, your derivation is technically correct. But imo something subtle has to be amiss. i gave my hunch then - strictly local equivalence only, for some given r.
What you mean by "locally"? As usually understood, every spacetime is locally Minkowskian except at singularities and therefore spatial slices (whether isochronous or not) are locally Euclidean. If you allow "locally" to include less local measures then you will soon have locally non-Euclidean (or non-Minkowskian) geometries in every curved space (or spacetime).

As you know I have not pretended to be conversant with the mathematical structure of GR. And simply offer in article a layman outsider's imo fresh insight not some high powered mathematically dense tensor based proof. But note from the literature that true conformal flatness (4D) is equivalent to having zero Weyl curvature. And that Weyl curvature is an intensive property existing 'at a point' thus having local existence. Analogously to say non-zero divergence or curl in EM, though existing 'at a point', the measurable effects vanish as one restricts closer and closer to a point volume. In EM that corresponds to saying that strictly locally there is just uniform fields E, B. Similarly in gravity case, locally Minkowski, and non-zero Weyl curvature = locally non-Euclidean = conformally non-flat, can coexist without contradiction depending on how both are used and interpreted.

The first example of physical manifestation of locally non-Euclidean I give in part 3, differential radial gap between nesting spherical surfaces, was first explained to me years ago by a GR pro in PhysicsForum., but only in terms of proper area/volume ratio. Quickly figured out for myself the simpler equivalent second example rather easily. So Anyway if ISM as its name implies truly is coordinate measured isotropic that means a locally spherical wavefront light pulse is also initially spherical as determined by a coordinate observer. Which via elementary geometric reasoning automatically implies conformal flatness = zero Weyl curvature actual.
But in SM case that same locally spherical wsavefront is determined to be oblate spheroidal (oblate axis directed radially from central gravitating mass source) by that same coordinate observer. The signature of conformal non-flatness = locally non-Euclidean geometry. Manifested locally by, as given in article, say measuring radial gap between closely spaced concentric great circles.

My argument then, as given numbers of times, is that ISM, which is a particular manifestation of your derived final result, is either truly equivalent to SM thus not actually isotropic thus not actually conformally flat, or it is and thus cannot represent the same physical metric as SM. Regardless that both can be strictly locally Minkowski.
Contradiction is not obvious until you can prove that 1 = 2. You have claimed that that some space or spacetime is or is not conformally flat depending on how you look at it but you have not proved anything (which you could do e.g. by computing the Cotton tensor).

What I write above and have elsewhere is imo a logically sound deduction. For strict formal mathematical proof well as you evidently have a much better grounding in formalities of GR, maybe you could do the honors and apply said Cotton tensor to both SM and ISM and let's see what then comes out! But as I say, evidently the determining factor is zero or not Weyl curvature: https://en.wikipedia.org/wiki/Conformally_flat_manifold (not clear to me one can ever ignore temporal component in shell or similar situation).
Radius is a little bit difficult. The centre of the Swartschild solution is singular. If you replace the interior part with a non-vacuum solution, you must integrate through heterogenous space. It is easier to use two spheres at constant radii so that their areas are A and A+dA so that both are in the vacuum region, and check the distance between these spheres.

Agree it is tricky - to the extent that one needs to integrate over a proper radial path if comparing appreciably radially separated spherical surfaces. And in ISM case that almost certainly requires numerical integration thus messy and somewhat ad hoc. That's something I realized soon after posting that bit and is why haven't got round to attempting it. But otherwise there is no call to evaluate anywhere other than exterior region as the problem is comparing SM and ISM which both apply to exterior vacuum region. I fall back on that the very form of ISM implies conformal flatness thus cannot claim to be physically equivalent to SM.

Moreover, as posted elsewhere:
If in a given coordinate system one encounters paradox, fleeing to another to escape it is senseless logic. It avoids that paradox existing in any supposedly sound coordinate system i.e. SM - the unique solution of EFE's for spherically symmetric static mass distribution, undermines such there and then. Period. And we are not talking about anything remotely like a so-called coordinate singularities situation such as 'black hole' 'event horizon' in SM. On the contrary the issues appear in the extreme weak gravity arena.

And further quoting from there:
What is my point there? Pretty obvious really. Demanding both external SM and internal MM are true inevitably forces a vanishing of functional dependence on gravitational potential for radial spatial metric factor $1/\sqrt{{-g}_{rr }}$ in interior. But not $\sqrt{{g}_{tt }}$ for clocks. Despite them being functionally identical in the SM exterior region. I say that is physical nonsense, an artifact that signifies the force fit, though non-drastic locally in terms of metric curvatures, is wrong in principle. There is more to a physically sensible metric transition than just local curvature concerns take care of. The shell wall cannot sensibly be acting as a magic membrane that vanishes one component's potential dependence on just one side. I say choose instead a conformally flat exterior metric and there is no such issue to consider. That alone is imo enough to choose the latter and reject SM.

But I go on in the next paragraph to summarize part 2 of article - the second and even stronger reason to reject SM. Direct summation over shell yields a predicted internal spatial metric, evaluated in particular for case of field point at r = 0 center, that IS a depressed function of Newtonian potential, in direct contradiction to above 'consistency requirement'. Any theory that predicts such paradox just has to be wrong somewhere. And once again, perform the same direct evaluation assuming conformally flat exterior metric, there is no paradox, no issue, everything naturally fits. Hence imo one has to reject SM thus GR as fatally flawed and choose the paradox free alternative - a genuinely conformally flat exterior metric. Of which Yilmaz gravity is a prime example.
Q-reeus

Posts: 272
Joined: Sat Jun 07, 2014 11:18 pm

### Re: Spherically symmetric space is conformally flat

Q-reeus wrote:And as acknowledged in earlier post, your derivation is technically correct.

I.e., any spherically symmetric space is conformally flat (except possibly at the centre).
But imo something subtle has to be amiss. i gave my hunch then - strictly local equivalence only, for some given r.
What you mean by "locally"? As usually understood, every spacetime is locally Minkowskian except at singularities and therefore spatial slices (whether isochronous or not) are locally Euclidean. If you allow "locally" to include less local measures then you will soon have locally non-Euclidean (or non-Minkowskian) geometries in every curved space (or spacetime).

As you know I have not pretended to be conversant with the mathematical structure of GR.

That's OK here, the intended topic of this thread does not include GR.
And simply offer in article a layman outsider's imo fresh insight not some high powered mathematically dense tensor based proof. But note from the literature that true conformal flatness (4D) is equivalent to having zero Weyl curvature.

In three dimensions conformal flatness is equivalent to having zero Cotton tensor.
And that Weyl curvature is an intensive property existing 'at a point' thus having local existence.

So is the Cotton tensor.
Analogously to say non-zero divergence or curl in EM, though existing 'at a point', the measurable effects vanish as one restricts closer and closer to a point volume. In EM that corresponds to saying that strictly locally there is just uniform fields E, B. Similarly in gravity case, locally Minkowski, and non-zero Weyl curvature = locally non-Euclidean = conformally non-flat, can coexist without contradiction depending on how both are used and interpreted.

Indeed. It all depends on how one interpreters "local". As a first approximation the space is Euclidean (or spacetime Minkowskian) and Cotton (or Weyl) tensor cannot be measured. As a second approximation they and other curvature tensors can be measured, and flatness and conformal flatnes become meaningful questions.
The first example of physical manifestation of locally non-Euclidean I give in part 3, differential radial gap between nesting spherical surfaces, was first explained to me years ago by a GR pro in PhysicsForum., but only in terms of proper area/volume ratio. Quickly figured out for myself the simpler equivalent second example rather easily. So Anyway if ISM as its name implies truly is coordinate measured isotropic that means a locally spherical wavefront light pulse is also initially spherical as determined by a coordinate observer.

A sufficiently small region is always flat so a wavefront (from a point source) always starts as spherical. When curvature becomes significant one must be careful with words like "spherical" as their meaning is not always obvious and also because their implications are not always the same as in a flat spacetime.
Which via elementary geometric reasoning automatically implies conformal flatness = zero Weyl curvature actual.

That is true with four (or more) dimensions but not with three (or less). Note that in Swatzschild's metric the Weyl tensor is non-zero and the spacetime is not conformally flat.
But in SM case that same locally spherical wsavefront is determined to be oblate spheroidal (oblate axis directed radially from central gravitating mass source) by that same coordinate observer.

It starts as spherical in the sense that its physical size is the same in all dimensions. There are no coordinate observers as coordinates are unphysical and therefore not physically observable. Of course, if the coordinates are attached to physical phenomena, those phenomena can be observable but they are equally observable when coordinates are defined differently.
The signature of conformal non-flatness = locally non-Euclidean geometry.

Not true. Whether the geometry is locally Euclidean or not depends on how locally you examine it (unless the space really is Euclidean). For example, the surface of a sphere is conformally flat but non-Euclidean (unless oserved too locally).
Manifested locally by, as given in article, say measuring radial gap between closely spaced concentric great circles.

My argument then, as given numbers of times, is that ISM, which is a particular manifestation of your derived final result, is either truly equivalent to SM thus not actually isotropic thus not actually conformally flat, or it is and thus cannot represent the same physical metric as SM.

As long as you need to say "or" you don't understand the situation sufficiently well. Anyway, it is easy to prove that they are equivalent.
Regardless that both can be strictly locally Minkowski.
Contradiction is not obvious until you can prove that 1 = 2. You have claimed that that some space or spacetime is or is not conformally flat depending on how you look at it but you have not proved anything (which you could do e.g. by computing the Cotton tensor).

What I write above and have elsewhere is imo a logically sound deduction.

A deduction should be a sequence of steps, not leaps. Otherwise its soundness is not obvious. Your deductions tend to have too long leaps near the end.
For strict formal mathematical proof well as you evidently have a much better grounding in formalities of GR, maybe you could do the honors and apply said Cotton tensor to both SM and ISM and let's see what then comes out! But as I say, evidently the determining factor is zero or not Weyl curvature: https://en.wikipedia.org/wiki/Conformally_flat_manifold (not clear to me one can ever ignore temporal component in shell or similar situation).
Radius is a little bit difficult. The centre of the Swartschild solution is singular. If you replace the interior part with a non-vacuum solution, you must integrate through heterogenous space. It is easier to use two spheres at constant radii so that their areas are A and A+dA so that both are in the vacuum region, and check the distance between these spheres.

Agree it is tricky - to the extent that one needs to integrate over a proper radial path if comparing appreciably radially separated spherical surfaces.

If spheres are too tricky, someting else might be more convincing. Sum of angles of a triangle is another demonstration of curvature. In order to prove that two presentations do not present the same space (or spacetime) it is sufficient to calculate some observable in both and get a different value. The hardest part is to prove that the calculated thing is the same in both cases, which may involve claculation of related observables and getting the same value. Good observables include angles, distances, curvature tensors, and for spacetime durations and speeds.
And in ISM case that almost certainly requires numerical integration thus messy and somewhat ad hoc. That's something I realized soon after posting that bit and is why haven't got round to attempting it. But otherwise there is no call to evaluate anywhere other than exterior region as the problem is comparing SM and ISM which both apply to exterior vacuum region. I fall back on that the very form of ISM implies conformal flatness thus cannot claim to be physically equivalent to SM.

Coordinates are unphysical so the choice of coordinates cannot have physical consequences.
Moreover, as posted elsewhere:
If in a given coordinate system one encounters paradox, fleeing to another to escape it is senseless logic.

Not at all. In order to determine whether the coordinate system is paradoxical it is useful to present the situation in another coordinate system. If the paradox disappears, one may suspect the original coordinate system, otherwise it may be better to search elsewhere.
It avoids that paradox existing in any supposedly sound coordinate system i.e. SM - the unique solution of EFE's for spherically symmetric static mass distribution, undermines such there and then. Period. And we are not talking about anything remotely like a so-called coordinate singularities situation such as 'black hole' 'event horizon' in SM. On the contrary the issues appear in the extreme weak gravity arena.

The rest is so far from the intended topic of this discussion that I don't want to comment.
And further quoting from there:
What is my point there? Pretty obvious really. Demanding both external SM and internal MM are true inevitably forces a vanishing of functional dependence on gravitational potential for radial spatial metric factor $1/\sqrt{{-g}_{rr }}$ in interior. But not $\sqrt{{g}_{tt }}$ for clocks. Despite them being functionally identical in the SM exterior region. I say that is physical nonsense, an artifact that signifies the force fit, though non-drastic locally in terms of metric curvatures, is wrong in principle. There is more to a physically sensible metric transition than just local curvature concerns take care of. The shell wall cannot sensibly be acting as a magic membrane that vanishes one component's potential dependence on just one side. I say choose instead a conformally flat exterior metric and there is no such issue to consider. That alone is imo enough to choose the latter and reject SM.

But I go on in the next paragraph to summarize part 2 of article - the second and even stronger reason to reject SM. Direct summation over shell yields a predicted internal spatial metric, evaluated in particular for case of field point at r = 0 center, that IS a depressed function of Newtonian potential, in direct contradiction to above 'consistency requirement'. Any theory that predicts such paradox just has to be wrong somewhere. And once again, perform the same direct evaluation assuming conformally flat exterior metric, there is no paradox, no issue, everything naturally fits. Hence imo one has to reject SM thus GR as fatally flawed and choose the paradox free alternative - a genuinely conformally flat exterior metric. Of which Yilmaz gravity is a prime example.
Mikko

Posts: 138
Joined: Mon Feb 17, 2014 1:53 am

### Re: Spherically symmetric space is conformally flat

Mikko wrote:I.e., any spherically symmetric space is conformally flat (except possibly at the centre).

Not true - as seemingly acknowledged by yourself below!:
"...Note that in Swatzschild's metric the Weyl tensor is non-zero and the spacetime is not conformally flat."

To resolve such apparent obvious contradiction I could guess you are meaning to imply above that non-zero Weyl curvature in SM is somehow an 'artifact' of SM which can be legitimately erased by resort to ISM. No other interpretation quite makes any sense to me. Assuming that guess is correct, it imo is equivalent to saying a non-zero curl E computed in say Cartesian coordinate system is just a non-physical artifact that can be banished by use of some other coordinate system. In which case we fundamentally disagree on some basics here.
What you mean by "locally"? As usually understood, every spacetime is locally Minkowskian except at singularities and therefore spatial slices (whether isochronous or not) are locally Euclidean. If you allow "locally" to include less local measures then you will soon have locally non-Euclidean (or non-Minkowskian) geometries in every curved space (or spacetime).

Well you quote me later on as explaining just what was meant by locally.
Q-reeus: "And that Weyl curvature is an intensive property existing 'at a point' thus having local existence."
Mikko: "Indeed. It all depends on how one interpreters "local". As a first approximation the space is Euclidean (or spacetime Minkowskian) and Cotton (or Weyl) tensor cannot be measured. As a second approximation they and other curvature tensors can be measured, and flatness and conformal flatnes become meaningful questions."

Yes and no and maybe with you it seems - I honestly cannot discern a consistent position.
Q-reeus: "Which via elementary geometric reasoning automatically implies conformal flatness = zero Weyl curvature actual."
Mikko: "That is true with four (or more) dimensions but not with three (or less). Note that in Swatzschild's metric the Weyl tensor is non-zero and the spacetime is not conformally flat."

Have already commented on last bit, and as for 3D vs 4D, well SM or ISM line element has four metric components - one temporal and three spatial.
Q-reeus: "But in SM case that same locally spherical wsavefront is determined to be oblate spheroidal (oblate axis directed radially from central gravitating mass source) by that same coordinate observer."
Mikko: "It starts as spherical in the sense that its physical size is the same in all dimensions. There are no coordinate observers as coordinates are unphysical and therefore not physically observable. Of course, if the coordinates are attached to physical phenomena, those phenomena can be observable but they are equally observable when coordinates are defined differently."

Of course there are coordinate observers - that concept is used all the time in mainstream GR articles. And equally such cannot by definition be locally measuring distant phenomena. That's what SM is all about - telling one what a coordinate observer must infer as to how spherically symmetric static gravity distorts spacetime from Euclidean.
Given transverse metric components are 'Euclidean' in SM, coordinate oblateness of initial wavefront does automatically imply non-Euclidean local geometry - non-zero Weyl curvature for SM. Conversely coordinate predicted initially spherical wavefront in ISM automatically implies the opposite - zero Weyl curvature. Which prediction is imo an actual artifact of SM ->ISM transformation. And this is becoming repetitious.
Not true. Whether the geometry is locally Euclidean or not depends on how locally you examine it (unless the space really is Euclidean). For example, the surface of a sphere is conformally flat but non-Euclidean (unless oserved too locally).

We have already gone over that issue via examples and analogies (and again above), hence this is going in circles.
As long as you need to say "or" you don't understand the situation sufficiently well. Anyway, it is easy to prove that they are equivalent.

I disagree. That 'or' means either case renders ISM useless as argument against paradox appearing in SM. It doesn't matter which position one takes. And given you claim as per title that "any spherically symmetric space is conformally flat", yet simultaneously acknowledge above "...Note that in Swatzschild's metric the Weyl tensor is non-zero and the spacetime is not conformally flat.", I am grappling with a cloud so to speak.
If spheres are too tricky, someting else might be more convincing. Sum of angles of a triangle is another demonstration of curvature. In order to prove that two presentations do not present the same space (or spacetime) it is sufficient to calculate some observable in both and get a different value. The hardest part is to prove that the calculated thing is the same in both cases, which may involve claculation of related observables and getting the same value. Good observables include angles, distances, curvature tensors, and for spacetime durations and speeds.

Will look at getting specific results for radial parameter-to-circumference ratios. Whether or not such confirms or undermines actual equivalence of SM and ISM, that will not and cannot overturn that paradox is present using SM "the unique solution of EFE's for spherically symmetric static mass distribution." - as declared for instance in the two lines below (1.4), top of p5 here:
http://www.diva-portal.org/smash/get/di ... TEXT01.pdf
And many other such mainstream GR articles say the same. Hence if paradox exists in SM setting, which for sure is so, SM and GR go down together. And repeated below since evidently continued repetition is needed here.
Coordinates are unphysical so the choice of coordinates cannot have physical consequences.

And as stated before I say the logical resolution is that SM <-> ISM transformation is a fairly meaningless and misleading formalism that simply allows a nominal 1:1 correspondence for and strictly at any chosen parameter r. Otherwise you have this conundrum of there being a physically real Weyl curvature and not depending on use of SM or ISM. Which is nonsense. Either Weyl or if you like Cotton curvature as proper intensive property is there or not. Just like curl or divergence in EM. And is ultimately irrelevant to thrust of my article even if not to the narrow focus of this thread.
Not at all. In order to determine whether the coordinate system is paradoxical it is useful to present the situation in another coordinate system. If the paradox disappears, one may suspect the original coordinate system, otherwise it may be better to search elsewhere.

Only if 'the chart fails to cover the manifold' to use GR parlance, in some severe 'coordinate singularity' sense as for instance standard external SM fails to handle below the imo fictitious dreaded black hole event horizon. Otherwise there is no legitimacy at all in that argument. And has nothing to do with that necessarily external and internal shell metrics are different. That applying to *any* choice of external metric be it SM, ISM, Yilmaz conformal gravity alternative, or whatever. To repeat:
It avoids that paradox existing in any supposedly sound coordinate system i.e. SM - the unique solution of EFE's for spherically symmetric static mass distribution, undermines such there and then. Period. And we are not talking about anything remotely like a so-called coordinate singularities situation such as 'black hole' 'event horizon' in SM. On the contrary the issues appear in the extreme weak gravity arena.

The rest is so far from the intended topic of this discussion that I don't want to comment.

Your choice, but imo it bears heavily on legitimacy of your position as per title.
Q-reeus

Posts: 272
Joined: Sat Jun 07, 2014 11:18 pm

### Re: Spherically symmetric space is conformally flat

Too late to edit my last post, so here is roughly how i would have edited it:
Mikko wrote:I.e., any spherically symmetric space is conformally flat (except possibly at the centre).

Not true - as seemingly acknowledged by yourself below!:

"...Note that in Swatzschild's metric the Weyl tensor is non-zero and the spacetime is not conformally flat."

To resolve such apparent obvious contradiction I could guess you are meaning to imply above that non-zero Weyl curvature in SM is somehow an 'artifact' of SM which can be
legitimately erased by resort to ISM. No other interpretation quite makes any sense to me. Assuming that guess is correct, it imo is equivalent to saying a non-zero curl E
computed in say Cartesian coordinate system is just a non-physical artifact that can be banished by use of some other coordinate system. In which case we fundamentally
disagree on some basics here.

[color highlighting added here] A partial retraction is in order. I had failed to notice the distinction made there between conformal flatness (or not) in a space vs a spacetime. So ok if restricted to 3D space curvature, Weyl curvature is by definition zero, but not necessarily so for that associated with Cotton tensor. In either case, appropriately compare Weyl curvatures for 4D case or Cotton curvatures for 3D case. In both cases SM will yield non-zero result, and ISM may or may not. Still looking around for an official source presenting a simple formula for calculating Weyl curvature in both SM and ISM.

Whatever is found, it will not alter that, working consistently with SM, the paradoxical findings in parts 1 and 2 of article are there. And as per part 3, 'vanishing' the paradox via use of ISM can in no way be considered a resolution. It just shows that SM and ISM cannot be truly equivalent. If they were, paradox would be present or not in both cases. Logical necessity.

And I had made a slip-up in part 3 where expression for Yilmaz metric incorrectly identifies the $\phi}$ exponent there with Newtonian potential but in fact the relation is
$\phi=2{\phi}_{n}/{c}^{2}$, where ${\phi}_{n}$ is the Newtonian potential. Will correct in an upcoming 4th version.
Q-reeus

Posts: 272
Joined: Sat Jun 07, 2014 11:18 pm

### Re: Spherically symmetric space is conformally flat

Q-reeus wrote:So ok if restricted to 3D space curvature, Weyl curvature is by definition zero, but not necessarily so for that associated with Cotton tensor. In either case, appropriately compare Weyl curvatures for 4D case or Cotton curvatures for 3D case. In both cases SM will yield non-zero result, and ISM may or may not. Still looking around for an official source presenting a simple formula for calculating Weyl curvature in both SM and ISM.

Weyl curvature is not a property of the coordinate system but of the space, so calculation from either coordinate system gives the same result. For Swarzschild's solution that result is non-zero. There are non-vacuum solutions with zero Weyl tensor. The 3D spatial geometry is spherically symmetric and therefore the Cotton tensor is zero. Cotton tensor is not necessary for a proof of conformal flatness, and I did my proof without it; but a proof of conformal non-flatness seems to require it.
Will correct in an upcoming 4th version.

You should also correct the errors about conformal flatness. They are not essential, just distracting.
Mikko

Posts: 138
Joined: Mon Feb 17, 2014 1:53 am

### Re: Spherically symmetric space is conformally flat

Mikko wrote:The 3D spatial geometry is spherically symmetric and therefore the Cotton tensor is zero.

Can you quote the literature supporting that claim?
Q-reeus

Posts: 272
Joined: Sat Jun 07, 2014 11:18 pm

### Re: Spherically symmetric space is conformally flat

Q-reeus wrote:
Mikko wrote:The 3D spatial geometry is spherically symmetric and therefore the Cotton tensor is zero.

Can you quote the literature supporting that claim?

You already found some web pages showing that conformally flat 3D space implies zero cotton page, and in the first message I have shown that spherical symmetry implies conformal flatness. There are many pages about Cotton tensor and conformal flatness but not so many about conformal flatness of spherically symmetric space.
Mikko

Posts: 138
Joined: Mon Feb 17, 2014 1:53 am

### Re: Spherically symmetric space is conformally flat

Mikko wrote:You already found some web pages showing that conformally flat 3D space implies zero cotton page,...

If you are referring to the Wiki article https://en.wikipedia.org/wiki/Conformally_flat_manifold (post #4), yes it specifies that. What it doesn't say there is what you think it implies - that *any* spherically symmetric space is conformally flat.
...and in the first message I have shown that spherical symmetry implies conformal flatness.

That is obviously how you interpret it, but I stick by earlier responses in posts #2, #4, #6, #7 (grrrr... can we not have post numbering incorporated here!?)
Look, you are claiming SM has 3D conformal flatness = zero Cotton tensor. We agree I hope that zero Cotton tensor for 3D case is synonymous with locally Euclidean. Well here's your challenge. In part 3 of article, I gave two specific examples of spatially non-Euclidean character of SM. See if you can refute my more easily determined 2nd example - that the proper measure of differential radial gap between closely spaced concentric great circles (centered about a central point mass) is greater by factor $\1/\sqrt{{g}_{tt }}$ than for matter-free Euclidean case. It has to be so.
There are many pages about Cotton tensor and conformal flatness but not so many about conformal flatness of spherically symmetric space.

And I will guarantee you will never find a page in regular literature equating conformal flatness to *any* spherically symmetric space.
Q-reeus

Posts: 272
Joined: Sat Jun 07, 2014 11:18 pm

### Re: Spherically symmetric space is conformally flat

Just to be clear that where I twice wrote "*any* spherically symmetric space" in above post, the meaning was in the general sense i.e. *every*. Apart from the one unambiguous case of spherically symmetric space with zero conformal flatness - empty i.e. matter-free, Yilmaz gravity equivalent to SM is as per article locally Euclidean thus conformally flat.
Q-reeus

Posts: 272
Joined: Sat Jun 07, 2014 11:18 pm

### Re: Spherically symmetric space is conformally flat

Q-reeus wrote:
Mikko wrote:There are many pages about Cotton tensor and conformal flatness but not so many about conformal flatness of spherically symmetric space.

And I will guarantee you will never find a page in regular literature equating conformal flatness to *any* spherically symmetric space.

Some pages mention the result without proof or reference.
Mikko

Posts: 138
Joined: Mon Feb 17, 2014 1:53 am

### Re: Spherically symmetric space is conformally flat

Mikko wrote:
Q-reeus wrote:
Mikko wrote:There are many pages about Cotton tensor and conformal flatness but not so many about conformal flatness of spherically symmetric space.

And I will guarantee you will never find a page in regular literature equating conformal flatness to *any* spherically symmetric space.

Some pages mention the result without proof or reference.

Umm - references please, preferably to freely accessible online material. Or a direct cut-and-paste quote, in full context, from mainstream textbook. I extremely doubt the eventuality for reasons already given - SM is for sure not only 4D conformally non-flat = non-zero Weyl curvature, but spatial part also conformally non-flat - non-zero Cotton curvature. I gave you the physical example demonstrating that to be so.

It's like this: On an ideal spherical non-rotating planet, a being climbs a radial pointing step-ladder X proper meters, where X << R the planet's surface radius (strictly speaking, in GR, R is 'just a parameter' defined by C = 2*pi*R). At that proper elevation X there exists a railing of constant nominal radius R+X - i.e. a great circle. What is the circumference C' there, as determined by the being measuring proper distance along the railing? If you get $C'= 2\pi(R+X)$ as first-order in metric approximation, start again. SM is locally spatially non-Euclidean. Correct answer is
$C'= 2\pi(R+\sqrt{{g}_{tt}}X)$

And just as not one of the assortment of fools and knaves (not all so categorized) I endured in that linked ATM thread ever took up my challenge to refute my two main findings - disappearing functional dependence on gravitational potential for just one metric component, or conflicting predictions for internal spatial metric, so it it has been here (not that I'm implying folks here are all fools and/or knaves! ).
Q-reeus

Posts: 272
Joined: Sat Jun 07, 2014 11:18 pm

### Re: Spherically symmetric space is conformally flat

Q-reeus wrote:
Mikko wrote:
Q-reeus wrote:And I will guarantee you will never find a page in regular literature equating conformal flatness to *any* spherically symmetric space.

Some pages mention the result without proof or reference.

Umm - references please, preferably to freely accessible online material.

Some typical examples are:
http://arxiv.org/abs/1405.1682
http://www.phy.olemiss.edu/~luca/Topics ... om_3D.html
http://homepage.univie.ac.at/piotr.chru ... Energy.pdf (page 5)
All these mention it as if it were in common textbooks but I don't know any such textbook.
Mikko

Posts: 138
Joined: Mon Feb 17, 2014 1:53 am

### Re: Spherically symmetric space is conformally flat

Mikko wrote:Some typical examples are:
http://arxiv.org/abs/1405.1682
http://www.phy.olemiss.edu/~luca/Topics ... om_3D.html
http://homepage.univie.ac.at/piotr.chru ... Energy.pdf (page 5)
All these mention it as if it were in common textbooks but I don't know any such textbook.

Will have to concede these GR experts are indeed agreeing with your thread title. The last one, whether or not you used it directly as basis for post #1, seems to employ the same reasoning - it's always possible to transform a spherically symmetric metric into an isotropic form. Hmm....

This begs the question - what does 3D conformal flatness, in particular relating to exterior SM or alternative, actually physically mean? My understanding is that it implies a differential excursion from any given radial value yields the first proper Euclidean relation for C' I gave last post. But the actual relation is given by my second expression. A manifestly locally non-Euclidean first order in metric relation. What relation for C' do you find in the scenario I gave? How do you explain it?
Q-reeus

Posts: 272
Joined: Sat Jun 07, 2014 11:18 pm

### Re: Spherically symmetric space is conformally flat

Q-reeus wrote:This begs the question - what does 3D conformal flatness, in particular relating to exterior SM or alternative, actually physically mean?

Probably not very much. There are no exactly spherical non-rotating bodies big enough to have significant gravity. In less symmetric situation it is not clear how one should choose the time coordinate. The spactime curvature does not depend on choice of coordinates so it can be considered more important. Spacetime is conformally flat only in some special situations.
Mikko

Posts: 138
Joined: Mon Feb 17, 2014 1:53 am

### Re: Spherically symmetric space is conformally flat

Mikko wrote:
Q-reeus wrote:This begs the question - what does 3D conformal flatness, in particular relating to exterior SM or alternative, actually physically mean?

Probably not very much. There are no exactly spherical non-rotating bodies big enough to have significant gravity. In less symmetric situation it is not clear how one should choose the time coordinate. The spactime curvature does not depend on choice of coordinates so it can be considered more important. Spacetime is conformally flat only in some special situations.

Which all deftly avoids my last point - SM deviates from locally 3D flatness by a first order in metric term. Asking you again - what value do you obtain for C' as per great circle railing scenario? It should be what I gave in second expression. How do you reconcile it with spatial conformal flatness?
Q-reeus

Posts: 272
Joined: Sat Jun 07, 2014 11:18 pm


text:
# tactical vim

nerdy bits & pieces

## windows xp OTF font install registry fix for nvidia

I can’t take credit for this but wish to spread the information as it took awhile to discover this gem; Phinney on Fonts has a writeup about fixing various font install problems, particularly with OpenType (OTF) fonts in Windows XP. The #4 fix on his list is pure gold.

The problem: double-clicking on an OTF file to display it, or attempting to install the font, in Windows XP results in an error dialog similar to “The requested resource c:\some\path\to\font.otf is not a valid font file.

The diagnosis: “Windows expects the device driver to be a certain size (maximum). In this case, the Nvidia driver is slightly larger than expected. A simply registry entry will resolve the issue in WinXP.”

The fix: save the below as as xpfontfix.reg, double-click on it to merge it into your registry and then reboot the machine.

Windows Registry Editor Version 5.00

[HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Control\Session Manager\Memory Management]
"SessionImageSize"=dword:00000020

So far this has fixed a number of machines having the problem, giving it a two thumbs up over here.

format: plain


text:
### Is Matter around Us Pure - Solutions

CBSE class IX Science

NCERT Solutions
Chapter 2
Is Matter around Us Pure

(Page No. 15)

1. What is meant by a Pure substance?

Ans: Such substance that has a uniform composition i.e. has particles with identical properties is called pure substance. eg. sugar, salt, water, nitrogen etc.

2. List the points of differences between homogeneous and heterogeneous mixtures.

Ans.

(Page No.18)

1. Differentiate between homogeneous and heterogeneous mixtures with examples.

Ans.

2. How are sol, solution and suspension different from each other?

Ans.

3. To make a saturated solution, 36 g of sodium chloride is dissolved in 100 g of water at 293 K. Find its concentration at this temperature.

Ans.

Mass of solute (sodium chloride) =36g
Mass of solvent (water) =100g
Mass of solution = Mass of Solute + Mass of solvent
= 36g +100g = 136g

(Page No. 24)

1. How will you separate a mixture containing kerosene and petrol (difference in their boiling points is more than 25${}^{\circ }$C), which are miscible with each other?

Ans. We can separate a mixture containing kerosene and petrol by distillation technique since difference in their boiling points is more than 25${}^{\circ }$C. Therefore, they can be easily separated by the technique of simple distillation.

2. Name the technique to separate

(i) butter from curd,

(ii) salt from sea-water,

(iii) camphor from salt.

Ans. (i) centrifugation method.

(ii) evaporation method or crystallisation method

(iii) sublimation method.

3. What type of mixtures are separated by the technique of crystallisation?

Ans. From impure samples of solids, pure solid crystals can be obtained by the method of crystallization for eg to obtain pure sugar from impure sample of the same.

(Page No. 24)

1. Classify the following as chemical or physical changes:

• cutting of trees,

• melting of butter in a pan,

• rusting of almirah,

• boiling of water to form steam,

• passing of electric current through water and the water breaking down into hydrogen and oxygen gases,

• dissolving common salt in water,

• making a fruit salad with raw fruits, and

• burning of paper and wood.

Ans. Cutting of trees = chemical change

Melting of butter in a pan = physical change

Rusting of almirah = chemical change

Boiling of water to form steam = physical change

Passing of electric current through water and the water breaking down into hydrogen and oxygen gases = chemical change

Dissolving common salt in water = physical change

Making a fruit salad with raw fruits = physical change

Burning of paper and wood = chemical change

2. Try segregating the things around you as pure substances or mixtures.

Ans. Distilled water, diamond, graphite, raw rubber are pure substances that can be found around us. In contrast, curd, ice cream, cooking oil , vulcanized rubber are the some of the examples of mixtures.

(Chapter – end)

1. Which separation techniques will you apply for the separation of the following?

(a) Sodium chloride from its solution in water.

(b) Ammonium chloride from a mixture containing sodium chloride and ammonium chloride.

(c) Small pieces of metal in the engine oil of a car.

(d) Different pigments from an extract of flower petals.

(e) Butter from curd.

(f) Oil from water.

(g) Tea leaves from tea.

(h) Iron pins from sand.

(i) Wheat grains from husk.

(j) Fine mud particles suspended in water.

Ans. (a) Evaporation

(b) Sublimation

(c) Filtration

(d) Chromatography

(e) Centrifugation

(f) Separating funnel

(g) Filtration

(h) with the help of a magnet

(i) Blowing air or sieving

(j) using alum

2. Write the steps you would use for making tea. Use the words solution, solvent, solute, dissolve, soluble, insoluble, filtrate and residue.

Ans. Take some amount of solvent (water) in a pan and after heating it add little amount of solute (sugar) to the solvent. Solute will dissolve completely in the solvent forming true solution, then add tea leaves that are insoluble along with another soluble liquid milk. After boiling allow filtration with a sieve so the filtrate you obtain is tea while the residue has tea leaves that are thrown away.

3. Pragya tested the solubility of three different substances at different temperatures and collected the data as given below (results are given in the following table, as grams of substance dissolved in 100 grams of water to form a saturated solution).

(a) What mass of potassium nitrate would be needed to produce a saturated solution of potassium nitrate in 50 grams of water at 313 K?

(b) Pragya makes a saturated solution of potassium chloride in water at 353 K and leaves the solution to cool at room temperature. What would she observe as the solution cools? Explain.

(c) Find the solubility of each salt at 293 K. Which salt has the highest solubility at this temperature?

(d) What is the effect of change of temperature on the solubility of a salt?

Ans. (a) At 313 K temperature the amount of potassium nitrate required was 62g in 100ml of water so in 50g water we will need to dissolve = 62 X 50/100= 31g potassium nitrate.

(b) When a saturated solution of potassium chloride at 353 K is cooled, the solubility of potassium chloride in water decreases. As a result,the amount of potassium chloride which exceeds its solubility at lower temperature separates out as crystals.

(c) Solubilities are (in 100g of water) 32,36,35,37 respectively for the mentioned salts and the highest solubility is of ammonium chloride at this temperature.

(d) Solubility of salts is directly proportional to the temperature i.e. if temperature increases then solubility will increase and if the temperature decreases solubility will also decrease.


text:
# nLab quantum linear group

Quantum linear semigroups are bialgebras which are deformations of bialgebras of coordinate functions on the groups of $n\times n$-matrices for some $n$. They belong to the class of matrix bialgebras. The usual notation for the one-parametric version is $\mathcal{O}(M_q(n))$ or sometimes simply $\mathcal{M}_q(n)$.

Suppose we are given $n\times n$ matrices $P = (p_{ij})$ and $Q = (q_{ij})$ with invertible entries in the ground field $F$, for which there exist $q$ such that

$p_{ij} q_{ij} = q^2, q_{ij} = q^{-1}_{ji},\,\,\,\,\,i \lt j,\,\,and \,\,\,\,\,\, q_{ii} = p_{ii},\,\,\,\,\,\,\, for\,\,\, all\,\,\,\, i.$

The multiparametric quantized matrix bialgebra (synonym: multiparametric quantum linear semigroup) $\mathcal{O}(M_{P,Q}(F,n)):= F \langle T^i_j , i,j = 1,\ldots, n\rangle/I$, where $I$ is the ideal spanned by the relations

(1)$\array{ T^k_i T^k_j = q_{ij} T^k_j T^k_i, & i \lt j \\ T^k_i T^l_i = p_{kl} T^l_i T^k_i, & k \lt l \\ q_{ij} T^k_j T^l_i = p_{kl} T^l_i T^k_j, & i\lt j,\,\,\,\,k\lt l \\ T^k_i T^l_j - q_{ij} q^{-1}_{kl} T^l_j T^k_i = (q_{ij}-p_{ij}^{-1}) T^k_j T^l_i,& i\lt j,\,\,\,\,k\lt l }$

$\mathcal{M} = \mathcal{O}(M_{P,Q}(F,n))$ is a bialgebra with respect to the “matrix” comultiplication which is the unique algebra homomorphism $\Delta : \mathcal{M} \to\mathcal{M} \otimes\mathcal{M}$ extending the formulas which are written in the matrix form as $\Delta T^i_j = \sum T^i_k \otimes T^k_j$ with counit $\epsilon T^i_j = \delta^i_j$ (Kronecker delta). This means that it is a matrix bialgebra with basis $\{T^i_j\}_{i,j=1,\ldots,n}$, in fact a free matrix bialgebra over $F$.

In these conventions, the 1-parametric version $\mathcal{O}(M_q(F))$ is obtained as a special case when $P = Q$ and $q_{ij} = q$ for $i \lt j$ and $q_{ij} = q^{-1}$ for $i \gt j$.

Quantum linear groups are Hopf algebras which are quantum deformations of Hopf algebras of coordinate functions on the general linear group or special linear group. There exist one parametric and many parametric versions as well as super analogues. They belong to the class of matrix Hopf algebras.

The usual notation for one-parametric versions is $\mathcal{O}(GL_q(n))$, $\mathcal{O}(SL_q(n))$ and variants thereof.

## Bibliography

• related $n$Lab pages: quantum group, quantum Gauss decomposition, quantized function algebra, general linear group
• Yu. I. Manin, Quantum groups and non-commutative geometry, CRM, Montreal 1988.
• Yu. I. Manin, Multiparametric quantum deformation of the general linear supergroup, Comm. Math. Phys. 123 (1989) 163–175.
• B. Parshall, J.Wang, Quantum linear groups, Mem. Amer. Math. Soc. 89(1991), No. 439, vi+157 pp.
• E. E. Demidov, Multiparameter quantum deformations of the group $GL(n)$, (Russian) Uspehi Mat. Nauk 46 (1991), no. 4 (280) 147–148; translation in Russian Math. Surveys 46 (1991) no. 4, 169–171.
• M. Hashimoto, T. Hayashi, Quantum multilinear algebra, Tohoku Math. J. 44 (1992) 471-521.
• M. Artin, W. Schelter, J. Tate, Quantum deformations of $GL_n$, Commun. Pure Appl. Math. XLIV, 879–895 (1991)
• Zoran Škoda, Localizations for construction of quantum coset spaces, math.QA/0301090, in “Noncommutative geometry and Quantum groups”, W.Pusz, P.M. Hajac, eds. Banach Center Publications vol.61, pp. 265–298, Warszawa 2003.
• Z. Škoda, Every quantum minor generates an Ore set, International Math. Res. Notices 2008, rnn063-8; math.QA/0604610
• Siân Fryer, From restricted permutations to Grassmann necklaces and back again, arxiv/1511.06664

Last revised on February 10, 2016 at 09:20:01. See the history of this page for a list of all contributions to it.


text:
# Publications and preprints

#### Publications

• On the homology of the commutator subgroup of the pure braid group. Proceedings of the American Mathemathical Society 149:6 (2021) 2387-2401. arXiv
• Splitting of the homology of the punctured mapping class group. Journal of Topology 13:3 (2020) 1230-1260. arXiv
• Topological complexity of unordered configuration spaces of surfaces, with David Recio-Mitter. Algebraic & Geometric Topology 19 (2019) 1359–1384. arXiv

#### Preprints

• Deloopings of Hurwitz spaces, 2021. arXiv
• A coordinate-free definition of Hurwitz spaces, 2021.
arXiv
• Partially multiplicative quandles and simplicial Hurwitz spaces, 2021.
arXiv
• Parametrised moduli spaces of surfaces as infinite loop spaces, with Florian Kranhold and Jens Reinhold, 2021. arXiv
• Mapping class group actions on configuration spaces and the Johnson filtration, with Jeremy Miller and Jennifer Wilson, 2021. arXiv
• Vertical configuration spaces and their homology, with Florian Kranhold, 2021. arXiv
• An upper bound on the topological complexity of discriminantal varieties, 2020. To appear in Homology, Homotopy and Applications. arXiv
• Braid groups, mapping class groups and their homology with twisted coefficients, 2019. To appear in Mathemathical Proceedings of the Cambridge Philosophical Society. arXiv


text:
# Understanding Edge Effects in Empirical Mode Decomposition

|   Source   |   Minimap

It has been a little over three years since I started working on a Python implementation of the Hilbert Huang Transform. When I first presented it at SciPy India 2011 (video) it was just a collection of small scripts, without packaging, testing or even docstrings. Over time PyHHT has garnered some interest, and I have, since the last few weeks, found the time to regularly work on it. I have been able to come up with a decent implementation of empirical mode decomposition, pretty much in line with everything described in the original paper by Huang et al. The other parts of HHT, like the Hilbert spectrum and its time frequency representations are right around the corner.

This blogpost, and many more that will follow, will hopefully help me track the progress of PyHHT. Who knows, doing this might even help me with the documentaiton later on. Currently PyHHT is under heavy development, and I cannot comment on either the stability or the correctness of the entire Python module. However, early on when developing more sohpisticated modules, I realized that it would have to be very modular, so as to enable better and more fine-tuned testing. The library is organized primarily into three modules:

1. pyhht.emd - The main classes and functions that perform EMD.
2. pyhht.utils - miscellaneous utility functions that help the EMD algorithms.
3. pyhht.visualization - pretty plotting for IMFs (and for the Hilbert spectrum, coming soon.)

Out of this, the EMD algorithm itself is fairly straightforward. Although it does have a lot of heuristics that make its results more "reasonable". As an example of these heuristics, this post discusses the effects that boundary conditions of the EMD algorithm might have on the resulting intrinsic mode functions.

Let us constuct a simple signal with two pure sinusoids with consummate frequencies:

$$s(t) = \sin (2\pi f_{1}t) + \sin (2\pi f_{2}t)$$

where $f_{1} = 10 Hz$ and $f_{2} = 5 Hz$

To highlight the idea of how EMD processes the residue (or the trend) in a signal, let's add a linear ramp to the signal. (For the sake of simplicity, let's just add the timestamps to the signal.)

In [4]:
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
%matplotlib inline

Click me to hide the output
In [5]:
mpl.style.use('ggplot')
mpl.rcParams['figure.figsize'] = (10.0, 8.0)

Click me to hide the output
In [6]:
n_points = 1000
f1, f2 = 5, 10
time_samples = np.linspace(0, 1, n_points)
mode1 = np.sin(2 * np.pi * f1 * time_samples)
mode2 = np.sin(2 * np.pi * f2 * time_samples)
trend = time_samples

signal = mode1 + mode2 + trend

plt.plot(time_samples, signal)

Click me to hide the output
Out[6]:
[<matplotlib.lines.Line2D at 0x10753f050>]

We would expect an ideal EMD algorithm to produce two IMFs, one corresponding to each of our unimodal signals, and then to extract the residue too. Let's write a naive sifting loop.

In [7]:
from scipy.signal import argrelmin, argrelmax
from scipy.interpolate import splrep, splev

tol = 0.05

def ndiff_extrema_zcrossing(x):
"""Get the difference between the number of zero crossings and extrema."""
n_max = argrelmax(x)[0].shape[0]
n_min = argrelmin(x)[0].shape[0]
n_zeros = (x[:-1] * x[1:] < 0).sum()
return abs((n_max + n_min) - n_zeros)

def sift(x):
"""One sifting iteration."""
maxima = argrelmax(x)[0]
minima = argrelmin(x)[0]
x_upper = np.zeros((maxima.shape[0] + 2,))
x_upper[1:-1] = maxima
x_upper[-1] = x.shape[0] - 1
x_lower = np.zeros((minima.shape[0] + 2,))
x_lower[1:-1] = minima
x_lower[-1] = x.shape[0] - 1
tck = splrep(x_upper, x[x_upper.astype(int)])
upper_envelop = splev(np.arange(x.shape[0]), tck)
tck = splrep(x_lower, x[x_lower.astype(int)])
lower_envelop = splev(np.arange(x.shape[0]), tck)
mean_amplitude = np.abs(upper_envelop - lower_envelop) / 2
local_mean = (upper_envelop + lower_envelop) / 2
amplitude_error = np.abs(local_mean) / mean_amplitude
return x - local_mean, amplitude_error.sum()

def emd(x, n_imfs):
imfs = np.zeros((n_imfs + 1, x.shape[0]))
for i in xrange(n_imfs):
mode = x - imfs.sum(0)
while (ndiff_extrema_zcrossing(mode) > 1):
mode, amplitude_error = sift(mode)
if amplitude_error <= tol:
break
imfs[i, :] = mode
imfs[-1, :] = x - imfs.sum(0)
return imfs

Click me to hide the output

Let's extract and visualize the IMFs to see if they meet our expectations.

In [8]:
n_imfs = 2
imfs = emd(signal, n_imfs)
plt.subplot(n_imfs + 1, 1, n_imfs + 1), plt.plot(time_samples, imfs[-1, :]), plt.ylabel('residue')
for i in range(n_imfs):
plt.subplot(n_imfs + 1, 1, i + 1), plt.plot(time_samples, imfs[i, :])
plt.ylabel('imf {}'.format(i + 1))

Click me to hide the output
/Users/jaidevd/anaconda/lib/python2.7/site-packages/ipykernel/__main__.py:30: RuntimeWarning: divide by zero encountered in divide

On closer inspection of the residue, it seems like it contains the true residue (the ramp) superimposed on an IMF that we haven't yet sifted out. Let's see what happens when we increment the expected number of IMFs.

In [9]:
n_imfs = 3
imfs = emd(signal, n_imfs)
plt.subplot(n_imfs + 1, 1, n_imfs + 1), plt.plot(time_samples, imfs[-1, :]), plt.ylabel('residue')
for i in range(n_imfs):
plt.subplot(n_imfs + 1, 1, i + 1), plt.plot(time_samples, imfs[i, :])
plt.ylabel('imf {}'.format(i + 1))

Click me to hide the output
/Users/jaidevd/anaconda/lib/python2.7/site-packages/ipykernel/__main__.py:30: RuntimeWarning: divide by zero encountered in divide

We've finally reached a stopping point beyond which sifting is not possible. While the decomposition itself is valid, the inferences we draw from it can be quite misleading. In a broad sense, here is what is wrong with the decomposition:

1. Qualitatively, the residue is supposed to be a somewhat linear, monotonically increasing function.
2. The number if IMFs should be two.
3. The IMFs should at least appear to be pure sinudoids.

To get to the heart of the problem, let's go back to our original signal and try to step through the sifting process. The first thing we do to an input signal is to construct upper and lower envelopes by interpolating with cubic splines its local maxima and local minima respectively, and then subtracting the mean of these envelopes from the signal.

In [10]:
def get_envelopes(x):
"""Compute the upper and lower envelopes of a signal."""
maxima = argrelmax(x)[0]
minima = argrelmin(x)[0]
x_upper = np.zeros((maxima.shape[0] + 2,))
x_upper[1:-1] = maxima
x_upper[-1] = x.shape[0] - 1
x_lower = np.zeros((minima.shape[0] + 2,))
x_lower[1:-1] = minima
x_lower[-1] = x.shape[0] - 1
tck = splrep(x_upper, x[x_upper.astype(int)])
upper_envelop = splev(np.arange(x.shape[0]), tck)
tck = splrep(x_lower, x[x_lower.astype(int)])
lower_envelop = splev(np.arange(x.shape[0]), tck)
return upper_envelop, lower_envelop

upper, lower = get_envelopes(signal)
plt.plot(time_samples, signal, 'b', label='signal')
plt.plot(time_samples, upper, 'r--', label='upper envelope')
plt.plot(time_samples, lower, 'g--', label='lower envelope')
plt.legend()

Click me to hide the output
Out[10]:
<matplotlib.legend.Legend at 0x10e749950>

If we zoom in on the left and right edges of the plot, here's what we see.

In [11]:
plt.subplot(121)
plt.plot(time_samples[:100], signal[:100], 'b',
time_samples[:100], upper[:100], 'r--', time_samples[:100], lower[:100], 'g--')
plt.subplot(122)
plt.plot(time_samples[900:], signal[900:], 'b',
time_samples[900:], upper[900:], 'r--', time_samples[900:], lower[900:], 'g--')

Click me to hide the output
Out[11]:
[<matplotlib.lines.Line2D at 0x10facf050>,
<matplotlib.lines.Line2D at 0x10fb622d0>,
<matplotlib.lines.Line2D at 0x10fb15090>]

It turns out that the envelopes constructed here are not true envelopes, in that the signal is leaking out of their bounds. This is because the naive interpolation method assumes the endpoints of the signal to be knots in the interpolation. Consequently, the splines cannot numerically "compensate" for the slope of the signal near the endpoints.

But if we don't consider the endpoints of the signals as the endpoints of the splines, what is the alternative? It's not like we have any more data, so we just make some. The solution, simply put, consists of:

1. selecting $k$ maxima and $k$ minima closest to each endpoint, and extending them symmetrically beyond the edges,
2. constructing the upper and lower envelopes with these extended extrema
3. slicing off these envelopes with the indices of the original signal

This is how it looks when $k = 2$:

In [12]:
from pyhht.utils import boundary_conditions
tmin, tmax, xmin, xmax = boundary_conditions(signal, time_samples, nbsym=2)

Click me to hide the output
In [13]:
plt.plot(time_samples, signal, 'b')
plt.plot(tmin, xmin, 'go', label='minima')
plt.plot(tmax, xmax, 'ro', label='maxima')
plt.vlines(tmax[2], -2, 3, 'r','dashdot', label='axis of symmetry')
plt.vlines(tmin[-3], -2, 3, 'g', 'dashdot', label='axis of symmetry')
plt.legend(loc=4)

Click me to hide the output
Out[13]:
<matplotlib.legend.Legend at 0x10ff50bd0>

The dotted lines represent the axes about which the extrapolated extrema are symmetric. Now we can use these new data points to reconstruct our envelopes, as follows.

In [14]:
tck = splrep(tmin, xmin)
lower_env = splev(time_samples, tck)
tck = splrep(tmax, xmax)
upper_env = splev(time_samples, tck)

plt.plot(time_samples, signal, 'b')
plt.plot(time_samples, upper_env, 'r-.')
plt.plot(time_samples, lower_env, 'g-.')

Click me to hide the output
Out[14]:
[<matplotlib.lines.Line2D at 0x110334ad0>]

It can be seen that the splines don't terminate at the endpoints of the signal itself. As before, here are the plots of the edges of this interpolation.

In [15]:
plt.subplot(121)
plt.plot(time_samples[:100], signal[:100], 'b',
time_samples[:100], upper_env[:100], 'r--', time_samples[:100], lower_env[:100], 'g--')
plt.subplot(122)
plt.plot(time_samples[900:], signal[900:], 'b',
time_samples[900:], upper_env[900:], 'r--', time_samples[900:], lower_env[900:], 'g--')

Click me to hide the output
Out[15]:
[<matplotlib.lines.Line2D at 0x11062b890>,
<matplotlib.lines.Line2D at 0x1106b8650>,
<matplotlib.lines.Line2D at 0x1106b8d10>]

The "leakage" of the signal beyond the envelopes is far less drastic, and we're much closer to a true envelope now. It's still not a perfect envelope, but I'm not entirely sure on whether that is a function of the sampling rate (I doubt it) or the order of symmetry, or anything else. I suspect it might have to do with the the degree of the interpolating polynomial. Perhaps a higher order polynomial might eliminate this little leakage, but I'm sure it won't work well for the rest of the signal, the bulk of which is far more critical to the decomposition. Anyhow, let's see how EMD fares with this added heuristic. For this we will have to redefine the sift and emd functions from before.

In [16]:
from pyhht.utils import extr

def sift(x, ts):
"""One sifting iteration."""
tmin, tmax, xmin, xmax = boundary_conditions(x, ts)
tck = splrep(tmin, xmin)
lower_envelop = splev(ts, tck)
tck = splrep(tmax, xmax)
upper_envelop = splev(ts, tck)
mean_amplitude = np.abs(upper_envelop - lower_envelop) / 2
local_mean = (upper_envelop + lower_envelop) / 2
amplitude_error = np.abs(local_mean) / mean_amplitude
return x - local_mean, amplitude_error.sum()

def emd(x, ts, n_imfs):
imfs = np.zeros((n_imfs + 1, x.shape[0]))
for i in xrange(n_imfs):
ix = 0
mode = x - imfs.sum(0)
nmin, nmax, nzero = map(len, extr(mode))
while abs((nmin + nmax) - nzero) > 1:
mode, amplitude_error = sift(mode, ts)
if amplitude_error <= tol:
break
imfs[i, :] = mode
imfs[-1, :] = x - imfs.sum(0)
return imfs

n_imfs = 2
imfs = emd(signal, time_samples, n_imfs=n_imfs)
plt.subplot(n_imfs + 1, 1, n_imfs + 1), plt.plot(time_samples, imfs[-1, :]), plt.ylabel('residue')
for i in range(n_imfs):
plt.subplot(n_imfs + 1, 1, i + 1), plt.plot(time_samples, imfs[i, :])
plt.ylabel('imf {}'.format(i + 1))

Click me to hide the output

This decomposition is a clearly better approximation of the original components. You can easily tell the two unimodal components from earlier, and even the residue is much closer to the linear trend that we had included in the signal.

The interesting thing to note here is that this signal was very straightforward, and even then the edge effects of sifting were severe. There's no telling how these might affect the decomposition of real world data.

There are many more heuristics involved in a good implementation of empirical mode decomposition. Following posts will deal with them in more detail.


text:
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# If 2^m = 5 and 5^n = 3, then 3^(1/n)/2^

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If 2^m = 5 and 5^n = 3, then 3^(1/n)/2^ [#permalink]

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12 Feb 2017, 06:47
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If $$2^m= 5$$ and $$5^{1/n} = 3$$, then $$\frac{3^n}{2^{-2m}}$$ =

A) 1/25
B) 1/5
C) 5
D) 25
E) 125

* kudos for all correct solutions

EDIT: Sorry - wrong forum
[Reveal] Spoiler: OA

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Joined: 02 Aug 2009
Posts: 5647
Re: If 2^m = 5 and 5^n = 3, then 3^(1/n)/2^ [#permalink]

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12 Feb 2017, 08:38
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GMATPrepNow wrote:
If $$2^m= 5$$ and $$5^{1/n} = 3$$, then $$\frac{3^n}{2^{-2m}}$$ =

A) 1/25
B) 1/5
C) 5
D) 25
E) 125

* kudos for all correct solutions

EDIT: Sorry - wrong forum

Hi,
Let's convert the equations given to fit into $$\frac{3^n}{2^{-2m}}$$ ..

$$2^m= 5, ..... 2^{2m}=5^2=25$$ and $$5^{1/n} = 3......(5^{1/n})^n=3^n.....3^n=5$$...

$$\frac{3^n}{2^{-2m}}=3^n*2^{2m}$$ ..
Substitute the values in $$3^n*2^{2m}=5*25=125$$

E
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Re: If 2^m = 5 and 5^n = 3, then 3^(1/n)/2^ [#permalink]

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12 Feb 2017, 08:53
Expert's post
Top Contributor
GMATPrepNow wrote:
If $$2^m= 5$$ and $$5^{1/n} = 3$$, then $$\frac{3^n}{2^{-2m}}$$ =

A) 1/25
B) 1/5
C) 5
D) 25
E) 125

Given: $$2^m= 5$$
We need to determine the value of $$2^{-2m}$$
So, take $$2^m= 5$$ and raise both sides to the power of -2
We get: (2^m)^(-2) = 5^(-2)
Apply Power of a Power rule to get: 2^(-2m) = 5^(-2)

Given: $$5^{1/n} = 3$$
We need to determine the value of 3^n
So, take $$5^{1/n} = 3$$ and raise both sides to the power of n
We get: [5^(1/n)]^n = 3^n
Apply Power of a Power rule to get: 5^1 = 3^n

So, (3^n)/2^(-2m) = (5^1)/[(5^(-2)]
= 5^3
= 125

RELATED VIDEO FROM OUR COURSE

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Re: If 2^m = 5 and 5^n = 3, then 3^(1/n)/2^   [#permalink] 12 Feb 2017, 08:53
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text:
# Fermats little theorem question

Repeated squaring

Hi
I have tasked with solving the following equation:
$$x^6 \ \equiv 24 \ \mathbf{mod} \ 68$$
I'm told now that I need to use a method called "Repeated squaring to solve the equation above".
Any hints/idears on how I do this will be appriciated very much !
/Bob

Last edited:

Bob19 said:
Hi
I have tasked with solving the following equation:
$$x^6 \ \equiv 24 \ \mathbf{mod} \ 68$$
I'm told now that I need to use a method called "Repeated squaring to solve the equation above".
Any hints/idears on how I do this will be appriciated very much !
/Bob
This should help:

http://web.usna.navy.mil/~wdj/book/node27.html [Broken]

Alex

Last edited by a moderator:


text:
# Contour Plot in Cylindrical Coordinates

I'm trying to render an equation in the cylindrical coordinates, and I used the following codes:

With[{λ = 1, f = 50, R = 25},
ContourPlot[
-Mod[(Pi/λ)*(Sqrt[ρ^2 + f^2] - f), Pi] /. {ρ -> Norm[{x, y}], ϕ -> ArcTan[x, y]},
{x, -R, R}, {y, -R, R},
Contours -> 20, ContourLines -> False,
RegionFunction -> (Sqrt[#1^2 + #2^2] < R &),
ColorFunction -> "Rainbow", Frame -> False]]

However, I got a pattern which seems not smooth because points were sampled equally in the Cartesian coordinates. How can I get a smooth contour plot? Furthermore, can I plot such a pattern in the 3D coordinates? Thanks a lot.

Here is an example of the output that I would like:

-
I added your graphic as an example for you, as I know the site does not let you do that yet. – Mr.Wizard Mar 8 '13 at 1:06
Thank you so much, Mr. Wizard. – Tony Dong Mar 8 '13 at 1:09
Tony, if my post fully answers your question please consider Accepting it. If not, please tell me in what way it is lacking. – Mr.Wizard Mar 9 '13 at 0:34
@ Mr. Wizard, I'll do that. You were good but the 3d plot was not exactly what I want. Probably I didn't describe it clearly. I intend to plot the same pattern in 3D coordinates, say, the target-sheet-shaped preservers. It seems to shear and/or rotate 2d graphics with certain viewpoint. Thanks anyway. – Tony Dong Mar 9 '13 at 2:29

First, your function can be simplified:

FullSimplify[
-Mod[(Pi/λ)*(Sqrt[ρ^2 + f^2] - f), Pi] /.
{ρ -> Norm[{x, y}], ϕ -> ArcTan[x, y]},
Element[x | y, Reals]
]

-Mod[(Pi*(-f + Sqrt[f^2 + x^2 + y^2]))/λ, Pi]

Second, you appear to want a DensityPlot rather than a ContourPlot.

Third, you can get smoother graphics by increasing PlotPoints:

With[{λ = 1, f = 50, R = 25},
DensityPlot[
Mod[(Pi*(-f + Sqrt[f^2 + x^2 + y^2]))/λ, Pi],
{x, -R, R}, {y, -R, R},
ColorFunction -> "Rainbow",
Frame -> False,
PlotPoints -> 100,
ExclusionsStyle -> Black,
RegionFunction -> (Sqrt[#1^2 + #2^2] < R &)
]
]

Or as a Plot3D:

With[{λ = 1, f = 50, R = 25},
Plot3D[Mod[(Pi*(-f + Sqrt[f^2 + x^2 + y^2]))/λ, Pi],
{x, -R, R}, {y, -R, R},
ColorFunction -> "Rainbow",
PlotPoints -> 25,
RegionFunction -> (Sqrt[#1^2 + #2^2] < R & )
]
]

Incorporating R.M's suggestions from the comments:

jet[u_?NumericQ] /; 0 <= u <= 1 :=
Blend[{{0, RGBColor[0, 0, 9/16]}, {1/9, Blue}, {23/63, Cyan}, {13/21, Yellow},
{47/63, Orange}, {55/63, Red}, {1, RGBColor[1/2, 0, 0]}}, u]

With[{λ = 1, f = 50, R = 25},
ArrayPlot[Table[
If[Norm[{x, y}] < R, Mod[(Pi*(-f + Sqrt[f^2 + x^2 + y^2]))/λ, Pi], None],
{x, -25, 25, 0.1}, {y, -25, 25, 0.1}
], ColorFunction -> jet, Frame -> False
]
]

-
Thanks Mr. Wizard, I can actually get the same figure as you shown. But I wanna cut out the corners, say, just show the circular patterns. – Tony Dong Mar 8 '13 at 0:18
@Tony Sorry, I removed one too many options in my attempt to simplify things. Please look again and tell me if this helps. – Mr.Wizard Mar 8 '13 at 0:20
That's exactly what I want. Thank you so much. – Tony Dong Mar 8 '13 at 0:22
@Tony I just noticed that ϕ (phi) does not appear to be used in the function. What am I missing? What do you intend? – Mr.Wizard Mar 8 '13 at 0:26
@TonyDong You don't see the exclusions in MATLAB only because you're doing a finite sampling without any consideration for the function's discontinuities. DensityPlot as Mr.Wizard used here is "aware" of the discontinuity, which is why we have to deal with Exclusions. If you instead sample the domain and plot it using ListDensityPlot, then you'll reproduce MATLAB's figure. Also, see this answer if you want to reproduce the jet colorscheme. – R. M. Mar 8 '13 at 4:44


text:
The box on the right side of this SERP is known as the Knowledge Graph (also sometimes called the Knowledge Box). This is a feature that Google introduced in 2012 that pulls data to commonly asked questions from sources across the web to provide concise answers to questions in one central location on the SERP. In this case, you can see a wide range of information about Abraham Lincoln, such as the date and place of his birth, his height, the date on which he was assassinated, his political affiliation, and the names of his children – many of which facts have their own links to the relevant pages.

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PageRank (PR) is a quality metric invented by Google's owners Larry Page and Sergey Brin. The values 0 to 10 determine a page's importance, reliability and authority on the web according to Google. This metric does not, however, directly affect a website's search engine ranking. A website with a PR 2 could be found on the first page of search results while a website with a PR 6 for the same keyword may appear on the second page of search results. To know more information about best seo services please visit our website.
Influencer marketing: Important nodes are identified within related communities, known as influencers. This is becoming an important concept in digital targeting. It is possible to reach influencers via paid advertising, such as Facebook Advertising or Google Adwords campaigns, or through sophisticated sCRM (social customer relationship management) software, such as SAP C4C, Microsoft Dynamics, Sage CRM and Salesforce CRM. Many universities now focus, at Masters level, on engagement strategies for influencers.

Search engines are smart, but they still need help. The major engines are always working to improve their technology to crawl the web more deeply and return better results to users. However, there is a limit to how search engines can operate. Whereas the right SEO can net you thousands of visitors and increased attention, the wrong moves can hide or bury your site deep in the search results where visibility is minimal.
[41] Katja Mayer views PageRank as a social network as it connects differing viewpoints and thoughts in a single place. People go to PageRank for information and are flooded with citations of other authors who also have an opinion on the topic. This creates a social aspect where everything can be discussed and collected to provoke thinking. There is a social relationship that exists between PageRank and the people who use it as it is constantly adapting and changing to the shifts in modern society. Viewing the relationship between PageRank and the individual through sociometry allows for an in-depth look at the connection that results.
“I have had the good fortune to work with SEO expert Nick Stamoulis and his team at Brick Marketing and it was clear form the get go that Nick had superior knowledge of marketing on the internet. The SEO campaign and technical writing that Brick Marketing provides has been impressive, where in the past we struggled to find the high levels of competence in these two skill sets. Were only just beginning with Brick Marketing and look forward to growing with their expert guidance. Thank you Nick Stamoulis and the Brick Marketing team!”

2. Domain authority and page authority. Next, you should learn about domain authority and page authority, and how they predict your site’s search rankings. Here’s the basic idea; your site’s domain authority is a proprietary score, provided by Moz, of how “trustworthy” your domain is. It’s calculated based on the quantity and quality of inbound links to your website. The higher it is, the higher all your pages across your domain are likely to rank in organic search results. Page authority is very similar, but page-specific, and you can use it to engineer a link architecture that strategically favors some of your pages over others. Authority depends on the authority and volume of inbound links.

To a spider, www.domain.com/, domain.com/, www.domain.com/index.html and domain.com/index.html are different urls and, therefore, different pages. Surfers arrive at the site’s home page whichever of the urls are used, but spiders see them as individual urls, and it makes a difference when working out the PageRank. It is better to standardize the url you use for the site’s home page. Otherwise each url can end up with a different PageRank, whereas all of it should have gone to just one url.
People tend to view the first results on the first page.[5] Each page of search engine results usually contains 10 organic listings (however some results pages may have fewer organic listings). The listings, which are on the first page are the most important ones, because those get 91% of the click through rates (CTR) from a particular search. According to a 2013 study,[6] the CTR's for the first page goes as:
There are simple and fast random walk-based distributed algorithms for computing PageRank of nodes in a network.[31] They present a simple algorithm that takes {\displaystyle O(\log n/\epsilon )} rounds with high probability on any graph (directed or undirected), where n is the network size and {\displaystyle \epsilon } is the reset probability ( {\displaystyle 1-\epsilon } is also called as damping factor) used in the PageRank computation. They also present a faster algorithm that takes {\displaystyle O({\sqrt {\log n}}/\epsilon )} rounds in undirected graphs. Both of the above algorithms are scalable, as each node processes and sends only small (polylogarithmic in n, the network size) number of bits per round.
The PageRank formula also contains a damping factor (d) . According to the PageRank theory, there is an imaginary surfer who is randomly clicking on links, and at some point he gets bored and eventually stops clicking. The probability that the person will continue clicking at any step is a damping factor. Thus, this factor is introduced to stop some pages having too much influence. As a result, their total vote is damped down by multiplying it by 0.85 (a generally assumed value).
Link page A to page E and click Calculate. Notice that the site’s total has gone down very significantly. But, because the new link is dangling and would be removed from the calculations, we can ignore the new total and assume the previous 4.15 to be true. That’s the effect of functionally useful, dangling links in the site. There’s no overall PageRank loss.
The linking page’s PageRank is important, but so is the number of links going from that page. For instance, if you are the only link from a page that has a lowly PR2, you will receive an injection of 0.15 + 0.85(2/1) = 1.85 into your site, whereas a link from a PR8 page that has another 99 links from it will increase your site’s PageRank by 0.15 + 0.85(7/100) = 0.2095. Clearly, the PR2 link is much better – or is it? See here for a probable reason why this is not the case.
Digital marketers monitor things like what is being viewed, how often and for how long, sales conversions, what content works and doesn’t work, etc. While the Internet is, perhaps, the channel most closely associated with digital marketing, others include wireless text messaging, mobile instant messaging, mobile apps, podcasts, electronic billboards, digital television and radio channels, etc.

Numerous academic papers concerning PageRank have been published since Page and Brin's original paper.[5] In practice, the PageRank concept may be vulnerable to manipulation. Research has been conducted into identifying falsely influenced PageRank rankings. The goal is to find an effective means of ignoring links from documents with falsely influenced PageRank.[6]
Facebook Ads and Instagram Ads take relevance and ad engagement into consideration. Ads that perform well are given a higher relevance score and are given more impressions at a cheaper price than ads with low relevance. Similarly, AdWords assigns ads a quality score based on factors like keyword relevance and landing page quality that can affect how much you pay for each click.
In 2007, Google announced a campaign against paid links that transfer PageRank.[29] On June 15, 2009, Google disclosed that they had taken measures to mitigate the effects of PageRank sculpting by use of the nofollow attribute on links. Matt Cutts, a well-known software engineer at Google, announced that Google Bot would no longer treat nofollowed links in the same way, to prevent SEO service providers from using nofollow for PageRank sculpting.[30] As a result of this change the usage of nofollow led to evaporation of PageRank. In order to avoid the above, SEO engineers developed alternative techniques that replace nofollowed tags with obfuscated Javascript and thus permit PageRank sculpting. Additionally several solutions have been suggested that include the usage of iframes, Flash and Javascript.[31]