Copy of the Reed chapter from Steve Bannister and University of Utah

This is an interesting paper and will take a while to digest it. It is pleasing to note that he included the B(3) field in his studies (p561).

Doug

On Thu, Jan 30, 2014 at 1:29 AM, <EMyrone> wrote:

I am most grateful to Steve for this, not least because it contains very nice illustrations of Beltrami flow and goes in to higher topology electrodynamics and phenomena in plasma physics not explicable by the standard model. I agree that I should not put it on the blog because of copyright and is being distributed to scholars for their private use. This is from M. W. Evans Ed., "Modern Nonlinear Optics", a special topical issue of I. Prigogine and S. A. Rice (Eds.) "Advances in Chemical Physics" (Wiley Interscience, New York, 2001, second edition), volume 119(3). This will be in any good library. It is an award winning production in six volumes and two editions. The first edition was edited by Stanislaw Kielich and myself.

To: EMyrone
Sent: 29/01/2014 20:15:34 GMT Standard Time
Subj: Reed chapter

Hello Myron. I don’t know if this duplicates the link that was sent out earlier, so I thought, since my research dept. found it, I would forward it on. Since I got this through the University, it probably should not go on the blog, but just to people who may need it.

Best,

Steve

Recent Progress

Many thanks, ‘t Hooft has certainly taken very heavy implied criticism in the profession, as of course have those responsible for UNCC: Buckingham, Barron, Lakhtakia, Almeida and some retired staff members of physics who panicked when B(3) pseudocriticism started to arrive from Buckingham and Barron. There was a battle with Buckingham consisting of thirty nine exchanges between Oxford and Cambridge men as he tried to block my reply to his post doc Barron in his own journal. This was finally brought to an end by Mansel Davies, and Barron’s ridiculous paper was not published. Buckingham then attacked again in “Science”, and again I was not allowed to reply. A reply was allowed by Alwyn van der Mewe in “Foundations of Physics Letters” and of course accepted by the entire profession as the scientometrics show. Prof. Emeritus van der Merwe comes out of this in a very fine way. They then put the boot in his journal and appointed ‘t Hooft. They may as well have appointed a dalek: exterminate, exterminate. So ‘t Hooft had the comical arrogance to “unpublish” fifteen papers (UFT1 – UFT15) refereed forty or fifty times, with the result that they have been read a few hundred thousand times. So I allowed debate, the other side did not because they had no idea what they were doing. They just put the boot in like bad prop forwards. As Gareth knows, Buckingham started to interfere in my work when I was a Ramsay Memorial Fellow. This was all totally unethical and Buckinhgam, Barron and Atkins deserve international condemnation. I am on quite friendly terms with the students at UNCC, which studies AIAS work quite regularly. For almost twenty years, those responsible for UNCC have not received a single voice in support of what they did, recorded very accurately in the UNCC Saga so that we may try to learn from history to be a little bit more tolerant of new thought. They caused my first wife and myself an immense amount of damage, as one or two people know here, and of course have never congratulated me on any achievement since I resigned in protest. They are not typical of the City of Charlotte, nor are they typical of North Carolina, nor do they represent the American tradition of toleration and democracy. I was naturalized a U. S. Citizen at Cornell five years after UNCC, appointed to the British Civil List ten years after UNCC and three years later raised to Armiger (the untitled nobility). They are also suspected of sending childish hate mail similar to Lakhtakia, so they are completely and utterly discredited, not least among their own students.

This is really interesting. Well done to the three of you yet again. It is clear by now that B(3) was a “missing link” – its neglect (lack of understanding) was preventing progress and created a real “black hole” – an age of speculative nonsense. Now physics is strictly logical again and science in general can move forward. As for ‘t Hooft – the best service he can make to real science is to climb down and recognise the progress. Otherwise he must disappear over the course of time, with his worthless NP, down one of those fictional black holes.

Sent from Samsung Mobile

Excellent!

In a message dated 30/01/2014 08:28:42 GMT Standard Time,

This is indeed a highly interesting development. With “graphics” I meant the copy of the slide, obviously it came through. When I have time I will make some 3D plots of vector fields, in Gnuplot this can be done by one line, if field data is provided in a file.

Horst

EMyrone@aol.com hat am 30. Januar 2014 um 09:10 geschrieben:

This is incisive and interesting as usual. I could not see the graphics from here, maybe they can be send in a pdf or different format, but I can see the Maxima output. These new solutions generalize the B Cyclic Theorem, which is the complex circular frame itself, and not only Lorentz covariant but generally covariant because ECE is a theory based on generally covariant Cartan geometry. The result we are looking for is of course a longitudinal vortex of infinitesimally small radius emerging from helicoidal or more general Beltrami flow of the free electromagnetic field. The vacuum potentials recently developed by Doug and yourself would also have the same characteristics. The infinitesimally thin vortex would be the B(3) field in vacuo. The Beltrami structures are obvious from plane waves, but what we are finding now is that there is a much richer structure of B field components which are all solutions to the higher topology Beltrami equation. The early “criticisms” of B(3) were vindictive rubbish, they did not even understand that B(3) is a result of higher topology. These were Lakhtakia, Buckingham and Barron initially, then from the Buckingham environment, e.g. Rikken and van Enk. These papers would often appear behind my back with no warning, in a completely unethical way, and every single one of them was replied to in great detail to the evident satisfaction of the entire international community of scientists and engineers. These were followed later with the UNCC and ‘t Hooft conspiracies, generally regarded as scraping the bottom of the dustbin in the history of physics and completely ignored by the profession as acute personal animosity. I will study these new solutions today to see if I can reduce them to something like the B Cyclic Theorem. The Nobel Prize nominations for B(3) appeared after these early personal attacks had been dealt with. The ‘t Hooft conspiracy was intended purely for the purpose of preventing me getting a Nobel Prize, because that would be a challenge to mediaeval dogma. Both UNCC and ‘t Hooft have been implicitly criticised heavily for years through the readership of my “UNCC Saga” and “Diplomatic Objection to ‘t Hooft” by Gareth Evans. Now we are making strong progress forward, we have always made strong progress forward. The appearance of stalkers and hate groups destroyed the credibility of these “critics” completely. We must sit back and enjoy the great success of AIAS, and above all, enjoy our work like a cookie in a Chinese restaurant. Not many groups get half a dozen Nobel Prize nominations and a nomination is as nice as a prize. I did no lobbying at all for the Nobel Prize, all the nominations arrived spontaneously.

To: EMyrone@aol.com
Sent: 29/01/2014 19:18:37 GMT Standard Time
Subj: Re: Analytical Solutions of the Beltrami equation

I hope the graphics come through.
The following should hold:

My result is:

This means: there is a sign error in the slide set, and the Z component contains second derivatives, not squared first derivatives. Nevertheless these equations should allow us to construct quite general solutions of the Beltrami equation. A suitable combination of phi and w has to be found. See the example in section 5 of the attached. I did not succeed in adding a z dependence to w, that requires further trials.

Horst

Remarks by Dr. Horst Eckardt: Numerical Analysis of the Beltrami equation

Excellent!

In a message dated 30/01/2014 08:28:42 GMT Standard Time, mail@horst-eckardt.de writes:

This is indeed a highly interesting development. With “graphics” I meant the copy of the slide, obviously it came through. When I have time I will make some 3D plots of vector fields, in Gnuplot this can be done by one line, if field data is provided in a file.

Horst

EMyrone@aol.com hat am 30. Januar 2014 um 09:10 geschrieben:

This is incisive and interesting as usual. I could not see the graphics from here, maybe they can be send in a pdf or different format, but I can see the Maxima output. These new solutions generalize the B Cyclic Theorem, which is the complex circular frame itself, and not only Lorentz covariant but generally covariant because ECE is a theory based on generally covariant Cartan geometry. The result we are looking for is of course a longitudinal vortex of infinitesimally small radius emerging from helicoidal or more general Beltrami flow of the free electromagnetic field. The vacuum potentials recently developed by Doug and yourself would also have the same characteristics. The infinitesimally thin vortex would be the B(3) field in vacuo. The Beltrami structures are obvious from plane waves, but what we are finding now is that there is a much richer structure of B field components which are all solutions to the higher topology Beltrami equation. The early “criticisms” of B(3) were vindictive rubbish, they did not even understand that B(3) is a result of higher topology. These were Lakhtakia, Buckingham and Barron initially, then from the Buckingham environment, e.g. Rikken and van Enk. These papers would often appear behind my back with no warning, in a completely unethical way, and every single one of them was replied to in great detail to the evident satisfaction of the entire international community of scientists and engineers. These were followed later with the UNCC and ‘t Hooft conspiracies, generally regarded as scraping the bottom of the dustbin in the history of physics and completely ignored by the profession as acute personal animosity. I will study these new solutions today to see if I can reduce them to something like the B Cyclic Theorem. The Nobel Prize nominations for B(3) appeared after these early personal attacks had been dealt with. The ‘t Hooft conspiracy was intended purely for the purpose of preventing me getting a Nobel Prize, because that would be a challenge to mediaeval dogma. Both UNCC and ‘t Hooft have been implicitly criticised heavily for years through the readership of my “UNCC Saga” and “Diplomatic Objection to ‘t Hooft” by Gareth Evans. Now we are making strong progress forward, we have always made strong progress forward. The appearance of stalkers and hate groups destroyed the credibility of these “critics” completely. We must sit back and enjoy the great success of AIAS, and above all, enjoy our work like a cookie in a Chinese restaurant. Not many groups get half a dozen Nobel Prize nominations and a nomination is as nice as a prize. I did no lobbying at all for the Nobel Prize, all the nominations arrived spontaneously.

To: EMyrone@aol.com
Sent: 29/01/2014 19:18:37 GMT Standard Time
Subj: Re: Analytical Solutions of the Beltrami equation

I hope the graphics come through.
The following should hold:

My result is:

This means: there is a sign error in the slide set, and the Z component contains second derivatives, not squared first derivatives. Nevertheless these equations should allow us to construct quite general solutions of the Beltrami equation. A suitable combination of phi and w has to be found. See the example in section 5 of the attached. I did not succeed in adding a z dependence to w, that requires further trials.

Horst

Remarks by Dr. Horst Eckardt: Numerical Analysis of the Beltrami equation

This is incisive and interesting as usual. I could not see the graphics from here, maybe they can be send in a pdf or different format, but I can see the Maxima output. These new solutions generalize the B Cyclic Theorem, which is the complex circular frame itself, and not only Lorentz covariant but generally covariant because ECE is a theory based on generally covariant Cartan geometry. The result we are looking for is of course a longitudinal vortex of infinitesimally small radius emerging from helicoidal or more general Beltrami flow of the free electromagnetic field. The vacuum potentials recently developed by Doug and yourself would also have the same characteristics. The infinitesimally thin vortex would be the B(3) field in vacuo. The Beltrami structures are obvious from plane waves, but what we are finding now is that there is a much richer structure of B field components which are all solutions to the higher topology Beltrami equation. The early “criticisms” of B(3) were vindictive rubbish, they did not even understand that B(3) is a result of higher topology. These were Lakhtakia, Buckingham and Barron initially, then from the Buckingham environment, e.g. Rikken and van Enk. These papers would often appear behind my back with no warning, in a completely unethical way, and every single one of them was replied to in great detail to the evident satisfaction of the entire international community of scientists and engineers. These were followed later with the UNCC and ‘t Hooft conspiracies, generally regarded as scraping the bottom of the dustbin in the history of physics and completely ignored by the profession as acute personal animosity. I will study these new solutions today to see if I can reduce them to something like the B Cyclic Theorem. The Nobel Prize nominations for B(3) appeared after these early personal attacks had been dealt with. The ‘t Hooft conspiracy was intended purely for the purpose of preventing me getting a Nobel Prize, because that would be a challenge to mediaeval dogma. Both UNCC and ‘t Hooft have been implicitly criticised heavily for years through the readership of my “UNCC Saga” and “Diplomatic Objection to ‘t Hooft” by Gareth Evans. Now we are making strong progress forward, we have always made strong progress forward. The appearance of stalkers and hate groups destroyed the credibility of these “critics” completely. We must sit back and enjoy the great success of AIAS, and above all, enjoy our work like a cookie in a Chinese restaurant. Not many groups get half a dozen Nobel Prize nominations and a nomination is as nice as a prize. I did no lobbying at all for the Nobel Prize, all the nominations arrived spontaneously.

To: EMyrone@aol.com
Sent: 29/01/2014 19:18:37 GMT Standard Time
Subj: Re: Analytical Solutions of the Beltrami equation

I hope the graphics come through.
The following should hold:

My result is:

This means: there is a sign error in the slide set, and the Z component contains second derivatives, not squared first derivatives. Nevertheless these equations should allow us to construct quite general solutions of the Beltrami equation. A suitable combination of phi and w has to be found. See the example in section 5 of the attached. I did not succeed in adding a z dependence to w, that requires further trials.

Horst

beltrami.pdf

Daily Report 28-29/1/14

On 28/1/14 there were 2695 hits from 530 distinct visits, spiders from baidu, google, MSN, softlayer and yandex. Auto1 199, Auto2 89, Book of Scientometrics 80, CEFE 63, Evans Equations 43 (English), numerous (Spanish), Englynion (Book of Poetry in Welsh and English) 27, Second Book of Poetry 15, Autosonnets 12 to date in January 2014. University of Quebec Trois Rivieres OO574, Technical University Berlin Proof1; Department of Astronomy University of Illinois UFT149; New Mexico State University general; United States Social Security Administration general; Indian Institute of Technology Delhi UFT43; Indian Institute for Plasma Research UFT228; Department of Geophysics Autonomous National University of Mexico Essay 87(Sp); Research Institute for Applied Mathematics and Robotics Autonomous National University of Mexico UFT145(Sp); King’s College Cambridge UFT2. Updated usage file attached for January 2014.

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Remarks by Norman Page on plasma jets

I think we are just about to unravel a new class of B(3) type solutions from vacuum Beltrami electrodynamics. I will work some more on this tomorrow. We can also animate the vacuum fields and Horst Eckardt and Douglas Lindstrom are first class with graphical and numerical analysis. I am not sure whether “Nature” deserves us, being tightly controlled by the standard crowd and their mediaeval dogma. I could try, but will probably get a very rude noise by return e mail. We have our own excellent journal run by Victor Riecansky, and worldwide interest much larger than “Nature”. All our UFT papers are also in Google Scholar, so our open source method is fast, efficient, and mainstream science.

To: EMyrone@aol.com
Sent: 29/01/2014 17:55:20 GMT Standard Time
Subj: Re: 257(2) : Extended Beltrami Electrodynamics for Vacuum Fields

On 1/29/2014 6:10 AM, EMyrone wrote:

These are illustrated with plane waves but more generally they are given by Eqs. (55) to (58), which are four equations in four unknowns which can be solved numerically in the general case (more general than vacuum plane waves). They ought to produce helicoidal vacuum flow of the type sketched in Figure (1), so that the B(3) field emerges as the field line along Z, well known in Beltrami hydrodynamics and aerodynamics. This is yet another way of proving the B(3) field and its hugely successful ECE theory.

257(2) looks great as far as I can follow the maths. You should send this to Nature with one of the images I linked to as an illustration. Best Regards Norman

Remarks by Norman Page, Sometime Oxford University: Plasma Jets and B(3)

This is a beautiful image, we were just discussing the transition to turbulence. Donald Reed also discusses Beltrami structures in plasma. The standard model has no explanation, yet again. We should look up the analytical equation of the Beltrami Rankine vortex. This is very probably the B(3) field driving the electrons.

t
To: EMyrone@aol.com
Sent: 29/01/2014 16:54:56 GMT Standard Time
Subj: Re: 257(2) : Extended Beltrami Electrodynamics for Vacuum Fields

Using Occam’s razor doesn’t this image show the plasma jets as the B3 field emerging from a Beltrami – Rankine vortex.The rotation curve of the galaxy depends on the Reynolds number ie when turbulent flow takes over.
https://www.google.com/search?q=galactic+jets+ images& amp;rlz=1T4GGNI_enUS554US555&tbm=isch&imgil=KyRwAZdH_i4zTM%253A%253Bhttps%253A%252F%252Fencrypted-tbn0.gstatic.com%252Fimages%253Fq%253Dtbn%253AANd9GcSvdDgQ33BphVJ3sXEGYF7FThC7S2TEscbkgmELUGtWV8nUxTDAOA%253B1600%253B1105%253BqXoZEUJVc2SAWM%253Bhttp%25253A%25252F%25252Fcandels-collaboration.blogspot.com%25252F2012%25252F09%25252Funcovering-role-of-black-holes-in.html&source=iu&usg=__26zQBWxiOU8hH1MHckrZHCkRjlE%3D&sa=X&ei=HCXpUs7qCMOS2QXNhIEo&ved=0CEIQ9QEwCg&biw=1122&bih=829#facrc=_&imgdii=_&imgrc=KyRwAZdH_i4zTM%253A%3BqXoZEUJVc2SAWM%3Bhttp%253A%252F%252F4.bp.blogspot.com%252F-HsgFbH9ph7o%252FUF-242L5SzI%252FAAAAAAAAPkY%252FVKoEWIm2J80%252Fs1600%252FAGN_wJets.big.cut.jpg%3Bhttp%253A%252F%252Fcandels-collaboration.blogspot.com%252F2012%252F09%252Funcovering-role-of-black-holes-in.html%3B1600%3B1105

Just paste this address in to see image. Regards Norman

On 1/29/2014 6:10 AM, EMyrone wrote:

These are illustrated with plane waves but more generally they are given by Eqs. (55) to (58), which are four equations in four unknowns which can be solved numerically in the general case (more general than vacuum plane waves). They ought to produce helicoidal vacuum flow of the type sketched in Figure (1), so that the B(3) field emerges as the field line along Z, well known in Beltrami hydrodynamics and aerodynamics. This is yet another way of proving the B(3) field and its hugely successful ECE theory.

Analytical Solutions of the Beltrami equation

Very interesting! It would be especially important to see how this structure goes over into vacuum plane waves and B(3) or into a helicoidal vacuum structure for electromagnetism as sketched in to 257(2). In order to make them propagate along Z the phase omega t – kappa Z must be incorporated. It seems that Beltrami flow is of interest to geology and geography, and is thought to cause rivers to meander. Some of the Beltrami structures are very intricate and there is a transition to turbulence at some point.

Sent: 29/01/2014 15:35:23 GMT Standard Time
Subj: Re: Analytical Solutions of the Beltrami equation

I found the example in
http://www.google.de/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CDYQFjAA&url=http%3A%2F%2Fwww.phys.sinica.edu.tw%2F~heptheory%2F2013_RS_talk_files%2F2013.0724.BeltramiAcademiaSinica24July2013f.pdf&ei=CxnpUqq-LrPfygO3t4CADw&usg=AFQjCNHL4jpjGuYbgpxdz9PCBqVbkrEwUg&bvm=bv.60157871,d.bGQ&cad=rja

The factor of 2 can be omitted. However B fulfils only the Beltrami equation with T1=T2=T3, see below. It is also interesting that B can be found by a differential equation.

EMyrone@aol.com hat am 29. Januar 2014 um 16:03 geschrieben:

Google “Helicoidal solution of the Beltrami Equation”, first site, “Generalized Helicity Conservation and Beltrami … ” gives many interesting solutions. For example:

B sub X = T2 sin 2TZ + T3 cos 2TY
B sub Y = T3 sin 2TX + T1 cos 2TZ
B sub Z = T1 sin 2TY + T2 cos 2TX

where T1, T2 and T3 are arbitrary functions.

In a message dated 29/01/2014 14:19:17 GMT Standard Time, writes:

I just wrote to Myron that it is difficult to find an analytical example (in cylinder coordinates) for the helical structure in the figure of the Reed article. If we want to investigate such structures it probably cannot be reduced to 2D because of the rotation in Z direction. However a Beltrami equation without this property should be possible to be formulated in 2D spherical coordinates.

Horst

Doug Lindstrom hat am 29. Januar 2014 um 14:35 geschrieben:

There is a good chance that flexpde can handle it using complex variables. It would be worth a try using radial symmetry to reduce the problem to 2D. Doug

On Wednesday, January 29, 2014, Horst Eckardt < mail> wrote:

PS:
Doug, is there a chance to solve the Beltrami equation numerically by the FEM method? I guess it has to be in 3D.

Horst

Horst Eckardt < mail@horst-eckardt.de> hat am 29. Januar 2014 um 08:42 geschrieben:

Myron, Doug,
The review article is good, nethertheless I have an understanding problem with interpretation of the Beltrami equation. There are interesting diagrams how such a flow can be imagined, but to my understanding the curl of a vector is always perpendicular to the vector itself. This seems also to be the case in the longitudinal flux field examples. How can the curl be parallel to the vector?

Horst

EMyrone@aol.com hat am 28. Januar 2014 um 18:53 geschrieben:

Many thanks, this is an important review by Donald Reed.

To: EMyrone@aol.com
Sent: 28/01/2014 13:45:47 GMT Standard Time
Subj: Re: Fwd: Beltrami fields Forgot the link – Norman

I’ll send the link again . I think this is from the Reed paper you refered to . Here is the bit on B3 Regards Norman.

“Evans/Vigier Longitudinal B(3) Field and Trkalian Vector Fields
Developed over the past decade, concurrently with both Hillion/Quinnez and Rodrigues/Vaz SU(2) EM field models, but based upon a different non-Abelian gauge group, is the so-called Evans/Vigier longitudinal B(3) field representation [89-93]. In this model, a Yang-Mills gauge field theory [94] with an internal O(3) gauge field symmetry [95] is invoked to account for various magneto-optical effects which are claimed to be a function of a third magnetic field vector component that has been termed B(3). O of the central theorems of O(3) electrodynamics is the B-cyclic theorem:
B(1) x B(2) = i B(o) B(3)*, (93)
A conjugate product which relates three basic magnetic field components in vacuo defined as:
B(1) = �(�)√2 ( � � + �)exp(� ∅) (93a)
B(2) = B(0)√2 (-i i + j)exp(-i ∅) (93b)
B(3) = B(0)k, (93c)
where ∅ = ωt − ��, a phase factor, and i, j, k are the three unit vectors in the direction of the axes x, y, and z, respectively. Although the existence of the B(3) field has been a subject of controversy both pro and con over recent years, Evans recently claimed [96] that these magnetic field components encompassed by the relations (93a-c), along with the electric field components as well as the components of the magnetic vector potential (A), are themselves components of a Beltrami-Trkalian vector field relations (assuming the Coulomb gauge div A = 0). This is readily verified in the case of (93ab), since they present the form of the circularly-polarized solution to the Moses eigenfunctions of the curl operator we have discussed formerly in connection with turbulence in fluid dynamics.
Associated with the above developments, is the increasing importance given to hypercomplex formalisms for modeling the symmetries in elementary particle physics and quantum vacuum morphology. As discussed in former papers [97,98], the author believe that the most appropriate algebra for describing a hypothesized vortical structure for quantum-level singularities, as well as their macroscopic counterparts (Beltrami-type fields), is the biquaternion algebra (hypercomplex numbers of order 8) – the Clifford algebra of order 3, represented by the Pauli algebra Cl(3,0) such as previously examined in the Rodrigues/Vaz model. For instance, it is known that in a macroscopic Euclidean context, biquaternions are required to describe the kinematics/dynamics for the most general twisting (screw) movement of a rigid body in space [99,100]. It is therefore suggested that the most suitable formalism for screw-type EM fields of the Beltrami variety should transcend a traditional vectorial treatment, encompassing a para-vectorial hypercomplex formalism akin to the Clifford (Dirac) algebra used effectively to describe the electron spin in a relativistic context [101].
It is a conclusion in this regard that the founders of vector field analysis were remiss in failing to take into consideration account of the significance of the Beltrami field topology in addition to the traditional solenoidal, lamellar and complex-lamellar fields. An inclusion of the thorough examination of the Beltrami condition in the development of the vector calculus, would possibly have brought attention to the important intimate association of this field configuration with non-Abelian mathematical structure. If the history of vector analysis had taken this path, it is possible that the architects of vector field theory and classical electrodynamics, would not have been so quick to indiscriminately sever its connection from the natural quaternion-based foundation. Perhaps the recent work by Hillion/Quinnez, Rodrigues/Vaz, Evans, etc. [102,103] showing the necessity of considering non-Abelian models in electromagnetism, will be instrumental in helping to set the future of classical EM theory and vector field theory in general, on a firmer foundation.”

http://vixra.org/pdf/1207.0080v1.pdf

On 1/28/2014 3:38 AM, EMyrone@aol.com wrote:

On 28/01/2014, at 7:00 PM, EMyrone@aol.com wrote: 
To Norman Page: I tried this link but I got "requested resource not available". Can you give me an idea of what it says about B(3)? Anything rude and out dated can be disgarded by now as a case of severe indigestion. The known animosity merhcants included Lakhtakia, Rodrigues, Bruhn, 't Hooft, Buckingham, Barron, Atkins and to a lesser extent Hehl (answered in comprehensive detail in UFT89). All of these have been disgarded entirely by the profession for several years now because they are so obviously biased and nasty. All criticisms of B(3) were answered years ago in all detail adn all these answers are well known. It was the other side who didn't allow discussion, not me, as Gareth Evans pointed out immediately. So the Nobel Prize committee should get it right and get their facts straightened out. After the Higgs thing they are very nearly a laughing stock, and that is a pity. The Nobel Prize should be awarded for service to humankind as well as for the individual subjects: physics, chemis try, pe ace, literature, medicine and economics. What use is a non existent boson? From: norpag@att.net To: EMyrone@aol.com Sent: 27/01/2014 18:11:09 GMT Standard Time Subj: Beltrami fields Forgot the link - Norman http://vixra.org/pdf/1207.0080v1.pdf 

On 28/01/2014, at 7:00 PM, EMyrone@aol.com wrote:

To Norman Page:

I tried this link but I got “requested resource not available”. Can you give me an idea of what it says about B(3)? Anything rude and out dated can be disgarded by now as a case of severe indigestion. The known animosity merhcants included Lakhtakia, Rodrigues, Bruhn, ‘t Hooft, Buckingham, Barron, Atkins and to a lesser extent Hehl (answered in comprehensive detail in UFT89). All of these have been disgarded entirely by the profession for several years now because they are so obviously biased and nasty. All criticisms of B(3) were answered years ago in all detail adn all these answers are well known. It was the other side who didn’t allow discussion, not me, as Gareth Evans pointed out immediately. So the Nobel Prize committee should get it right and get their facts straightened out. After the Higgs thing they are very nearly a laughing stock, and that is a pity. The Nobel Prize should be awarded for service to humankind as well as for the individual subjects: physics, chemistry, peace, literature, medicine and economics. What use is a non existent boson?

Analytical Solutions of the Beltrami equation

Google “Helicoidal solution of the Beltrami Equation”, first site, “Generalized Helicity Conservation and Beltrami … ” gives many interesting solutions. For example:

B sub X = T2 sin 2TZ + T3 cos 2TY
B sub Y = T3 sin 2TX + T1 cos 2TZ
B sub Z = T1 sin 2TY + T2 cos 2TX

where T1, T2 and T3 are arbitrary functions.

In a message dated 29/01/2014 14:19:17 GMT Standard Time, writes:

I just wrote to Myron that it is difficult to find an analytical example (in cylinder coordinates) for the helical structure in the figure of the Reed article. If we want to investigate such structures it probably cannot be reduced to 2D because of the rotation in Z direction. However a Beltrami equation without this property should be possible to be formulated in 2D spherical coordinates.

Horst

Doug Lindstrom hat am 29. Januar 2014 um 14:35 geschrieben:

There is a good chance that flexpde can handle it using complex variables. It would be worth a try using radial symmetry to reduce the problem to 2D. Doug

On Wednesday, January 29, 2014, Horst Eckardt < mail> wrote:

PS:
Doug, is there a chance to solve the Beltrami equation numerically by the FEM method? I guess it has to be in 3D.

Horst

Horst Eckardt < mail@horst-eckardt.de> hat am 29. Januar 2014 um 08:42 geschrieben:

Myron, Doug,
The review article is good, nethertheless I have an understanding problem with interpretation of the Beltrami equation. There are interesting diagrams how such a flow can be imagined, but to my understanding the curl of a vector is always perpendicular to the vector itself. This seems also to be the case in the longitudinal flux field examples. How can the curl be parallel to the vector?

Horst

EMyrone@aol.com hat am 28. Januar 2014 um 18:53 geschrieben:

Many thanks, this is an important review by Donald Reed.

To: EMyrone@aol.com
Sent: 28/01/2014 13:45:47 GMT Standard Time
Subj: Re: Fwd: Beltrami fields Forgot the link – Norman

I’ll send the link again . I think this is from the Reed paper you refered to . Here is the bit on B3 Regards Norman.

“Evans/Vigier Longitudinal B(3) Field and Trkalian Vector Fields
Developed over the past decade, concurrently with both Hillion/Quinnez and Rodrigues/Vaz SU(2) EM field models, but based upon a different non-Abelian gauge group, is the so-called Evans/Vigier longitudinal B(3) field representation [89-93]. In this model, a Yang-Mills gauge field theory [94] with an internal O(3) gauge field symmetry [95] is invoked to account for various magneto-optical effects which are claimed to be a function of a third magnetic field vector component that has been termed B(3). O of the central theorems of O(3) electrodynamics is the B-cyclic theorem:
B(1) x B(2) = i B(o) B(3)*, (93)
A conjugate product which relates three basic magnetic field components in vacuo defined as:
B(1) = �(�)√2 ( � � + �)exp(� ∅) (93a)
B(2) = B(0)√2 (-i i + j)exp(-i ∅) (93b)
B(3) = B(0)k, (93c)
where ∅ = ωt − ��, a phase factor, and i, j, k are the three unit vectors in the direction of the axes x, y, and z, respectively. Although the existence of the B(3) field has been a subject of controversy both pro and con over recent years, Evans recently claimed [96] that these magnetic field components encompassed by the relations (93a-c), along with the electric field components as well as the components of the magnetic vector potential (A), are themselves components of a Beltrami-Trkalian vector field relations (assuming the Coulomb gauge div A = 0). This is readily verified in the case of (93ab), since they present the form of the circularly-polarized solution to the Moses eigenfunctions of the curl operator we have discussed formerly in connection with turbulence in fluid dynamics.
Associated with the above developments, is the increasing importance given to hypercomplex formalisms for modeling the symmetries in elementary particle physics and quantum vacuum morphology. As discussed in former papers [97,98], the author believe that the most appropriate algebra for describing a hypothesized vortical structure for quantum-level singularities, as well as their macroscopic counterparts (Beltrami-type fields), is the biquaternion algebra (hypercomplex numbers of order 8) – the Clifford algebra of order 3, represented by the Pauli algebra Cl(3,0) such as previously examined in the Rodrigues/Vaz model. For instance, it is known that in a macroscopic Euclidean context, biquaternions are required to describe the kinematics/dynamics for the most general twisting (screw) movement of a rigid body in space [99,100]. It is therefore suggested that the most suitable formalism for screw-type EM fields of the Beltrami variety should transcend a traditional vectorial treatment, encompassing a para-vectorial hypercomplex formalism akin to the Clifford (Dirac) algebra used effectively to describe the electron spin in a relativistic context [101].
It is a conclusion in this regard that the founders of vector field analysis were remiss in failing to take into consideration account of the significance of the Beltrami field topology in addition to the traditional solenoidal, lamellar and complex-lamellar fields. An inclusion of the thorough examination of the Beltrami condition in the development of the vector calculus, would possibly have brought attention to the important intimate association of this field configuration with non-Abelian mathematical structure. If the history of vector analysis had taken this path, it is possible that the architects of vector field theory and classical electrodynamics, would not have been so quick to indiscriminately sever its connection from the natural quaternion-based foundation. Perhaps the recent work by Hillion/Quinnez, Rodrigues/Vaz, Evans, etc. [102,103] showing the necessity of considering non-Abelian models in electromagnetism, will be instrumental in helping to set the future of classical EM theory and vector field theory in general, on a firmer foundation.”

http://vixra.org/pdf/1207.0080v1.pdf

On 1/28/2014 3:38 AM, EMyrone@aol.com wrote:

On 28/01/2014, at 7:00 PM, EMyrone@aol.com wrote: 
To Norman Page: I tried this link but I got "requested resource not available". Can you give me an idea of what it says about B(3)? Anything rude and out dated can be disgarded by now as a case of severe indigestion. The known animosity merhcants included Lakhtakia, Rodrigues, Bruhn, 't Hooft, Buckingham, Barron, Atkins and to a lesser extent Hehl (answered in comprehensive detail in UFT89). All of these have been disgarded entirely by the profession for several years now because they are so obviously biased and nasty. All criticisms of B(3) were answered years ago in all detail adn all these answers are well known. It was the other side who didn't allow discussion, not me, as Gareth Evans pointed out immediately. So the Nobel Prize committee should get it right and get their facts straightened out. After the Higgs thing they are very nearly a laughing stock, and that is a pity. The Nobel Prize should be awarded for service to humankind as well as for the individual subjects: physics, chemis try, pe ace, literature, medicine and economics. What use is a non existent boson? From: norpag@att.net To: EMyrone@aol.com Sent: 27/01/2014 18:11:09 GMT Standard Time Subj: Beltrami fields Forgot the link - Norman http://vixra.org/pdf/1207.0080v1.pdf 

On 28/01/2014, at 7:00 PM, EMyrone@aol.com wrote:

To Norman Page:

I tried this link but I got “requested resource not available”. Can you give me an idea of what it says about B(3)? Anything rude and out dated can be disgarded by now as a case of severe indigestion. The known animosity merhcants included Lakhtakia, Rodrigues, Bruhn, ‘t Hooft, Buckingham, Barron, Atkins and to a lesser extent Hehl (answered in comprehensive detail in UFT89). All of these have been disgarded entirely by the profession for several years now because they are so obviously biased and nasty. All criticisms of B(3) were answered years ago in all detail adn all these answers are well known. It was the other side who didn’t allow discussion, not me, as Gareth Evans pointed out immediately. So the Nobel Prize committee should get it right and get their facts straightened out. After the Higgs thing they are very nearly a laughing stock, and that is a pity. The Nobel Prize should be awarded for service to humankind as well as for the individual subjects: physics, chemistry, peace, literature, medicine and economics. What use is a non existent boson?

Numerical Solution of the Beltrami equation

Looks good, the more complete problem is as just defined in Note 257(2).

PS: Beltrami equation

There is a good chance that flexpde can handle it using complex variables. It would be worth a try using radial symmetry to reduce the problem to 2D. Doug

On Wednesday, January 29, 2014, Horst Eckardt <mail> wrote:

PS:
Doug, is there a chance to solve the Beltrami equation numerically by the FEM method? I guess it has to be in 3D.

Horst

Horst Eckardt <mail@horst-eckardt.de> hat am 29. Januar 2014 um 08:42 geschrieben:

Myron, Doug,
The review article is good, nethertheless I have an understanding problem with interpretation of the Beltrami equation. There are interesting diagrams how such a flow can be imagined, but to my understanding the curl of a vector is always perpendicular to the vector itself. This seems also to be the case in the longitudinal flux field examples. How can the curl be parallel to the vector?

Horst

EMyrone@aol.com hat am 28. Januar 2014 um 18:53 geschrieben:

Many thanks, this is an important review by Donald Reed.

To: EMyrone@aol.com
Sent: 28/01/2014 13:45:47 GMT Standard Time
Subj: Re: Fwd: Beltrami fields Forgot the link – Norman

I’ll send the link again . I think this is from the Reed paper you refered to . Here is the bit on B3 Regards Norman.

“Evans/Vigier Longitudinal B(3) Field and Trkalian Vector Fields
Developed over the past decade, concurrently with both Hillion/Quinnez and Rodrigues/Vaz SU(2) EM field models, but based upon a different non-Abelian gauge group, is the so-called Evans/Vigier longitudinal B(3) field representation [89-93]. In this model, a Yang-Mills gauge field theory [94] with an internal O(3) gauge field symmetry [95] is invoked to account for various magneto-optical effects which are claimed to be a function of a third magnetic field vector component that has been termed B(3). O of the central theorems of O(3) electrodynamics is the B-cyclic theorem:
B(1) x B(2) = i B(o) B(3)*, (93)
A conjugate product which relates three basic magnetic field components in vacuo defined as:
B(1) = �(�)√2 ( � � + �)exp(� ∅) (93a)
B(2) = B(0)√2 (-i i + j)exp(-i ∅) (93b)
B(3) = B(0)k, (93c)
where ∅ = ωt − ��, a phase factor, and i, j, k are the three unit vectors in the direction of the axes x, y, and z, respectively. Although the existence of the B(3) field has been a subject of controversy both pro and con over recent years, Evans recently claimed [96] that these magnetic field components encompassed by the relations (93a-c), along with the electric field components as well as the components of the magnetic vector potential (A), are themselves components of a Beltrami-Trkalian vector field relations (assuming the Coulomb gauge div A = 0). This is readily verified in the case of (93ab), since they present the form of the circularly-polarized solution to the Moses eigenfunctions of the curl operator we have discussed formerly in connection with turbulence in fluid dynamics.
Associated with the above developments, is the increasing importance given to hypercomplex formalisms for modeling the symmetries in elementary particle physics and quantum vacuum morphology. As discussed in former papers [97,98], the author believe that the most appropriate algebra for describing a hypothesized vortical structure for quantum-level singularities, as well as their macroscopic counterparts (Beltrami-type fields), is the biquaternion algebra (hypercomplex numbers of order 8) – the Clifford algebra of order 3, represented by the Pauli algebra Cl(3,0) such as previously examined in the Rodrigues/Vaz model. For instance, it is known that in a macroscopic Euclidean context, biquaternions are required to describe the kinematics/dynamics for the most general twisting (screw) movement of a rigid body in space [99,100]. It is therefore suggested that the most suitable formalism for screw-type EM fields of the Beltrami variety should transcend a traditional vectorial treatment, encompassing a para-vectorial hypercomplex formalism akin to the Clifford (Dirac) algebra used effectively to describe the electron spin in a relativistic context [101].
It is a conclusion in this regard that the founders of vector field analysis were remiss in failing to take into consideration account of the significance of the Beltrami field topology in addition to the traditional solenoidal, lamellar and complex-lamellar fields. An inclusion of the thorough examination of the Beltrami condition in the development of the vector calculus, would possibly have brought attention to the important intimate association of this field configuration with non-Abelian mathematical structure. If the history of vector analysis had taken this path, it is possible that the architects of vector field theory and classical electrodynamics, would not have been so quick to indiscriminately sever its connection from the natural quaternion-based foundation. Perhaps the recent work by Hillion/Quinnez, Rodrigues/Vaz, Evans, etc. [102,103] showing the necessity of considering non-Abelian models in electromagnetism, will be instrumental in helping to set the future of classical EM theory and vector field theory in general, on a firmer foundation.”

http://vixra.org/pdf/1207.0080v1.pdf

On 1/28/2014 3:38 AM, EMyrone@aol.com wrote:

On 28/01/2014, at 7:00 PM, EMyrone@aol.com wrote: 
To Norman Page: I tried this link but I got "requested resource not available". Can you give me an idea of what it says about B(3)? Anything rude and out dated can be disgarded by now as a case of severe indigestion. The known animosity merhcants included Lakhtakia, Rodrigues, Bruhn, 't Hooft, Buckingham, Barron, Atkins and to a lesser extent Hehl (answered in comprehensive detail in UFT89). All of these have been disgarded entirely by the profession for several years now because they are so obviously biased and nasty. All criticisms of B(3) were answered years ago in all detail adn all these answers are well known. It was the other side who didn't allow discussion, not me, as Gareth Evans pointed out immediately. So the Nobel Prize committee should get it right and get their facts straightened out. After the Higgs thing they are very nearly a laughing stock, and that is a pity. The Nobel Prize should be awarded for service to humankind as well as for the individual subjects: physics, chemis try, pe ace, literature, medicine and economics. What use is a non existent boson? From: norpag@att.net To: EMyrone@aol.com Sent: 27/01/2014 18:11:09 GMT Standard Time Subj: Beltrami fields Forgot the link - Norman http://vixra.org/pdf/1207.0080v1.pdf 

On 28/01/2014, at 7:00 PM, EMyrone@aol.com wrote:

To Norman Page:

I tried this link but I got “requested resource not available”. Can you give me an idea of what it says about B(3)? Anything rude and out dated can be disgarded by now as a case of severe indigestion. The known animosity merhcants included Lakhtakia, Rodrigues, Bruhn, ‘t Hooft, Buckingham, Barron, Atkins and to a lesser extent Hehl (answered in comprehensive detail in UFT89). All of these have been disgarded entirely by the profession for several years now because they are so obviously biased and nasty. All criticisms of B(3) were answered years ago in all detail adn all these answers are well known. It was the other side who didn’t allow discussion, not me, as Gareth Evans pointed out immediately. So the Nobel Prize committee should get it right and get their facts straightened out. After the Higgs thing they are very nearly a laughing stock, and that is a pity. The Nobel Prize should be awarded for service to humankind as well as for the individual subjects: physics, chemistry, peace, literature, medicine and economics. What use is a non existent boson?