I can say that in many ways I hope you wrong. I published a paper last year on the categorical equivalency between Tsirelson bound on quantum mechanics and spacetime structure. My goal is to illustrate how divergences in quantum gravity can be absorbed into unobservable nonlocal hidden variables. It is a sort of renormalization procedure.
Your recent drop here is not due to me. I am tabulating plausible future scores on a copy of the FQXi essay page, and have graded rather few so far.
Cheers LC
Dear Peter,
Thanks for reading and commenting on my essay. I have read and enjoyed your essay and just rated it highly. It was very informative. Unfortunately, my education in QM is not sufficient enough to break through the barriers I specified via way of Dennis Polis in my paper, but I am glad you are trying and may have succeeded. Describing life in this way well be an important part of the answer to consciousness, and indeed the question to which this contest is founded.
All the best with your future efforts in this and other intellectual regards.
Jack
Philosopher.io
Dear Peter,
In your previous comment to me you say "On RNA, I identify a 'mutation' (evolution) not creation mechanism, equivalent to people having to decide if they're spinning clockwise or anti clockwise with Earth when standing exactly on the equator. Both answers may result in that case."
This is a very good example of something that I have found concerning most people in this world and that is that they are extremists. When confronted with an observation, most will look for the most immediately conveniently found answer and accept that as the true answer and tend to reject all other answers. In the above example you say both answers may result and you are right in that observation because it would be likely that any who were standing looking to the north when the sun came up would give one answer and those who were looking south would give the other answer. A true scientist on the other hand, would look for all of the observational information and then give a complete answer based on all of it. First after seeing the sun come up and go down several times he would come to the conclusion that either the sun was going around the earth or the earth was rotating on its axis. To determine which, he would look at the background stars and see that they also seemed to be moving in coordination with the sun's movements. This would mean that either the sun and all of the stars, etc. were revolving around the earth or the earth was rotating on its axis. The logical conclusion would be that since the sun and all of the stars would almost certainly contain much more mass than the earth, the earth must be rotating on its axis. If he then stood facing north when the sun came up he would see the sun come up on his right side. If the sun was considered to be relatively stationary in comparison to the earth during one rotation time, he would come to the conclusion that the earth was rotating clockwise when looked at from his current position or if he were to back up off of the earth and move down until he was above the South Pole. If he then went back to the equator and turned around so he was facing south, when the sun came up it would come up on his left side. From this he would see that the earth was traveling to his left. From this he would come to the conclusion that it was rotating counterclockwise when looked at from his current position or if he were to back up off of the earth and move down until he was above the North Pole. If he then stood facing east, he would see that as he continued to travel around the curve of the earth, he would be traveling down compared to his current position which he would consider to be the top for reference purposes relative to his standing position on the earth with the earth under him. If he then turned to face west he would see that the earth in front of him was coming up over the curve of the earth toward him forcing him to move backwards compared to his initial position. From all of this information put together he could truthfully say that the earth was rotating clockwise and counterclockwise and was traveling up and down and also to the right and to the left depending on the given input parameter conditions. Not to say that there are not any other possible parameter variations or ways of looking at them.
Of course, we do not always have time to analyze all observable details, but when confronted with another possibility than what is currently believed, most people will just deny it and never even check it out to see if it has merits. That reaction is one of the things you have to be prepared for in this world when you try to give a new concept or even a different way of looking at an old one. So when that happens to you, just smile within yourself and think, "That's earthlings for you." Wait a minute that somehow doesn't seem quite acceptable. How about, "That's humons for you." Still could use some tweaking. Maybe, "That's man for you." I may have to consider that a little more yet. Just be glad when you find any of the few who are able and willing to think, regardless of where they are from or who or what they are. For any who can hear it, think of what God has to go through, he has to try to reach and convince all of us of his love for all of us, most of who are not just denying what he is telling us, but are actively trying to prove that he doesn't even exist.
Sincerely,
Paul
Hi Peter,
I have missed your intellectual contribution on the main forum but I don't blame you as many of the recent discussions have not been of the standard when you actively participated.
I am also happy you put in an essay despite not being fairly treated in the past. I also only decided to put in a contribution almost at the last minute.
You may wish to view and comment later although I see from one of your responses above that you may not be a fan of a single universe.
I see your continuing effort in this essay to demystify quantum mechanics. This is an area that still contains too much magic for my liking so illuminating contributions like yours are very much welcome.
All the best in your endeavors,
Akinbo
Stefan,
Sorry, I've done that so many times I forget how invisible it can be. It was supposed to be implicit in the essay but the work limit cut it to the bone. Now this is representation remember (as you can't absorb and re-emit stuff!);
1. Take one spinning sphere and a dynamometer (dym) or 2 able to record linear momentum AND rotation (or 'curl'). Link its output to a standard pair of photomultiplyers (or avalanch photodiodes), or just tabulate & feed it in later.
2. Find a girl called Alice or play both roles if you want. The 'entangled' pair are antiparallel so say horizontal & opposite polar axes as the diagram in the essay. But you don't have to go 2 light yrs from yourself so only need one sphere to represent both (AND detector field electrons).
3. Now just touch anywhere on the surface of the rotating sphere with the dym. Then go 180o round the other side (the condition when Bob & Alice's dials are set THE SAME) and repeat. I think all can see that the findings at 180o will be the exact OPPOSITE. (i.e. for equators one is UP and the other DOWN, for poles S= clockwise N= anti..).
4. Now either Bob or Alice can rotate their dials (=field angle). If one is REVERSED it simply means you don't need to walk round and the finding is identical. (that may need a bit of thought- just imagine Alice has a separate sphere and flips it 180o
5. At each POLE you'll find linear momentum is zero. But 'curl' (rotation) is at Max, and the INVERSE at the equator, where 'curl' is 'uncertain'. You can stay at a pole and Alice can go just HALF way round (turn her dial 90o) or ANY angle.
6. Now you'll find that the AMPLITUDE of each property (linear/curl) changes NON-LINEARLY; by the cosine of the latitude on the sphere (angle from centre, over 90o) Lets say from zero to 1. AND of course from zero to MINUS 1 round the other side (180o from 1).
7. So what we have when plotted with 'angle' on the x axis is TWO cosine curves, offset by 90o[/sup). (We already know the linear momentum from geophysics, and 'curl' is simply the inverse). But as fermions are quite small we need amplification, so we send the signals into the photomultiplier (pm) tubes, one of which which will produce a 'click' at a threshold amplitude. (Normally '2-channel' set in opposite directions).
8. Now these pm's and pd's are quite clever. Ihe input amplitude is magnified by cascade interactions in the field (pair re-emissions) which lets say is 'pre charged' so there's no amplitude loss.
9. Now when we draw cascades on a sheet of paper we get a simple mathematical 'doubling' progression. However nature ISN'T 2D! Reality is 3D(+t) so we'll get a CONE, which outputs the SQUARE of the input amplitudes! Which in fact we already know from Quantum Chromodynamics (which I found afterwards, costing 2yrs as QCD was a theory I'd only skimmed!)
And that's it. Shocking I know but that has done the 'impossible'. Plot the 'click' outputs and you'll get a pair of orthogonal curves changing by the SQUARE of the cosine of the angle of the detectors. QAM is simply complex OAM and NO NON-LOCALITY IS NEEDED. Einstein AND Bell were correct (just not entirely). The video shows how the so called 'measurement problem' and most other 'weirdness' resolves.
However I've found wide cognitive dissonance in academia and publishing. The essay was a self referring test of that, so far confirming it. It's simply human nature, which is what I identify we need to evolve to advance.
Do ask questions, you should have quite a few.
Peter
Lawrence,
Oh dear. That's the exact inverse of good science! And that may be the greatest problem we have. If I derive a hypothesis I'll go to the ends of the earth to DISprove it! That's the 'gold standard' scientific method, (which I'm sure is why the pure triple filtered science I end up with works!)
At present if a non eminent academic publishes a paper or even just teaches students something he will feel WEDDED TO WHAT HE WROTE! That's the worst way to do science, and, I suggest, the biggest block we have on advancing understanding.
May I suggest you goal should be to find conclusively IF "divergences in quantum gravity can be absorbed into unobservable nonlocal hidden variables" or NOT! the not being as valuable a finding. In fact may I also suggest it'd be useful to re-write that sentence in English (as arXiv now demand!) so it actually means something to your average ('Sci-Am level') reader!
I hold you in higher regard than to downmark essays Lawrence. Unfortunately it seems that doesn't go for all.
Best
Peter
I am not sure what you see as a problem. The goal is to see if this can be mathematically realized. As for disproving that this is the domain of experiment. You generally do not disprove a theory with a theory.
The idea is that the metric structure is categorically equivalent to the Tsirelson bound, then for Einstein spaces where R_{ij} = Ag_{ij}, for A a constant the quantized curvature or expectation is similarly bounded and divergence removed. This requires going beyond the Weyl tensor spacetime physics (which is where I have worked) to the full Riemann = Weyl Ricc domain. This is rather tough as it is similar to gauge theories with sources that are notoriously difficult to work.
LC
Dear Peter,
thanks for your answer.
I like your creativity and that you do not give up with your goal of demystifying QM. So please don't take it personal that i am not convinced.
I thought you present a quantum mechanical experiment (at the microscale). Your experiment may be creative and intelligent form the point of view to build up some analogies between your experiment and the quantum world, as it should be according to your theory (here the first critique: Your experiment and the assumed properties of quantum particles build up a circular reasoning). But a spinning macroscopical sphere is not necessarily the same as a particle. Surely, you want to convince the audience that it indeed is in some sense, but i sincerely think that one cannot call this an experiment regarding what you want to prove - because you don't experiment with the objects you want to find out something, but with some macroscopic replacements.
If i do not understand you wrong, what you want to achieve is to map the probability distribution curve (which is nicely sinusodial, or cosinusoidal) onto some physical processes, and what would be better than using a sphere for this. I can understand that you are fascinated with this approach and i surely cannot disprove it (without delphing into your arguments or due to the fact that i do not have access to the mechanisms you describe).
I therefore do not judge your approach, i am simply not convinced. Especially some macroscopic spheres do not necessarily say something about microscopic particles. I think if you really want to establish a new interpretation of QM, then you had to develop some theory which is able to predict something different from QM. Yes, i know, you will say you exactly had already done this, but as i remember, your predictions were all just retrodictions. Besides that, probing the behaviour and properties of the quantum world, i think, necessitates to operate at the quantum level and not with macroscopic spheres.
I will not downrate you according to my opinions, because nonetheless i think you are a creative thinker and i think i understand the fascination you have for your approach. Therefore i will not rate your essay.
Best wishes,
Stefan Weckbach
Stefan,
You'd need a deep understanding of QM, which you agree you lack, and to overcome cognitive dissonance to be 'convinced'. I understand & predicted that. But anyway;
1. QM suggests ABSOLUTELY NO classical mechanism can reproduce those orthogonal cos2 curves! ('QM's predictions') My 'surrogate' simplification shows that to be false. How close to the 'actual' mechanism it is has no relevance in that respect. (I state regularly that rotating spheres are great simplification of particles!)
2. Using the simple explanation to help understanding means you jumped to a false conclusion. In fact the REAL mechanism of absorption/re-emission came FIRST. You may remember it as Discrete Field dynamics, localising 'c' on fermion field re-emissions. There is then no 'circular reasoning'.
3. You misunderstand QM, the formulation of which is ITSELF is purely based on experimental findings. That means the 'correct scale' experiments have all been done and all entirely support my (absorption momentum transfer) mechanism. I analyzed the key Aspect and Weihs experiments i.e. here; DOI: 10.13140/RG.2.1.3754.1287. or (note this was incomplete;)
Experiments at the quantum scale would only then repeat what's already been done. That's NOT the task. The ONLY problem was findings some classical mechanism able to reproduce the findings. Dirac's twin stacked spinor maths are just fine, but now we have an inkling of the physical process the formula REPRESENTS.
4. The other important part of all this is that ALL the 'spooky' interpretations fall away. Too shocking and unbelivable to accept I quite understand, but the fact is they do. See the definitions in the above paper.
a) 'Entanglement' effects simply come from the anti paralell axes.
b) 'Superposition' refers to the ACTUAL momentum transferred subject to detector electron orientation (everything is RELATIVE!).
c) Wavefunction collapse' simply comes from absorption and re-emission, which solves the 'measurement problem'.
d) Non-integer spins come purely from relative y or z axis rotations as the video shows. (very compressed one of some basics here;)
https://youtu.be/WKTXNvbkhhI 100 Second Video
I'll even explain what's really happening in the 'quantum eraser' experiment without needing 'backward causality' if you can't see it (it's not easy first time).
Again, I have no delusions that humankind is able to assimilate and rationalise this quite 'different' understanding of nature, even with Occams razor. However I do feel obliged to steer anyone interested 'back on track' when they depart. The track will remain invisible to most until well trodden. I thank you kindly for that interest.
Very best
Peter
Lawrence,
"I am not sure what you see as a problem"
I suggest the need for us to attempt falsification of our theories is far MORE important for theorists than experimentalists.
It seems using purely maths has led us away from that. Might that be a reason we're 'wandering' in such a dense forest of untenable and/or unfalsifiable theories!? Do you not also agree we need to remember at times that just because something is describable mathematically doesn't mean it can or does really happen 'physically'?
In experiments, it's invariably not the output data that's meaningfully important but the interpretation put on it, which is theoretical physics and too often based on previous flawed assumptions and interpretations. I suggest then that DISproving those assumptions, mainly theoretical or mathematical, may be far MORE more important that showing something MAY be possible.
Is that not reasonable?
Peter
Hi Akinbo
Great to hear from you. I only moved to QM as a test of 'discrete field' SR, and as the quanta ultimately drives everything. My SR solution passed the test, so exposing the error in the assumptions underlying QM, giving a unified description, and much more!
An earlier test addressed the flaws, paradoxes, anomalies & unexplained in 'Concordance' (doctrinal) cosmology. That too surprisingly succeeded. The more coherent model free of all the anomalies etc united the Hawking & Penrose models into a single 'recycling' mechanism, at work in the galaxy evolutionary cycle (giving the mass function growth you invoke by pair production). I'll confirm your correctness on that in your essay blog.
Unfortunately significant evidence suggests that some of your adopted starting assumptions were unreliable and probably wrong. Not necessarily the 'bi-bang per se as that needs little improved understanding (and some logic) to be a recycling (re-ionizing) mechanism.
Did you know the only original 'evidence' for accelerating expansion was cosmic redshift? It was only ever 'one possible interpretation', but repeated often enough, like all lies, it becomes 'truth'. I've shown an irrefutable mechanical production of redshift over time which is rather better as it's less problematic. Will the gatekeepers let it surface? Your guess will be correct!
VIDEO; Time Dependent Redshift; http://youtu.be/KPsCp_S4cUs let me know if you understand it all.
So just correcting the expansion rate alone removes a host of issues and allows a more coherent picture to emerge!
I enjoyed your essay and it's original approach, and am sure you'll enjoy mine, though it may be more testing than many!
Very Best
Peter
Dear Peter,
thanks for the answers. Your theory enables a similar kind of hidden-variable theory that has implications for the question of qualia. Let me be as original as you and claim that the impression of the color red is - at its fundamental level - not due to a certain wavelength, but due to the fact that a red stream of light consists factually of *red* photons. Via absorption an re-emission photons with all kinds of colors are produced naturally. Due to the process of absorption an re-emission, in the brain the impression of all colors is facilitated by the re-emission of the apropriate color mixture photon ensembles. These photons are projected in a cinema-like process in a holographical, encoded fashion onto the homunculus - what 'proves' that there must be some homunculus which can filter and decode a kind of holographical colored data-stream of photons. I am not sure if this also holds for all the other senses :-)
I ponder over wether your mechanism is strictly deterministic and wether or not the maths of your theory should lead to an infinitely precisely defined location for every particle at detection. Since you have reproduced the cosine-curves, one should expect that interference patterns and all the rest should be arise like a perfect printer plot, without some positive accidents in the destructive parts. Please explain.
Peter, i hope you can swallow some humour, not because your theory is silly as such, but how can it make a difference if it has no experimental consequences? The difference would only be in changing some worldviews. People would believe in absorption an re-emission and in your mechanisms instead of believing in entanglement etc. Would this improve the world or make it a better place, beyond changing some minds in the scientific community?
Jack,
Many thanks. You're far from alone in not understanding QM, Feynman said effectively (I paraphrase) that those who claim they do are delusional! It may be a GOOD thing you haven't been educated on it as you'd have had to swallow it hook line & sinker to pass the exams! That's not to say the maths don't model the findings ok, it's the 'interpretation' that went off the rails from the start.
Just to advise you I'm now scoring your essay, in line with my positive comments.
Best
Peter
Again you can't disprove a theory with a theory. A theory is falsifiable, while information that falsifies a theory is not. Hence you do not falsify theory with theory. Of course you can show a theory is inconsistent and thus false because it is internally flawed. Experimental data does support Bell theorem which rules out local hidden variable or classical-like physics underlying QM.
LC
Lawrence,
I entirely agree with all of that. Very well put. We can only EVER reveal inconsistencies, which is the task, but never 'prove' anything absolutely (even with experiments come to that!) so my point remains. I agree 'disprove' wasn't a perfect word but that is a little semantic.
And I entirely agree 'experimental data supports Bells theorem' which was anyway self evidently correct. And that same experimental evidence also entirely supports the model I identify, which in NOT a hidden variable theory, indeed is consistent with Bells predictions.
The only problem you have here Lawrence is Cognitive Dissonance. That's entirely to be expected. I'll give you a few more quotes of John Bell (which I didn't find till AFTER I derived the model and was trying to disprove it, including scouring every page of his; compilation; 'Thinkable and Unthinkable in QM')!.
"We are interested only in the possibility of hidden variables in ordinary QM and will use freely all the usual notions." p.2.
"..fundamental obscurity in quantum mechanics. Our theorists stride through that obscurity unimpeded... sleepwalking?" p.170
"The quantum phenomena do not exclude a uniform description of micro and macro worlds...systems and apparatus." p.171.
"I think that conventional formulations of quantum theory, and of quantum field theory in particular, are unprofessionally vague and ambiguous. Professional theoretical physicists ought to be able to do better." p.173.
"It may be that a real synthesis of quantum and relativity theories requires not just technical developments but radical conceptual renewal." p.172.
"...the new way of seeing things will involve an imaginative leap that will astonish us. In any case it seems that the quantum mechanical description will be superseded." p.27
"...the 'Problem of Interpretation of QM' has been encircled. And the solution, invisible from the front, may be seen from the back.." p.194.
Bell was very wise. A "radical leap that will astonish" is not one most readers of an essay contest are going to immediately be able to accept (for the reasons I identify in the essay). Indeed if anyone says; 'Yes' wow! I see; brilliant!' I'll most likely assume they're crazy! The mechanism none the less self evidently does what Bell predicted, which most people wrongly assumed he thought 'impossible'!
So did you actually watch the video? Classic QM It's just that from what you write it appears you didn't.
Best
Peter
yehuda
I liked Bohm but he didn't crack the problem. I find close analogies between the original strings and helical path ellipticities. But a more important point on self organisation;
I suggest the REAL start of conscious intelligence is when;
A BRAIN IS ABLE TO ORGANIZE AND ARRANGE STORED INPUT TO 'IMAGINE' FUTURE SCENARIOS, TRIGGERING MOTOR NEURONE RESPONSES, WHICH THEN LEADS TO CREATION OF WHAT WE TERM 'INTENT' and 'GOALS'.
Can anyone think of a better learning 'step point' for the definition?
Peter
Paul
Wow, that's longer than many essays! I'll try;
"If a Schrodinger sphere actually exists in nature, and not just as a mathematical construction, what is it composed of and how does it actually function to produce that helical path? How can it be observed? In your theory what limits the speed of light to C?
The sphere surface is the plane wavefront of any signal. Imagine a supernove in space. The light travels at c in all directions creating a growing sphere (just one of a packed sequence of them). If it hasn't reached an obsever he hasn't yet seen it!
Now image a rotating 'photon' or pair of charges at any point. The propagation (translation) at 'c' produces the helix we find in Photonics at ALL points on the sphere surface. If the rotational axis does NOT orthogonal to the sphere surface we get elliptical polarity. As the sphere expands the radius of each orbit increases. However; 'c' is LOCAL so although the 'sphere expansion' rate is c (or 2c considering the whole thing) the speed round each local orbital path CANNOT increase so the orbital time increases.
Simple geometry shows this can produces cosmic redshift - WITHOUT requiring acceleration expenasion of the universe!
Back to 'c'. Propagation speed modulated locally by fermion interactions. If ALL fermions re-emit at c then we'll always FIND c locally so NO PROBLEM EXISTS which needs paradoxical mathematical gymnastics to solve!
"....How would this develop the complex molecular structure of RNA?". The 4 dots preceeding that sentence represent a lot of doctrinal assumptions. As in computors, it only takes one tiny original input or design flaw to make EVERYTHING it produces from then on illogical gobbledygook full of inconsistencies, anomalies and paradoxes . That's what's happened. Revert to my 'Discrete Field Model' etc essays 5 years ago onwards and the adjusted input removing all those inconsistencies, anomalies and paradoxes is explained. Any 'shear' perturbation of the condensate produces identiacal 'pairs', with reverse spin orientations (the 'Higgs process'). RNA is at a much larger scale with more complex 'proteins' but the fundamental structure is fractal.
"Generally in a recycling universe the big crunch destroys everything created in the previous cycle. How does your theory work in that respect and if things in some way survive from one cycle to the next what is the observational evidence of that?"
The (smaller fractal) galaxy model shows us best. Not quite ALL matter is recycled. Most is re-ionized (solving that one!) o and the old is mixed with the new, i.e the hypervelocity stars already spat out whole on the axis of our own AGN WON'T be re-ionized this time round. Same with the outermost halo matter seen in 'ring' galaxies. i.e. Google Centaurus A. (note also the helical form of the superluminal (collimated) jet outflows) A mass of evidence exists, identified in this paper; http://www.hadronicpress.com/issues/HJ/VOL36/HJ-36-6.pdf or Academia, or DOI: 10.13140/RG.2.1.4540.5603.
Enough for now.
Peter
Dear Peter,
I highly appreciate the excellent essay in the quest for classical quantum mechanics. Your thoughts are very close to me.
And I strive to fully understand the mechanisms of the universe «We do know physical motion and interactions exist, but we won't know if any algorithm is correct until we fully understand the mechanisms».
And I'm using «Non-linear 'layered ... ' architecture».
And for me the main «a more important point on self organisation»
Few people are constructively trying to find the answer to this question.
You are considering neural networks, but how do think your how works self-organization at the micro and macro level in a dynamic universe?
By what mechanism «'matter' condenses from the sub-matter continuum condensate energy on perturbation»?
And why, as a result of chaotic collisions in accelerators, are formed exactly identical particles, atoms or electrons?
Kind regards,
Vladimir,
Thank you for understanding and agreeing these important advancements in understanding. I look forward to reading your essay.
"You are considering neural networks, but how do think your how works self-organization at the micro and macro level in a dynamic universe?"
I've identified fractal toroidal self organization, the largest local toroid (whithin oblate spheroids) being our current the universe itself and the smallest in 'matter' as fermions (all the same but dipoles) then probably the same at quark and smaller as 'dark energy' continuum/condensate states. I've studied and published on the very visible example of galaxies (Active Galactic Nuclii) which has wide implications, including recycling cosmology, resolving a whole tranche of anomalies as well as allowing unification of relativity and QM.
"By what mechanism «'matter' condenses from the sub-matter continuum condensate energy on perturbation»?" Most know it as the Higgs mechanism, but an additional macro 'spin state' as paired 3D vortices, which we know as electrons (wrongly thought of as different things 'positrons' when viewed from the other pole. The local concentrations of condensate energy cause a paucity in the vicinity which is maintained as an energy (or 'gravity') gradient or 'wave' until the 'matter' evaporates again.
"And why, as a result of chaotic collisions in accelerators, are formed exactly identical particles, atoms or electrons?" As above really, by 'shear' force, and all fermions, which also hold & dissipate (so conserve) the high 'input' energy. I discuss this is an earlier essay, indeed many important aspects in essays from 2010 onwards, mainly as 'discrete field' dynamics. I suggested it'd be 2020 before it was 'visible' to doctrine (I'm an optimist!)
Implicit derivation of Cosmic Redshift by local helical expansion without requiring accelerating inflation and other key aspects are in this video; Time Dependent Redshift http://youtu.be/KPsCp_S4cUs.
There are many other papers on various aspects of the model. Last years here was seminal and scored top (but wasn't in the prizes). I'm not a mathematician and most do like their numbers and symbols!
I'll put you essay at the top of my 'read' list.
Peter
Dear Peter,
Thanks for the excellent answers to the questions. I got your work on ResearchGate and saw a wonderful film. Now I need to process all the information received and see the old essay, it's not easy - I'm sorry that I'm not an English speaker.
I hope that the ideas in my essay will be interesting to you.
A direct link to the ResearchGate file in your message does not work. For download I must first copy the link, then remove the extra. I came across this, you need to erase all the gaps between [link and /link] when building links.
Kind regards,
