353. Vortex line representation. Cauchy invariant
APPLIED — By Dmitry Podolsky on April 13, 2009 at 12:08 pm
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Several days ago I’ve promised in comments to discuss dynamics of vortex lines in turbulent flows, today is probably a good day to start. And the natural starting point of course is the Kelvin theorem and Cauchy invariant.
Let us consider an ideal (inviscid, uncompressible) fluid described by the Euler equation
and incompressibility condition
.
A rather non-trivial fact (I bet you don’t know it 
) is that these two equations have infinite number of (non-local) integrals of motion. Of course, existence of these integrals does not make the Euler equation integrable – the latter can be clearly seen from the fact that there may exist solutions of the Euler equation that feature chaotic behaviour, turbulence. Nevertheless, it is interesting to understand the nature of these integrals, since it might also shed some light on physics of viscous fluid, described by Navier-Stokes equation.
Another video explaining how turbulence looks like in Lagrangian coordinates
The simplest way to derive these integrals of motion is to use Kelvin theorem. The latter states that velocity circulation
is conserved along an arbitrary time-dependent contour 
. Let us rewrite the expression above using Lagrangian markers 
instead of Eulerian coordinates 
. We have
,
where the integration is now taken along a static, time-independent contour 
. Since the contour can really be taken arbitrary, we apply the Stokes theorem and conclude that
is conserved in time for each point . The expression above is known as Cauchy invariant. Its physical meaning is actually simple – if coinside with Lagrangian markers corresponding to particles of the fluid in the initial moment of time,
,
initial vorticity of the flow. In other words, vorticity is frozen in into the motion of the fluid – fluid particles cannot leave the vortex line they belonged to in the initial moment of time, i.e., the only relevant degree of freedom for them is along the vortex line.
33 Comments
Hi Dmitry, when you say that a “chaotic” turbulence can’t be described by “integrable” functions, is that just an intuitive common-sense guess or is that something that you can actually prove?
Hi Lubos,
I think it’s a part of common lore that all Lyapunov exponents are negative (or zero) for integrable systems, while some of them are necessarily positive for chaotic systems. If the given system features chaos, it cannot be completely integrable. Basically, since vortex lines interact with each other through Coulomb interaction, I would expect that almost arbitrary configuration with non-zero vorticity and more than two vortex lines features chaos -I think, it’s a kind of trivial statement.
Cheers,
Dmitry.
Dear Dmitry,
you have surely tried
but I have learned nothing. Could you please define “chaos” for me? Is there some objective definition, or is “chaos” everything that you and the people who share the same “lore” with you don’t understand?
Concerning the talk about Lyapunov exponents, are you equating stability, predictability, and integrability? Sorry if I am picky but these three notions are at least a priori different for me – and in some cases they can be a posteriori different and very different.
Best wishes
Lubos
Sorry Lubos for saying that
, but if you don’t quite see the relation of chaotic behavior to Lyapunov exponents of the system, it would be good to check out the Wikipedia article on chaos 
Any good basic book on dynamical chaos will also help 
Hmm… interesting
Would you list those “some cases”?
Cheers,
Dmitry.
Ok, I am really sorry for being rude. The definition of chaos is strong sensitivity to the choice of initial conditions, and the relation to Lyapunov exponents of the system should be clear from that (Lyuapunov exponent is the rate of separation between liquid particles infinitely close to each other in the initial moment of time).
Unpredictability certainly holds for systems with positive Lyapunov exponents – in practice, you are not really able to follow the history of the given liquid particle with exponential precision. I think I do equate instability, unpredictability and non-integrability if Lyapunov exponents are consistently positive for all t. I would be happy if you give me a counterexample
, since I really doubt anything of stated above is currently questioned.
Cheers,
Dmitry.
Dear Dmitry, I think it’s almost clear that there are counterexamples to the claim that “chaos” with your definition above is the same thing as non-integrability.
N=0 QCD (or another generic gauge theory) and N=4 super Yang-Mills have the same Lyapunov exponents (don’t they?) but only the latter is arguably integrable. If integrability were the same thing as Lyapunov stability, don’t you think that one could simply see what happens with the latter, in order to establish whether or not N=4 is integrable (and it probably is)?
[If one measures the distances
, in the Hilbert space sense, supplemented by Schr?dinger's evolution, the distance stays constant. That's called unitarity. 
If one studies the expectation values of some classical quantities, things are different but it depends what classical quantities - i.e. what classical limit and approximations - we choose. The black hole evaporation will be unitary and thus distances-constant in the exact quantum sense but in the classical GR description, it will also be stable because things end up with a nicely universal thermal radiation. With some intermediate choices, picking a lot of classical degrees of freedom, the evaporation may start to look "chaotic" but it is never "quite chaotic" in the real world because the relevant Hilbert space is finite-dimensional for any finite black hole.]
But let me return to the N=0 QCD and N=4 SYM comparison: I don’t think that it is sensible, in any sense, to use the word “non-integrable” just because the functions that appear in a solution are unstable. For example, take one of the rolling tachyon solutions, and continue the solution so that it describes the runaway evolution. It is equally “integrable” because it differs from the previous integrable just by a sign flip. But it will give you unstable Lyapunov exponents if you do it right.
If one considers a quantum system, the Lyapunov stability depends on the eigenvalues, their organization, distances etc. while integrability – in the quantum sense – depends on the possibility to calculate these eigenvalues.
I feel that the way how you use the words “chaos” and “turbulence” are designed in such a way that no further insights about them are “desirable” because you secretly try to impose our current ignorance about any details what’s happening, and equate all “mysterious” adjectives with the very qualitative notion of ignorance. I don’t like it at all.
Something can “look” incomprehensible or “chaotic” but in science, one is only allowed to make well-defined propositions, and every statement about the equivalence of two words has to be supported by evidence, not by sentiments. Many things that are comprehensible today look incomprehensible in the past.
If you had good definitions of these adjectives and mathematical proofs of their equivalences, that would be interesting. But otherwise it looks like a game with prejudices to me. I
By the way, how do you define integrability in your sentence in the text? These are a priori vague words – because, in the case of integrability, it refers to some integrals of motion (foliations, whose required number is unclear) or, in the quantum/statistical context, functions that can be used to write the explicit result but one is never sure which functions are “simple enough” to deserve being called integrable. Look e.g. here
http://en.wikipedia.org/wiki/Integrable_system
Can you say which definition of integrability you refer to, or a different one? Sorry but I think that these “details” do matter. Otherwise it is just a game “how to impress oneself” by repeating those “noble” words such as “chaotic” or “turbulent”. There is nothing noble about these words. That’s why the words “chaos” etc. were used so many times e.g. in Alan Sokal’s hoax. In the corners he humiliated, it is a cheap buzzword used by imbeciles to look smart in the eyes of other imbeciles.
One more technical example. Consider the law of physics telling you that
z(x,y) = exp(y / zeta(1/2+ix)).
Is that integrable? Well, in my understanding, it’s fully solvable, so it probably is. Now, try to decide whether it is chaotic, unpredictable, turbulent, having infinitely many integrals of motion etc. You may make the formula a bit more complicated, add sums that will depend on zeta of multiples of x, and so on. Various well-defined properties will cease to hold at different moments.
So what I am really uncomfortable with is your (and not only your, it is common in people who like to talk about “chaos”) attempt to define a universal boundary between “chaos” (given by unpredictability, but assumed to be a universal, unchanging word) and “non-chaos”.
There’s no such a fixed boundary. As we understand more, different laws and situations become increasingly comprehensible, even if their evolution finely depends on initial conditions. Some of them may turn out to be exactly solvable in terms of quasi-elementary functions. As we learn more, the previous boundaries about different kinds of ignorance – that were blurred into a continuum of boundaries or one thick boundary – become sharply different.
Some people involved with “chaos theory” are even worse with this point. They try to “extend” the realm of phenomena and equations that belong into “chaos”, whatever it means. Because chaos is de facto defined by ignorance and unpredictability, this effort of theirs is nothing else than the attempt to push the scientific method away from an increasing collection of disciplines and objects to study.
If I summarize my point succintly, there exists no “chaos” of the type that would be a very important and objective word in science. And indeed, this is not just about this single word: I am also learning nothing whatsoever out of the random videos you are posting. All of them seem to be just some random videos. What’s really bad is that it doesn’t seem that we are expected to notice any patterns or probability distributions or sizes: we should just see that things look incomprehensible, right?
But even if we see nothing interesting going on in the videos, that doesn’t mean that the underlying science is unlearnable. After all, they were all produced by programs that were much shorter than the resulting video file, weren’t they?
Dear Lubos,
OMG, I cannot believe my eyes – I even apologized
Probably you’ve heard this many times, but it’s worth repeating – you would make a great pastor. I doubt that science was recently in need of pastors, though.
Leaving apart your very deep comments on philosophy of science (which, I should admit, do not sound very interesting/worth discussing to me) and complaints that you cannot understand anything on the videos I post (sorry, dude, but I did not hear a physical question you ask – just a lot of very general statements), let us focus on physics you discuss.
Do they?
The latter has a larger number of degrees of freedom, these degrees of freedom are coupled to each other. By the way, classical N=0 YM is a theory featuring chaos (check out the book by Smilga, which you have certainly heard of already at some point 
)
Is it, really? If so, can you give me a reference where I can read the proof of integrability?
Have you ever heard words “condensate fragmentation”?
I am afraid the only counterexample (N=0 YM vs. N=4 YM) you tried to present does not quite work.
Cheers,
Dmitry.
P.S. By the way, I have strong impression that you mix the notions of classical and quantum integrability
When I talk about chaos, I mean only classical theories – so, classical integrability/non-integrability.
Dear Dmitry,
I didn’t understand your “OMG = Oh My God?” comment. Was that a part of a prayer? What did the prayer have to do with an apology? Sorry, not getting this part.
Concerning the slightly more technical statements, you seem to be deliberately vague – that’s probably the very point of “chaos theory”. I am telling you that the signs of the Lyapunov exponents in N=0 and N=4 gauge theories are likely to be equal (and even Smilga is likely to agree with that, not sure about Marco Frasca haha), while the answers to the question whether N=0 and N=4 gauge theories are integrable are likely to differ.
I don’t have a complete proof of integrability – I am just indicating why your identification of the different adjectives seems sloppy to me, and is likely to be based on a mistake. If you want to show that it is not sloppy, you have to provide us with the proof: it is up to you. Meanwhile, we must stay with incomplete evidence, and my incomplete evidence
http://scholar.google.com/scho.....yang-mills
including many papers above 300 or 500 citations indicates that the N=4 theory is likely to be integrable. And even if it were not, the absence of a proof that it is not integrable is enough to see that your discourse about these matters is sloppy.
I was not mixing classical and quantum integrability: exactly because I appreciated different meanings of the word, I was just asking you what you meant by “integrability”. Indeed, when I say “integrability” without adjectives, I mean “quantum integrability”. I hope it’s not yet crime. The world is quantum, after all.
Most notions of classical integrability only apply in the classical limit, and all of these inevitably break down due to the limitations on scales imposed by the uncertainty principle and the finiteness of the Universe. Moreover, if the planar limit of N=4 super Yang-Mills is integrable, i.e. solvable, it makes the classical limit solvable, too. It’s just a limit, after all. The planar limit of gauge theory is already a classical theory, in other words. But this solvability is a different criterion than the stability conditions etc., isn’t it?
See my text against the religion of chaos
http://motls.blogspot.com/2009.....chaos.html
I don’t know what’s exactly “condensate fragmentation” but I don’t know how this buzzword itself could possibly weaken any of my arguments. If you think otherwise, could you please offer me an argument and not just the buzzword? I am unmoved by buzzwords and I am also baffled where your conclusion that my “N=0 vs N=4″ counterexample doesn’t work. As far as I can see, you have given no arguments for your conclusion whatsoever.
One more thing, dude. The fact that there are no physical questions one can ask about the videos you posted is exactly my point. One doesn’t learn anything out of them. The videos don’t answer any well-defined or semi-well-defined questions – and they don’t even inspire one to ask any questions. In this sense, they’re not scientifically interesting and maybe, they’re not relevant for science at all.
Best wishes
Lubos
Dear Lubos,
thank you for bringing controversy to the blog, it always lifts traffic
Here you go – OMG comment was made to indicate that the relation “physics/generic ranting using physics buzzwords” in your comment above is around 1/99.
Unfortunately, you are unable to go beyond the “likeliness” of both statements, that’s why I say that your counterexample doesn’t quite work
Also, observe
that citing the number of papers attempting to solve the problem but not quite getting the whole proof is not really an argument (unless, of course if you believe in democracy in science, do you?)
LS would be certainly able to cite hundreds of papers on LQG to support it, but this probably would hardly convince you that LQG is the way of the future, would it?
It is funny though that in turn you don’t want to provide us with the proof of your statement above – although it would be certainly much easier to present a working counterexample than to prove a general statement, wouldn’t it?
Well, unfortunately, dude, the context of our discussion is Navier-Stokes/Euler turbulence, where quantum effects are not that relevant, aren’t they? You actually remind me Laughlin who tries to use his quantum Hall effect studies to quantize gravity. The physics of quantum Hall effect is not necessarily relevant for an arbitrary phenomenon in Nature, though.
Well, that’s a second surprise for me today. When tachyon rolls down, condensate looses spatial homogeneity quite rapidly (we are both talking about classical problem, aren’t we?). Spatial distribution actually strongly depends on initial conditions, that’s exactly because Lyapunov exponents of the system are positive as you correctly pointed out. Lubos, if you will be unable to get this hint, I raise my hands – I have a lot of work to do today, while you don’t seem to have any – that’s unfair
However, dude, this just characterizes your inability to ask the relevant question
I’ll kindly teach you what to do in this situation – you just say “I” instead of “one”. It is simple really. You say things like “from my humble opinion”, “from my point of view”, “I think” etc., you don’t generalize and everybody is happy.
Remember I told you long time ago that Slavic people have long lost their ability to participate in civilized discussion, lost the very culture of human relations. That’s because of communism of course. Imagine that you start arguing like “according to Bohr, electron does not contain the whole universe”, and in reply tovarisch commissar squeezes your balls due to inability to support his position opposite to yours (except maybe by citing Lenin). This way, the community is quickly left without people able to talk like civilized persons.
However, communism is long gone, and we can (should!) learn to talk to each other and to our West-European brothers
in civilized way again. Lubos, your mind was poisoned by communists from your very childbirth! However, the hope is not yet lost forever (since sometimes you don’t talk like a troll). Actually, I believe that our future is bright.
Cheers,
Dmitry.
P.S. Mozhno prodolzhit’ konechno, hotya obsuzhdenie fiziki beznadezhno sorvano
A esli ser’ezno, esche raz tak sdelaesh’ – budu k eb… materi ubirat’ iz posta tvoi ssylki.
Dear Dmitry,
interesting. My feeling is that my comments contain the only physics in this thread – like the integrability of N=0 and N=4 gauge theories.
Well, you may think that the N=4 theory is not integrable. I think it is – and I’ve only explained you why I think that what you write is wrong, much like prety much everything that Lee Smolin is saying is wrong. The reason why I believe that N=4 is integrable is that I’ve evaluated the evidence, not because some papers exist.
The integrability is established at one loop and in various other “projections” and “perspectives”. I am not trying to convert someone, especially because I know it is impossible. I was just answering your question why I thought that what you were writing was crap. Once again, it is sloppy science even in the absence of complete evidence in either way because you are simply making assumptions you are not allowed to make.
I don’t mind resembling Laughlin in your eyes. If the question is whether the world is fundamentally a quantum world, and if Laughlin says yes, I am very happy to be with him – entirely – on the same ship. The “true”, infinite-hierarchy fractal structure of Navier-Stokes and other equations is a pure mathematical idealization that cannot be realized in the real world. You don’t seem to care whether assumptions are relevant in physics – and your assumption is fundamentally wrong in physics – but I do. Fine.
The questions about systems where such behavior can truly be defined by “limits” are purely mathematical problems. It’s OK, I have nothing against mathematical problems, but I prefer physics. And by the way, yes, I also disagree with your guess that the atomic perspective has no value for turbulence.
My comment that the videos didn’t inspire any well-defined scientific questions was not any “humble opinion” of mine – that’s why I didn’t identify it as a “humble opinion”. It was just a summary of the observed data. No one – I or anyone else – has asked any such question and you know that the reason is that these videos don’t lead to any such questions, qualitative or quantitative. Their being “ungraspable by words or formulae” seems to be the very point of these randomly looking videos.
Otherwise, I am in the Western region of a country in the Western European cultural space (where it has belonged for more than 1000 years). Learn your geography and history classes again. If you learned a different geography – from your Brezhev stalinist school system – where our country is a part of your Eastern Bear, you should better forget all the stuff that you have been taught at school because it’s stinky crap. Thank you for your consideration.
Best wishes
Lubos
Dear Lubos,
However, this physics does not quite have anything to do with Euler or Navier-Stokes turbulence. I really appreciate that you try to bring staff that you know on the table, but it is not directly relevant. That’s why I said you remind me of Laughlin
Actually, my belief (from 2003, I think – when I explained it to Marshakov in Toronto, he might remember it) is also that N=4 SYM should be integrable – you don’t really need to convert me. What I actually doubt is that Lyuapunov exponents of N=4 SYM have the same sign as ones of N=0.
It seems, your main mistake is that you always take yourself too seriously. In particular, you seem to often miss irony in my (and others’) words directed to you way. Apart from Laughlin, you also remind me of Don Kihot
Oh, so this is getting serious. If you really think that quantum gravity (or string theory
) corrections are important for all processes in quantum electrodynamics, I understand why you decided to leave science. You surprised me today for the third time.
I think I’ve explained you rather clearly the physics of the “Lagrangian turbulence” video. No second question followed, so I assumed that my explanation was clear enough for you. Apart from Laughlin and Don Kihot, you also remind me a 12-year old boy. After a lecture on special relativity given by an old gray professor the boy attacks the lecturer. He argues that the professor is wrong, pointing at the school physics textbook with the picture of Newton. He is absolutely mad, he shouts and calls professor “old f..cking idiot”, “marasmatic”, etc.
Do you find it ironic taking into account that you are 6 years older than me and were assistant prof. at Harvard once?
I went to school after Brezhnev’s death and started learning geography in 1986 with Gorbachev being our gensec at that time. Once again, I am afraid, you don’t know me well enough for the point you try to make to hurt my feelings
By the way, sure, Czechs are European, but some of them (you, for one
) were clearly so much influenced by the communist propaganda that they started talking like Vladimir Il’ich or Iosif Vissarionovich 
Cheers,
Dmitry.
Dmitry,
the simplest reason why I think you’re naive in thinking that atoms can’t influence turbulence is that turbulence has a scale-invariant behavior and some processes occur “uniformly” at each scale of the hierarchy.
Because the number of hierarchies goes like the logarithm of the size in atomic units, it may be damn important to know where the description breaks down, i.e. how large atoms are. The only way why you can so easily humiliate these important issues that you are a superficial person who only repeats confusion emitted by others – but who is actually not able to neutrally and scientifically think about similar issues.
Please learn to spell Don Quijote.
Well, if Gorbachev schools taught you that Czechia belonged to the Eastern civilization space, I can only say that Gorbachev hasn’t improved much about the schooling of little Soviets like you.
Well, it’s always better to be a 12-year-old genius who is right in every respect than his obnoxious would-be stalinist teacher.
Best wishes
Lubos
Dear Lubos,
please stop talking nonsense
Kolmogorov scaling breaks down at scale where viscosity becomes important. I think it would be useful for you to reread by posts on Kolmogorov turbulence or any other basic level book on it, since my posts don’t apparently go beyond the basic level. Viscosity is macroscopic parameter of the fluid, macroscopic in the same sense as, say, temperature. Different atomic structures will lead to the very same Navier-Stokes in the IR.
Cheers,
Dmitry.
P.S. Accept you correction regarding Don Quijote, thanks.
Viscosity can be calculated from the first principles, from basic atomic physics, and the typical scales, even under realistic assumptions, give the Kolmogorov distance scale just a few orders of magnitude away from the size of the atoms: the scale is often below a millimeter, which is technically included into microscopic realm in many cases.
At any rate, it is absurd to say that these processes have nothing to do with atoms. Viscosity and everything that follows from it is a process resulting from the interactions of atoms, and the size and other properties of atoms determine all the relevant macroscopic quantities.
After all, it is no coincidence that the bound on viscosity – the value for perfect fluids – is proportional to the entropy density, which is pretty much close to the atomic density. In other words, viscosity is at most just a few orders of magnitude from the density of atoms. Your thinking about these issues is stuck somewhere in the 18th century. In the real world, those classical idealizations have their limits, and the more you study things that change on short distance scales, the more important the atomic structure matters. You may be trying to solve an interesting math problem (not sure about your methods) but it is known not to be a fully relevant problem for physics.
We enters the chaos theory by looking into the transtion of simple iterative system into ergodic, unpredictable state, but still we do not understand the transit from laminar flow to turbulence. From this point I am not sure whether chaotic description of turbulence would help or not.
Dear Yi Zhang,
It certainly helps when we deal with weak turbulence – check out Zakharov, L’vov, Falkovich book. Basically, random phase approximation works since Lyupunov exponents for phases are positive, so that phases get randomized quickly.
Navier-Stokes (and Euler) turbulence is much more complicated than weak turbulence, since vorticity enters the equation, so no – and this point chaotic theory does not help much to understand transition to developed turbulence. But this wasn’t the topic of my post anyway.
Cheers,
Dmitry.
Totally agree.
Thanks.
Yi
Break the barrot!
bey burjuev:)
Dear Max,
this is not the way of the future, we are friends now, we are trying to learn things from our Western colleagues. Also, Lubos is not actually a burjui
Cheers,
Dmitry.
Are you(both Dmitry and Lubos) joking?
Or this is a sample of “civilized discussion”.:)
To theme:
Heuristic evidence: The enigmatic “information-like” behaviour of RG flow(one loop) (alike scale*log(scale) ).
Ergodicity appearance?
Cheers,
Max.
Dear Max,
Me – somewhat, Lubos – not sure
What do you mean by “information-like” behavior?
Cheers,
Dmitry.
Dear Dmitry,
I mean:
scale*log(scale)
Very like to behavior of the informational entropy in concentration measure regime.
Cheers,
Max.
Dear Max,
In reality, I have no idea. I think, there might be actually some connection but I’m not sure whether anybody is able to figure out this connection at the present moment. Gravity seems to have some intrinsic entropy associated with it in those particular problems where horizons naturally appear, and not-CFT – asymptotically free – quantum field theories (where coupling is running) might also probably have gravity duals. So what, you say? The answer is I don’t know
Cheers,
Dmitry.
I think so:
http://arxiv.org/abs/0801.3996
The calculations of the observables(partition function) may be considered as a probe statistics:
an iterative process of
gluing the Feynman tree diagram to the big horizon of the Universe(in perturbative language).
By means of strong or weak glue depends on measure(scale).
Thus the measure is being generated.
The entropy is an effective dimension(central charge analog)of this measure.
The Measure consentration(condensat by Witten)may appears, for example, in case of moduli (meta)stabilization (in the
chamber in moduli space surrounded by walls of marginal stability).
In my approach there is a Gravity dual because of measure-entropy-geometry(effective curvature) relationships(numerical).
Also there is an intermittency behaviour.
Alike a laminar-turbulence chain in (aero)hydrodynamics.
It is interestingly to discuss(construct) the moduli of hydrodynamics flow. I was hear about entropy/viscosity ratio(or something like).
Cheers,
Max.
Dear Max,
I have to admit – your thinking is too advanced for me to understand it (basically, I fail to understand every single sentence of your comment). For example, what the heck is
?
Witten has talked about many different condensates, what condensate in particular do you have in mind? What does
mean? How tree Feynman diagrams have anything to do with renormalization?
So far, my impression is that your picture cannot be right since renormalization is essentially UV (loop, not tree level!) effect, while horizon provides an IR cutoff.
Dmitry.
Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory
N. Seiberg, E. Witten
http://arxiv.org/abs/hep-th/9407087
for example.
I mean the process( time sequense) of the gluing of primitive tree diagrams( step by step).
In result is appearing the “big diagram” (with loops).
The “big diagram” increases and represents RG flow.
This is image of ergodic limit.
But I suppose a more interesting behaviour in pre-ergodic phase:
phase transitions, intermittency, memory…
P.S.
In my opinion you too strict in terminology.
It was my attempt to imagine this domain in more familiar form.
Math in my paper.
Just one link. If anyone has doubts about the borders of the Western cultural space, see the map at
http://en.wikipedia.org/wiki/Western_culture
Czechia has always been a part of it.
I attribute it to you being strongly influenced by Lenin’s (or any other communist’s) writings – one can compare Lenin’s conversation techniques to yours and find many surprising similarities. I think that is the main reason why you hate communism so much – it poisoned your brain to the level you cannot really overcome it.
Cheers,
Dmitry.
This is really cute. First of all, you have clearly no experience, genetic or social, with the Western culture, so you shouldn’t be surprised that what you write has nothing to do with the Western principles.
Second, all the amateur demagogists you list are far-left ideologues, pretty much communists. I don’t know how you could possibly link Western culture to any of these people, especially the ultracommunist bitch Hossenfelder. She has nothing to do with Western values whatsoever and if her country were correlated to her ideas, she should certainly live in North Korea.
By the way, I am surprised to find that my Alexa rank is higher than yours. Alexa ways are truly mysterious, taking into account that compete shows that your traffic is 10 times larger. Hmm.
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