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I wish to present the paper “6-th MILLENNIUM PRIZE PROBLEM (NAVIER – STOKES EQUATIONS) IS SOLVABLE BY CLASSICAL METHODS OF MATHEMATICAL PHYSICS” (arxiv.org/math-ph, endorsement code is: QB7N8D). It has been reported at the international mathematical conference (XII International Scientific Kravchuk Conference, v.I, 15-17 May, 2008, Kyiv. pp.197-198. ISBN: 978-966-432-031-0). The summary in Russian is published in materials of this conference.
The summary in English is published on your Almanack

You can endorse for math-ph. And you have papers about NAVIER – STOKES EQUATIONS. Therefore I ask to endorse my article for math-ph.

New version of this paper has alternative proof.
I wish to switch these results into the second section of the manual “Paradoxes of continuum mechanics and of contiguous areas of a knowledge. Section 1. New approaches to statements and solutions of some classical problems of mathematical physics” (in Russian) http://www.amazon.com/dp/3639135954 .
Therefore before issuing of the second section I would like to consider these results on pages of your Almanack.

In case of need I will send all text of this paper.

Yours faithfully,
Alexandr Kozachok

It would not surprise me that the Valmiki Ramayana is a later retelling of stories that were extant long before then. After all, every language in India has its own Ramayana, not all of them a retelling of Valmik. A story that arose as a local legend in Ayodhya could have become embellished and made into a pan-Indian tale. We see this process of local heroes becoming national Gods even now, as with the Ayyappa phenomenon in Kerala.

http://threeroyalwarriors.tripod.com/id10.html

There is evidence that there might’ve been the story of Rama and Sita being told in the time of Buddhist suttas (Pali texts, Samyukta Nikaya, mention the telling of the abduction of Sita). The Valmiki Ramayana (2:109:3 4)itself might be a post Buddhist text, it mentions that the wise (Buddha) should not associate with athiests.

Well one way to look at this is the Back bone - If you are familiar with Yoga this makes sense! because As above (skies) so below(body),
Viswa

regards sree

Can u send me the PDF version of any good malayalam
calendar 2008

Regards
Aneesh

The nonlinearities of the Navier-Stokes equations make them very different from the Laplace or other well known equation of mathematical physics.

Kozachok A.A., Kiev, Ukraine

Formulated by Clay Mathematics Institute the sixth Millennium Problems about existence and smoothness of solutions of the Navier – Stokes equations periodically was discussed at numerous forums (http://grani.ru/Society/Science/m.112524.html). On recognition of some commentators the complete presentation of problem’s solution can demand about thousand pages for mathematical formulas (http://lib.mexmat.ru/forum/viewtopic.php?t=4289). The author of Official Problem Description–Charles Fefferman has set the task about demonstration of existence and smoothness of the solution, instead of solution’s obtaining. However, the Navier-Stokes equations can be reduced correctly to more simple classical equations of mathematical physics . The problem of an existence proof of solutions of such equations is not so actual.
It is known, and are identical to infinitesimal magnitude and velocity of the relative modification of a volume element of the strained medium. Therefore divergency of acceleration , probably, there is a magnitude, identical to a acceleration of the relative modification of the same volume. In that case for incompressible liquid alongside with requirements it is necessary to accept .
The requirement for incompressible liquid is formulated by analogy and proved. Operation will convert the Navier – Stokes equations to the three-dimensional Laplace equation for pressure at some limitations of a mass force vector. The time enters into Laplace equation as parameter.
Laplacian of the Navier – Stokes equations (pressure – harmonic function) and a change of a variable (velocity on acceleration) allow to gain system of conventionally independent integro-differential equations with acceleration’s components . In that case components of the acceleration for ideal incompressible liquid are harmonic functions too. The change of a variable allows to use boundary conditions of an adhesion of a fluid absolutely correctly. According to this requirement vectors of acceleration on firm immobile boundary line are equal to null.
Conversion of the Navier-Stokes equations to more simple equations has actually removed a problem of an existence proof and smooth finish of their solution on a background. Such demonstration in view of that one of required variables is a harmonic function, it is possible to not fulfil. It is possible to use known effects about properties of harmonic functions or representation of the common decision of the Laplace equation(http://continuum-paradoxes.narod.ru, the link « Manual, p.1 », p. 58).
More in detail on a site http://continuum-paradoxes.narod.ru, the link “Russian pages”, “Sixth Millennium Problems (NAVIER-STOKES equations) is solvable by classical methods”.

Yours faithfully, Alexandr Kozachok

As far as I can make out, it was some combination of dead reckoning and celestial navigation. If you can sight a fixed star and a planet within a short time (about ten minutes) you can get longitude. I suspect that some version of this is what they did, and corrected the estimate by their knowledge of the ships speed and currents. The trouble is that on a moving ship, it is hard to take sightings accurately.

If you do find out more, please let me know.

Can anybody can help me?

Thanks

Pereira Goncalves
eng_pg@hotmail.com

Surely, myth and historical fact are hard to disentangle after all these centuries. I doubt if the Buddhist writers intended to snub anyone.

Rajesh,
I don’t find anything humorous about this situation. Humour is punishable by three years of rigorous imprisonment. Irony will only get you flogged. As for Kalam, how many regiments has he?

Let me assure the international public that the elections will be held on January 2008 as previously announced. But it may take a while to get there because I have issued a Presidential decree that this is year 64 in our Calendar. Only fitting that the Calendar be redefined for my birth date.For, I am the Savior of Pakistan. Christmas will now be celebrated on August 11. Back to the saving the country from itself.

And here’s Mushie’s videoconference with Kalam(when he was still Prez of India)

http://www.yaxis.in/2007/05/apj-abdul-kalam-pervez-musharraf.html

Thanks for this. I don’t know about enlightened moderation but I have been enlightened for sure.

For example, he will be ordering everyone to wear pink underwear from now on. To make it easy for the police to verify compliance, underwear shall now be worn on your head. He has given me a picture of himself so attired (see above). Now you know how to do it.

Remember that it is now punishable by up to three years in jail to ridicule public officials such as the President General. I am sure the evil-doers are trembling in their boots.

Keep up the good work. The nation needs you!

I’m deeply grateful to you for bringing such an esteemed personality to the blogosphere. Those of us who watch Pakistan closely have no doubt been very lightened.

General Musharraf’s readers would do well to check out this recent video clip.

Yours is the view of many modern American Evangelical and Pentecostal Christians. There was a time when such views would have been condemned as heretic. Both the Orthodox and Catholic hierarchies take a dim view of any idea that you can have a personal relationship with Jesus not mediated by the Church.

The Revelations are perhaps the most controversial of the Biblical texts.

Peter is a word of Indo-European origin; in Hindi, `Pathar’ means rock even now. Names of Greek origin were common in the Holy Land at the time of Jesus: Andrew is an example. So it is not strange that Jesus would chose a name that is of Greek and not of Semitic origin for his disciple. Since the earliest versions of the New Testament avaliable to us are in Greek, it is difficult to get to the original word as used by Jesus.

The pun only makes sense if Jesus meant Simon to be the Rock upon which he (Jesus) would build the church. The image is that Peter is the Rock while Jesus is the builder: that God would be the hand that shapes the material world into a Church.

In the first instance, I am heartened that, unlike the reality-averse brand of politically fueled pseudo-science pedaled by the likes of the current US administration, you do not seem to be denying the material fact of global warming, its overwhelmingly probable causes, nor its likely physical outcomes if present forms and patterns of production-consuption persist . Your brand of reasoned optimism appears to stem from a sense of how we might deploy our collective creative resources to meet the likely scenarios that we at this time, with our present forms of knowledge frame as catastrophic.

So, I am glad that you appear to acknowledge that the predictive science is sound. Insinuating that it is otherwise has been the insidious intent of the spin that corporations such as Shell have spent many millions of dollars propagating (’Global warming or hot air?” - no more than willful obfuscation in the guise of debate).

Of course, we can not know with mathematical certainty what outcomes await us, of course we are figuring systems of great complexity, on ranges of scale from the cosmic to the molecular, and of course one of the things that we do know with certainty is that there are certain things we cannot know with certainty. That said, because of the nature and scope of these predictions, and the powerful consensus behind them, it would be unspeakably irresponsible to adduce the existence of formal uncertainty, even slight substantive uncertainty, as reason for proceeding as if what will not necessarily be the case will not be the case.

I wanted to make clear at the outset that I am not reading your brand of intelligent optimism as the rhetoric of ‘the-market-will-deliver-us-from-a-situation-we-are-denying’. That said, I also think you underestimate the strength of the interests that deploy this rhetoric (this underestimation is disconcerting because in many ways your tone sounds superficially similar to such rhetoric). For instance, I grew up in the Niger Delta region of Nigeria, and there, for half a century, multinational oil companies (aligned with the interests of the world’s richest nations and in league with their colluding junior parters of the Nigerian military and their junior partners the local elites) have devastated a rich and diverse ecosystem, destroyed cultures and lives, and driven a people to armed resistance.

Perhaps your optimism about the possibilities of humans collectively striking out in new directions should be somewhat tempered by a figuring in of the forces of inertia that such interests as are vested in the current industrial-ideological status quo exert. Indeed, in the shadow of the Cheney-Bush position — so transparently serving the interests of those that profit so massively from the current hydro-carbon energy configuration — it should surely be reckoned that such forces are not merely forces of inertia, but interests that, as in the Niger Delta, are willing to waste entire regions, to actively murder opponents (should this sound to wildly conspiratorial, one need only research into the well-documented complicity of Shell in various massacres and judicial murders in the Niger Delta, particularly during the 1990s).

Yes it is true, as you point out, we humans are ingenious, adaptable, creative. And creativity is by definition unpredictable, so, who knows what novel insights, what elegant technological solutions, what marvelous inventions await us tomorrow that will transfigure our forebodings of doom into bright futures?

But there are myriad ethical and political questions that you do not seem to address seriously. Given that we are likely to create massive displaced aggregate populations over the next decades, do we not need to urgently consider their fates, not in some abstract frame of possible technological fixes, but in actual, morally informed institutions of policy formation and implementation. One can talk, ironically, about migration from Bangladesh to Siberia, but what practical, material considerations are we to make for these displaced millions? Is it ethically acceptable to sacrifice the actual homes, cultures, livelihoods, lives even, of millions (of mostly already poor) peoples to some enthusiastic, but vague, vision of human creativity? It is the failure to answer, or even pose, such questions in your post that I find alarming.

For this is what your post does not acknowledge, global warming is in many sense a current crisis, not a future threat. For many non-human life forms, for instance, it is a present danger - extinction rates have increased precipitously over the last few decades. But most of all, for all its insight into the potentials of creativity, what your post does not admit of is that we might rather change our minds - that is, change some of the underlying premises that drive our hyper-consumption, change our values, be creative poetically. We need to ask, ‘what is the good life?’, not merely, ‘how can we find new ways to power our air-conditioners?’

I can see that it may be possible to predict what happens to an average quantity over along period of time ( century) even if you cannot predict its instantaneous value over a shorter time ( month). This must be the weather vs climate distinction you are making.

But I remain skeptical of such extrapolations in the case of global climate, as many small factors we don’t yet know about might have a big impact over a long period of time. The argument is that the Earth’s climate is unstable and is sensitive to small changes in parameters such the percentage of CO_2 in the atmosphere. If that is the cause (sounds plausible to me) how do we know that it is not also sensitive to other parameters not under human control such as a small change in the Sun’s luminosity or some random fluctuation in the ocean currents? It is only the general skepticism of an outsider, as I am not a geophysicist. My attitude is that even if I find the extrapolations rather bold, I should accept it tentatively as it is made by people who have more specialized knowledge. So let us assume that the warming will occur.

This is similar to the predictions a generation ago that a global population explosion will occur. It did. What did not happen are the dire consequences predicted from this explosion. A main reason is the green revolution: the food production rose at an even greater pace than the population. Also, I argue, the new people were far more ingenious and productive than anyone had expected. They are more than paying for their consumption, which is why the calamities predicted did not happen.

So my expectation is that the world will warm. Delhi probably will be hotter by two degrees, sea levels probably will go up a few inches. But life will still be better because we will find a way to create the energy that will pay for the required air conditioning. The area lost to human habitation from warming will be replaced by others that are gained. In any case there are still many quite habitable parts of the world, even in crowded countries like India. Arizona today supports a large population which would have been impossible before air conditioning and artificial irrigation. May be Rajasthan would be an analogue in India. So the problems are technological, economic and political to live with the warming, not finding a way of stopping it. Population control was not the main solution to the last crisis.

The comment about Bangladeshis moving to Siberia was meant as a joke. They are more likely to emigrate to parts of India. The flow of humans across political borders is a natural phenomenon that cannot be stopped by any fence or law. Bangladeshis will continue to find a way into India and people of Mexican origin will be the majority in the Southern US very soon, no matter what the current populations in these countries think about it. And both India and the US will be better off for it. People trying to survive are more ingenious than those trying merely to preserve their standard of living. There will be tensions and struggles along the way though.

All of which really brings us to the energy shortage. There is no question that this is the great technological challenge today as food production was in Norman Borlaug’s time. To take the example of his work, the funding for his research in Mexico came ultimately from the US through international agencies. The amount needed was tiny compared to the billions spent on weapons research or even fundamental scientific research. Apparently that was enough. In addition to scientific skills, Borlaug also had the vision and charisma to convince the developing nations to adopt his ideas. People of that sort are rare: maybe one in a billion, but then we should expect to have six of them in the future. One was enough to solve the food crisis.

So what will replace petroleum? We don’t know now, any more than people in the nineteenth century knew what would replace whale oil. There are several different energy problems. What is a non-polluting way of producing electricity? What is a way of making energy available to remote villages? How to cool and heat homes in cities? Or fuel for vehicles? I don’t know answers any more than any one else. But answers will be found when needed.

It might be a combination of the ideas we hear about these days. Solar for remote villages and for communications. Nuclear for making electricity: only in the US is there a knee-jerk reaction against it. Synthetic fuels (NOT biofuels) for airplanes and such. Better batteries for cars. Given the economic incentives, these solutions will catch on, just as third world farmers quickly adopted Borlaug’s high yield varieties. Much of this is not glamorous research scientifically. I think it is unlikely that some fundamental breakthrough like fusion holds the key. It is probably sweating the details of the science we already know. Borlaug was not Mendel.

There already are signs that creative people are thinking up solutions. I ran across a remarkable wind farm in Tamil Nadu.
Although I was making a joke about him, Arnold Schwarzenegger is in fact a good example of a politician who sees the opportunity presented by the energy shortage. California might be a better source of funding for energy research than the Federal Government. It has an economy as big as most countries, so it can be a major player.

The green revolution didn’t happen in Africa because, unlike India or Mexico, it did not have responsible governments which would take up the cause. Times have changed a bit. Perhaps the future will be different. Borlaug has been active there too.

http://www.ifdc.org/New_Design/Whats_New/Borlaug%2520AFS%2520Release%2520Final.pdf

It is true that the green revolution would not have happened without US subsidy of agricultural research. But the amount needed was small. It was an even lower priority in the fifties than energy research is today.

Thanks for an insightful comment. I know my audience is small, but I hope they are all like you.

The people of Bangladesh already find it difficult to immigrate to India as do the people of Mexico to the US. That’s not because of lack of living space. What makes you think Siberia will accept the people of the inundated world?

You are optimistic that we’ll find a substitute for petroleum and still maintain the same energy wasting lifestyle. I can’t see any natural resources at this point of time that can provide a carbon free substitute, carbon sequestering will increase fuel costs significantly, solar energy has yet to reach high efficiencies and you can’t fly planes with it, wind energy depends on the vagaries of nature, and so on. And with the tiny amounts of money being spent on energy research I don’t see any change attitudes in the near future. I think the global warming and energy issues are serious problems that need to be solved soon.

Also, why didn’t the green revolution work out in Africa? And would the green revolution have been possible without serious agricultural research (and subsidies) in the West?

Thanks for your appreciation. It always helps to know someone out there is reading what I write. It is like working on your garden. You would do it anyway, but it is good to know others like it too.
By the way do you subscribe to this blog?
–Rajeev

In two dimensions, ( with periodic bounary conditions) it was done some years ago by Dowker. I have generalized it to fluids on a 2-sphere with viscosity and coriolis force to model the atmosphere. The idea is to produce an effective theory of large scale structures in the atmosphere like hurricanes and the jet stream. I have done some simple numerical work also, which confirms the general ideas. With some help I may be able to turn this into a more realistic model of the atmosphere, including effects of convection ( which I have not yet included).

The idea is to average over small scale fluid motion to get an effective theory for larg scale structures. This is analogous to the averaging over the moleculr motion to get fluid mechanics, but carried one more step. The main point is that this second average involves dynamical variables with a non-commutative algebra (matrices in the simplest case of two dimension) rather than just functions or vector fields. Still, it is a very practical way of describing large scale objects because there are very few variables in the effective theory.

I hope to write more about this in another post.

M

It is my impression still that the papers you referred to are all about the two dimensional discrete non-commutative sphere, applied to regularize quantum field theory. For the current paper about three dimensional fluids in the continuum, it remains an open question whether any of the effects you talk about can happen. Actually it is possibly equivalent to the question I raise at the end. Does this really regularize the potential classical divergence in Fluid mechanics? That is, does the regularized equation have smooth solutions? I don’t know the answer. Let us keep talking.

I am also writing another paper on fluid mechanics on a two dimensional discrete sphere where many of the issues you raise are more directly relevant. But there I know that the equations do have smooth solutions for all time with the regularization: there is no analogue for a non-commutative anomaly in the application of fuzzy methods to this classical theory.

I have to read and understand your papers before I can write a detailed response. Couple of small points though. I am using the three dimensional sphere and not the two-sphere. Also, I am not using a discrete version of the sphere; for example not only the standard . Does that make a difference?

Moreover, my fluid mechanics remains classical even with the non-commutativity of the regularization.

Thanks for a thoughtful comment.

M

Here’s how to circumvent the problems you address.

Any time you replace a differential operator by a difference operator or an integral operator, you are introducing a coupling between points that are not infinitesimally close. What I mean by non-locality is that points that are not infinitesimally close are coupled together in the evolution equation. Violations of causality are a bit different, when the time co-ordinate is also affected. These can happen if you modify ( or discretize) relativistic theories. Since incompressible flow anyway has signals propagating infinite speed, this is not the issue in Navier-Stokes or Euler equations.

The more important point is that you would like to keep regularity when you introduce a cutoff. If the cutoff procedure itself introduces discontinuities, it will be even harder to study the regularity of solutions this way.