## Another Millenium Problem May Have Been Solved 134

S3D writes

*"After recent verification of the proof of the Poincaré conjecture, another of the Clay Institute's Millenium Problems may have been solved. This new solution is for Navier-Stokes equations under physically reasonable conditions. Navier-Stocks equations describe the motion of fluid substances such as liquids and gases. Penny Smith has posted an Arxiv paper entitled 'Immortal Smooth Solution of the Three Space Dimensional Navier-Stokes System' which may prove the existence of such solutions."*
## Pretty nifty stuff (Score:2)

## Hm. (Score:5, Funny)

LOUD ones.

## Re:Hm. (Score:5, Funny)

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You sir have pwned me.

## pr0n (Score:2, Funny)

This new solution is for Navier-Stokes equations under physically reasonable conditions. Navier-Stocks equations describe the motion of fluid substances such as liquidswho needs a description of the motion of fluid substances? I want video, perferably in slow-motion and from multiple angles.

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## Smoother rendition possible... (Score:2, Funny)

Don't player hate, player appreciate baby.

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## Neat indeed (Score:3, Interesting)

Note: Not considering

P vs. NPas it is quite possibly unprovable.## Re: (Score:2)

It certainly couldn't ever be proven unprovable, like some things can be, since proving it unprovable would also prove there was no way to implement a conversion P = NP, and, therefore, P != NP.

Just because we can't prove it doesn't mean it's unprovable.

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How could it be unprovable?

Just because we can't prove it doesn't mean it's unprovable.

Godel's incompleteness theorems [wikipedia.org]

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## Re:Neat indeed (Score:5, Insightful)

Not necessarily -- it is conceivable that there exists a poly-time algorithm for an NP-complete problem,

butthere is no proof (within ZFC, say) that it is correct. The physical truth is certain -- but what we canknowabout the physical truth is limited.Now, I'm with you in believing that that's extraordinarily improbable, but math doesn't always respect what we consider to be likely.

In my opinion (as a complexity theory grad student), the "maybe P=NP is independent" speculation is bunk. There are genuine, interesting results talking about the limits of how we can resolve P vs. NP, but none of them come anywhere near logical independence, and giving up on a field-defining problem after 30-odd years is just very odd considering how long the really major open problems often take to solve. I believe the solution exists, and I hope it is found soon, but I will be unsurprised if it takes another 100 years or so while we get a better handle on what computation really means.

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Yes, that's conceivable but seems unlikely. A more likely scenario (and in fact my money is on it) is that we can eventually prove that ZFC can neither prove nor disprove P=NP, and in that case we don't know whether your scenario above is correct, or if on the contrary no such algorithm exists but ZFC is simply too weak to establish that.

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Yes, the axiom of choice would be the Z. Who was it that said that great quote, "The axiom of choice is obviously true, the well-ordering principle is obviously false, and who the hell knows about Zorn's lemma!"?

(the joke being, all three of those things are mathematically equivalent, if you accept the axiom of choice, you accept the other two).

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(And we won't even get into the correct attribution for CH/GCH.)

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You are fundamentally misunderstanding the incompleteness theorem. The incompletness theorem states that in a powerful axiomatic system, there are statements that are true, yet they cannot be proven using the system. That is, it is possible for there to be a polynomial algorithm for an NP-complete problem, but we cannot prove that it is polynomial-time. Conversely, it could be true that there is infact no such algorithm (ie NP != P), but we cannot prove this fact, even though it is still true. Pick up G

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So even if it does turn out to be independent of current accepted axioms, which I will admit I'm skeptical about, I feel that knowing that allows us to immediately add an axiom to make that view of math approach "physical computer hardware". Or, alternatively, to define a subset of all N

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AKA find index of Y in a sorted set X vs sort set X.

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Then look at this a finite state machine which takes a binary number and counts down.

Aka

11111 =

11110 =

11101 =

11100 =

11011 =

00000

Starting with a set of N binary digits counting down with this finite state machine takes at worst (N^2)-1 steps.

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There have been some problems (e.g. the weak pigeonhole principle) where it's been shown that any proof must be intractably large. There is some evidence http://citeseer.ist.psu.edu/cache/papers/cs/27779 / http:zSzzSzwww.wisdom.weizmann.ac.ilzSz~ranrazzSzp ublicationszSzPchina.pdf/raz02np.pdfthat [psu.edu] this is true of P!=NP.

If you think about this, there's a certain amount of poetic justice. NP-hard problems are solvable in principle, just not in practice. And the conjecture that P!=NP may be true in principle

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This Navier-Stokes thing seems to be more of an applied-math problem, and although I'm sure it's important, it's just not as exciting to me as the more abst

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> would be far more important.

I don't deny that applied math can be _important_. I only said that I find pure math more _interesting_. Plenty of things in this world are more interesting than important, or vice versa.

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Not really. Actually solving Navier-Stokes for concretely given boundary conditions is very much an applied math problem, maybe the most important one of them all, and it is done with computers and algorithms from numerical analysis. But the paper we're discussing here is pure math: she proves that for a certain class of boundary conditions a solution

must exist, without saying what it looks like or how to get it. It's of pure intellectua## Re: (Score:1)

P vs. NP on the other hand would be more than quite significant. It would turn the world upside down, inside out, and potentially make our mathematical abilities unstoppable. The day someone does an RSA challenge number by hand or in their head will be a grand day indeed.

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There are few results that depend on RH being false, however. The methods developed to actually prove RH will probably have more use as general methods than the simple fact of RH's truth will have.

In my opinion. I am not much of anything.

## Quite impressive (Score:5, Informative)

## Re:Quite impressive (Score:5, Informative)

## Re:Quite impressive (Score:5, Informative)

it's about the existence of a solution for certain boundary / initial conditions of the NSEs. This is still a very big deal because you can now expect correct results when doing numerical calculations. By the way you probably meant FEM (Finite Element Method), not "fractional element methods". FEM is rarely, if not at all used for solving the NSEs, you'd rather use Finite Volume Methods (applicable for structured and unstructured grids, as are FEM).

## FEM is used plenty for solving Navier-Stokes (Score:2)

The choice of method for solving the equations

doesseem to vary quite a bit between disciplines. Engineers tend to love FEM, while, say, atmospheric modelers seem to prefer finite-volume or finite-difference approaches.## Re: (Score:1, Offtopic)

Here's an example. Two board, one 3m and one 2m are laying crisscross in an alley, with one end in each corner of the alley, and laying the other end on the opposite wall.

Their intersection is exactly 1m from the ground, how wide is the alley?

This problem is very easy to find a numeric solution, but suprisingly difficult to fi

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You take a quartic equation [nerdparadise.com] and choose to call "exact" what is called "a solution by radicals".

Yes, a solution by radicals can be hard to find even when it turns out to exist. (Indeed quartics weren't solved by radicals until Ferrari in 1540.)

But the question whether a solution by radicals exists has nothing to do with whether a solution (period) exists. Indeed polynomial of higher degree have the latter (Gauss' fundamental theorem of algebra) but not always the f

## Re:Quite impressive (Score:4, Funny)

## A series of tubes? (Score:1)

You mean the series of tubes that make up the Internet?

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but rather the PROOF of THE EXISTENCE OF A FORMAL SOLUTION. You still have to find it,

either analytically or (most probably) numerically.

Bottom line: about this a mathematician gets horny, an engineer says SO WHAT!!!

Ciao

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existenceof solutions. At times, we even care about theuniquenessof such solutions. Or, how about convergence## Whuh? (Score:5, Funny)

## Why the constant Slashdot self-hate? +5 funny (Score:1)

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May be if you dropped watching so much porno and picked up a math book instead... :-)

## Someone had better tell the Formula One teams (Score:3, Interesting)

## Re:Someone had better tell the Formula One teams (Score:4, Informative)

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verygood results. I've worked with Comsol and Floworks simulations designing a variety of things - but mostly cooling loops. This is where the problem lies - these simulations are very computer intensive and even a simple simulation such as a cooling loop through copper (one bend) can take over a day to converge to a solution (and i would make all sorts of assumptions to cut the time down, like perfectly smooth walls). A desktop computer wouldn't even be## I solve 3 millennium problems before breakfast (Score:3, Insightful)

Wait for the peer review to begin. I've not seen anyone familiar with the field say anything about the paper yet, only then does it gain credibility.

FatPhil

## blink blink ! (Score:2)

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One of the things that I understood was a real problem with NS is that not only were there no existence proofs, but there were no uniqueness proofs. Does nayone know if the uniqueness question has been answered?

## Millenium? (Score:1)

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## I am not amused. (Score:1)

landnboth appear twice. But then, this is Slashdot; correct spelling may not be a reasonable expectation.## Cancel the survival gear! (Score:1)

## The toughest millenium problem of all... (Score:4, Funny)

thatone would be a million dollars ofanybody'smoney...## Re:The toughest millenium problem of all... (Score:5, Funny)

A mille

nnium ismille + annus: a thousand years.A mille

nium ismille + anus: a thousand assholes.If you get it wrong, you're anal; if you get it right, you're annual.

## What is the geometry? (Score:3, Informative)

Abstract of this postIt is a big deal for the mathematicians. That is allThe N-S Eqn has been "solved" in 2D using Velocity Potential, Stream Function approach. But in 3D stream function does not exist and the method does not extend. But in practice the only problem that is really "solved" even in 2D was was this driven cavity problem, a box with a moving wall.

Take the much more simple to solve for a hundred years, the Heat Equation. Analytical solutions exist for simple domains like a semi infinite plate or a box with Dirichlet boundaries. But in practice ANSYS sells numerical solutions to Heat Equations and the industry has been buying millions dollars worth every year. Similarly FLUENT (Recently acquired by ANSYS) does not have to worry its market has fallen out of the bottom. For real life geometries we will be using numerical solutions of NS Eqn for the foreseeable future.

Further though I could not see any geometry restrictions in the paper, it appears as though they have just proved solutions exist, and not actually solved it. Depending on the assumptions made and terms neglected, engineers may be able to build better turbulence ing out of this.

Caveat: Though I started out in CFD I have not read CFD papers for some 12 years. and frankly I dont understand much of the math in this paper.

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Caveat: Though I started out in CFD I have not read CFD papers for some 12 years. and frankly I dont understand much of the math in this paper.That's OK - this is slashdot.

Most commenters won't have even read the article, let get as far as failing to understand it!

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It is a big deal for the mathematicians. That is allI wouldn't go so far as to say it is only interesting for mathematicians. Fluid dynamics and Navier-Stokes especially, is what, for example, many 3D engines use to simulate water by now. Granted, they use simplified equations, usually only taking the surface into consideration, but any breakthrough in the theory their models are based on might have implications for those models as well. I'd say let's wait until a) those new findings have been properly p

## An important step (Score:5, Informative)

Back to the paper... While I am not a mathematician, the paper appears kind of rough to me - lots of punctuation errors, commas in the wrong place, unclosed parehtneses... I suspect this paper has not been fully through the peer review process. I don't know how the mathematicians do it, but I would say this paper is a draft (not discrediting the work - I am not quallfied to judge it - but it looks rough).

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I suspect this paper has not been fully through the peer review process. I don't know how the mathematicians do it, but I would say this paper is a draft (not discrediting the work - I am not quallfied to judge it - but it looks rough).Not that I think you are making an attack on mathematicians here, but I just want to comment on this for anyone that might construe it as such.

Mathematicians do subject papers to full peer review before being published in any reputable journal, but the arXiv is not a journal

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Proof: This is less than proved in the brilliant paper of [H]. QED.## Re: (Score:2)

## a fitting tribute (Score:2)

## Going to need to follow how this reviews (Score:2)

## What the World Needs (Score:2)

## Engineer here.. and can I say WOW! (Score:1)

## You'll have to tame your enthusiasm (Score:2)

If this holds up, the methods used are doubtless going to lead to better approximations and possibly - after a lot more research - to constructive methods. It's going to be exciting to see what ha

## An arxiv article does not a headline make (Score:2, Interesting)

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## Shoehorning in my field (Score:1)

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## another explanation (Score:1)

## Penny Smith's usenet posts! (Score:2)

## MOD PARENT UP (Score:1)

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## Withdrawn (Score:4, Informative)

## Paper is withdrawn (Score:1)

## If it was right... (Score:1)

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http://en.wikipedia.org/wiki/Catastrophe_theory [wikipedia.org]

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