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Actually, my response was off-topic since the article only mentionned mersenne primes in passing. It certainly doesn't discuss any theorems. Instead of lashing at me gratuitously, maybe you should take a deep breath while reading the article yourself.

If I can bring your attention here, you'll see why I imposed my condition the way I did.

I say goodday!

Simple.
Look at equation (10) here: http://scienceworld.wolfram.com/physics/EhrenfestT heorem.html

<A> is conserved (i.e. d<A>/dt=0) if A commutes with the hamiltonian. That is, if HA=AH, or [H,A]=0.

One of their geometric criteria is compactness, which they define as having the smallest possible perimeter. Does this mean they're districting it in circles?

For a mathematician, this actually a very, very funny joke. Basically, the Banach-Tarski theorem says that you can take a unit sphere...

"Cut it up" into six pieces...

Rearrange the pieces...

And get TWO unit spheres - BOTH identical to the original. The proof hinges on the fact that the six "pieces" concerned are so complicated and "jaggedy" that they cannot be said to have an absolute volume. More information here.

That was the first thing I thought of -

followed by
Lagrange numbers... not sure if it has anything to do with that, either :)

Horrible, but to quote myself:

But in two sentences it appears you've demonstrated that your zealotry for Debain can outweigh your vision for what could be best for the community.

s/Debian/yourself/

I'm not sure if I should be happy or upset that you're helping to prove that point. I'd like to thank the AC for his contribution to this thread which hit it on the mark and was about as good of a re-reply that I could have made. The heart of the post was not about you, Bruce Perens the individual, it was about you, the individual fronting the UserLinux project who also happens to be Bruce Perens. It's notable that your reaction was to focus in on yourself instead of the greater project. What really gets me is that you're a respectable and intelligent person, and if you're having trouble understanding that viewpoint then there is a lot to worry about.

If I were the one who stood up and announced I was developing a new Linux initiative with the same forward looking and positive goals as UserLinux nobody would listen to me (well, my mother maybe, but she's obligated, no?). But being who you are you've got street cred which nobody can deny you. This needs no explanation though, as you were quick to beat your own chest and list your accomplishments (which I promptly skipped over) in your reply. It seems the longer the list of accomplishments, the greater the lack of modesty.

You have so far given many reasons to respect your large scale, early work with Debian, and in turn have earned respect for yourself. But my original point still stands that there is so far no reason to hold UserLinux itself in higher regard then any other Debian based distrobution or even Fedora as the thread originator suggested. Frankly, your curriculum vitae is moot in the sense that it is yours and not that of UserLinux, a so far nonexistant, hypothetical product. It gives you the community standing and experience needed to lead the project, but those are still descriptors of singular you, not future efforts.

You have become a strange loop in your efforts in that: Bruce Perens does not need to prove himself, UserLinux itself does need to prove itself, currently UserLinux is Bruce Perens, Bruce Perens needs to prove himself.

Man, I thought Inna Gadda Da Vida was long.

In related news astronomers create a simulation of the universe as long and as big as the universe itself.

Of course, it didn't occur to me to take a look at the Science section before submitting my own copy of this story (which, since it has several other useful links in it, follows):

Michael Shafer, a graduate student at Michigan State University, took time out for a "short victory dance" upon learning his computer had discovered the 40th known Mersenne prime as part of The Great Internet Mersenne Prime Search. The number itself is 2**20996011-1 and when expressed in base 10, has 6,320,430 digits (zipped copy). However, this is not necessarily the 40th Mersenne prime; there could be another between the previous largest known prime (M39=2**13466917-1, also discovered by GIMPS) and this one. Also worth noting is the still-standing USD$100,000 EFF prize for the discover of the first prime of at least 10 million (decimal) digits. GIMPS clients are available for various operating systems as well as information on how GIMPS would distribute the prize. A press release on the achievement is available as well as several articles. Of course, this also means there's a new largest known even perfect number in town.

Of course, it didn't occur to me to take a look at the Science section before submitting my own copy of this story (which, since it has several other useful links in it, follows):

Michael Shafer, a graduate student at Michigan State University, took time out for a "short victory dance" upon learning his computer had discovered the 40th known Mersenne prime as part of The Great Internet Mersenne Prime Search. The number itself is 2**20996011-1 and when expressed in base 10, has 6,320,430 digits (zipped copy). However, this is not necessarily the 40th Mersenne prime; there could be another between the previous largest known prime (M39=2**13466917-1, also discovered by GIMPS) and this one. Also worth noting is the still-standing USD$100,000 EFF prize for the discover of the first prime of at least 10 million (decimal) digits. GIMPS clients are available for various operating systems as well as information on how GIMPS would distribute the prize. A press release on the achievement is available as well as several articles. Of course, this also means there's a new largest known even perfect number in town.

Of course, it didn't occur to me to take a look at the Science section before submitting my own copy of this story (which, since it has several other useful links in it, follows):

Michael Shafer, a graduate student at Michigan State University, took time out for a "short victory dance" upon learning his computer had discovered the 40th known Mersenne prime as part of The Great Internet Mersenne Prime Search. The number itself is 2**20996011-1 and when expressed in base 10, has 6,320,430 digits (zipped copy). However, this is not necessarily the 40th Mersenne prime; there could be another between the previous largest known prime (M39=2**13466917-1, also discovered by GIMPS) and this one. Also worth noting is the still-standing USD$100,000 EFF prize for the discover of the first prime of at least 10 million (decimal) digits. GIMPS clients are available for various operating systems as well as information on how GIMPS would distribute the prize. A press release on the achievement is available as well as several articles. Of course, this also means there's a new largest known even perfect number in town.

I understand it's possible all of mathematics could be a joke, but from what I have studied and know it would be highly unlikely for that to be true.

It does get a little hairy when you start reducing it to as basic a set of concepts as you can. You start getting hung up on certain things. The Axiom of Choice is a fine example. Almost all modern mathematics requires it to be true. It feels like it ought to be true. Then again youy can do nasty things like the Banach Tarski Paradox if you assume it true. Ouch.

Jedidiah

I understand it's possible all of mathematics could be a joke, but from what I have studied and know it would be highly unlikely for that to be true.

It does get a little hairy when you start reducing it to as basic a set of concepts as you can. You start getting hung up on certain things. The Axiom of Choice is a fine example. Almost all modern mathematics requires it to be true. It feels like it ought to be true. Then again youy can do nasty things like the Banach Tarski Paradox if you assume it true. Ouch.

Jedidiah

I don't think you know what algebra is, sir.

http://www.m-w.com/
Main Entry: buzzword
Pronunciation: 'b&z-"w&rd
Function: noun
Date: 1946
1 : an important-sounding usually technical word or phrase often of little meaning used chiefly to impress laymen
2 : a voguish word or phrase -- called also buzz phrase

I know that this is Slashdot and that around here the looks of a mathematician are more important than her work, but if anyone is interested, here are a few pointers to get to know more.

First, a short description of Hilbert's problems at Wolfram: Hilbert's Problems -- from MathWorld.

Then, a link to a text of Hilbert's original lecture in Paris in 1900.

Next, a quote of the 16-th problem as laid out by Hilbert. (Sorry, no fancy LaTeX here.)

16. Problem of the topology of algebraic curves and surfaces

The maximum number of closed and separate branches which a plane algebraic curve of the n-th order can have has been determined by Harnack. There arises the further question as to the relative position of the branches in the plane. As to curves of the 6-th order, I have satisfied myself--by a complicated process, it is true--that of the eleven branches which they can have according to Harnack, by no means all can lie external to one another, but that one branch must exist in whose interior one branch and in whose exterior nine branches lie, or inversely. A thorough investigation of the relative position of the separate branches when their number is the maximum seems to me to be of very great interest, and not less so the corresponding investigation as to the number, form, and position of the sheets of an algebraic surface in space. Till now, indeed, it is not even known what is the maxi mum number of sheets which a surface of the 4-th order in three dimensional space can really have.

In connection with this purely algebraic problem, I wish to bring forward a question which, it seems to me, may be attacked by the same method of continuous variation of coefficients, and whose answer is of corresponding value for the topology of families of curves defined by differential equations. This is the question as to the maximum number and position of Poincare's boundary cycles (cycles limites) for a differential equation of the first order and degree of the form dy/dx = Y/X where X and Y are rational integral functions of the n-th degree in x and y. Written homogeneously, this is X(y dz/dt - z dy/dt) + Y(z dx/dt - x dz/dt) + Z(x dy/dt - y dx/dt) = 0, where X, Y, and Z are rational integral homogeneous functions of the n-th degree in x, y, z, and the latter are to be determined as functions of the parameter t.

Finally, I'll quote the abstract from Miss Elin Oxenhielm's article On the second part of Hilbert's 16th problem :

Let k be an integer such that k is larger than or equal to zero, and let H be the Hilbert number. In this paper, we use the method of describing functions to prove that in the Lienard equation, the upper bound for H(2k+1) is k. By applying this method to any planar polynomial vector field, it is possible to completely solve the second part of Hilbert's 16th problem.

Author Keywords: Second part of Hilbert's 16th problem; Hilbert number; Lienard equation; Describing function; Limit cycle; Polynomial vector field

To get the full text of the article you must apparently have a subscription of pay a $30 fee. It is easily available if you follow the directions from the author's page as I did.

Hope this helps

Now allow me for a few comments: solving one of Hilbert's problem is a huge achievement, even it's only part of one. What is even more stricking is that it's coming from a woman. Don't get me wrong, I'm no sexist, quite the contrary. What I mean is that only very few women made it to be recorded in the history of the mathematical science at large: other than Hypatia of Alexandria; Maria Gaetana Agnesi; Sophie Germain; Ada Byron, Lady Lovelace; Sofia Kovalevskaya; Emmy Noether, not many names come to mind. It would be really nice to add another one, to begin, and then work up from there.

Xavier

Understanding a problem and solving it are two very different things. Take Fermat's Last Theorem. Anyone with a reasonable grasp of high school mathematics can understand the problem.

However, a solution eluded mathematicians for hundreds of years and in the end required mathematics well beyond high school level.

You can find a list of all 23 problems here. Some of them have already been solved.

I wouldn't really agree with that..

But it does seem true that math is "the young man's game".
(To quote the great mathematician GH Hardy)

Some of history's great mathematicians never lived to see their 30th birthday. Galois, and Abel for instance.

There are counterexamples, of course, the chemist Joel Hildebrand published his last research paper at over 100 years of age.

I wouldn't really agree with that..

But it does seem true that math is "the young man's game".
(To quote the great mathematician GH Hardy)

Some of history's great mathematicians never lived to see their 30th birthday. Galois, and Abel for instance.

There are counterexamples, of course, the chemist Joel Hildebrand published his last research paper at over 100 years of age.

Yes.. Here's a bio..

Certainly not Theorem - the definition of a theorem is that it can be proven to be true. And not Axiom either - an axiom is typically a statement that provides the base for a mathematical system (like "There exists and empty set"). Postulate is, AFAIK, just a synonym for axiom.

One could call it "Moore's conjecture", "Moore's oberservation", or "Moore's prediction" if one wants to be strict. The use of the term Law is not completely wrong however, there are other examples of the term being used for things that are just heuristically observed, like Zipf's Law for instance.

Certainly not Theorem - the definition of a theorem is that it can be proven to be true. And not Axiom either - an axiom is typically a statement that provides the base for a mathematical system (like "There exists and empty set"). Postulate is, AFAIK, just a synonym for axiom.

One could call it "Moore's conjecture", "Moore's oberservation", or "Moore's prediction" if one wants to be strict. The use of the term Law is not completely wrong however, there are other examples of the term being used for things that are just heuristically observed, like Zipf's Law for instance.

Certainly not Theorem - the definition of a theorem is that it can be proven to be true. And not Axiom either - an axiom is typically a statement that provides the base for a mathematical system (like "There exists and empty set"). Postulate is, AFAIK, just a synonym for axiom.

One could call it "Moore's conjecture", "Moore's oberservation", or "Moore's prediction" if one wants to be strict. The use of the term Law is not completely wrong however, there are other examples of the term being used for things that are just heuristically observed, like Zipf's Law for instance.

Look, it's not my numbers, my part is making an optimistic assumption of hardware capable of computing 10 billion positions per second. With these conservative estimates and optimistic assumptions it's still impossible.

Remember we're talking about playing perfect games, not just being better than the best human. To find a perfect game, you can't just ignore large parts of the search tree and only look a few moves ahead like an ordinary chess program does, you'll have to do an extensive search of each and every move possible before you play your first move, because there can be ways to win playing white or draw playing black which don't look very promising in the beginning.

If you don't understand how something works, you can not duplicate it, although you can produce something that resembles it.

That is besides the point. We don't want to mimic the brain or its biological processes pre ce, we want to duplicate its product, intelligence; we don't have to care about how the human brain produces intelligence (even though it could prove a good starting point).

Modern psychiatry and psychology distinguishes various types of intelligence in humans, so it's fair to try and duplicate only one of those (autonomous) intelligences. It is very well possible to write a program that is capable of learning from its mistakes in playing games (the world champion in checkers is such a program), and it is also possible to write a program that is capable of autonomously learning the rules of various (board) games. If we combined those two concepts, we would have something that duplicates "game intelligence" in humans quite well (except for being awfully slow). Would this be a thinking machine by your standards?

The actual patent is number 5373560. I have to admit that I don't fully understand the content of the patent. Does anyone else know?

--
"she said Jesus was my age when he got nailed" - Dark Star, 'I Am The Sun'

why not just use the standard deviation?

Orbital
Mind-Control
L^HMasers!

A square numbers is one which is equal to n^2 when n is a whole number. You are almost talking about primes numbers but you made a mistake, 1 is not a prime. A prime p has to be divisible by two distinct (ie. different) numbers, 1 and itself. 1 and 1 are not distinct (there are some good reasons for this mentioned in the links bellow).

Square Numbers

Prime Numbers

A square numbers is one which is equal to n^2 when n is a whole number. You are almost talking about primes numbers but you made a mistake, 1 is not a prime. A prime p has to be divisible by two distinct (ie. different) numbers, 1 and itself. 1 and 1 are not distinct (there are some good reasons for this mentioned in the links bellow).

Square Numbers

Prime Numbers

Using the oh-so cliche E=mc^2 you quickly find that by going the speed of light matter turns into energy

How does E=mc**2 mean going the speed of light turns matter into energy? There is no velocity dependence in that equation. As a matter of fact, that equation is the energy of a mass m at rest in the absence of an external potential.

The real answer is that as one travels faster, the energy needed to maintain a certain accleration increases. Ah, ha, here we are. So you see, there is nothing about matter turning into energy there. In a way, as you approch light speed it is like you are getting more matter (in that your apparent mass is increasing).

No. Except tiny amount of energy emitted via radio waves ALL electric energy in a chip is eventually transformed into heat First Law of Thermodynamics.

However, you are right than only a fraction of the heat can be transformed into work via steam engine. A reversible heat engine that has hot reservoir at 370K and cold reservoir at 300K has maximum efficiency of 1-300/370= 18.9% Efficiency of an engine. Silicon chips are too cold to be an effective heat engine.

No. Except tiny amount of energy emitted via radio waves ALL electric energy in a chip is eventually transformed into heat First Law of Thermodynamics.

However, you are right than only a fraction of the heat can be transformed into work via steam engine. A reversible heat engine that has hot reservoir at 370K and cold reservoir at 300K has maximum efficiency of 1-300/370= 18.9% Efficiency of an engine. Silicon chips are too cold to be an effective heat engine.

As far as I can tell - from various news reports - there have been a number of suspicious results. Suspicious in the sense of pointing to software flaws as opposed to corruption.

A number of results have thrown up the same odd set of figures a number of times.

I just wonder if this isn't a place where Benfords Law could be applied ?

ac

See also here. I don't quite know why that guy is offering a prize. It's well understood as coming from the properties of the j-function.

Very briefly: you may have sketched the function y^2=P(x) in your life where P(x) is a cubic. If you allow x and y to be complex numbers you get a 2D surface. That 2D surface is basically a twisted up torus (minus a point at or two corresponding to when x and y go to infinity) and the j function gives a way of specifying exactly what torus. It also plays an important role in string theory. But the full explanation of why you get all these near integers is quite long and involved.

That's what I thought too, but the only graphs I could found online so far related to Lotka was something about "Lotka-Volterra equations," which didn't seem to resemble the Lotka Curve concept in the article. I'm not even sure it's the same Lotka person.

I like the idea! But I suggest rule 110, instead of rule 30. Rule 110 (and its inverses) is the only example of "class 4" behavior in Wolfram's 1D automata. It develops into discrete structures that move around and interact with each other, whereas rule 30 is just an example of "class 3" behavior -- it generates apparent randomness.

Or, we could always just use the goatse guy.

actually, I think rule 90 would be a better representation, for a few reasons: The design is a fractal triangle, which represents both order and mystery the closer you examine it. It is also associated with the logical XOR operation, which is a subtle reference to encryption/decryption fun :) And finally, the design is both visually pleasing and mathematicly beautiful.

I personally think rule 30 would be a better logo, but may we should pick something from the Game of Real Life.

http://mathworld.wolfram.com/Calabi-YauSpace.html

The above-linked page has a rough definition of the kind of space a 10-dimensional string theory might live on. I don't know why they take a product $M \times V$ --- why not allow a more general bundle of C-Y 3-folds?

The imaginary version of string theory which exists only in my mind has the universe as a bundle of Calabi-Yau 3-folds over $M$ (a real 4-manifold with a Minkowski metric, or something like that... anyway, a $(3,1)$ form --- that's a Minkowski metric, right???). That's 10 dimensions. The 11- or other-dimensional kinds of string theory, I have no idea.

Being an algebraic geometer (in training), I think of a C-Y 3-fold as a smooth projective complex 3-dimensional variety, or manifold, with trivial canonical bundle.

Well, that's the space. Then all kinds of crap happens *in* the space, with strings and stuff, and that's a whole nother story...

Brute force is killing thought. We do not learn from randomly testing cases.

I agree that random testing of cases doesn't solve anything. But there are problems that can be solved by reducing the problem to a set of special cases which can then be checked by a computer to verify our claim. The magic tour problem for 4k x 4k boards was proved this way.

Of course, mathematicians usually prefer a completely analytic solution, like was the case with the computerized proof of the 4-color theorem.

Brute force is killing thought. We do not learn from randomly testing cases.

I agree that random testing of cases doesn't solve anything. But there are problems that can be solved by reducing the problem to a set of special cases which can then be checked by a computer to verify our claim. The magic tour problem for 4k x 4k boards was proved this way.

Of course, mathematicians usually prefer a completely analytic solution, like was the case with the computerized proof of the 4-color theorem.

340 billion billion billion billion = 340*(10^9)^4) = 340*10^36 = 340 undecillion.

I had a Profy in our freshmen year by the name Dalton. He would always refer to "Quantum Numbers" as "Condom Numbers"

You may have misheard him say Condon, as in the Condon-Shortley phase convention for spherical harmonics.

I had a Profy in our freshmen year by the name Dalton. He would always refer to "Quantum Numbers" as "Condom Numbers"

You may have misheard him say Condon, as in the Condon-Shortley phase convention for spherical harmonics.

It's kind of like the Coriolis Force. It's one of those "fake" forces, but something causes the effects described by the Coriolis Force.

You may want to read up on Reductio ad absurdum. Whatever your personal distaste for it, it has fine credentials as a valid and useful form of argument.

-- MarkusQ

I'm not going to bother, because about 20 people are about to tell you why pi is has no end.

From Mathworld: A number which is not the root of any polynomial equation with integer coefficients, meaning that it is not an algebraic number of any degree, is said to be transcendental. This definition guarantees that every transcendental number must also be irrational, since a rational number is, by definition, an algebraic number of degree one.

Once again: Irrational numbers have decimal expansions that neither terminate nor become periodic. Every transcendental number is irrational.

Pi, of course, is the most famous transcendental number of them all.

http://mathworld.wolfram.com/Dodecahedron.html

I thought the pictures were cool!

Large or not, it's still finite. In fact, it's rather pitiful as far as large numbers go. You want to see a really large number? Take a look at the best known upper bound on Graham's number.

It is a mix of Hexagonal and Penatagonal shapes, more commonly seen as a C60 (carbon60) or Bucky-Ball.

Icosahedra and dodecahedra are strongly related, so that's why a soccer ball looks a bit dodecahedral (having, as it does, 12 pentagons). In fact, if you keep on truncating a icosahedron's corners more and more deeply, you end up with a dodecahedron. Duality is very cool!