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Maxwell Equations! They just work! They were formulated before relativity theory was out, yet they proved relativity invariant. As soon as you write them for real media instead for the void they work a little worse, though...

The answer is simple. The most beautiful equations, hands down, are those from which all of mathematics can be derived. These are the axioms of ZFC set theory. What could possibly be more beautiful or more important than that? And it's a shame so few people know about them. See Zermelo-Fraenkel Axioms and Metamath Proof Explorer.

The answer is simple. The most beautiful equations, hands down, are those from which all of mathematics can be derived. These are the axioms of ZFC set theory. What could possibly be more beautiful or more important than that? And it's a shame so few people know about them. See Zermelo-Fraenkel Axioms and Metamath Proof Explorer.

"The difficulty of formal logic was demonstrated in the monumental Principia Mathematica (1925) of Whitehead and Russell's, in which hundreds of pages of symbols were required before the statement 1 + 1 = 2 could be deduced."

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

I've been slowly trying to figure out the Atiyah-Singer index formula. See here and here. Does someone know of a good discussion of this formula?

You should try the general formula for the solutions of a cubic equation. Or how about the quartic?

Once you get to the quintic (where the maximum power of x is 5) you find there is no general solution, which quite frankly comes as a relief...

You should try the general formula for the solutions of a cubic equation. Or how about the quartic?

Once you get to the quintic (where the maximum power of x is 5) you find there is no general solution, which quite frankly comes as a relief...

Earth is the cradle of humanity, but one cannot remain in the cradle forever.
-- Konstantin Tsiolkovsy

Proof: Assume a cow that is topologically equivalent to a Klein bottle. The rest is left as an exercise for the butcher.

If my memory serves me well, the process of transcription usually produces a fairly good "copy" of the DNA sequence, while translation seems to have a few unknowns in how he sequence is transformed into AA chains. And then the way in which the proteins fold, and hence gain their function is still up for grabs.

Translation is fairly well understood. There are specific enzymes (aminoacyl tRNA synthetases) that conjugate the appropriate amino acid to a specific tRNA. These tRNA have an anticodon that base pairs with the mRNA codon in the ribosome which hooks all the amino acids together and kicks out the spent tRNA. What's less well understood is the regulation of translation, but most genes are transcriptionally regulated and you could say the same thing about translation.

As relates to the article. I simply don't see a reason to expect that humans need a complex genome to be complex organisms. This is seen most clearly in Wolframs recent work on cellular automata, where certain extremely simple programs behave in a facinatingly complex manner.

I can't remember by whom or when though.

It was actually finally proved by Andrew Wiles in 1995.

See here towards the bottom of the page for who did what when. It' was quite a convoluted precess getting there.

That's quite simple. Your understanding of how calculators do it seems to be wrong. Nothing that I know of creates a derived equation and uses that. They use numerical methods that make use of the original equation to compute the derivative.

The HP-48 calculator or Mathematica both do the thing that "nothing you know of" does. We could also discuss why the numerical approach may be incorrect.

That was the old view. There were some problems with their use of infinitesimals, but those problems have been cleared up more recently. The modern version of calculus via infinitesimals is known as nonstandard analysis. The landmark work on the subject is Robinson's 1966 book "Non-standard analysis".

Moreover, that sort of hen-pecking at Newton and Leibniz is not really productive. No one cares more about precision and correctness in definitions than mathematicians, and yet mathematicians still assign credit to those two.

Have you read the Principia? I have only read portions, but Newton does some pretty amazing stuff in there, besides just the use of calculus and the derivation of the inverse square law for gravity. For example, he proves that there is no closed form for elliptic integrals of a certain kind.

Well, that's not true. Speckle interferometry can get to 70 milliarcseconds at 1.2 microns wavelength, and I'm working on an AO system that can get down to 85 milliarcseconds. What you may mean is that the Strehl ratio is nowhere near as good, which is very true.

If you are talking about the visible bands though, then it is true that hardly anyone has done well in that wavelength regime, and there I've heard the AEOS telescope on Maui can get halfway decent performance around a micron wavelength.

Dr Fish

By the Axiom of Foundation such a set does not exist in ZF so there is no such problem.

Actually, he could be serious. There's a very simple formula for any arbitrary digit of pi in base 16.

"All of a sudden, causality decided to give physical laws and time the finger,

Actually physical laws appear to have given causality the finger quite a while ago.

My personal choice is Mathematica. A study done four years ago used a dataset with known results to check a number of popular statistics packages. Mathematica was the only package that got everthing correct:

http://www.wolfram.com/news/statistics.html

In QM, you measure a property of an object by applying an "operator" (you put in a function, and it spits out another function) to its wavefunction. Heisenberg said[*] that certain pairs of operators don't commute (meaning order is important - AB != BA), and so some pairs of properties can't be measured together.

That's correct; here's an example (another more technically involved example was posted by Wass about Quantum Mechanics using Commutors). The position operator is the same as multiplying by the position of the system. The actual position could be denoted as, x and the operator called position could be denoted as, x. So (g(x) is just there so you can see what goes on the other side of the operator - i.e. nothing):

g(x) x f(x) = g(x) * x * f(x)

Another operator is called the (one-dimentional) momentum operator. Say the acutal momemtum is written as p and the operator as, p. Well, unlike the position operator, p is defined as k d/dx where k is a constant (k = h-bar * (-1)^.5 ) and d/dx is the gradient along the x-direction (AKA derivative of whatever's on the right hand side). So:

g(x) p f(x) = g(x) * k * f'(x)

What Heisenberg says is that these operators don't communte. This is easy to see by an example:

g(x) p x f(x) = g(x) * p (x * f(x)) = g(x) * k * (x * f'(x) + f(x))

g(x) x p f(x) = g(x) * x (k * f'(x)) = g(x) * x * k * f'(x)

Since p x and x p give two different equations, they don't commute.

You say that you work for a large financial company. You might check with the company's research group: they likely already use one of Maple/Mathematica/Matlab; so you could potentially be best off using what they use.

You might want to try Jython and the Numerical Python for Jython.
I have not used either for a long time, but use plain Python and Numerical Python a lot; sure beats Matlab and Mathematica for most things. Right now for solving optimization problems with 10k+ s.t. constraints.

Pedantic? You want pedantic?
You CAN'T HANDLE pedantic...

Not all icosahedrons are 20-sided. To wit:
http://mathworld.wolfram.com/Icosahedron.html

Try googling harmonic series limit or just look at formula three on this page

This page gives an explanation why, and links to a strategy for a proof.

Wolfram has a fuller explanation

Try googling harmonic series limit or just look at formula three on this page

This page gives an explanation why, and links to a strategy for a proof.

Wolfram has a fuller explanation

This is called the Look and Say Sequence. John H. Conway, creator of the Game of Life (the cellular automaton, not the board game), has studied this sequence extensively, including the sequences resulting from starting with digits besides 1.

No, this is wrong. Google's f(x) is a function. There is nothing wrong with it whatsoever. From the statement of the problem one can make several reasonable assumptions: 1) f is a function; 2) the domain of f contains the set {1, 2, 3, 4, 5}; and 3) the range of f contains the set {7182818284, 8182845904, 8747135266, 7427466391}. None of these assumptions are violated by google's expected result of f(5) = 5966290435, and the posted method of obtaining that result. f is still a function in this case.
I think that you and J.C. Roberts do not understand the definition of a function and that this is where your confusion is coming from. Here is a good definition to help you out.

public static double calcE ( double x ) {
x=x+1;
double n = x;
for ( int i=0; i<20; i++ ) {
double m = Math.pow(x,(double)i+1)/(i+1);
m = m* Math.pow( -1.0,(double)i);
n = n + m;
}
return n;
}

Is it me or is this kind of question not the stuff of genius. I mean it's just a case of writting a program to brute force the answer. The only leap is figuring out the 49 / sum of digits bit.

Clever maths stuff doesn't (usually) require brute force. Things like the proof of infinite primes and proof of the irrationality of 2^0.5 - now they are clever. Next time I suggest they have a bill board asking for the proof of Goldbach Conjecture

Also interesting to note: Use of Quantum Computing has been suggested as a way to speed up crafted brute-force attacks on existing encryption schemes. All existing encryption schemes rely on the principle of even probability key distribution to reduce attack vectors to simple or crafted brute-force attacks (trying some or all of the possible keys in the key-space.) With conventional computing, this means trying each key one at a time, until the correct key is found. Symmetric key encryption is generally fast, but can also be broken faster (months or years, for the average case). Public key encryption is slower, but takes decades or centuries to break (again, for the average case).

With Quantum Computing, however, every key of length n can be tried at the same time by a n-qbit computer. So if you have a 128-qbit Quantum Computer, you can try every 128 bit key all at once.

Fortunately (or unfortunately?), the last I heard (2 years ago) was that the most qbits formed was 7, and that was in a lab using chemical injections. Anyone know the current upper limit?

Furthermore, anything more complex than the two-body problem is chaotic and incapable of exact solution, so it's up to the two-body problem to carry us along.

Not quite; the restricted three-body problem, where one of the masses is infinitessimal compared to the other two, can be solved analytically. The solutions reveal the existence of five points where the net effective force on the massless third body vanishes -- these points being, of course, the Lagrange points familar to students of orbital mechanics.

I'm surprised that the reviewer found so much of the material new; do college physics courses these days not include classical mechanics and the like?

Measure the speed of light in two directions: parallel and perpendicular to the direction of motion of the Earth in its orbit. Compare the two to discover whether or not the Earth's velocity is added to that of light.

And guess what? It's been done.

This one seems the most similar to what I learned.

And in which decade were you in 8th grade?

The 80's. Specifically 1988, slightly after the backlash against New Math.

None of the younger math teachers even knew such an algorithm existed.

Now, I find that hard to believe. At the very least, every one of them should have been able to use Newton's iteration, which is taught in basic college math courses.

At least, with pencil and paper, it's as mistake-proof as long division.

I don't consider long-division mistake proof.

You define it differently, pretty much as a function with a finite number of terms using the field operations. Fair enough, and I follow what you're saying from there. But that's definitely different from what I learned from taking real anaylsis and complex analysis.

Here's a token MathWorld definition, though I know that's not definitive. I'm well aware the definitions can change significantly from area to area in mathematics.

Anyway, regardless of semantics, great explanations, and thanks for sharing!

The method presented in the paper looks a lot like James Cockle and Robert Harley's differential resolvent, which was new in 1862. This page gives an overview of some of the known methods for solving quintic and higher degree equations. Apparently, about twenty years ago Hiroshi Umemura found a general analytical solution for a polynomial equation of arbitrarily high degree involving Siegel modular forms, which are generalizations of the elliptic modular functions Charles Hermite used in 1858 as a solution to the quintic. Note: these don't violate Abel's Impossibility Theorem as they are not solutions in radicals.

It appears that for integer solutions no such algorithm exists. That is [*] in their celebrated work Matiyasevich, Putnam and Julia Robinson have shown that no algorithm exists which can tell given an equation whether it has an integral solution or not.

MathWorld article on Diophantine Equations[*]

I have one name for you - Dirichlet

I believe it means that there will never be a magic equation to 4th order polynomials and above like one learned in algebra to the 2nd order polynomial.

Bergman (1957/58) considered an irrational base, and Knuth (1998) considered transcendental bases. This leads to some rather unfamiliar results, such as equating pi to 1 in "base pi," pi = {10}_pi.

There's a (not yet /.-ed) link: http://hal.ccsd.cnrs.fr/docs/00/00/32/74/PDF/The-r oots-of-any-polynomial-equation.pdf Also I'm afraid that it's been done before http://library.wolfram.com/examples/quintic/main.h tml#diffeq

Even if it were a closed formula it would not neccesarily conflict with Abel's proof. Abel proved that the roots of the general polynomial cannot be found using only a finite number of additions, subtractions, multiplications, divisions, and radicals. See Abel's impossibility theorem.

ii) Student mistaken; popular media talking out of arse.

iii) Abel's theorem holds ("you cannot solve all polynomial equations by radicals"); student solves all polynomial equations not using radicals but using differential equations and power series; popular media like /. do not understand that this method is known for more than hundred years and that there is no inconsistence.

/graf0z.

ps: a link provided by the author himself: solving the quintic

And the ONLY thing your computer can calculate are recursively enumerable sets which to be frank are boring.

The author of the parent post claims to explain the conjecture from the point of view of a "MS Mathematics". This would be fine if the explanations had not been copied directly from MathWorld.

Quote from the parent post:

The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical (and was known even to 19th century mathematicians), n = 3 has remained open up until now, n = 4 was proved by Freedman in 1982 (for which he was awarded the 1986 Fields Medal), n = 5 was proved by Zeeman in 1961, n = 6 was demonstrated by Stallings in 1962, and n >= 7 was established by Smale in 1961 (although Smale subsequently extended his proof to include all n >= 5).

Now please compare this with the middle paragraph from http://mathworld.wolfram.com/PoincareConjecture.ht ml. The one that starts with "The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical (and was known even to 19th century mathematicians), [...]"

This is just an example. Other paragraphs can be found in MathWorld's pages about the Poincaré conjecture, definition of manifold and compact manifold, homeomorphic, etc.

Now I don't mind if some useful information is posted on Slashdot. But some obvious plagiarism like that without crediting any sources definitely deserves some Overrated treatment...

The author of the parent post claims to explain the conjecture from the point of view of a "MS Mathematics". This would be fine if the explanations had not been copied directly from MathWorld.

Quote from the parent post:

The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical (and was known even to 19th century mathematicians), n = 3 has remained open up until now, n = 4 was proved by Freedman in 1982 (for which he was awarded the 1986 Fields Medal), n = 5 was proved by Zeeman in 1961, n = 6 was demonstrated by Stallings in 1962, and n >= 7 was established by Smale in 1961 (although Smale subsequently extended his proof to include all n >= 5).

Now please compare this with the middle paragraph from http://mathworld.wolfram.com/PoincareConjecture.ht ml. The one that starts with "The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical (and was known even to 19th century mathematicians), [...]"

This is just an example. Other paragraphs can be found in MathWorld's pages about the Poincaré conjecture, definition of manifold and compact manifold, homeomorphic, etc.

Now I don't mind if some useful information is posted on Slashdot. But some obvious plagiarism like that without crediting any sources definitely deserves some Overrated treatment...

The author of the parent post claims to explain the conjecture from the point of view of a "MS Mathematics". This would be fine if the explanations had not been copied directly from MathWorld.

Quote from the parent post:

The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical (and was known even to 19th century mathematicians), n = 3 has remained open up until now, n = 4 was proved by Freedman in 1982 (for which he was awarded the 1986 Fields Medal), n = 5 was proved by Zeeman in 1961, n = 6 was demonstrated by Stallings in 1962, and n >= 7 was established by Smale in 1961 (although Smale subsequently extended his proof to include all n >= 5).

Now please compare this with the middle paragraph from http://mathworld.wolfram.com/PoincareConjecture.ht ml. The one that starts with "The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical (and was known even to 19th century mathematicians), [...]"

This is just an example. Other paragraphs can be found in MathWorld's pages about the Poincaré conjecture, definition of manifold and compact manifold, homeomorphic, etc.

Now I don't mind if some useful information is posted on Slashdot. But some obvious plagiarism like that without crediting any sources definitely deserves some Overrated treatment...

The author of the parent post claims to explain the conjecture from the point of view of a "MS Mathematics". This would be fine if the explanations had not been copied directly from MathWorld.

Quote from the parent post:

The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical (and was known even to 19th century mathematicians), n = 3 has remained open up until now, n = 4 was proved by Freedman in 1982 (for which he was awarded the 1986 Fields Medal), n = 5 was proved by Zeeman in 1961, n = 6 was demonstrated by Stallings in 1962, and n >= 7 was established by Smale in 1961 (although Smale subsequently extended his proof to include all n >= 5).

Now please compare this with the middle paragraph from http://mathworld.wolfram.com/PoincareConjecture.ht ml. The one that starts with "The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical (and was known even to 19th century mathematicians), [...]"

This is just an example. Other paragraphs can be found in MathWorld's pages about the Poincaré conjecture, definition of manifold and compact manifold, homeomorphic, etc.

Now I don't mind if some useful information is posted on Slashdot. But some obvious plagiarism like that without crediting any sources definitely deserves some Overrated treatment...

The author of the parent post claims to explain the conjecture from the point of view of a "MS Mathematics". This would be fine if the explanations had not been copied directly from MathWorld.

Quote from the parent post:

The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical (and was known even to 19th century mathematicians), n = 3 has remained open up until now, n = 4 was proved by Freedman in 1982 (for which he was awarded the 1986 Fields Medal), n = 5 was proved by Zeeman in 1961, n = 6 was demonstrated by Stallings in 1962, and n >= 7 was established by Smale in 1961 (although Smale subsequently extended his proof to include all n >= 5).

Now please compare this with the middle paragraph from http://mathworld.wolfram.com/PoincareConjecture.ht ml. The one that starts with "The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical (and was known even to 19th century mathematicians), [...]"

This is just an example. Other paragraphs can be found in MathWorld's pages about the Poincaré conjecture, definition of manifold and compact manifold, homeomorphic, etc.

Now I don't mind if some useful information is posted on Slashdot. But some obvious plagiarism like that without crediting any sources definitely deserves some Overrated treatment...

Do you often take other people's writings, and portray them as your own?

Except, of course, that mathematicians have read it, and it seems, in all those pages, there isn't actually a proof. (See the bottom of the front page of Mathworld)

As opposed to Perelman, who appears to have actually proved a larger conjecture, of which the Poincaré conjecture is a specific case.

A June 8 Purdue University news release reports a proof of the Riemann Hypothesis by L. de Branges. However, both the 23-page preprint (from 2003) cited in the original release and a 124-page preprint (from 2004) cited in a back-dated modified release seem to lack an actual proof. Furthermore, a counterexample to de Branges's approach by Conrey and Li has been known since 1998. The media coverage therefore appears to be much ado about nothing.