Wikipedia:Reference desk/Archives/Mathematics/2011 September 20

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September 20

When will Goldbach's conjecture be tested.

Since the "odds" of a Goldbach's conjecture violation are 10^-3700 (see page history for deleted post), when will it be tested up to 10^4000 to ensure that no counterexamples are found in the expected range? Hcobb (talk) 03:36, 20 September 2011 (UTC)

I don't think that will ever be possible, as that's way more than all the atoms in the universe. StuRat (talk) 04:19, 20 September 2011 (UTC)

That's not how it works. The figure of 10^-3700 is given for an even number of a specific magnitude, which as I understand is on the order of a few thousands. For larger numbers the probability is smaller. By the time you get to the even number 10000 the probability is maybe 10^-10000 (made up number) and so on. These very low probabilities mean the opposite of what you're implying - it means we're more confident of Goldbach's conjecture, not less confident and in need to look harder to be convinced. -- Meni Rosenfeld (talk) 07:41, 20 September 2011 (UTC)

Your first post [1] was removed because it didn't ask a question. Goldbach's comet says: "the probability of zero pairs for any one E, in the range considered here, is of order 10^-3700." The range in the graphs of the article is 1,160,000 to 1,540,000. My own calculations also indicate that for the middle of this range around 1,350,000 the odds are of order 10^-3700. Your question apparently assumes that the odds are the same for each even integer but in fact the odds decrease quickly when the integer grows. My quick estimate says the odds are 10^-15 for 1,000, 10^-75 for 10,000, 10^-440 for 100,000, 10^-2900 for 1,000,000, and 10^-20500 for 10,000,000. It is far smaller for numbers beyond the search limit of 26×10¹⁷ a week ago.[2] The sum of the odds taken over all numbers above 26×10¹⁷ is extremely small so we think Goldbach's conjecture is highly likely to be true. However, the calculation of the odds makes assumptions that seem reasonable and fits existing data very well but could theoretically turn out to be wrong. PrimeHunter (talk) 10:33, 20 September 2011 (UTC)

Even 10^10000 is only a ten thousand digit number and will handily fit in the CPU cache of any laptop. (It's a little more than 4KB.) So why not build a quantum prime solver of this modest size? (Assuming quantum computers actually work of course.) Hcobb (talk) 10:55, 20 September 2011 (UTC)

We don't have useful quantum computers and I don't know whether there is an algorithm which would aid Goldbach's conjecture significantly. If we could instantly determine whether a number is prime then the project reaching 26×10¹⁷ would still have to examine each even number and only become a little faster. By the way, existing computers and algorithms can handle individual numbers the size of 10¹⁰⁰⁰⁰ = 47717 + 10¹⁰⁰⁰⁰-47717. I just found the probable prime 10¹⁰⁰⁰⁰-47717 in 15 minutes on a PC. It is almost certainly prime but it would take much longer to prove primality. PrimeHunter (talk) 11:44, 20 September 2011 (UTC)

Intergrating ring of charge in 1/r field

Re #Distribution of charge in 1,2,3 dimensions on symmetrical bodies (second part) - I got stuck (no suprises there) - I was trying to sum (integrate) the combined effect of a ring (circle or radius r) of "charge" generating a 1/r field at distance r , at an offset f from the centre of the ring.

I got the integral between 0 and pi of (f-rcosθ)r/(f²+r²-2frcosθ) , which I boiled down to the integral of

1/2f (1 + (f2 - r2)/(f2+r2-2frcosθ) )

At which point I got stuck. Clues appreciated. Thanks. Imgaril (talk) 10:30, 20 September 2011 (UTC)

Laplacian correction for discriminative naive Bayes

In a naive Bayesian classifier with discriminative learning, is it appropriate to increase the amount of Laplacian correction with the logarithm of the dimensionality, as the Weka package DMNBtext does? If so, does this still hold as dimensions with less and less entropy (e.g. rare words being encountered for the first time) are added? I ask this because I'm testing a straight conversion of DMNBtext in a chat-spam filter, and finding that its probabilities are too close to 50% (i.e. it's not confident enough in its classifications). NeonMerlin 12:17, 20 September 2011 (UTC)

I don't know enough to give a straight answer, but maybe I can offer an opinion if you clarify the context and notation.

Do I understand correctly that DMNBtext assumes that in addition to the observed words, there are $\log(d)$ prior words divided evenly among the d words in the vocabulary, for each example? This does sound like a lot, especially if there aren't many real words in each example. I'm not sure about the theory that supports the logarithmic value, but it undoubtedly based on a specific prior on the distribution of word frequencies, which may not match the real distribution.

Also, why would it give 50% rather than whatever the prior class frequencies are?

AFAIK Naive Bayes is a generative model, what do you mean by using it in discriminative learning? -- Meni Rosenfeld (talk) 14:53, 20 September 2011 (UTC)

Is this function chaotic

I was interested in chaotic functions a while back, so I tried to make one. I think I may have stumbled on one that is chaotic, though I am not sure.

$f(a,x)=a\times \left({\frac {x}{a}}-2\times \left\lfloor {\frac {x}{2\times a}}+{\frac {1}{2}}\right\rfloor \right)\times (-1)^{\left\lfloor x/(2a)+1/2\right\rfloor },$

$g(x)=(\left\vert x\right\vert \%1)\times {\left\lceil \left\vert x\right\vert \right\rceil }+0.5\times {\left\lfloor \left\vert x\right\vert \right\rfloor }^{2}+0.5\times \left\lfloor \left\vert x\right\vert \right\rfloor ,$

$h(a,x)=f\left(a,{\frac {g(x)}{f(a,x)+1}}\right)$

--Melab±1 ☎ 20:01, 20 September 2011 (UTC)

Are the following formulas what you wanted? The formulas you typed above have syntax errors so they aren't showing up.

$f(a,x)=a\times \left({\frac {x}{a}}-2\times \left\lfloor {\frac {x}{2\times a}}+{\frac {1}{2}}\right\rfloor \right)\times (-1)^{\left\lfloor x/(2a)+1/2\right\rfloor },$

$g(x)=(\left\vert x\right\vert \%1)\times {\left\lceil \left\vert x\right\vert \right\rceil }+0.5\times {\left\lfloor \left\vert x\right\vert \right\rfloor }^{2}+0.5\times \left\lfloor \left\vert x\right\vert \right\rfloor ,$

$h(a,x)=f\left(a,{\frac {g(x)}{f(a,x)+1}}\right)$

Also, does the percent sign mean modulo?

– b_jonas 20:34, 20 September 2011 (UTC)

Knuth's The Art of Computer Programming chapter 3.1 concludes that you cannot generate random numbers using random methods, which I think might imply that this function is not chaotic. – b_jonas 10:06, 21 September 2011 (UTC)

Is there supposed to be a recurrence relation here? e.g. x(i)=h(a,x(i-1))

Yaris678 (talk) 12:43, 21 September 2011 (UTC)

Chaos theory deals with something quite different from noise though it may sometimes appear to be noise, an yes a recurrence relation would be a good idea for the question! It can be quite reasonable to say something exhibits both chaos and random noise. Dmcq (talk) 18:20, 21 September 2011 (UTC)

Who mentioned noise? Yaris678 (talk) 19:24, 21 September 2011 (UTC)

Above see b_jonas that I was replying to. Random means noise. Chaos isn't random. Dmcq (talk) 19:49, 21 September 2011 (UTC)

I think b_jonas meant pseudorandom. Pseudorandom number generators makes use of chaos to appear random. Yaris678 (talk) 21:01, 21 September 2011 (UTC)

Yes, those are what the equations were supposed to look like and the percent sign is modulo. There is no recursion. Just specify $a$ in $h(a,x)$ and $x$ is the input. --Melab±1 ☎ 20:43, 21 September 2011 (UTC)

In that case I would say that the function is not chaotic. Chaos theory studies the behaviour of dynamical systems that are highly sensitive to initial conditions. Even something that seems static, like the Mandelbrot set, is defined by a recurrence relation. Yaris678 (talk) 11:49, 23 September 2011 (UTC)