Wikipedia:Reference desk archive/Mathematics/2006 October 9

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1 Presentation for B(2,4)
2 triple integrals
3 Prime Factor
4 What does adition, multiplication ... means?
5 Genetic algorithms
6 Name of function?
7 Cardinality of infinite sets

[edit] Presentation for B(2,4)

I need to find or calculate the finite presentation for the Burnside group B(2,4); Googling finds a GAP package that might be useful for me but it doesn't seem to work properly. Dysprosia 02:01, 9 October 2006 (UTC)

[edit] triple integrals

hoe do you find the volume of a sphere by converting rectangular to spherical coordinates209.88.91.213 12:28, 9 October 2006 (UTC) takunda. emai address ( togie@myway.com)

See the Multiple integral article, esp. section Change of variables, subsection Spherical coordinates. --CiaPan 15:45, 9 October 2006 (UTC)

[edit] Prime Factor

What are the prime factor of 188198812920607963838697239461650439807163563379417382700763356422988859715234665485319060606504743045317388011303396716199692321205734031879550656996221305168759307650257059

I don't think anyone has enough time to answer this. - Rainwarrior 14:13, 9 October 2006 (UTC)

And by that I mean read "Integer factorization". This is a problem computer scientists have been working on for a long while but have found no solution. Unless you know how that number was put together from primes, it's really hard to break it apart. Your number could be factored but it will probably take a lot of calculation time. Maybe hours, maybe days, I dunno. - Rainwarrior 14:21, 9 October 2006 (UTC)

This number is 174 digits long. It would probably take a month of processing in an average high-end computer to find out. ☢ Ҡi∊ff⌇↯ 14:25, 9 October 2006 (UTC)

After taking a further look at it, it would take much more than a month, and given that it is a semiprime of two very large primes it would be about as hard as a number of that length can be (which is why it had a $10,000 reward). (The more factors a number has the faster it breaks down, as once you find one, the numbers get smaller and the computation speeds up.) - Rainwarrior 19:50, 9 October 2006 (UTC)

The number is RSA-576. It was factored in 2003 using the General Number Field Sieve. Fredrik Johansson 15:37, 9 October 2006 (UTC)

Ahh, neat. I was wondering if it had already been factored by someone. I tried google, but it doesn't work for hideously long strings like that. - Rainwarrior 15:43, 9 October 2006 (UTC)

I just googl'd for the last digits, "650 257 059". The answer is given in a German paper. -- DLL ^{.. T} 18:27, 9 October 2006 (UTC)

Or if you had any idea to suspect it might be an RSA-xxx number, the 2-log is 575.6, so in binary the number has 576 bits. --Lambiam ^Talk 05:39, 10 October 2006 (UTC)

[edit] What does adition, multiplication ... means?

What does it means? I am not interested in a example or circular definion.

Have you checked our articles on addition and multiplication? ☢ Ҡi∊ff⌇↯ 15:33, 9 October 2006 (UTC)

Metamath gives rigorous definitions of its addition and multiplication operators. Guaranteed not to be circular (but also quite difficult to understand). —Keenan Pepper 23:01, 9 October 2006 (UTC)

[edit] Genetic algorithms

can you please give me an example of a genetic algorithm and how exactly do they enhance the bounded rationality concept applied to economics ?

This has nothing to do with Mathematics, I suggest you put this question to the Computer/IT section of the reference desk. 202.168.50.40 23:06, 9 October 2006 (UTC)

My apologies , I tought since bounded rationality is directly related to the game theory which is mathematics I can put the question here . Thank you anyway .

Though it is not a part of the classic mathematics, it can be applied mathematics, as long as you are willing to consider the computer/it science as applied math. Anyway, I don't know about bounded rationality, but for economy and A.I. related stuff, including perceptrons and genetic algorithms, you might want to check: http://www.smartquant.com/references/NeuralNetworks/neural19.pdf#search=%22Forecasting%20Financial%20Markets%20Using%20Neural%20Networks%3A%20An%20Analysis%20Of%20Methods%20And%22

[edit] Name of function?

Is there a name for the function ${x \over x^2+1}$ ? The function is "intersting" because $\lim_{x\rightarrow 0} = \lim_{x\rightarrow \infty} = \lim_{x\rightarrow -\infty} = 0 \,$ . I was wondering if this unique property gave this function a special name. - SigmaEpsilon → ΣΕ 18:57, 9 October 2006 (UTC)

What is unique here? There are infinitely many functions satisfying those limits. Yours is a simple example of a rational function; I don't think there is a more specific name. Fredrik Johansson 19:04, 9 October 2006 (UTC)

Technically, any function of the form ${Ax \over Bx^n+C}$ would still work. The above-mentioned fn is just the simplest case of this class of functions. Furthermore, the function is negative for all negative numbers, and positive for all positives. I just happen to find it to be an interesting function. I thought perhaps some other mathemetician also found it interesting, and named it after him/herself... - SigmaEpsilon → ΣΕ 20:00, 9 October 2006 (UTC)

I think you intend A and B to be positive, and n to be a positive and even integer. If so, then please write this information! JoergenB 08:58, 10 October 2006 (UTC)

I am fairly certain that it is uncommon for a mathematician to name something after himself (his name is given to the item by others). This function is in some ways interesting, but not enough to deserve a special name. -- Meni Rosenfeld (talk) 21:24, 9 October 2006 (UTC)

It is rather uncommon. Instead, even if the concept is named after the inventors/discoverers by others, they themselves usually avoid this terminology.

An example: The Stanley-Reisner rings are fairly important objects, in the cross-field of combinatorics and commutative algebra. The 'Stanley' is Richard (P.) Stanley, who IMO (and in the opinion of many others) is one of the world's leading (now living) combinatorians. However, Richard himself never calls the Stanley-Reisner rings anything else than face rings. (Just noted that these articles seem not to be written??) JoergenB 09:10, 10 October 2006 (UTC)

Calling the function in question f, it further has the interesting property that for x ≠ 0 we have f(x) = f(1/x). Together with its continuity at 0 this gives the equality of the three limits as a special case. This property of being invariant under taking the multiplicative inverse of the argument is still not "very unique". --Lambiam ^Talk 21:32, 9 October 2006 (UTC)

I believe this function is interesting in that the power of the higest exponent in the numerator is one less than the highest exponent in the denominator. In this case, this function would possibly the simplest representation of a function with the x-axis as a horizontal asymptote. But maybe I'm looking to hard for patterns, lol. --Ķĩřβȳ ♥Ťįɱé Ø 09:22, 10 October 2006 (UTC)

Wow. I didn't expect so many comments. As JoergenB mentioned, I should have noted that A, B, amd C should be positive, and n should be even. Anyway, I've just always liked this (class of) funtion(s), and wondered if there was any special name for it. - SigmaEpsilon → ΣΕ 15:24, 10 October 2006 (UTC)

OK. Now that we agreed on the properties of A, B, C, and n, we may note that the function $f(x) = {Ax \over Bx^n+C}$ has the property that $f(-x) = -f(x)\,$ for any real number x. There are many other functions with this property, e. g. the sine function. These functions are called odd. JoergenB 16:38, 10 October 2006 (UTC)

[edit] Cardinality of infinite sets

Can somebody explain to me how the sets of whole numbers and integers are cardinal (have the same size)? Can you present an explanation of the bijection between the two (i.e., have a strict definition of how to pair one number from wholes into integers)? Thanks. --Fbv65e del / ☑t / ☛c || 23:09, 9 October 2006 (UTC)

Going from the integers to the non-negative integers:

for n < 0, map n to 2|n|–1

for n ≥ 0, map n to 2n

It is easy to verify that this is a bijection. --Lambiam ^Talk 23:30, 9 October 2006 (UTC)

Thanks, perfect and simple answer! --Fbv65e del / ☑t / ☛c || 23:32, 9 October 2006 (UTC)

Any 2 sets are cardinal if their contents can have one to one correspondence with each other. Rational numbers and whole numbers are cardinal, irrational numbers are not cardinal with either. There is no possible way to have an sequence that has every irrational number in it. But with rational numbers, one can start with the sequence $\frac{0}{0}, \frac{1}{0}, \frac{1}{1}, \frac{0}{1}, \frac{2}{0}, \frac{2}{1}, \frac{1}{2} \frac{3}{0}, \frac{3}{1},$ , etc. etc. and that can correspond one to one with the whole numbers and the integers if you ignore the nonsensical numbers of the sequence that divide by zero. --Ķĩřβȳ ♥Ťįɱé Ø 09:31, 10 October 2006 (UTC)

While you are correct, all irrational numbers are not countable, it is also inexact. The fact is that many irrational numbers are countable, namely all algebraic numbers (such as the square root of two). Instead of saying that irrational numbers are uncountable, it is better to say that the trancendental numbers are uncountable. Minor point, but still. Oskar 17:02, 10 October 2006 (UTC)

Change "namely" to "for example". There are many more numbers than the algebraic ones that are countable. Fredrik Johansson 04:45, 11 October 2006 (UTC)

Well, yeah, but usually in an informal "hierarchy" of numbers, it goes integers $\supset$ rationals $\supset$ algebraic, and those that doesn't fit are the trancendental ones. Oskar 12:35, 11 October 2006 (UTC)

You are wrong again: integers $\subset$ rationals $\subset$ algebraic.
By the way, the set $\{ \pi, 2\pi, 3\pi, \dots\}$ contains only transcendental numbers, anyway it is countable. --CiaPan 15:24, 11 October 2006 (UTC)

Kirby, your example seems quite mysterious to me. I tried to discover what you mean by etc, etc, but I failed to find a rule describing your 'sequence'. Possibly you missed some members of it? I'd suggest to use an explicit definition instead of, or in addition to, the initial part of the sequence. Also when we talk about numbers try to give sequences containing numbers only, without 2/0 or 0/0 symbols.

One possible approach is this: take every positive integer as a fraction's denominator, and give them all non-negative enumerators less than or equal to the denominator's square:

$\frac 0 1, \frac 1 1, \frac 0 2, \frac 1 2, \frac 2 2, \frac 3 2, \frac 4 2, \frac 0 3, \frac 1 3, \frac 2 3, \frac 3 3, \frac 4 3, \frac 5 3, \frac 6 3, \frac 7 3, \frac 8 3, \frac 9 3, \frac 0 4 \cdots$

Another one could be like this: for every positive integer sum take all pairs of non-negative integer enumerator and positive denominator, which give that sum:

$\frac 0 1, \frac 0 2, \frac 1 1, \frac 0 3, \frac 1 2, \frac 2 1, \frac 0 4, \frac 1 3, \frac 2 2, \frac 3 1, \frac 0 5, \frac 1 4, \frac 2 3, \frac 3 2, \frac 4 1, \frac 0 6 \cdots$

CiaPan 18:12, 10 October 2006 (UTC)

Oskar, I'm sorry to say this but your reply seems to indicate a flaw in your understanding (or a very poor choice of words). Being countable is not a property of a number but rather of a set of numbers (or other objects). There's no such thing as an irrational number being countable or not. The set of rational numbers is countable; the set of algebraic numbers, although larger, is also countable. The set of irrational numbers is not countable (it is "too large"); The set of transcendental numbers is also not countable.

On a different note, I'll say I have never before seen the phrase "A and B are cardinal" used for the purpose of stating they have the same cardinality (or any other purpose, for that matter). -- Meni Rosenfeld (talk) 20:24, 10 October 2006 (UTC)

Yes, poor choice of words, nothing else. I meant what you meant. Oskar 12:35, 11 October 2006 (UTC)

Okay, no problem then. -- Meni Rosenfeld (talk) 13:49, 11 October 2006 (UTC)

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