Wikipedia:Reference desk/Archives/Mathematics/2008 February 5

= February 5 =

Top x% contribute y%
I know there's a saying something along the lines of "90% of the world's wealth is held by 10% of its population", or possibly the US's wealth, or something else. Is there a name for a statistic that, for example, given X_1, X_2, ..., X_n from some random distribution, will tell you what percentage of the total comes from the m largest values? Confusing Manifestation (Say hi!) 01:30, 5 February 2008 (UTC)


 * Try Pareto principle. AndrewWTaylor (talk) 11:24, 5 February 2008 (UTC)


 * I don't know the answer to your question, but I wanted to mention that for any integrable distribution on an interval there will always be exactly one n with the property that the bottom n% of the interval contains (100−n)% of the total. (This is because $$f(x) = (x{-}a) \int_a^x p(u) \, du - (b{-}x) \int_x^b p(u) \, du$$ is continuous and monotonic on [a,b] and negative at x=a and positive at x=b, therefore $$f^{-1}(0)$$ exists, and $$n = 100\frac{f^{-1}(0)-a}{b-a}$$.) So n is a well-defined and potentially interesting measure of the degree to which the distribution is biased toward one end or the other, and I wouldn't be too surprised if it did have a name. -- BenRG (talk) 14:52, 5 February 2008 (UTC)


 * The Lorenz curve will answer such issues on income distribution. Pallida  Mors  17:21, 5 February 2008 (UTC)

Σ notation
Is there any way using the summation operator ($$\sum$$) that one can cause it to skip at regular intervals such as the sequence: $$1^3+3^3+5^3+7^3+9^3...$$ ? If so how would one write this?Zrs 12 (talk) 02:43, 5 February 2008 (UTC)


 * There are two ways to do this. Way #1 involves writing the terms of summation as expressions in the integers, so in this case, t_n = (2n + 1)^3, giving:

$$\sum_{i=1}^n \left(2 i + 1\right) ^ 3$$
 * The other way is a little bit more versatile, and basically involves writing specific conditions in the operator. For example (my poor LaTeX notwithstanding),

$$\sum_{\begin{array}{c}i = 1 \\ i \mbox{ odd} \end{array}}^n i^3$$
 * Note that in the first example, the terms go from 1 to 2n+1, whereas in the second, they go from 1 to n (assuming n is odd), so always keep an eye on that. For more complicated syntax, you can also define a set, S, say, that contains all of the indices you want to sum over, and you can then write

$$\sum_{i \in S} x_i$$
 * Where the xi are the terms. Confusing Manifestation (Say hi!) 04:26, 5 February 2008 (UTC)
 * You mean $$\sum_{i=0}^n \left(2 i + 1\right) ^ 3$$. --wj32 t/c 08:56, 5 February 2008 (UTC)
 * More likely $$\sum_{i=0}^{n-1} \left(2 i + 1\right) ^ 3$$ or $$\sum_{i=1}^n \left(2 i - 1\right) ^ 3$$, if you want to preserve the number of terms. &mdash; Lomn 14:50, 5 February 2008 (UTC)


 * You can get scriptsize multiline subscripts to a big Σ by using \begin{smallmatrix}...\end{smallmatrix}:
 * $$\sum_{\begin{smallmatrix}i = 1 \\ i~\mathrm{odd} \end{smallmatrix}}^n i^3.$$
 * --Lambiam 16:05, 5 February 2008 (UTC)

One can also write
 * $$\sum_{\text{odd }i=1}^n i^3 $$
 * $$\sum_{\text{odd }i=1}^n i^3 $$

which is perhaps not all that widespread, but nonetheless it will be understood and has a certain advantage of simplicity. Michael Hardy (talk) 22:53, 5 February 2008 (UTC)

HELP!!! I dont understand prime numbers.
Could anyone please explain prime numbers to me, i am totally confused. I've learnt that it is a number having no factors except itself and one, In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself.Here are some prime numbers: 2,3,5,7,11,13,17,19,23,29,31,37 etc... I have no idea what "natural number divisors" mean Please can someone explain it to me???

367	373	379	383	389	397	401	409 —Preceding unsigned comment added by 220.239.2.52 (talk) 22:24, 5 February 2008 (UTC)


 * A natural number divisor of a number is a whole number greater than zero (1, 2, 3, 4 etc.) that divides a number without a remainder. For instance, 2 is a NND of 4 (2 fits into 4 twice). 7 is an NND of 21 (7 fits into 21 three times). Prime numbers are numbers that only have the number 1 and itself as a NND. Take 11 for instance - can you think of a number apart from 1 and 11 that divides it without leaving a remainder? There isn't one, and that's why it's prime. Damien Karras (talk) 22:34, 5 February 2008 (UTC)


 * You quoted text from prime number which just has a more formal way of saying "a number having no factors except itself and one", without risk of ambiguity. Your own formulation makes it a little unclear whether 1 is a prime. It isn't, which is made explicit by demanding two distinct divisors. And in some contexts a number is said to have negative factors. For example 5 having the factors 5, 1, -1, -5, since 5 = (-1) &times; (-5). This ambiguity is avoided by demanding natural divisors (natural numbers are by definition positive). By the way, editors of prime number don't agree on the best way to formulate the definition and it has changed many times. PrimeHunter (talk) 22:50, 5 February 2008 (UTC)
 * Quibble: Natural numbers are by definition nonnegative. There are two distinct usages that differ on the question of whether zero is a natural number. Some of the sillier religious wars in WP math articles are fought over this difference in usage. --Trovatore (talk) 23:05, 5 February 2008 (UTC)
 * Ah, but when you study abstract algebra, -5 is considered prime in certain contexts. The definition of prime really does depend on context (the term is used for things other than integers, as well). --Tango (talk) 14:08, 6 February 2008 (UTC)
 * Of course, in the ring of the integers, -5 is prime. That has nothing to do with my point, which was about zero, not about the negatives. --Trovatore (talk) 18:31, 6 February 2008 (UTC)
 * In the ring of integers, the principal ideal (-5) generated by -5 is prime, not the element. Tesseran (talk) 19:50, 9 February 2008 (UTC)


 * That's not true. The word prime is used to describe certain ideals, but it's also used to describe individual elements. Prime_element Black Carrot (talk) 04:12, 10 February 2008 (UTC)

Predicate logic, bis
Are the statements

1. $$\forall x (P(x) \to Q(x))$$

and

2. $$\forall x P(x) \to \forall x Q(x)$$

logically equivalent?

I do not know, but was hoping 1. would "distribute", allowing me to write:

$$\forall x \neg P(x) \lor \forall x Q(x)$$

I could then write 2. as:

$$\exists x \neg P(x) \lor \forall x Q(x)$$

Which are not logically equivalent? Damien Karras (talk) 22:53, 5 February 2008 (UTC)


 * They are not logically equivalent; 1 implies 2 but not vice versa. Note that 2 is true if $$\forall x P(x)$$ is false; that is, it's sufficient that there be at least one thing that does not have P. But that is not sufficient to make 1 true. --Trovatore (talk) 22:59, 5 February 2008 (UTC)


 * Thanks! I therefore assume that quantifiers can distribute? Is there a formal definition? Damien Karras (talk) 06:24, 6 February 2008 (UTC)


 * It wouldn't be a definition; it would be a theorem. You can prove that 1 implies 2, using the definitions of $$\forall$$ and $$\to$$ and previously proved theorems, thus establishing "1 implies 2" as a theorem. But there are lots and lots of things like that which you can prove. I don't know that this particular one has a name. —Bkell (talk) 12:18, 6 February 2008 (UTC)


 * Okay, what I mean to ask: Is $$\forall x (P(x) \lor Q(x)) \equiv \forall x P(x) \lor \forall x Q(x)$$? Damien Karras (talk) 18:27, 6 February 2008 (UTC)


 * No. Come on, translate it into words, and you should be able to figure out why. --Trovatore (talk) 18:34, 6 February 2008 (UTC)


 * With that in mind I can create a countermodel where P(x) : x is positive and Q(x) : x is negative and the universe is all nonzero integers, which makes $$\forall x P(x) \lor \forall x Q(x)$$ false and $$\forall x (P(x) \lor Q(x))$$ true. Putting it into words does nothing for me. :( But I get the idea... Damien Karras (talk) 18:58, 6 February 2008 (UTC)


 * The cases of quantifiers distributing over logical connectives can be derived from two basic ones:
 * $$\forall x (P(x) \land Q(x)) \quad \equiv \quad (\forall x \,P(x)) \land (\forall x \,Q(x));$$
 * $$\exists x (P   \land Q(x)) \quad \equiv \quad              P     \land (\exists x \,Q(x)).$$
 * In the last equivalence, proposition P is required to be independent of x.
 * For example, you can derive:
 * $$\begin{align} \forall x (P(x) \to Q) \quad

& \equiv \quad \forall x (\neg P(x) \lor Q) \\ & \equiv \quad \neg \exists x \,\neg(\neg P(x) \lor Q) \\ & \equiv \quad \neg \exists x (P(x) \land \neg Q) \\ & \equiv \quad \neg((\exists x P(x)) \land \neg Q) \\ & \equiv \quad \neg(\exists x P(x)) \lor Q \\ & \equiv \quad (\exists x P(x)) \to Q. \end{align}$$
 * --Lambiam 19:31, 6 February 2008 (UTC)