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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:

$\sum _{i=1}^{n}\left(2i+1\right)^{3}$

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 x_i are the terms. Confusing Manifestation(Say hi!) 04:26, 5 February 2008 (UTC)

You mean $\sum _{i=0}^{n}\left(2i+1\right)^{3}$. --wj32 ^t/c 08:56, 5 February 2008 (UTC)

More likely $\sum _{i=0}^{n-1}\left(2i+1\right)^{3}$ or $\sum _{i=1}^{n}\left(2i-1\right)^{3}$, if you want to preserve the number of terms. — 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

$\sum _{{\text{odd }}i=1}^{n}i^{3}$

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) × (-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#Divisibility.2C_prime_and_irreducible_elements 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. $\forall xP(x)\to \forall xQ(x)$

logically equivalent?

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

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

I could then write 2. as:

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

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 $\forall xP(x)$ 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 $\forall x(P(x)\lor Q(x))\equiv \forall xP(x)\lor \forall xQ(x)$? 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 $\forall xP(x)\lor \forall xQ(x)$ 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:

${\begin{aligned}\forall x(P(x)\to Q)\quad &\equiv \quad \forall x(\neg P(x)\lor Q)\\&\equiv \quad \neg \exists x\,\neg (\neg P(x)\lor Q)\\&\equiv \quad \neg \exists x(P(x)\land \neg Q)\\&\equiv \quad \neg ((\exists xP(x))\land \neg Q)\\&\equiv \quad \neg (\exists xP(x))\lor Q\\&\equiv \quad (\exists xP(x))\to Q.\end{aligned}}$

 --Lambiam 19:31, 6 February 2008 (UTC)

February 6

Trig Proof

I'm stuck on the following proof. Could anyone help me out? ${\frac {cosx+1+sinx}{cosx+1-sinx}}={\frac {1+sinx}{cosx}}$ --Sturgeonman (talk) 00:58, 6 February 2008 (UTC)

Got it. Never mind. -- Sturgeonman (talk) 01:34, 6 February 2008 (UTC)

BTW, you should put a backslash before common mathematical functions like \cos and \sin in LaTeX. It makes it look much better:

${\frac {\cos x+1+\sin x}{\cos x+1-\sin x}}={\frac {1+\sin x}{\cos x}}$ —Keenan Pepper 06:04, 6 February 2008 (UTC)

Two unrelated calculus questions

1) I'm being asked to evaluate the indefinite intergal ∫sin² 2x and for some reason, I don't know how to proceed ... all my calculations have been fruitless and led to culs-de-sac. Could anyone at least give me a hint in the right direction? (I don't think the [First] Fundamental Theorem of Calculus comes into play, since that's for definite integrals, if I'm not mistaken.)

2) For computing a Riemann sum using the midpoint method, my teacher recommends staying away from the formula and using a simple, straightforward nonformulaic approach. It's something like taking the midpoint of each interval and adding f(midpoint₁) + f(midpoint₂) + ... + f(midpoint_n-1) + f(midpoint_n). Will this give me the right answer, or do I have to multiply through by ∆x/2 or something like that, like when finding a trapezoidal sum?

Thanks! —anon —Preceding unsigned comment added by 141.155.57.220 (talk) 05:12, 6 February 2008 (UTC)

There is a version of the FTOC that applies to indefinite integrals, but it's true that in this case you need something a little more - and that something a little more is a trick called a double-angle formula. Put simply, do you know an identity that lets you write cos 2x in terms of sin x? And, having done so, can you rearrange that into something you can use in your first integral?

As for your second question, try thinking about it conceptually - a Riemann sum is essentially an approximation of the area of a bunch of rectangles. The area of a rectangle is the product of the two side lengths, so there's a reasonable chance you should be multiplying something by something to get your results. Confusing Manifestation(Say hi!) 05:25, 6 February 2008 (UTC)

The problem is not the 2x term, which can be dealt with by u-substitution. It's the squaring of the sine. There are power reduction formulae at the trig identities article. Black Carrot (talk) 09:08, 6 February 2008 (UTC)

I think you misunderstood ConMan's post. Note that he specified "cos 2x", which can be expanded using a $\sin ^{2}x$. This formula is memorized more commonly than the explicit formula for $\sin ^{2}x$ itself, and is thus a good starting point for the calculation. -- Meni Rosenfeld (talk) 23:09, 8 February 2008 (UTC)

(1) $\sin ^{2}(2x)={\tfrac {\sin ^{2}(2x)+\sin ^{2}(2x)}{2}}={\tfrac {1-\cos ^{2}(2x)+\sin ^{2}(2x)}{2}}={\tfrac {1-\cos(4x)}{2}}={\tfrac {1}{2}}-{\tfrac {1}{2}}\cos(4x)$
HTH. CiaPan (talk) 10:02, 6 February 2008 (UTC)

(2) Of course you should multiply each f(midpoint_i) by Δx, which is the n-th part of the integration interval. Putting all Δx-es outside the parens makes the integral approximation = IntervalLength/n × sum of f(midpoint_i). --CiaPan (talk) 09:57, 6 February 2008 (UTC)

February 7

Singular Value Decomposition

Suppose that I want to decompose ${\begin{bmatrix}3&0\\0&-2\end{bmatrix}}$ using singular value decomposition. So I define $B=A^{T}A={\begin{bmatrix}9&0\\0&4\end{bmatrix}}$. Since B is a diagonal matrix, the eigenvalues of B are simply the diagonal entries so I label the singular values of A as $\lambda _{1}=3,\,\lambda _{2}=2$. So the right and left eigenvectors are just $e_{1}$ and $e_{2}$ and their transpose. So the decomposition comes out to:

${\begin{bmatrix}3&0\\0&-2\end{bmatrix}}={\begin{bmatrix}1&0\\0&1\end{bmatrix}}{\begin{bmatrix}3&0\\0&2\end{bmatrix}}{\begin{bmatrix}1&0\\0&1\end{bmatrix}}$ which is obviously not true because of the minus sign in front of 2 in our original matrix A. So the question is how are these eigenvectors chosen. I know that this decomposition is not unnique. Since, there is an infinite number of left and right singular vectors to choose from, how do we choose the eigenvectors? I thought, the standard was to choose some unit vector but that still seems not to be enough. A Real Kaiser (talk) 04:41, 7 February 2008 (UTC)

Let's see. $A=VBU$. In the case of distinct singular values, V is determined uniquely up to multiplying by some diagonal matrix of unimodular values on the right. For square matrices and a fixed B and V, U is uniquely determined. In this case, once you had chose to use the identity on the right, you would have been forced to choose the diagonal matrix with 1 and -1 on the left. You must always use unit eigenvectors to make U and V unitary, and I believe the procedure is to choose V and then find U. This is in Johnson and Horn, Matrix Analysis, Theorem 7.3.5. (Sorry, I know that this could use some wikilinks) 134.173.92.17 (talk) 05:58, 7 February 2008 (UTC)

Real Analysis

On a completely different note, let $\Omega$ be a set and let F be a $\sigma$-algebra on $\Omega$. Let $\mu$ be any measure on $(\Omega ,\,F)$. In my real analysis class, our teacher told us that if ${A_{n}}$ is a nested decreasing sequence in F (meaning $A_{n}\subset A_{n+1}$) then we have that

$\mu (\cap _{n=1}^{\infty }A_{n})=\lim _{n\rightarrow \infty }\mu (A_{n})$

The question is that is this still true if $\mu (A_{1})=+\infty$? If not, can anyone provide a counterexample because I think that this would be false if the measure of $A_{1}$ is infinite to begin with. A Real Kaiser (talk) 05:00, 7 February 2008 (UTC)

Example 1.20 (c) in Rudin's Real and Complex Analysis, Third Edition: "Let $\mu$ be the counting measure on the set $\{1,2,3,\dots \}$, and let $A_{n}=\{n,n+1,n+2,\ldots \}$. Then $\bigcap A_{n}=\emptyset$ but $\mu (A_{n})=\infty$". —Preceding unsigned comment added by 134.173.92.17 (talk) 05:42, 7 February 2008 (UTC)

Just a small correction: If it's a nested decreasing sequence, then you mean $A_{n}\supset A_{n+1}$. —Bkell (talk) 06:14, 7 February 2008 (UTC)

That is of course what I meant. Thanks a lot. I actually have the book and this is exactly what I was looking for. Awesome!

A Real Kaiser (talk) 06:36, 7 February 2008 (UTC)

Points on a plane

If I give a number of random points and asked them to be expressed by an equation. (i.e. the points must be able to be crossed by an shape that is able to be defined by an equation. As in, a line, parabola, the edge of a square. etc.) What is the maximum number of points for which there is definitely an equation to express them? I know that it is possible to use a circle to express 3. Is it possible to express 4? Thanks ahead of time. 99.226.39.245 (talk) 05:20, 7 February 2008 (UTC)

Well, for any finite number of points, you can always find a polynomial that goes through all of those points. Just like how you need a straight line to go through two points, you need a parabola to go through three points. So, if you give me any arbitrary n points (that are distinct, of course), I can always find at least one polynomial of degree n-1 that goes through all of those points.A Real Kaiser (talk) 06:38, 7 February 2008 (UTC)

A Real Kaiser is correct, but the resulting polynomial may be very ugly. --Gerry Ashton (talk) 07:45, 7 February 2008 (UTC)

It would be best to read the article Polynomial interpolation (and probably Interpolation) for the actual equation. --Martynas Patasius (talk) 14:05, 7 February 2008 (UTC)

That article only talks about polynomials in one variable (ie. y=P(x)), which makes it impossible to have two points with the same x coordinate. Certainly in some cases it is possible to use a polynomial in both x and y to go through points in the same vertical line (eg. given points (1,0),(0,1) and (0,-1) the polynomial x^2+y^2=1 goes through them), is that always possible? --Tango (talk) 17:30, 7 February 2008 (UTC)

Sure. If you want to go through $(x_{1},y_{1}),\ldots (x_{n},y_{n})$ say, just use the equation $(x-x_{1})\ldots (x-x_{n})=0$ (say). It's not very edifying, but it's a polynomial equation, as desired. Algebraist 18:22, 7 February 2008 (UTC)

Or you could use $\prod _{i=1}^{n}((x-x_{i})^{2}+(y-y_{i})^{2})=0$, which gives you exactly the points you want. Algebraist 18:29, 7 February 2008 (UTC)

A more interesting version of your question is whether, given finitely many points in the plane, there exists an algebraic variety containing all of them. I strongly suspect that there is, but can't immediately see how to prove it and can prove it. Just transform co-ordinates by rotating the axes slightly until no two points share the same x co-ordinate and use the Kaiser's idea. Algebraist 18:34, 7 February 2008 (UTC)

Thank you! 99.226.39.245 (talk) 21:49, 10 February 2008 (UTC)

Fitting regular polygons inside each other

Is there any existing proof out there proving that it's impossible to fit EXACTLY (all corners must touch a side) a regular pentagon inside a square? or a regular pentagon inside a regular hexagon?

What about a general test to determine if a polygon with X equal sides can fit EXACTLY inside another polygon with (X+1) or (X-1)equal sides?

Oh, yeah is there any special name to the polygons with such relationship? --Kvasir (talk) 18:40, 7 February 2008 (UTC)

My initial thought for the first one (the others are probably similar) is that it can't be done because the circle circumscribing the pentagon would intersect the square 5 times. Assuming the square and pentagon are concentric (I doubt it will work in the non-concentric case, but can't prove it yet), the circle would intersect the square an equal number of times on each side, by symmetry, so that's 0, 4 or 8 intersection points. So you have to go with 8 (the only one bigger than 5), and those 8 points will have a rotational symmetry of order 4, so I think that would end up requiring the pentagon to have a 4-symmetry, which it doesn't. So basically, you end up needing an n-gon to have a rotational symmetry of order n-1, which is only the case for n=2, which isn't really a polygon. I'm not 100% sure on that last bit, and I can't think of how to approach non-concentric polygons, but I think that's almost an answer to your question. --Tango (talk) 21:19, 7 February 2008 (UTC)

I think you can fit a pentagon in a hexagon by putting a vertex of one in a vertex of the other. Black Carrot (talk) 22:00, 7 February 2008 (UTC)

OK how do you prove that such a hexagon exist Black Carrot? or any other polygon with n>=5?

I should rephrase the question. The polygons need not be concentric. I should illustrate it better as I have already tried it both ways:

A: This is to fit the largest possible (n+1)-gon inside n-gon. Start with a circle (n=1) of a given area, then insert a triangle (n=3, n=2 doesn't form a regular polygon), then a square (n=4) inside the triangle. But there is no way to fit a regular pentagon (n=5) inside the square where all 5 vertices touch the square. Has anyone proven that it's impossible? Has anyone proven that the fitted (n+1)-gon has the maximum area when it is fitted inside the n-gon this way?

B: This is to enclose an n-gon with the smallest possible (n+1)-gon. Start with a circle of a given area, then enclose it with a triangle where the circle intersect the triangle exactly 3 times. Then the triangle is fitted inside a square; and the square is fitted inside a pentagon. I can only get this far, I could not get the pentagon to fit inside any hexagon where all 5 vertices end up on the sides of hexagon. Similarly, has anyone proven that it's impossible to fit a regular pentagon inside a regular hexagon? Has anyone proven that the fitted (n+1)-gon has the minimum area?

A related part of the question would be a generic formula to get the area of n-gon that fits exactly inside (n-1)-gon or (n+1)-gon of a given area, if it exists. --Kvasir (talk) 22:12, 7 February 2008 (UTC)

Here's a simple proof that you can't fit a regular pentagon inside a square in the way you describe. Suppose a regular pentagon fits inside a rectangle (which might be a square) so that all five vertices of the pentagon touch a side of the rectangle. Then one side of the rectangle must touch two vertices of the pentagon, so one of the sides of the pentagon must be part of one of the sides of the rectangle. Then obviously the other three vertices of the pentagon must touch the other three sides of the rectangle. Now consider the distance between parallel sides of the rectangle. One set of parallel sides is separated by the distance from a vertex of the pentagon to the midpoint of the opposite side of the pentagon, and the other set of parallel sides of the rectangle is separated by the distance between two nonadjacent vertices of the pentagon. Plainly these distances are not equal, so the rectangle is not a square. (This would be a lot easier to describe with a picture, but hopefully this is clear.) —Bkell (talk) 06:01, 8 February 2008 (UTC)

I was imagining it wrong. I tried to draw it, and it didn't work. Incidentally, part of what Bkell suggested works for all polygons. To fit an n+1gon inside an ngon, they must share a side. That is, a side of the inner one must be part of a side of the outer one. Imagine this side at the bottom, flat to the ground. Both shapes have a vertical axis of symmetry. If the inner one is touching the outer one correctly on both sides, their axes of symmetry must match, so the inner one must be centered on the bottom of the outer one. That means there's only one diagram to work out for each n, which isn't bad. (BTW, this isn't true for a triangle on the inside, since it doesn't have two points to compare to each other, and it isn't true for a square on the inside if the outer shape has a flat top. Not that a pentagon does, but still.) Black Carrot (talk) 06:30, 8 February 2008 (UTC)

A similar argument applies if the two polygons share a vertex, call it A. Imagine A at the bottom. For each pair of symmetric vertices of the inner one, there's a circle centered at A and going through both of them that can only (I think) intersect the outer polygon in two points, which will be symmetric with each other. If the inner polygon hits both these points, its axis of symmetry must be vertical. Black Carrot (talk) 06:38, 8 February 2008 (UTC)

That last one works best if you choose the two vertices adjacent to A. Black Carrot (talk) 07:14, 8 February 2008 (UTC)

Ok this is what I should've uploaded at the start. Yes, this does illustrate why (n+1)-gon can't fit inside an n-gon except for a square inside a triangle as Bkell explained. What about the other way? fitting an n-gon inside an (n+1)-gon? As you can see I've gotten as far as a pentagon. Is there a proof that the pentagon is the smallest there is to enclose the square? How can one disprove or prove that a hexagon in this series exist? I'm surprised no Chinese or Greek mathematician has pondered over this before. --Kvasir (talk) 15:59, 8 February 2008 (UTC)

Bkell's argument doesn't prove it can't be done for an n+1gon in an ngon (other than pentagon in square), it just shows that there's only one diagram to consider. You'd still have to work out the measurements. Black Carrot (talk) 19:03, 8 February 2008 (UTC)

I suppose I'm not looking for THE proof per se, more like a reference of someone who have worked this out before or at least have a name of this series of polygons. Hey may be I can make a thesis out of it! I'm currently working out a proof whether or not a hexagon exist, and if it does, if a heptagon exist, and so on. So far i've proven that no hexagon exists if any of the spaces between the hexagon and the pentagon form a quadrilateral. If such a hexagon exists, there must be five triangles formed in the spaces between the hexagon and the pentagon. --Kvasir (talk) 06:32, 9 February 2008 (UTC)

I can't help you with references, but I'd be surprised if nobody's worked out the answer before. Since I ran out of ideas and had some time, I trig-bashed my way through the general case. If you want to know how, I can explain it, but it shouldn't be anything new to you. Unless I made a mistake, there's not way to fit more than four consecutive vertices of a regular polygon to the inside of another regular polygon with one side more or fewer. Black Carrot (talk) 09:15, 9 February 2008 (UTC)

You can consider this as a system of equations and do some simple analysis to see if you get an overdetermined system. An n-gons is completely determined by 4 parameters, say the position of centre of the polygon and the one of the vertices. Now consider each vertex, for it to fit in another given polygon the vertex must lie on one of the straight line which make the second polygons boundary, so it must satisfy the equation ax+by+c=0 of the straight line. So to fit an n-gon inside another polygon we have a system of n-equations in 4 unknowns. This indicates that you would expect a one parameter family of triangles to fit in any other polygon (3 equations in 4 unknown). You would expect to find a finite number of squares which fit (4 eqn in four unknowns) but not expect pentagons or hexagons to fit. There is of course a caveat in that symmetry may mean that some of the equations are degenerate. Indeed an n-gon will fit trivially inside a 2n-gon. --Salix alba (talk) 10:59, 9 February 2008 (UTC)

A regular-pentagon does not fit inside a square. Take a pentagon with the distance from the centre to each vertex is 1, and arrange it so that one edge in horizontal. Now calculate its height and width, for it to fit in the square in the diagram above the height and width must be equal. If θ=π/5 we find the height is 1+cos(θ)=1.809 and its width is 2 sin(2 θ)=1.902, these are quite close so thy may look fine visually but they don't actually fit. --Salix alba (talk) 11:29, 9 February 2008 (UTC)

Yeah i know that pentagon in the diagram doesn't fit inside the square, that's why it's coloured blue. --Kvasir (talk) 14:02, 9 February 2008 (UTC)

February 8

Ever any lotteries with an expected value greater than the ticket price?

I live in the UK and I like to think I'm rational. While it could be fun to gamble, I am only prepared to gamble where the expected value of the lottery is greater than the ticket price. Does this ever happen with any lottery available in the UK please? Unfortunately it seems to be impossible to get hold of the statistics to calculate this (and possibly the lotteries are set up to make this impossible - I'm not sure). I've no idea how many "rollovers" it would take. 80.0.102.226 (talk) 00:02, 8 February 2008 (UTC)

Yes, it occasionally happens. To make it simple, let's consider just the main (UK) Lotto game, and just the jackpot. The probability of a given ticket winning the jackpot is in fact 1 in 13,893,816. So if the predicted jackpot is in the region of £14 million or above, it makes mathematical sense to buy a ticket (the National Lottery website gives the predicted jackpot for the next draw - it has to be a double rollover to get to that kind of figure). Incidentally, always buy your ticket on Friday afternoon or Saturday - if you buy it any earlier in the week, you're more likely to be dead by Saturday evening than to win the jackpot! AndyofKent (talk) 02:51, 8 February 2008 (UTC)

But any calculation of payout would have to include the possibility that more than one person wins the jackpot, thereby splitting the amount of prize money. If you assume that each person chooses their lottery numbers independently and randomly, then presumably you could estimate this as something like (# of people buying tickets) / (# of combinations), but as many people have "methods" of selecting lottery numbers, if you can find a method of your own that selects a less-popular combination, you will greatly increase your expected payout. Given that the average lottery is designed to make more money than it pays out in jackpots, it would require fairly specific circumstances to have such an occurence. Confusing Manifestation(Say hi!) 03:52, 8 February 2008 (UTC)

Plus, even when this does happen, the expected utility of buying a lottery ticket is still negative. —Keenan Pepper 04:45, 8 February 2008 (UTC)

If the utility is negative, how come millions of people buy tickets? 80.0.121.236 (talk) 12:23, 8 February 2008 (UTC)

The utility is only negative if you base utility solely on net worth. By that argument it would be even more irrational to go to the movies, where the ticket price is higher and the chance of winning the jackpot essentially nil. For many people the attraction of a movie is that it lets them forget their own lives for a while and vicariously live the life of someone else who has amazing adventures and does great things. I think the people who buy lottery tickets do it for the same reason. Instead of £5 for a few hours' fantasy they spend £1 for a week's fantasy. It's a cost-effective way of satisfying a basic human need, and I think it's perfectly rational, though also somehow sad. -- BenRG (talk) 12:57, 8 February 2008 (UTC)

If you ignore rollovers, the expected value of a £1 ticket is 50p, simply because that's the amount that goes into the prize fund (the rest goes to good causes, admin, tax and, of course, profit), and all tickets have an equal chance of winning. So you expect to lose half your money. For the expected net gain to be positive, you would need enough rollovers for the total prize fund to be double what it would normally be - that's a pretty easy calculation to check, I think, as they give estimated prize funds and jackpots, so it's just basic arithmetic once you've found the right numbers. --Tango (talk) 11:32, 8 February 2008 (UTC)

There are also more minor prizes in addition to the jackpot - these would affect the expected value a lot. Very good point that the expected value of a ticket must be 50p on average for the Lotto - is it the same for Euromillions? And are these the only UK lotteries that have rollovers? And what is the normal prize fund - I think, maybe I'm wrong, that the lottery TV rollover adverts only quote the forecast jackpot size, rather than the total prize fund. I'm not sure if extra ticket sales would be reflected in a bigger jackpot for that draw or if the extra sales money goes to make the prize-money in the next draw. And do the smaller prizes vary in value, or are they fixed? Sorry, so many questions. 80.0.121.236 (talk) 12:13, 8 February 2008 (UTC)

I don't know about anything other than Lotto. With that, the prize fund for any given week is 50% of the total ticket sales for that week, plus any rolled over jackpots. Extra ticket sales due to there being a rollover would be included, but would actually reduce the expected net gain, since the effect of there being more tickets out there is more significant than the extra prize money. I haven't really watched the lottery for a while, but I know they used to give both the total prize fund and the jackpot fund before the draw - I would assume both numbers are available somewhere in advance. The only fixed prize is £10 for 3 numbers, all the others are done in terms of percentage - I think it's x% split between all the people that get 4 balls, y% split between all the people that get 5 balls, etc., with the x, y... being fixed (and published on the official website somewhere, as I recall). --Tango (talk) 14:03, 8 February 2008 (UTC)

"Extra ticket sales due to there being a rollover would be included, but would actually reduce the expected net gain, since the effect of there being more tickets out there is more significant than the extra prize money." Does this mean its impossible in practice to get an expected value greater than the ticket price? I'm curious to see any statistics that demonstrate that rollovers decrease the expected value due to the increase in ticket sales. Does anyone know where I can find the stats to calculate the expected value - I need a) total prize fund, and b) number of ticket sales - both figures now seem to be kept secret. 80.3.47.25 (talk) 23:23, 8 February 2008 (UTC)

The extra tickets will reduce the expected gain, but not by as much as the rolled over jackpot increases it, I would expect. Rollovers are still better value, I would think. --Tango (talk) 13:34, 9 February 2008 (UTC)

It isn't enough for the expected value to be greater than the ticket price. Imagine if, for one pound, you could by a ticket that would automatically win a £1.01 prize. Would you do it? Would you go through the work of buying a ticket for one measly pent? Also, AFAIK it's common for lotteries to pay in increments. It's generally excepted that money later is worth less than money now. — Daniel 23:45, 8 February 2008 (UTC)

It isn't enough for the cost of the ticket to be less than the chance of winnings multiplied by the winnings amount. The number of people playing becomes an important part too, as its more likly you will have to share the prize. I know that in Australia on new years eve there is usually a $31 million dollar draw. The amount you get for winning is only about 2 million because of the huge numbers of winners. While lotteries of $9 - $13 million have average wins of 4 to 5 million. Unfortuently most places don't post the number of people who entered, you have to go through there public tax records and work out how many tickets they must have sold etc. The best way to win lottery is just to pick a sequence that is unlikely to be picked by people because its seemingly rediculous like 1 2 3 4 5, that way if you win, you have more chance at being the only winner.--Dacium (talk) 22:26, 10 February 2008 (UTC)

You can quite easily find out the number of tickets sold - take the total prize fund and divide by how much of each ticket goes into it (after minusing off any rolled over jackpots). As for the best way to win the lottery: Don't play. I win 50p twice a week on the lottery using that method. (Incidentally, I think I read somewhere that a very large number of people buy a ticket with 1,2,3,4,5,6 on the UK lottery each week, so if it ever comes up they won't win much at all.) --Tango (talk) 23:44, 10 February 2008 (UTC)

$\lambda$ calculus

I was reading the article on lambda calculus, and it appeared to me that it is used almost exclusively for programming. Is this true, or does it have real mathematical applications? Thanks, Zrs 12 (talk) 03:27, 8 February 2008 (UTC)

Definitely has "real mathematics" applications. The lambda calculus predates electronic computers, and was used by its inventor Alonzo Church and his students to derive important results in mathematical logic in the 1930s. Although our lambda calculus article says it "can be thought of as an idealized, minimalistic programming language", this is a reversal of the actual historical development - programming languages such as Lisp were designed around the lambda calculus, which already existed. So it would be more accurate to say that programming languages are implementations of the lambda calculus. Gandalf61 (talk) 07:49, 8 February 2008 (UTC)

I think you've engaging in a bit of historical revisionism there. Lisp definitely was not designed around the lambda calculus — I'm not sure McCarthy had even heard of the lambda calculus when he started working on Lisp. The driving idea behind Lisp was symbolic computation. At the time electronic computers were seen as replacements for human computers, and the idea that programs might work with words instead of numbers was new. All of Lisp's other innovations were accidental byproducts. The lambda keyword was introduced to the language very early on, but Lisp did not at the time contain anything resembling Church's lambda calculus. It used dynamic scoping, not by choice but because nobody involved understood scoping. It had destructive update. It had statement labels and goto. It had car, cdr, ctr and cpr. The funarg problem was noticed by an outsider. McCarthy saw it at the time as a bug, not the fundamental design flaw it actually was, and Steve Russell fixed it with a trick called the "funarg device", now known by the less embarrassing name of "lexical closures". Lexical closures are a great implementation trick, but unfortunately they've permanently contaminated the semantics of the language because of their interaction with destructive update of bindings. Church understood the trickiness of lexical scoping, but McCarthy's team doesn't seem to have benefited from that. I don't think Church's work had any influence beyond the name lambda, which has pointlessly frightened generations of students. They should have called it function. It wasn't until lexical scoping finally became standard in Lisp that you could reasonably claim that it even contained the lambda calculus as a sublanguage, and it's still a major hassle to program in that style in Common Lisp; you have to write #' and funcall all over the place and you can't rely on tail recursion. The language was pretty clearly not designed for that kind of programming. Even Scheme's embedded lambda calculus isn't very faithful to the original, with its eager evaluation order and lack of automatic currying (and logical inconsistency, but that's a different story). I'm sorry to rag on Lisp so much; I actually like it a lot, but it gets romanticized these days in a way that has little bearing on reality. Historically speaking, Lisp is very thoroughly in the Perl school of "design". -- BenRG (talk) 19:17, 8 February 2008 (UTC)

Very interesting. The second paragraph of our Lisp (programming language) article says "Lisp was originally created as a practical mathematical notation for computer programs, based on Alonzo Church's lambda calculus" - so maybe it needs some expert attention to correct what appears to be an urban myth. Gandalf61 (talk) 11:09, 9 February 2008 (UTC)

Why does mathematics use the idea of 'truth'?

I don't get why mathematics uses the idea of statements being 'true' or 'false' rather than just being 'consistent' or 'unconsistent' or 'unattainable' commpared with the system in question.

What's truth got to do with it? What's truth but a second-hand emotion? —Preceding unsigned comment added by 212.51.122.27 (talk) 17:28, 8 February 2008 (UTC)

Well, to a Platonist, mathematical objects exist just as other objects do, and mathematical statements are statements about these objects which are true or false just as statements about regular objects are. From the point of view of other philosophical positions (such as formalism), 'true' is a defined term like any other which is used because it is useful. Results such as Gödel's completeness theorem tell us that the semantic notion 'true in every model of a theory T' coincides with the syntactic notion 'provable in T', and so to a certain extent we can equate truth with provability (is this what you mean by being 'consistent' or 'unconsistent'?). I have no idea what you mean when you call truth a second-hand emotion. Algebraist 17:46, 8 February 2008 (UTC)

What's Love Got to Do with It (song) --LarryMac | Talk 17:49, 8 February 2008 (UTC)

I think "true/false" and "consistent/inconsistent" are different. "True" means "provable" (these kinds of statements will always be of the form "A=>B", it's generally meaningless to say simply "B is true" in maths, although we don't always explicitly state the assumptions). "False" means "the negation is provable". "Consistent" means "does not contradict", it's not necessarily true, but there is nothing stopping you assuming it's true (for example, the axiom of choice is consistent with the ZF axioms, you can't use them to prove it or it's negation [ie. it's independent], so it's not "true" or "false", but you don't get a contradiction if you simply assume it's true). "Inconsistent" means you do get such a contradiction - A is inconsistent with B if A=>C and B=>not C for some C. That doesn't say anything about the truth or A, B or C, just their relationship with each other, a priori, either A or B can be true (given appropriate assumptions), just not both. --Tango (talk) 18:05, 8 February 2008 (UTC)

No, "true" does not mean "provable". The most important take-home message from the Gödel incompleteness theorems is that you cannot identify provability and truth (at least without losing properties of truth that we would like it to have, such as excluded middle).

Pre-Gödel formalists tried to identify provability with truth. Gödel showed that that sort of formalism was non-viable. Post-Gödel formalism simply does away with the notion of truth. The only major school that still identifies provability with truth is intuitionism, but it gets away with it by declining to fix a single formal notion of provability. --Trovatore (talk) 18:15, 8 February 2008 (UTC)

While you are of course correct, I feel I ought to defend my post by pointing out that modern formalists still admit truth (as a defined term) in the sense of true-in-a-model, just not in the sense of true-in-a-theory. Algebraist 18:31, 8 February 2008 (UTC)

I think this is getting a bit off track. Head-in-the-skies philosophy is important, but keep in mind, there is such a thing as being actually true. It's hard to dispute a winning Nim strategy, or the fact that every square is a sum of consecutive odd numbers. One artificial construction may be as good as another in some sense, but that doesn't change what's real. (This is not, BTW, Platonism. I'm not saying circles "exist", I'm saying that what exists exists.) Both those examples are discrete math with small numbers, which is the area of math most easily accepted as not made up. a(b+c) = ab+ac, no matter who you are. Black Carrot (talk) 18:40, 8 February 2008 (UTC)

Small detail - the name of the link references the scrollover text of the image. Black Carrot (talk) —Preceding comment was added at 18:41, 8 February 2008 (UTC)

BC, you make a good point, but it's surprisingly hard to draw the line. Where do you put Goldbach's conjecture, for example? The probabilistic evidence for it is extraordinarily convincing, yet it's entirely possible that it will never be proved from any "foundationally relevant" theory. That's a case where, if there were a counterexample, it could be checked by computer. If you're willing to take that as part of "reality", then what about the twin prime conjecture, where there's no such thing as checking an example or counterexample by computer? Before you know it you're having Woodin cardinals for breakfast and expressing your opinion on the truth or falsity of the continuum hypothesis before lunch. --Trovatore (talk) 18:45, 8 February 2008 (UTC)

Oh come on, it's not a slippery slope. There are obvious limits. I'm not saying the line is easy to draw, or even that there could ever be a single well-defined boundary, just that it's silly to paint everything with the same brush. Some things are confusing and uncertain, sure, but most of them are things nobody's heard of and nobody cares about. Number theory is uncertain, but arithmetic isn't. Black Carrot (talk) 18:58, 8 February 2008 (UTC)

By the way, you can't use probabilistic evidence for the Goldbach conjecture, no matter how you slice it. Within the range of numbers the vast bulk of humanity actually uses, it's been proven beyond reproach. Beyond that, it hasn't been shown probable at all. Black Carrot (talk) 19:00, 8 February 2008 (UTC)

No, that isn't true; the probabilistic argument is not simply the absence of a counterexample within numbers that can be checked. It's the fact that the bigger the numbers get, the harder it is for there to be a counterexample, because you have more primes available from which to make sums. This argument is so convincing that in my opinion the truth of the Goldbach conjecture is settled, to the point that a formal proof would not change its epistemic status much -- after all, the theory from which you prove it could also turn out to be inconsistent.

What I'm saying about the "slippery slope" is that it's extremely hard to do anything in mathematics without accepting, in one way or another, the real existence of (at least some) abstract objects. And once you've done that, the case against even-more-obviously-abstract objects is harder to make. --Trovatore (talk) 19:17, 8 February 2008 (UTC)

I don't find that argument convincing, though. As certain as it may be that almost all of them are sums of pairs of primes, even that almost all of them are sums of pairs of primes in many ways, that doesn't rule out, or even make less likely, that one or a few somewhere up the line won't be. It's not a matter of how many of them there are, it's how they fit together, and that's what's hard to prove.

Yes, it's hard to draw a line, but that's true of anything. In any area of study, there are some things that are too convincing for argument and a whole lot of fringe ideas that are just silly, and everything in between. I'm just saying that you don't have to ignore one end of the spectrum in favor of the other. Black Carrot (talk) 19:27, 8 February 2008 (UTC)

I think it does make it less likely. So much less likely that, as I say, the question is effectively settled. A counterexample to Goldbach would require an unbelievably massive "conspiracy" of prime numbers "avoiding" summing to the counterexample. A similar conspiracy resulting in a proof of 0=1 from Peano arithmetic is just as believable.

I think there is no "silly" end to this spectrum, because full-on hardcore realism about large cardinals is just fine, and it allows you to decline to draw an arbitrary line somewhere in the middle. --Trovatore (talk) 19:33, 8 February 2008 (UTC)

I have no problem with the prime numbers conspiring. They've done it before, and they'll do it again, they're fickle like that. There's nothing fundamental to my view of the world that says they're well-behaved. You might view them differently. Peano arithmetic, though, is abstracted barely an inch and a half from its roots in bean-counting. If there's something wrong with that, the universe is broken. The only claim it makes that isn't obviously true is that there are infinitely many whole numbers. If a paradox comes out of that, fine. Maybe there's a cap on how far you can push the pattern, but for sufficiently small numbers it's impossible for it to be wrong.

Every crackpot has a reason. I think most people would agree that a system with more patches than an old sweater, that reasons glibly about imaginary objects nobody could have conceived of a hundred years ago, is a bit fringy. I'm not saying it can't be justified philosophically, in the same way that you can justify any crime by a sufficiently extreme plea to moral relativism, I'm saying that that's what it takes to justify it. It doesn't take mental gymnastics to find a foundation for arithmetic. Black Carrot (talk) 20:07, 8 February 2008 (UTC)

There are no "patches" in set theory. That's a misunderstanding of the antinomies. The antinomies did not come from an informal notion of set, but from the wrong informal notion (the conflation of the extensional and intensional concepts). --Trovatore (talk) 20:15, 8 February 2008 (UTC)

Fair enough. I'm sorry, bad example. I think you should reread the discussion that prompted my first comment, though, and compare it to our discussion since then. I don't think you actually disagree with what I was trying to say. I was only arguing for a more balanced viewpoint. Especially in front of the OP, who needs it. Black Carrot (talk) 21:15, 8 February 2008 (UTC)

There is a big difference between finding a counterexample for something that all evidence points towards being true and finding a counterexample for something that's been proven true. One is very unlikely, the other is impossible, that being what "proven" means. By the standard definitions of 0, 1 and equals, $0\neq 1$. The only way you could change that would be to change the definitions, and that's moving the goalposts. --Tango (talk) 21:26, 8 February 2008 (UTC)

No, sorry, Tango, you're laboring under a common misconception here. Just because something has been proved does not imply that we know apodeictically that it is true. Possibly the assumptions we used to prove it were false. In the case of sufficiently simple arithmetic statements like Goldbach, for the assumptions to prove such a statement true even if it were actually false, the assumptions would have to be more than false--they'd actually have to be mutually inconsistent. But we do not know beyond all doubt that that doesn't happen. --Trovatore (talk) 21:30, 8 February 2008 (UTC)

Word. Could you point me to a good source on the words extensional, intentional, and antinomy? Our articles on them suck could be more detailed. Black Carrot (talk) 21:35, 8 February 2008 (UTC)

I'm afraid I'm caught short here -- can't think of a good source at the moment. --Trovatore (talk) 21:38, 8 February 2008 (UTC)

But mathematical theorems always take the form "A=>B", the assumptions are, by definition, true, because that are assumed to be. If those assumptions don't actually hold for what we intuitively think of as numbers, then the theorem isn't very useful, but it's still true. The statement explicitly said "Peano arithmetic" - that Peano arithmetic implies $0\neq 1$ is not disputable. That Peano arithmetic is an accurate description of what we intuitively know about arithmetic is possibly disputable, but that's another matter. --Tango (talk) 22:38, 8 February 2008 (UTC)

No, that's the shallow sort of formalism, and is quite demonstrably wrong. Peano arithmetic does not define what is true about the naturals; it's a collection of statements that we believe are true about the naturals, and from which we can derive others that -- assuming we were correct in the first place -- must also be true. But maybe we were wrong in the first place.

I agree that Peano arithmetic implies 0≠1, but that isn't the question -- the question is whether it implies 0=1. It does not follow, merely because it implies the first, that it does not imply the second. --Trovatore (talk) 22:53, 8 February 2008 (UTC)

Has Peano arithmetic not been proven consistent? If not, then I need to do some more reading on the subject before continuing this discussion. --Tango (talk) 23:10, 8 February 2008 (UTC)

Ok, after a quick bit of reading, it seems it depends on what you actually mean by "proven consistent"... I think this is all getting a bit too deep for this time of night... (it's 2313hrs here). --Tango (talk) 23:13, 8 February 2008 (UTC)

Depends on what you mean by "proven". Sure, it's been proved, but not without assumptions. Maybe those assumptions are wrong.

Gödel's second incompleteness theorem is taken by some to eliminate all hope of proving that PA is consistent "by finitistic methods", as Hilbert wanted. However there has never been a good definition of "finitistic", so this is a little hard to pin down absolutely. As I recall Gödel himself disclaimed this interpretation of his theorem. From the other side of the question, Gentzen proved that PA is consistent by analyzing possible proofs of a contradiction and performing induction -- but it was transfinite induction up to a certain infinite ordinal number. Taking the two results together, the status of Hilbert's second problem is particularly muddled -- has it been resolved positively, resolved negatively, or is it as yet unresolved? --Trovatore (talk) 23:16, 8 February 2008 (UTC)

I can't help thinking of "That that is is that that is not is not that that is is not that that is not that that is not is not that that is is that not it".  :) -- JackofOz (talk) 23:23, 8 February 2008 (UTC)

Power reduction writ large

Does anyone know of a relative of the addition theorem on circles rather than spheres? In particular, I'd like to have the coefficients in the expansion
$P_{l}(\cos \theta )\equiv \sum _{j=0}^{l}a_{j}\cos j\theta$
The extension to $\cos(\theta -\phi )$ would then be easy — apply the sum formula to each term and get a different sum of cosines and a sum of sines. Of course, one way to approach this would be to apply the power reduction rules repeatedly, but I'm not sure how (or if it's even always possible) to write $\prod _{i=1}^{n}\sin ^{p_{i}}j_{i}\theta \prod _{i=1}^{n}\cos ^{q_{i}}k_{i}\theta$ as a sum of only first-power sines and cosines. --Tardis (talk) 20:46, 8 February 2008 (UTC)

It occurs to me that it would be sufficient to know how to evaluate $\int _{0}^{2\pi }\cos(j\theta )\cos ^{k}\theta \,d\theta$ for $j,k\geq 0\in \mathbb {I}$. Any thoughts on that? --Tardis (talk) 05:12, 9 February 2008 (UTC)

February 9

Centre of gravity of a country and capital cities

Is there a standard, generally accepted, method of determining the centre of gravity of an irregular shape such as that of most countries (as defined by their internationally recognised borders)? How would this work if we have to take far-distant offshore island components (e.g. Hawaii in the case of the USA), non-contiguous components (e.g. Alaska), and exclaves, into account? Has any research been done on the capital cities that are furthest from or nearest to their country's centre of gravity (however that might be calculated)? Thanks. -- JackofOz (talk) 00:19, 9 February 2008 (UTC)

There is a standard definition of the centre of gravity of a body, yes. The problem is that the natural definition of centre of gravity will give you something inside the earth, which is presumably not what you're after (as an extreme example, if your country covered the entire earth's surface, its centre of gravity would have to be the centre of the earth). This problem shouldn't be too extreme for actually existing countries, especially reasonably small ones: you could just take the point on the earth's surface closest to the true centre of gravity. I know very little about computers, but I suspect that with one you could very easily calculate the centre of gravity of any given country, with or without exclaves and such. I have never seen this concept elsewhere (except in the motel in the centre of America featured in American Gods) and am unaware of any research into centres (centroids?) of countries, distance from capitals, etc. Algebraist 00:53, 9 February 2008 (UTC)

I have seen many claims of a particular town being the "geographical center" of a country. How they've arrived this I don't know. --Kvasir (talk) 06:43, 9 February 2008 (UTC)

From that disambig and the linked articles, it seems to be standard to use the centroid=centre of mass, but unfortunately none of these pages explain how they get around the problem that the earth isn't flat. Algebraist 10:33, 9 February 2008 (UTC)

Yes, I noted that too. Presumably what they're actually calculating is the centroid of a 2-dimensional map projection of the country, rather than its real 3-dimensional surface. And, as we know, map projections, while useful, are inherently erroneous because it's not possible to accurately map 3 dimensions onto 2. The best we can do is minimise the error. When it comes to relatively small countries, eg. Switzerland, the variation from the "true" centroid would be small and probably insignificant for most purposes; but with the Russias, Canadas, Brazils and Australias of the world, it would be a different story. That's in absolute distance terms; but when compared with the extreme internal distances in such countries, it would still be relatively insignificant. Wouldn't it? -- JackofOz (talk) 10:48, 9 February 2008 (UTC)

That'll depend on what projection you use. I expect you can get the errors small by choosing the right projection, I'm just not sure what the right projection is. Algebraist 11:59, 9 February 2008 (UTC)

Can't you just find the centre of gravity inside the Earth, draw a line through it and the centre of the Earth and take the point where it intersects the surface as the centre of the country? --Tango (talk) 13:38, 9 February 2008 (UTC)

That's what I suggested above. What I don't know is whether anyone is actually using it. Like JackofOz, I suspect not. Algebraist 14:27, 9 February 2008 (UTC)

OK. Thanks for the comments so far. Here's a slightly different question: In front of me is a map of Afghanistan. I have no issues with whatever projection was used; I'm just interested in the irregular shape I've got. How do I go about working out its centroid? Algebraist suspected above that with a computer I could very easily calculate this. How, exactly? -- JackofOz (talk) 23:28, 10 February 2008 (UTC)

Have you tried reading Centroid? There is a formula in there - for an irregular shape like a country you'll end up doing numerical integration. --Tango (talk) 23:40, 10 February 2008 (UTC)

I don't know how to do it with a computer, but I can do it with a piece of string. Take a cardboard cut-out of the country, and suspend it from a point on its edge such that it can swing freely. Attach the string to the same point, and weight down the end of the string. Then, trace the string onto the cutout - the centre of mass will be somewhere on this line. Repeat the process, suspending the cutout at a different point, and the intersection of the lines will be the centre of mass. Confusing Manifestation(Say hi!) 23:41, 10 February 2008 (UTC)

Algebra

We have an equation. Let's say it's $x^{2}-y^{2}=1$ . If we reduce this equation, we come out with $y=x-{\sqrt {1}}$ . Now, if we plug a number in for x, let's say 2, we get $y=2-{\sqrt {1}}=1$. Now, if we substitute back into the original equation, it comes out $2^{2}-1^{2}=1$. This is obviously incorrect for the fact that $2^{2}-1^{2}=3$. Would someone please explain to me why this does't work? Thanks, Zrs 12 (talk) 02:13, 9 February 2008 (UTC)

You have rearranged $x^{2}-y^{2}=1$ to give $y^{2}=x^{2}-1\Rightarrow y={\sqrt {x^{2}-1}}$, which is not equal to $x-{\sqrt {1}}$ if $x\neq 1$. -mattbuck (Talk) 02:17, 9 February 2008 (UTC)

Oh, okay. It's because ${\sqrt {x^{2}-1}}\neq {x}-{\sqrt {1}}$ if $x<1<x$? Zrs 12 (talk) 02:32, 9 February 2008 (UTC)

If $x<1<x$, then God didn't make little green apples and it don't rain in Indianapolis in the summertime. --Trovatore (talk) 03:30, 9 February 2008 (UTC)

$x<1<x$ means $x<1$ and $1<x$, which is never true. So your statement is vacuously true, but not the same as mattbuck's. -- BenRG (talk) 05:53, 9 February 2008 (UTC)

More generally, the reason is that you can't distribute the square root operation over addition to each term underneath the square root. In other words, ${\sqrt {a+b}}\neq {\sqrt {a}}+{\sqrt {b}}$. --Kinu ^t/_c 02:56, 9 February 2008 (UTC)

You should say $x>1$ or $x<1$. Visit me at Ftbhrygvn(Talk|Contribs|Log) 07:36, 9 February 2008 (UTC)

Do some countries have different order of operations

or is it the same in all parts of the world.

also, do some parts of the world use math that is 'false' im a different part of the world and vice versa, like one ton plus one ton doesn't equal two tons? —Preceding unsigned comment added by 212.51.122.26 (talk) 11:31, 9 February 2008 (UTC)

While different countries could conceivably have different conventions, basic arithmetic is pretty absolute. 1+1=2, wherever you go. You just have to take one bean and another bean, put them in a pot together and count them and you'll find you have 2 beans - there's no convention involved in that. The order of precedence of operations is purely convention, and different countries could teach it differently, I don't know of any that do, though. --Tango (talk) 13:44, 9 February 2008 (UTC)

Thank you. But your bean example implies that 'reality' makes sense everywhere, not only in Western countries. But isn't the absolute, objective reality you describe -- wherein you can be sure there will be two beans in a pot after adding one to an empty one followed by another one without taking out the first -- limited to the Western tradition???? Surely it's not the same in a Buddhist temple...? —Preceding unsigned comment added by 212.51.122.26 (talk) 13:54, 9 February 2008 (UTC)

Not really. Reality is reality. 1+1=2 in your bedroom, on Mars and indeed in a Buddhist Temple. -mattbuck (Talk) 14:00, 9 February 2008 (UTC)

There is perhaps a possibility for cultural differences. The statement "1+1=2" requires a concept of "oneness" and "twoness" as abstract entities separate from any physical meaning. For example, I might have two books on a shelf, but that's two books; and I might have two beans in a pot, but that's two beans. In our culture we have developed an abstract idea of "twoness" which is shared by the two books on the shelf and the two beans in the pot, but which does not require beans or books or in fact any physical manifestation. We are able to use this concept to express general statements such as "1+1=2" which can be interpreted for books or beans or whatever, rather than having to make individual statements such as "one book and one book make two books" or "one bean and one bean make two beans".

This abstraction is so fundamental to us that most of us don't even recognize it. But there are some languages in the world (I think Japanese and some Polynesian languages, for example) in which the way you say "two books" is quite different from the way you say "two beans". This is because these languages developed from roots that came before people learned to abstract the concept of "twoness" away from real physical objects. —Bkell (talk) 15:15, 9 February 2008 (UTC)

Japanese has a system of counters which I think was borrowed from Chinese. English also has nouns that require counters, like "clothing". You can't have two clothings, but you can have two articles of clothing; "article" is the counter. I suppose this distinction (between mass nouns and count nouns) reflects some aspect of human psychology, but I'm darned if I can imagine what it is. -- BenRG (talk) 15:39, 9 February 2008 (UTC)

We also have several words for "twoness" in English, with rules for which word is correct depending on context. You can have a brace of pheasants or a brace of ducks, but not a "brace of pigs". You can have a couple of minutes or a pair of shoes, but not a "pair of minutes" or a "couple of shoes" (unless you mean two shoes from different pairs). The word "pair" can also refer to just one object, as in a pair of trousers, a pair of scissors, a pair of tweezers or a pair of glasses. However, I don't think these peculiarities of language cause English speakers to have any confusion about the concept of "twoness". Gandalf61 (talk) 15:42, 9 February 2008 (UTC)

If you don't have a concept of "twoness" then you can't have a concept of arithmetic, so the question is pretty redundant. I know there are cultures which only have names (and even concepts) for small numbers ("1,2,3,4,many") - I would assume they don't really have a well developed idea of addition. --Tango (talk) 18:38, 9 February 2008 (UTC)

Order of operations is a concept from grade-school arithmetic that doesn't have any exact counterpart in (higher) mathematics. I wouldn't be surprised at all if the rules are different in different places. Where I come from they teach that A × B ÷ C × D means ((A × B) ÷ C) × D, but a mathematician is more likely to interpret $AB/CD\,\!$ as ${\frac {AB}{CD}}$. In grade-school arithmetic A ÷ B ÷ C means (A ÷ B) ÷ C, but a mathematician would never write $A/B/C\,\!$; it's a meaningless string of symbols unless the / is meant to stand for something other than division. To a mathematician A + B − C + D doesn't mean ((A + B) − C) + D, it means A + B + (−C) + D, and you can do the operations in any order because addition is associative. In fact the whole concept of "doing operations from left to right" is almost never found in mathematics. Lambda calculus is the only clear exception that comes to mind. The set difference operator might count, but I don't know how often it's chained that way.

Regarding whether 1+1=2, it's important to understand that concepts like 1 and addition began as abstractions of certain things in the real world. They aren't the right concepts for everything. If you combine "one" of something with "one" of something and don't get "two" of something, that doesn't mean there's something wrong with the idea 1+1=2, it means you tried to apply it where it (evidently) doesn't apply. But these concepts are useful in surprisingly many situations. They've been independently rediscovered many times by different civilizations. They're useful all over the world because the laws of physics are the same all over the world. In fact the laws are the same not just all over the world, but for billions of light years in every direction and billions of years into the past, for reasons no one understands. -- BenRG (talk) 15:24, 9 February 2008 (UTC)

On the subject of 'doing operations from left to right', you might find reverse Polish notation interesting. Algebraist 15:59, 9 February 2008 (UTC)

Many older, and some modern texts indeed use $AB/CD\,\!$ as ${\frac {AB}{CD}}$. However, with the advent of computer algebra systems which rigorously follow a set order of operations, it seems to me that a lot of people try to avoid use of things such as $AB/CD\,\!$ because it’s ambiguous in that it actually means ${\frac {AB}{C}}\cdot D$, but many times that’s not the intention. GromXXVII (talk) 16:10, 9 February 2008 (UTC)

I know Mathematica parses A B / C D as (A B / C) D, but I don't think that's what A B / C D means. I think A B / C D means "A times B over C times D", and has the same ambiguity as the English phrase (and for the same reason). I'm not even sure mathematicians want this kind of thing to be unambiguous. Formal precision distracts from the interesting aspects of a problem. If they wanted to be exact in every detail they'd be computer programmers. -- BenRG (talk) 14:39, 10 February 2008 (UTC)

Those are bad examples - multiplication and division commute with each other and are associative, as are addition and subtraction, as they're just inverses (addition and subtraction are the same operation, just one of them uses the negative of the number). It's when you mix them that it becomes ambiguous. "1 + 2 * 3" could be any number of things, by convention, it's taken to mean "1 + (2 * 3)". That applies as much in school as it does in higher level maths - the only difference is mathematicians usually just use juxtaposition to denote multiplication, so it becomes much clearer which operations take precedence, "x+yz" clearly means "x + (y * x)", you wouldn't read it as "(x + y) * z", even if you hadn't learnt BODMAS. --Tango (talk) 18:38, 9 February 2008 (UTC)

I don't know what it means to say that two binary operations commute with each other. The unary operations and commute, but and do not. Division and subtraction are not associative:  --Lambiam 00:51, 10 February 2008 (UTC)

The key point is that division and subtraction aren't really binary operations in their own right, they are simply inverses of multiplication and addition. "subtracting x" just means "adding -x", viewed that way, it clearly commutes with addition since it is addition and addition is commutative. --Tango (talk) 13:25, 10 February 2008 (UTC)

Also, division isn't a binary operator at all outside of arithmetic (and programming languages). In typeset formulas the division slash is only a vertical-space-saving variant of the horizontal bar. There's no general rule governing how much of the formula on each side is gobbled up by the slash. Ambiguous cases like "1/2 (x + y)" are pretty common in practice. I'd probably guess that that means "(1/2) (x + y)", and I'd probably guess that "x/2y" means "x/(2y)". Google Calculator parses "1 km / 2 km" as "(1 km) / (2 km)" but "1 / 2 km" as "(1 / 2) km". -- BenRG (talk) 14:39, 10 February 2008 (UTC)

What is division outside arithmetic? It's an arithmetic operation. Don't get hung up on the notation. ${\frac {ab}{cd}}$ means (ab)/(cd) - a horizontal bar has implied brackets. --Tango (talk) 20:27, 10 February 2008 (UTC)

Well, my examples were meant to illustrate something different, but I think even in yours there's a big difference between BOMDAS and precedence. Given a + b + c · d, I think BOMDAS would say to multiply c and d first, but precedence just says that it means a + b + (c · d), and you can perfectly well start by adding a and b. -- BenRG (talk) 14:39, 10 February 2008 (UTC)

BODMAS is the rules of precedence of arithmetic operations. What order you do parts of the sum that are independent of each other obviously doesn't matter, it's when they depend on each other that it matters. If your sum is "(a+b)+(c.d)" then obviously it makes no difference if you do the (a+b) or the (c.d) bit first, as long as you do them both before the addition in the middle. If you did the middle addition before the multiplication, you would get the wrong answer. (The fact that (a+b)+(c.d)=a+(b+(c.d)) is simply associativity of addition, which had nothing to do with precedence since only one operation is involved.) --Tango (talk) 15:26, 10 February 2008 (UTC)

My pocket is a set.

My pocket is a set.

I add the first ten fibonacci numbers one at a time. [1]

How many elements does my pocket have?

[1] If this formulation is nonsensical after all, then "it is equal to the union of ten single-element sets, each of whose sole element is a successive number of the fibonacci sequence". or if I'm still not phrasing it right, could I get a little help? —Preceding unsigned comment added by 79.122.91.85 (talk) 23:46, 9 February 2008 (UTC)

Are you asking how many elements are in $\bigcup _{n=1}^{10}\{F_{n}\}$? If so, the answer is 9 (since $F_{1}=F_{2}$, using the convention in Fibonacci number). -- Meni Rosenfeld (talk) 23:59, 9 February 2008 (UTC)

Are you quite sure that your pocket is a set ? Perhaps it is more like a multiset (my pockets certainly are). This may be the source of your confusion. Gandalf61 (talk) 15:07, 10 February 2008 (UTC)

I'd say my pockets are like sets. They often contain very similar objects, but never perfectly identical ones (other than things like air molecules, which I don't count among the contents). —Keenan Pepper 17:19, 10 February 2008 (UTC)

But if hypothetically you did find two identical objects and put them in your pocket, and then reached into your pocket a short time later, you would be surprised if you only found one object in there (unless there is perhaps a hole in your pocket ?).

My semi-serious point is that the analogy set up by the initial statement "my pocket is a set" is subtly misleading. Gandalf61 (talk) 18:33, 10 February 2008 (UTC)

February 10

LOGICAL SYSTEMS+NEW PRINCIPLES+ATTEMPT TO SOLVE COLLATZ CONJECTURE.

LOGICAL SYSTEMS+NEW PRINCIPLES+ATTEMPT TO SOLVE COLLATZ CONJECTURE.

First i would like to say that am honoured to share my thoughts with great people in here who always provide help.I will not say iam right or wrong.I hope this post will be aspark to good mindes.

I have viewed the laws of logic and I think that there is asort of aconfusion about how the logical system works? and how the logical process conducts the conclusion?there is also aconfusion about the limits of the logical laws, specially the mixing between the law of contradiction and the law of noncontradiction. In this post I will be trying to show that all laws of logic are serving two simple prima principles and we will call them "principle of interior contradiction”and”principle of exterior contradiction”. Now let, A, be acase we want to prove,the logical process that followed by any logical or mathematical system to prove A,is 1=(A,exists,end of proof), 2=(A,doesnot exist), we know that the pair(1,2) is called"contradiction law".we have to notice that the whole idea to build aproof start with that pair. now,the pair(1,2)leads to alogical processes depend on aseries of steps of "the noncontradiction law "until we end to astate we cannot step over with anew contradiction or astep of afinal contradiction which is the conclusion,(A,exists,end of proof),the problem is ,whenever the noncontradiction law stops working during the process then we have to create or find anew contradiction other wise the statmment,A,stills open or unsolvable or we get aparadox. We will call the relation between the law of contradiction=pair(1,2)and the law of noncontradiction that works in the logical process as"EXTERIOR CONTRADICTION PRINCIPLE "or the ECP. Now the essential question,(I) is the contradiction law exists?(II)or it doesnot exists?if it exists then we will never be able to proof or know any judgement that the logical statement,A,exists or true or false .if it doesnot exists then why we need the proof in the first place?because we will know for sure any judgement about the statement,A.now we will call the the pair(I,II)”THE INTERIOR CONTRADICTION PRINCIPLE”or the ICP.

The ICP. and the ECP.are the two principles. that all the logical and mathematical systems and processes depend on. In asipmle word,(ICP) →(ECP) →(conclusion).or in another word,(contradiction law) →(prosseses of steps of noncontradiction law) →(final contradiction,or what we call it the conclusion).

Law of identity. P ≡ P The proof, P,exists

1- P ≡ P,end of proof 2- P no ≡ P (1,2) → if (P no ≡ P)→ (P ≡anything ,anything includes, P)→( P ≡ P),end of proof. (1,2) → if (P no ≡ P) →( P ≡nothing) → (P,doesnot exist) →(final contradiction=conclusion).as amatter of facts all logical laws are eventually serving the LIC.

law of the excluded middle. Eventhough , some systems of logic have different but analogous laws, while others reject the law of excluded middle entirely. We will trying here to prove that this law is serving the LIC. The law states, "P ∨¬P" 1-p,exists. 2-p,doesnot exist. (the pair (1,2)=contradiction law).it sems that law of the excluded middle is identical to law of contradiction or another forming to the law of contradiction.

THE LOGICAL SPACE. The logical spaces is asystem of operations ,relations,groups,variables,constants and operators where the logical statement exists.logical space is not aring or field.it is agroup of tools that we can use to create either ATRUE logical statement or FALSE logical statement.ofcourse we can shrink or expand the logical space depend on what we have or create of tools.

For example,the statement , 1/x ‹x ,x≠0 .this statement exists in the logical space (1,=,/,x,‹,G)where ,G,is the real numbers› 1.

assume that ,A, is alogical statement exists in the logical space Q, and K,is adifferent logical space 1-A,is true in,Q 2-A,false in,K The pair(1,2) is not acontradiction. This means that we will not be able to prove the existance of,A,is true. in the example above. we either testing all the numbers of,G,which is impossible,to prove the statement, 1/x ‹x, x≠0, or we start with the contradiction 1-A,is true in,Q 2-A,false in,Q (1,2) →assume 1/x ‹x ,x≠0,is true in another space,such as, (1,=,/,›,x,G) →(1/x›x)→(1›x)→(1›1) → contradiction of identity law=final contradiction=conclusion.

We also can find anothers spaces to serve the case. One could say ,1/x ‹x ,x≠0 , in (‹,G)is so obvious and doesnnot need any proof,even if it does,we can prove it by ashorter way like,assume, (1/x ›x)→(1/2›2)→(contradiction).or (1/x =x)→(1/2=2)→contradiction of law identity. The answer is yes,it is the same above but all the mathematical process in the argument is going intuitively while we want it to goes more axiomatically by defining what logical space the statement exists?and if it is atrue? or false?.when we are unable to know the limits where contradiction and noncontradiction lwas work,when we are unable to define the logical statement and in which logical space it exists,then we will get an open problems or paradox.

The important thing here is to notice that the time is not actually amthematical operator we depend on.instead of time we use more well defined operators.for example 1 ≠0 , in the space(G,›),where ,G,is any group.not now not any time,but it is simple to notice that ,(1,could=0),in any space hase the operator,d/dx,i.e,d/dx(1)=0.

THE ONE DIRECTION INFINITY PRINCIPLE,THE STEADY STATEMENT Assume he statement ,(a1,exists)→(a2,exists)→ …….an,exists).the qustion is there astatement,a∞ exists? Obviousely the possiblity of existance of any of the statements,an, is,1,which leads us to say the possibility of existance , a∞,is also 1.we will call, a∞,asteady statement.

THE TWO DIRECTIONS INFINITY.

Assume he statement ,(a1,exists)→(either,a2,exists,or,a3,exists). (a2,exists)→(either,a4,exists,or,a5,exists). (a3,exists)→(either,a6,exists,or a7,exists)……etc. Obviousely the possiblity of existance , a∞,is,1\2.

THE MULTI DIRECTIONS INFINITY.

Assume he statement ,a1,exists→(either,a1,or,a2,or,………. an,exists) Obviousely the possiblity of existance , a∞,is,0.

COLLATZ CONJECTURE AND STEADY STATEMENT. Post:the staedy statement exists in collatz formula and it equals,1. collatz formula, A-F(n)=n/2……if,n≡0(mod2). B-F(n)=3n+1…if,n≡1 (mod2).

Proof.

(some,A→ A),(some,A→ B),BUT(all,B→A,in the space(1,=,2k,/,N),Where,k,apositive integer and,N,is the natural numbers set.by using the one direction infinity principle,there exists asteady statements,k∞,that makes 2k=1,is atrue statement in the space above ,namely,2k. end of proof.Husseinshimaljasimdini (talk) 10:17, 10 February 2008 (UTC)husseinshimaljasimdini

You should talk to User:BenCawaling. He also claims to have a proof of the Collatz conjecture. —Keenan Pepper 17:22, 10 February 2008 (UTC)

I don't think your system will work. You should probably test-drive this idea before using it on something new. Try to prove and disprove things that are already known. Try to prove, for instance, that there are only finitely many prime numbers, or that there are infinitely many positive integers less than 10,000. If you succeed, it might make it clearer what's wrong. Black Carrot (talk) 01:15, 11 February 2008 (UTC)

multiplicative inverse

Hello, please help me! I have the number 2008. The sum of any natural numbers shall give the number 2008. But the multiplicative inverse of these numbers shall give 1. Which numbers must be taken? --85.178.52.149 (talk) 10:19, 10 February 2008 (UTC)

OK, just so I have this clear, you want numbers x such that $\sum \limits _{i}x_{i}=2008$ and $\sum \limits _{i}{\frac {1}{x_{i}}}=1$, is this correct? If so, there's no way to work it out without using trial and error - n equations can only ever determine at most n variables, so if we treat the $x_{i}$ as the variables, then we have two equations and so could only work it out for i = 1 or 2. Do you have more information? -mattbuck (Talk) 10:25, 10 February 2008 (UTC)

E.g.:

• 3 + 44 + 44 + 526 + ... = 2008
• 1/3 + 2/44 + 1/526 + ... = 1

Everything clear? --85.178.52.149 (talk) 10:30, 10 February 2008 (UTC)

Yes, but as I said, it's not possible to solve it explicitly - you need to just do trial and error. -mattbuck (Talk) 10:49, 10 February 2008 (UTC)

I will give you (or someone else) a barnstar if you will get out the answer. Is the price worth enough? ;) --85.178.52.149 (talk) 10:55, 10 February 2008 (UTC)

If I give you the answer then you learn nothing. If I give you a hint and you follow it and work out the answer for yourself then you learn something. So here is a hint:

• If we take a repetitions of the number b then their sum is ab and the sum of their reciprocals is a/b.
• So if we can make b equal to a then the sum of the reciprocals will be 1. But this will only work if the sum ab is a square number. For example, 9 = 3x3 = 3+3+3 and 1/3 + 1/3 + 1/3 = 1.
• 2008 isn't a square number, so we can't use this idea directly. But suppose we can make b equal to 2a. Then a/b = 1/2 and ab = 2a². In other words, if the sum is twice a square then we can partition it so that the sum of the reciprocals is 1/2. For example 18 = 2x3^2 = 6+6+6 and 1/6 + 1/6 + 1/6 = 1/2.
• So if we could find two numbers each of which is twice a square and which sum to 2008, then we could partition each of these numbers into a set of smaller numbers whose reciprocals sum to 1/2. Then when we take all the smaller numbers together, they will sum to 2008 and their reciprocals will sum to 1. For example 26 = 18+8 = (6+6+6) + (4+4).
• If 2008 is the sum of two numbers that are twice squares, then 1004 would be the sum of two squares. Unfortunately, 1004 isn't the sum of two squares, so this doesn't work either.
• But, going back to our original line of thought, suppose we make b equal to four times a. Then a/b = 1/4 and ab=4a². So if we can make 2008 equal to the sum of four numbers, each of which is four times a square, then we are done.
• To find four numbers, each of which is four times a square, which sum to 2008 then we need to find four squares whose sum is 502. And Lagrange's four-square theorem says that any integer can be expressed as the sum of at most four squares, so this approach is looking hopeful.

I think that is enough of a hint. I'll let you take it from there. (Incidentally, there are many solutions to this problem - I have just found 6 or 7 different solutions using this method). Gandalf61 (talk) 14:55, 10 February 2008 (UTC)

At first, thank you! --- I have now this: 20^2 + 10^2 + 2*2^2 = 502. But it doesn't work because i got two times the same number. Can you help me again? --85.178.31.13 (talk) 17:16, 10 February 2008 (UTC)

I think you mean 20^2 + 10^2 + 2*1^2 = 502. Having 1^2 appear twice is fine - this still leads to a solution to the original problem. Gandalf61 (talk) 17:23, 10 February 2008 (UTC)

No, it doesn't work. 2*1^2 is not agreeable because 1^1 is 1, the reciprocal is 1, too! So the solution is wrong (I had written 2*2^2, because I wanted to get 508. This was a mistake by me). 2*2^2 would also be a problem: 2*4 means 4*(2*4) at the end. But the reciprocals would be then 2 (and this is surely wrong). --85.178.31.13 (talk) 18:11, 10 February 2008 (UTC)

You don't want to take the reciprocal of 1, you want to take the reciprocal of 4*1. --Tango (talk) 18:54, 10 February 2008 (UTC)

?? The reciprocal must be at the end only 1. --85.178.31.13 (talk) 19:32, 10 February 2008 (UTC)

Yes, and you'll end up with 1/4 four times, which adds up to 1. Read through Gandalf61's hint again, it might take a few tries to get your head round it, but it does work. --Tango (talk) 20:22, 10 February 2008 (UTC)

Parallelogram

Can you get the circumference if you only have the diagonals e and f (nothing else is given)? Could you write down the formula to get this? --85.178.52.149 (talk) 10:46, 10 February 2008 (UTC)

I wouldn't have thought so. There are 3 variables in a parallelogram - the height, the overhang, and the length of the long side. Three pieces of information cannot be uniquely determined with only two equations. -mattbuck (Talk) 12:18, 10 February 2008 (UTC)

Ok...maybe you can do something with their values; e is 7, f is 9. Maybe now (all sides are natural numbers)? --85.178.52.149 (talk) 12:28, 10 February 2008 (UTC)

Well, if we take h to be height, a to be the length of the long side and b to be the overhang:

$e={\sqrt {h^{2}+(a-b)^{2}}}$ and $f={\sqrt {h^{2}+(a+b)^{2}}}$ which gives $4ab=f^{2}-e^{2}$, but again, you can't get further unless you assume both a and b are integers, in which case you would at least get a finite number of possibilities which you could then work out, remembering that ${\sqrt {b^{2}+h^{2}}}$ must be an integer. -mattbuck (Talk) 13:06, 10 February 2008 (UTC)

You'll need 1 more information since you can draw a number of parallelograms with 7 and 9 as the length of diagonals. —Preceding unsigned comment added by Ftbhrygvn (talk • contribs) 15:27, 10 February 2008 (UTC)

Why does logic work?

What is it about the universe that makes logic work. I don't see why it needs to.

Also, does logic work everywhere, or are there some places (maybe the bermuda triangle?) where logic doesn't hold, and something that would of necessity follow, elsewhere, there it does not??? —Preceding unsigned comment added by 79.122.117.186 (talk) 11:34, 10 February 2008 (UTC)

I think this is a prime example of the possible application of the anthropic principle - put simply, if the universe weren't logical, we wouldn't be asking the question. -mattbuck (Talk) 12:16, 10 February 2008 (UTC)

It depends on what you mean by "logic". If you're asking why the world seems to make as much sense as it does, I guess we just got lucky. Not extremely lucky, considering how much of it nobody understands even now, but still. If you're asking why a particularly popular logical system works so well, it's because it was designed by some very smart philosophers to mimic the real world (which, as I pointed out, usually makes sense) and has been improved over many years. If there was anything wrong with it, someone would have noticed by now. In fact, flaws have been pointed out every once in awhile, and the system has been improved more than once. You've heard of it because it works, and it works because people have fixed it until it does. It applies, by definition, to everything it applies to, and nothing else. The particular system I linked to, for instance, only applies to statements that are either completely true or completely false. Black Carrot (talk) 00:57, 11 February 2008 (UTC)

Evaluating a definite integral

I'm curious about the function

$w(n)=\int _{-\infty }^{\infty }(1-(1-\alpha )^{n}-\alpha ^{n})\,dx$ ,

where $\,\alpha =\Phi (x)$, i.e. the cumulative standard normal distribution, and n is an integer greater than one. I suspect that the usage of α without any indication that it is a function of x may be non-standard (at least, it had me confused). However, I wanted to reproduce it exactly as it appears in the paper in which I found it: Tippet, LHC. On the Extreme Individuals and the Range of Samples Taken From a Normal Population. Biometrika vol 17, 364-387, 1925. It is my understanding that this function gives the expected value of the range of a sample of size n, taken from a standard normal distribution. In the field of Statistical process control, the function w is known as d2.

These are my questions:

• For n = 2, it is easy to show that the integral is numerically equal to 2/sqrt(π) within machine precision, and I feel reasonably certain that 2/sqrt(π) is indeed the exact value. I would like to know how one determines whether this is the case. As I am neither a statistician nor a mathematician, I would need the details spelled out.
• Can this integral be expressed in terms of simple functions for values of n greater than 2? If so, how?
• Is my suspicion avove, that the notation today would be considered non-standard, correct? If so, what would standard notation be?

Thanks. --NorwegianBlue ^talk 14:45, 9 February 2008 (UTC)

I haven't done much probability, so I'm not really sure what's going on there, but it does seem odd to me to be integrating the CDF - normally you integrate the density function to get the CDF, so I'm pretty much lost. As for the standard notation, wouldn't that just be:

$w(n)=\int _{-\infty }^{\infty }(1-(1-\Phi (x))^{n}-\Phi (x)^{n})\,dx$?

It's not particularly unusual not to use a notation that doesn't explicitly state a dependence, although when you're only dealing with functions one variable, I can't see much point in not being explicit. --Tango (talk) 13:40, 10 February 2008 (UTC)

Thanks! Well, the function works, and there's no doubt that it's the CDF that is to be integrated. Below is a very crude implementation in R. The standard normal density in R is dnorm, while the distribution function is pnorm:

     d2 = function(n, distribution=pnorm, lolim=-12, hilim=12)
     {
         dx = 0.01
         steps = seq(lolim, hilim, by=dx)
         alpha=distribution(steps)
         rectangles = (1 - (1-alpha)^n - alpha^n)*dx
         return(sum(rectangles))
     }

And here's the output when testing it:

     > format(d2(2), digits=15)
     [1] "1.12837916709551"
     > format(2/sqrt(pi), digits=15)
     [1] "1.12837916709551"

--NorwegianBlue ^talk 14:08, 10 February 2008 (UTC)

A little more experimenting shows that d2(3)=3/sqrt(pi). However d2(4) = 2.058751, while 4/sqrt(pi) = 2.256758. Can anyone see a pattern? --NorwegianBlue ^talk 00:32, 11 February 2008 (UTC)

I can't offer much, but Mathematica confirms that $w(2)={\tfrac {2}{\sqrt {\pi }}}$ and $w(3)={\tfrac {3}{\sqrt {\pi }}}$. It is also worth noting that $w(1)=0$. It doesn't readily succeed in calculating $w(4)$ symbolically. The numerical value is 2.058750746..., for which Plouffe's inverter gives no match. So I doubt there is any significantly simpler way to express it. -- Meni Rosenfeld (talk) 09:05, 11 February 2008 (UTC)

Thanks a lot, Meni. --NorwegianBlue ^talk 13:02, 11 February 2008 (UTC)

Are mathematicians racist against non-Jews?

sorry, I doin't mean to be incendiary, but I wonder if anyone has heard of it being more difficult for a non-Jew to become a mathematician for whatever reason? —Preceding unsigned comment added by 79.122.117.186 (talk) 11:42, 10 February 2008 (UTC)

You what? I can't say I've known anyone ever ask about religion here in the UK. Where exactly are you asking about? -mattbuck (Talk) 12:13, 10 February 2008 (UTC)

united states. —Preceding unsigned comment added by 79.122.117.186 (talk) 12:35, 10 February 2008 (UTC)

Why do you ask? What makes you think there would be any problem? Is there even a higher proportion of Jews in the US Mathematical community than in the general population? --Tango (talk) 13:31, 10 February 2008 (UTC)

Yes, there was a period of about 20 years when only Jews were allowed to become mathematicians. Fortunately the Supreme Court overturned the law. Strad (talk) 16:39, 10 February 2008 (UTC)

This is the most nonsensical statement I have ever heard. Phils 16:51, 10 February 2008 (UTC)

Also consider the fact, that only Roman Catholics may become professonal popes. The VEOB (Vatican Equal Opportunity Board) appears to be rather conservative, when it comes to gender, race and creed. But then again, I only ever applied for a missionary position...

PS: Re Strad´s comment: Me thinks your instrument has been tuned by the late Beethoven (one of his many places of abode is just next door but one to the block I live in). --62.47.136.108 (talk) 18:10, 10 February 2008 (UTC)

From my experience the mathematical community is particularly indifferent to questions of race/religion. Its how good at maths they are which is the most important thing. There are quite a number of notable jewish mathematicians see List of Jewish American mathematicians.--Salix alba (talk) 20:08, 10 February 2008 (UTC)

The question was about racism towards *non*-Jews. I agree with the rest of your point, though - my Maths department barely has two people from any one country! I once got all the appropriate flags off commons for a poster I was making about the department (as part of a summer job) - it took quite a while! --Tango (talk) 20:12, 10 February 2008 (UTC)

Maybe the question should be do Jewish get an easier ride due to the large list of very notable jewish mathematicians? BTW I find the whole suggestion of this topic rediculous.--Dacium (talk) 22:20, 10 February 2008 (UTC)

my question (not the subject, the actual body of the question) asks "is it more difficult for non-Jews" -- how is this not the same as asking "do Jewish get an easier ride"???? I mean, if it's easier for Jews then it means it is harder for non-Jews. You can't say "It's not any harder to be a NASCAR driver if you're a woman, it's just easier if you're a man..." That's a completely nonsensical sentence!!!! —Preceding unsigned comment added by 212.51.122.22 (talk) 22:41, 10 February 2008 (UTC)

As far as I know, there's no bias against non-Jews. In fact, the only bias I've ever seen mention of until today was in one of Feynman's several autobiographies, where it claimed that Jews had a harder time because they were unpleasant to work with. With the sole exception, naturally, of Feynman himself. Black Carrot (talk) 01:18, 11 February 2008 (UTC)

Inverse of sum of two matrices

I have two large matrices A and B, and I have to find the inverse of A + c*B, for many different values of c. Is there a more efficient way of doing this rather than finding the inverse of the sum A + c*B every time. Thanks, deeptrivia (talk) 19:46, 10 February 2008 (UTC)

Well, you can do it in terms of c and plug the values in at the end. It might end up being quite a mess (for n-by-n matrices, you'll potentially be dealing with (n-1)-degree polynomials in c divided by n-degree polynomials in c - if A and B are nice, some of it might cancel out, though). I don't know any formula for directly finding the inverse of a sum of matrices - matrix addition doesn't behave very nicely - but that doesn't mean there isn't one out there somewhere. If you know any nice properties of A and B, it might make things easier. The best option with anything to do with matrix inverses is just to do it on a computer, really. --Tango (talk) 20:20, 10 February 2008 (UTC)

I don't know if this is in any way helpful, but , and for sufficiently small c you can use  --Lambiam 20:51, 10 February 2008 (UTC)

Does that first formula have a name? I'm curious to see the derivation. The binomial approximation is only useful if an approximate answer is sufficient, which depends on what deeptrivia is trying to do - plugging it all into Mathematica or Maple seems like an easier approach if you need more than the first order term, anyway. --Tango (talk) 21:00, 10 February 2008 (UTC)

Such CASes are no match for C code optimized for this purpose, which could be what deeptrivia is after. If performance is really crucial and c is assumed to be small, a power expansion may be the best way to go. Of course, you don't want to calculate it naively - a good way to calculate $(I+X)^{-1}$ is to iterate $Y_{n+1}=I-XY_{n}$, and an even better way is probably to calculate $(I+X)^{-1}=(I-X)(I+X^{2})(I+X^{4})(I+X^{8})\cdots$. -- Meni Rosenfeld (talk) 08:52, 11 February 2008 (UTC)

If by "many", we're talking thousands, maybe. It's easier just to plug the general case into a CAS, get a general answer, and plug the values for c in at the end - substituting in a value for a variable is a pretty quick function on any platform, it's just evaluating n^2 rational functions. --Tango (talk) 11:05, 11 February 2008 (UTC)

Yes, n^2 rational functions, each of which having $O(n)$ terms, totaling an $O(n^{3})$ calculation for each new c, not to mention the expensive preprocessing. A calculation along the lines suggested by Lambiam is, if I'm not mistaken, $O\left(n^{2.376}\log {\frac {\log \epsilon }{\log {\|cA^{-1}B\|}}}\right)$ (where ε is the tolerable error) per c. We can hybridate the approaches - invert symbolically, and then expand each term in a way that will facilitate the calculation. It all depends on how many cs we want (for a huge number, preprocessing cost is irrelevant), the size of the matrices and the error tolerance. -- Meni Rosenfeld (talk) 12:03, 11 February 2008 (UTC)

I don't know if it has a name, but here is a derivation:

(A + cB)⁻¹ = (I(A + cB))⁻¹

                  = (AA⁻¹(A + cB))⁻¹

                  = (A(A⁻¹A + cA⁻¹B))⁻¹

                  = (A(I + cA⁻¹B))⁻¹

                  = (I + cA⁻¹B)⁻¹A⁻¹.

 --Lambiam 02:14, 11 February 2008 (UTC)

is {2,2} a set?

Sorry if I've asked this before, but is {2, 2} a set or a mistake. I know it's a legitmate mmultiset... —Preceding unsigned comment added by 212.51.122.22 (talk) 20:49, 10 February 2008 (UTC)

It is a permissible notation for the same set as {2}. So it is a set, yes, with exactly one element.  --Lambiam 20:52, 10 February 2008 (UTC)

So is the answer to "How many ways can you type a set if you only have braces commas spaces and the number 2" infinite, since you can type {}, {2}, {2,2}, {2,2,2} etc and in all these cases you're typing a set??? (And the answer to that question is not, say, 2, namely {} and {2}, the other typings not being instancse of typing a set...) —Preceding unsigned comment added by 212.51.122.22 (talk) 22:39, 10 February 2008 (UTC)

{2,2} is notation for a set - the set containing the number "2". It's important to distinguish between the object and the notation. {2} and {2,2} are clearly different notations, but they refer to the same set. It's also important to note that 2,{2},{{2}},{2,{2}} etc. are all distinct mathematical objects (in fact, they are all sets using the most common model of natural numbers where 2={0,1}={{},{{}}}). --Tango (talk) 23:21, 10 February 2008 (UTC)

You don't even have to get into trickery with sets of sets to get different sets using the symbols listed: sets such as {2, 22, 222} are permitted by the condition that "you only have braces, commas, spaces, and the number 2". -- AJR | Talk 01:23, 11 February 2008 (UTC)

I disagree, there is a difference between the number 2 and the digit 2. Taemyr (talk) 09:30, 11 February 2008 (UTC)

Agreed. The question explicitly said "number 2". The fact that in the base ten positional notation for numbers, the same shape is drawn when denoting other numbers is irrelevant. --Tango (talk) 10:59, 11 February 2008 (UTC)

February 11

Indefinite integral of erf^3

Does anyone know of a reference to express the following integral in closed form?

$\int \operatorname {erf} ^{3}(x)\,\mathrm {d} x\,\!$

erf is the error function. I can get the square with a CAS, but I have found nothing through either googling, CASs, MathWorld, etc. Any help would be appreciated. Baccyak4H (Yak!) 04:39, 11 February 2008 (UTC)

Is this equation dimensionally balanced?

This popular article http://news.bbc.co.uk/1/hi/magazine/7238637.stm includes an equation which looks to me as if it is not dimensionally balanced. In other words dimensional analysis would cast doubt on its validity. Am I right? 80.3.47.33 (talk) 18:02, 11 February 2008 (UTC)

Mandelbrot and Julia sets

Map of 221 Julia sets

Can anyone get me a picture similar to this, but centered, and with a separate picture of a Mandelbrot set (just the black part) the same size and in the same place to use as a mask? I only need the main cardioid part. (Technically, I only need half, as it's symmetric). I'm trying to make my own version of quantum weather butterflies. — Daniel 18:05, 11 February 2008 (UTC)