Wikipedia:Reference desk/Archives/Mathematics/2011 May 15

= May 15 =

Ellipse
The equation $$ax^2+bxy+cy^2=1$$ describes an ellipse, but it is not a standard ellipse because the ellipse's axes are not necessarily parallel to the x and y axes, i.e. it has been rotated. How do you read the angle of rotation from the equation? Widener (talk) 03:13, 15 May 2011 (UTC)
 * If the major axis forms an angle of $$\theta$$ with the x-axis, then θ minimizes the value of $$ax^2+bxy+cy^2$$ with the substitution $$(x,y)=(\cos\theta,\sin\theta)$$. Substituting and differentiating gives $$(c-a)\sin(2\theta)+b\cos(2\theta)=0$$ which means $$\theta=\tfrac12\tan^{-1}\left(\frac{b}{a-c}\right)$$. This gives the result up to a multiple of $$\pi/2$$. -- Meni Rosenfeld (talk) 08:17, 15 May 2011 (UTC)
 * Did this reply help you? -- Meni Rosenfeld (talk) 15:56, 16 May 2011 (UTC)
 * Yes. Why wouldn't it? You explained the answer to my question. Widener (talk) 11:48, 17 May 2011 (UTC)
 * It's common that even an answer as straightforward as this leaves an OP with something to be desired. I don't presume to divine what an OP would think upon seeing an answer of mine. I appreciate having explicit closure for the exchange. -- Meni Rosenfeld (talk) 10:01, 18 May 2011 (UTC)

Multiplication of Cardinals and Order Preservation.
See Multiplication. If a, b, and c are cardinal numbers, does the following still hold?

" Multiplication by a positive number preserves order: if a > 0, then if b > c then ab > ac. Multiplication by a negative number reverses order: if a < 0 and b > c then ab < ac."

Thanks in advance. voidnature 08:24, 15 May 2011 (UTC)
 * [ec]Are you referring to cardinal numbers? I don't know of a way to multiply a cardinal number with a negative number. For multiplication by a cardinal number a, this will hold if either $$a 0, then if b > c then ab > ac". Thankyou. voidnature 08:36, 15 May 2011 (UTC)
 * Ok, so you need either $$a 0, then if b > c then ab > ac" hold? Thankyou. voidnature 08:58, 15 May 2011 (UTC)
 * Every cardinal multiplied by 0 is 0, so you're basically asking, "if a>0 and b>0 are cardinal numbers, is ab>0?" The answer is yes, because the cartesian product of two nonempty sets is nonempty. -- Meni Rosenfeld (talk) 09:05, 15 May 2011 (UTC)
 * "the cartesian product of two nonempty sets is nonempty": can you give a proof please? voidnature 09:09, 15 May 2011 (UTC)
 * $$a\in A,\ b\in B \Rightarrow (a,b)\in A\times B$$. -- Meni Rosenfeld (talk) 10:17, 15 May 2011 (UTC)

Why is S5 a modal companion of CPC?
Why is S5 a modal companion of CPC? It seems like this should imply that S5 implies the translation of excluded middle, which seems to be $$\Box p \lor \Box \lnot p$$, which seems to say there are no contingent propositions - but surely S5 allows for contingent propositions? 88.104.173.35 (talk) 19:27, 15 May 2011 (UTC)

Plane partitioning algorithm
Oh, this is on the tip of my tongue: I hate when that happens. Begins with "L", I think... I'm trying to remember the name of an algorithm which takes an array of points on a plane and partitions the plane such that each point is surrounded by a polygon - I think the margins of which fall equidistantly with another point (or is it some other definition of "influence"?). The resulting diagram looks like a honeycomb made by tipsy bees. What is that algorithm? (it's not a BSP or its ilk)-- Finlay McWalter ☻ Talk 22:22, 15 May 2011 (UTC)
 * Voronoi diagram perhaps?--RDBury (talk) 23:39, 15 May 2011 (UTC)
 * Yes, that's it (and no L in sight)! Thanks. -- Finlay McWalter ☻ Talk 23:45, 15 May 2011 (UTC)
 * “L, I KNOW it begins with L!” &#x2013; b_jonas 18:46, 16 May 2011 (UTC)