Wikipedia:Reference desk/Archives/Mathematics/2011 May 15
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May 15
editEllipse
editThe equation 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 with the x-axis, then θ minimizes the value of with the substitution . Substituting and differentiating gives which means . This gives the result up to a multiple of . -- 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)
- Yes. Why wouldn't it? You explained the answer to my question. Widener (talk) 11:48, 17 May 2011 (UTC)
Multiplication of Cardinals and Order Preservation.
editSee Multiplication#Properties. 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 or b is finite. -- Meni Rosenfeld (talk) 08:34, 15 May 2011 (UTC)
- Yes, I am referring to cardinal numbers. My question is actually concentrated on this bit :"if a > 0, then if b > c then ab > ac". Thankyou. voidnature 08:36, 15 May 2011 (UTC)
- Ok, so you need either or b finite. Otherwise, a counterexample is . However, it will always be true that . I think the axiom of choice might be needed for some of these results. -- Meni Rosenfeld (talk) 08:49, 15 May 2011 (UTC)
- Oops, sorry, c has to be zero. So I don't need to worry about . So my question should be If a and b are cardinal numbers and c is 0, does "if 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)
- Oops, sorry, c has to be zero. So I don't need to worry about . So my question should be If a and b are cardinal numbers and c is 0, does "if a > 0, then if b > c then ab > ac" hold? Thankyou. voidnature 08:58, 15 May 2011 (UTC)
- Ok, so you need either or b finite. Otherwise, a counterexample is . However, it will always be true that . I think the axiom of choice might be needed for some of these results. -- Meni Rosenfeld (talk) 08:49, 15 May 2011 (UTC)
Why is S5 a modal companion of CPC?
editWhy 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 , 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
editOh, 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)