Wikipedia:Reference desk/Archives/Mathematics/2012 April 19

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April 19

Mercator and stereographic projection

Hi. If you consider the Mercator projection as a C-valued function f of lattitude $\theta$ and longitude $\phi$, and rotate the map 90 degrees to the right so the meridian $\phi =0$ is the real axis, you get

$f(\phi ,\theta )=\ln(\sec \theta +\tan \theta )-i\phi$.

Similarly if you consider the stereographic projection from the North pole as a C-valued function g of lattitude $\theta$ and longitude $\phi$, you get that

$g(\phi ,\theta )=(\sec \theta +\tan \theta )\cos \phi +i(\sec \theta +\tan \theta )\sin \phi =(\sec \theta +\tan \theta )\exp(i\phi )$.

Now consider $F=g\circ f^{-1}$. Were F = exp (which is what I'm supposed to prove), you would expect that $F\circ f=g$, but in fact $F_{0}\circ f=g$ for $F_{0}(z)=\exp({\bar {z}})$. Can someone give me a hint about where I've gone wrong in my definitions of f and g? Thanks for any help. —Anonymous Dissident^Talk 04:52, 19 April 2012 (UTC)

there is a kind of fuction named L-fuction,what meanings the "L"?

p-adic number what meanings the "adic",there is not in EN dictionary — Preceding unsigned comment added by Cjsh716 (talk • contribs) 07:26, 19 April 2012 (UTC)

"-adic" comes from the common ending in words like "dyadic" - pertaining to two, twofold; "triadic" - pertaining to three, threefold; "pentadic" - pertaining to five, fivefold etc. In a similar way, "n-ary" is used as a generalisation of "binary", "trinary" etc. Gandalf61 (talk) 10:31, 19 April 2012 (UTC)

I'm also curious where the "L" in L-function comes from. Rckrone (talk) 13:55, 20 April 2012 (UTC)

I believe it is an allusion by Dirichlet to the Legendre symbol. That seems to be pure OR on my part as I can't find anything about it. Dmcq (talk) 12:07, 22 April 2012 (UTC)

Interpretation of Beta hyperparameters

Our article Conjugate prior states that for Bernoulli and Binomial distributions the alpha and beta hyperparameters of the Beta conjugate prior may be interpreted as "a-1 successes, b-1 failures". I don't understand this however. Previously I have always interpreted the parameters as corresponding to "a successes, b failures", and I can't work out the motivation for this alternative interpretation. Certainly in the Bernoulli case, the 'most likely' value of the probability parameter is a / (a + b) (i.e. the ratio of successes to trials), and this seems to favour the latter interpretation? Thanks --Iae (talk) 20:22, 19 April 2012 (UTC)

Also I have had practical applications before where I define one or both of my prior Beta hyperparameters to be less than one, to show that my prior knowledge should have very low weight. The interpretation of this as meaning a negative number of successes (or failures) seems unnatural. --Iae (talk) 20:25, 19 April 2012 (UTC)

The expression ${\binom {n}{i}}p^{i}(1-p)^{n-i}$ is the probability that i is the number of successes in n Bernoulli trials each having the success probability p. The same expression is the likelihood that the success probability has the value p when n trials has produced i successes. This defines a beta distribution with mean value ${\frac {i+1}{n+2}}$ and standard deviation ${\sqrt {\frac {(i+1)(n-i+1)}{(n+2)^{2}(n+3)}}}$. Before any observation we have n=i=0 and the prior distribution is uniform on 0≤p≤1. Then p=0.5(3). (concise notation). Bo Jacoby (talk) 10:07, 22 April 2012 (UTC).

If we assume that the absolute prior is uniform (which is maximum entropy), then a,b correspond to a-1/b-1 success/failure (since a=1, b=1 is uniform). Alternatively, we could say that even a uniform distribution represents some information - that we have supposedly seen a success and a failure, and thus understand that either is possible and we can't guess the exact probability - and then it would correspond to a/b success/failure. -- Meni Rosenfeld (talk) 20:22, 23 April 2012 (UTC)

I don't understand why "even a uniform distribution represents some information". Before any observation is made we only know how to distinguish success from failure, but that does not give us any information about the success probability. p_deduction=i/n is the probability that a random one of the n already observed trials was a success. p_induction=(i+1)/(n+2) is the probability that the next trial will be a success. Bo Jacoby (talk) 12:07, 24 April 2012 (UTC).

square footage

How do I figure the square footage of a room? — Preceding unsigned comment added by 71.171.25.31 (talk) 20:59, 19 April 2012 (UTC)

I believe that would be a reference to the square footage, or the area, of the floor of the room? Would that be correct? Bus stop (talk) 21:07, 19 April 2012 (UTC)

If the room can be approximated by a polygon, you can use the generalized polygon area formulae for any room shape. Nimur (talk) 21:54, 19 April 2012 (UTC)

As most rooms are rectangular, multiplying the length by the width, both measured in feet, normally does the trick. --Tagishsimon (talk) 22:54, 19 April 2012 (UTC)

And some are a number of rectangles. You can add and subtract individual rectangles to get the total:

□ □ □
□ □ □
□ □ □ □
□ □ □ □
□ □ □ □

If each box is 3.5 feet square, that would make the room 5×3.5, or 17.5 feet long, and 4×3.5, or 14 feet wide. This gives us 17.5×14 or 245 square feet total. From this we must subtract the missing rectangle, which is 1 box by 2 boxes, or 3.5×7 feet, for a total of 24.5 square feet. 245 - 24.5 = 220.5 square feet.

If all else fails, you can draw the room on graph paper, where one square equals one square foot, and then total up the squares. StuRat (talk) 06:04, 22 April 2012 (UTC)