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December 14

Slight variation on the angle sum formulae

I was looking over the angle sum formulae on List_of_trigonometric_identities#Angle_sum_and_difference_identities specifically at the one for arccos, Can the formulae be expressed more generally for say $A\arccos \alpha \pm B\arccos \beta$ ? I guess in theory its simply a case of repeated use of the formulae but my trig skills are severely lacking, Thanks--137.108.145.10 (talk) 12:28, 14 December 2010 (UTC)[reply]

Am i being an idiot or is it just

A*B\arccos(\alpha \beta \mp {\sqrt {(1-\alpha ^{2})(1-\beta ^{2})}})

--137.108.145.10 (talk) 17:53, 14 December 2010 (UTC)[reply]

No, it's not that. That would be

AB\arccos \alpha \pm AB\arccos \beta

. I'm not sure if there is a nice expression for what you want. --COVIZAPIBETEFOKY (talk) 18:07, 14 December 2010 (UTC)[reply]

Combining two low resolution images

Can mathematics give any clues about the improvement in resolution you could get from combining two different low resolution digital photos of a painting? Are there any general principals? If the (square?) pixels were not in perfect alignment but overlaped, what would be the best mathematical procedure to improve the resolution? Thanks 92.15.11.6 (talk) 18:28, 14 December 2010 (UTC)[reply]

This is called Super-resolution. Our article doesn't say much but you may find useful information in the references.

I think you need the photos taken from slightly different locations for it to work. The general approach is to express what each pixel of the low-res images would be for a given high-res object, and choose the values that give the best fit to the input low-res images. One implementation is to use sum of squared errors as a measure of fit, so the solution consists in solving a system of linear equations. -- Meni Rosenfeld (talk) 19:42, 14 December 2010 (UTC)[reply]

Registax can do this. 92.28.247.44 (talk) 22:39, 16 December 2010 (UTC)[reply]

Cluster points of uncountable sets

Does an uncountable subset of the reals have to have uncountably many cluster points? —Preceding unsigned comment added by 199.17.35.133 (talk) 20:15, 14 December 2010 (UTC)[reply]

Sure. Start with an uncountable set; it has to have a cluster point because there must be at least one interval [n,n+1] within which it has uncountably (and therefore infinitely) many points, and [n,n+1] is compact. Remove that cluster point; there are still uncountably many points left, so there must remain a cluster point. Iterate through all countable ordinals (what happens at limit ordinals should be obvious); at each step, you've removed only countably many points, so there are uncountably many left, so you can keep going. --Trovatore (talk) 20:49, 14 December 2010 (UTC)[reply]

My proof goes like this. We show by contradiction that for any such uncountable S, there exist two disjoint intervals in the reals that each contain uncountably many points from S. To do this, suppose otherwise, and follow along the proof of the Bolzano-Weierstrass Theorem, to show that S has only countably many points, which is untrue. Now that we know each uncountable S has uncountably many points in each of two disjoint intervals, we show it has uncountably many cluster points (in the same way one shows the Cantor set is uncountable). Eric. 82.139.80.124 (talk) 21:10, 14 December 2010 (UTC)[reply]

It's good to get in the mental habit of transfinite induction on problems like this, though. In this particular example it doesn't really buy you much, because the points removed don't really "interact" much (or their interaction is ultimately irrelevant). But it's a way of thinking that often makes the answer obvious after a small amount of thought. --Trovatore (talk) 22:43, 14 December 2010 (UTC)[reply]

Hmm, now that I think about it a little more carefully, my argument doesn't work as stated, because it's not clear what's meant by "removing a cluster point". --Trovatore (talk) 22:55, 14 December 2010 (UTC)[reply]

logical terminology

An implication like $A\Rightarrow B$ is equivalent to $\neg B\Rightarrow \neg A$ and this is called the contrapositive. The formula $(\forall x)\phi (x)$ is equivalent to $\neg ((\exists x)(\neg \phi (x)))$ . Is there a word like "contrapositive" that describes switching a quantifier like that? Thanks. 67.117.130.143 (talk) 20:45, 14 December 2010 (UTC)[reply]

Not exactly answering your question, and maybe you already thought of this this, but you're talking about something which already is basically a contrapositive.

(\forall x)\phi (x)

is the same as

x\in U\Rightarrow \phi (x)

, where U is the universe of discourse. Then the contrapositive of this is

\neg \phi (x)\Rightarrow \neg x\in U

, which means

\neg ((\exists x)(\neg \phi (x)))

. But I've never heard of a precise name for what you're talking about. Staecker (talk) 13:29, 15 December 2010 (UTC)[reply]

It's a variant of De Morgan's laws.—Emil J. 14:15, 15 December 2010 (UTC)[reply]