Wikipedia:Reference desk/Archives/Mathematics/2010 February 22

= February 22 =

Modal logic
From the article, just trying to understand the semantics.

We have a non-empty set (frame) $$G$$, and $$v, w \in G$$ are worlds. We have a accessibility relation $$R$$, such that $$v R w$$ means that $$v$$ is possible in $$w$$.

As an example:

$$p$$ = it is snowing

$$w$$ is $$\Box p$$. $$v$$ is $$\Diamond p$$.

Therefore $$v R w$$.

Does this mean $$R$$ is like an equality relation, or a kind of "is a subset of" relation? —Preceding unsigned comment added by 81.149.255.225 (talk) 10:29, 22 February 2010 (UTC)
 * It does not make sense to say that "v is possible in w". You can ask whether a statement (formula) is possible in a world, but there is no concept of a world being possible in another world. The intuitive meaning of the accessibility relation v R w is actually that every statement that is true in w is possible in v. However, it's not really defined like that, it's a primitive notion: you are given a set of worlds, an accessibility relation, and truth values of atomic statements in each world, and all this together determines the truth values of compound statements (which may involve the $$\Box$$ and $$\Diamond$$ operators), see Kripke semantics. Your example does not make much sense as is, but even if we read it as "$$\Box p$$ is true in w and $$\Diamond p$$ is true in v", this is insufficient information to determine whether v R w or not. The accessibility relation can be an arbitrary binary relation, in general it does not have to be transitive or reflexive, hence neither equality nor "subset of" is an adequate analogy.—Emil J. 14:52, 22 February 2010 (UTC)


 * The double turnstile ⊨ is read 'models' rather than 'is', where models here means the right hand side proposition is true for the world given by the left hand side. p is a proposition and w is a world. I think your business about inclusion is where you are thinking of R as implementing the most common system, what the article calls the strongest logic S5, but this isn't necessarily so. Dmcq (talk) 15:16, 22 February 2010 (UTC)

What more do we need to know in order to Identify a function
If we know f(n)=(1/n^2) for all integers n, and f is analytic in the complex plane except for possibly at some singularities, then what more would we need to know in order to establish g exactly? Obviously you could have g(z)=cos(2piz)/(z^2) or something similar so it needn't necessarily be g(z)=1/z^2 at this point - but I can't see how we can classify all possible g which take the appropriate values at integers, in order to work out what more information we need to know to identify g. Could anyone suggest anything?

Thanks all very much! 82.6.96.22 (talk) 13:34, 22 February 2010 (UTC)
 * Your condition on g is equivalent to g(z) = 1/z2 + h(z), where h is a function meromorphic  in C (assuming that's what you mean by "analytic except for possibly at some singularities") which vanishes in all integer points. You can thus ignore the 1/z2 part, and concentrate on the (less messy, if not really easier) task of classifying such h.—Emil J. 14:59, 22 February 2010 (UTC)

Centre of a group algebra
Why do the conjugacy class sums of a group, G, form a basis of the center of the group algebra, FG, for some field F? I've tried it out for a few concrete examples, and it worked, but I can't really see why. The closest I could come was taking h to be in the centre - then hz = zh for all z in the algebra, so $$h = z^{-1}hz$$, which introduces conjugation, but that's only true when z has an inverse in the algebra. I know also that the class sums are invariant under conjugation by the elements of G (i.e. the basis elements of FG), but I can't seem to string it altogether. The notes I'm working with only dedicate one line to the explanation, so I feel I'm missing something pretty obvious. Thanks, as always! Icthyos (talk) 18:02, 22 February 2010 (UTC)
 * I think you're over thinking the problem. Let z be in the center of the group algebra and write
 * $$z = \sum z_g g$$
 * Then
 * $$z^h = \sum z_g g^h = \sum z_{g^{h^{-1}}} g$$
 * where the second sum is obtained by reindexing the first one. But z=zh for all h so matching coefficients
 * $$z_g = z_{g^{h^{-1}}}$$
 * for all h. In other words the value of zg depends only on the conjugacy class of g. So z can be written
 * $$z = \sum z_C C$$
 * where the sum runs over the conjugacy classes C of the group. The Cs are obviously linearly independent and they are in the center, so they form a basis. The proof is kind of obvious if you've been working with these sums long enough, but probably not if you never seen them before. My group theory book (W.R. Scott) gives the proof in 7 lines, slightly different than the one here.


 * A-hah, I see now - thanks for the help. Icthyos (talk) 21:37, 22 February 2010 (UTC)

Math Help
Yes- I realize you won't help me with my homework, but I have no idea where to start here.

Can someone give me the equations?

Arriana bought two kinds of stamps, 50cent stamps, and 65cent stamps. She bought 40% more 50cent stamps then 65 cent stamps, spending a total of $50.56. How many of each type did she buy? There is a 7% tax on stamp sales.

Thanks in advance. —Preceding unsigned comment added by 174.112.38.185 (talk) 21:55, 22 February 2010 (UTC)


 * Start by giving a name to everything you dont know. these are called variable for some reason
 * x for the number of 50c stamps
 * y for the number of 65c stamps
 * and then write them in like 40% more 50c stamps means take y and 40% of y and you get x or shorter
 * (1 + 40/100)y = x
 * All you do is write down the x ory instead of the actual amounts in each of the other statements as well and you've got some equations. Dmcq (talk) 22:07, 22 February 2010 (UTC)


 * (ec)Variables are useful here. When you want to manipulate some number, but you don't know what the number is, place a variable in its stead;  this gives you information about the variable which later may allow you to determine its value.
 * Let x be the number of 65 cent stamps that she bought. Then answer the following questions in terms of x.  How much did the 65 cents stamps she purchased cost?  How many 50 cent stamps did she purchase?  How much did they cost?  What is the total cost (before tax) of all the stamps she purchased?  How much did she spend (after tax)?
 * You know that the answer to the last question is $50.56 from the problem statement. Once you've answered the last question in terms of x (using your previous answers), you will have an equation for x.  This will allow you to calculate x and solve the problem from there.  Eric.  131.215.159.171 (talk) 22:11, 22 February 2010 (UTC)
 * Everyone acts as if mathematics is the only way to solve these problems, when sometimes the humanities will suffice. (I suppose that's what you get when you ask such a question in a mathematics forum.  When your only tool is a hammer, ...)  In this case a little detective work will do.  I followed Arriana to the post office when she mailed off her wedding invitations, and she crossed a name off her list as she put a stamp on each envelope.  I pulled that list out of the trash after she left and found out that the total number of guests receiving invitations happens to be the same as the lim sup of the largest finite subgroup of the mapping class group of a genus $$g$$ surface divided by $$g$$.  Moreover, the number of guests who lived locally, and not over the border in Canada (and thus requiring the higher international postage), was the square of the largest prime factor of the total number of guests. 58.147.60.130 (talk) 01:50, 23 February 2010 (UTC)


 * Indeed I must admit I was wondering why they couldn't just ask Arriana or look at the stamps, and why did they want to know anyway? Surely you'd only want to know how any stamps you've got left. Dmcq (talk) 12:00, 23 February 2010 (UTC)