@tb quite far... Shilyack does not mean anything good in Russian. We don't have such word - but based on its sound, if we would have it, it would not be nice. I regret my name is not Erl :)
Well, it started with someone pretending not to know what this two-letter abbreviation means then somebody throwing seven letters (two distinct ones) at me and then people permuting some letters in name.
you should be happy then, indeed :) sorry, but I have 52 minutes left to write an introduction. I have to submit paper today - but I am also assisting an exam for about 3 hours. See you later, or maybe tomorrow, @tb. And good luck
@DavidWallace: I agree with your answer. $\mathbb{Q}$ is closed under addition, and infinite sums are a limit. It would be nice if TonyK would give a reference to where Rudin "explains how to give meaning to certain infinite sums of rationals" rather than simply claiming it in contradiction to Mariano.
Look,he said it was 1977 that he read it. I think people are allowed to forget a few details in that many years.
You could argue that the whole "limit of a sequence of partial sums" business is just a way of adding infinitely many numbers. Then, you're arguing lingustics, not mathematics.
@robjohn Winters are not cold. It drops to say 7 degree celsius but nothing like what the visitors to our institute have to say, like -10 degrees or whatever... (And this drop is to very particular places, for instance, the place where I really come from stays the same in winter as it would in summer!)
@DavidWallace I am at Bangalore, while I really come from Trichy, Tamilnadu.
@robjohn This is for the summer holidays. I had received a fellowship from the Science Academies here. And, I had requested him to be my guide, to which he agreed.
@robjohn Yes, it is lesser known, and it is actually a 2nd tier city in India. So, many won't know it, even in India.
@DavidWallace It is not a coastal area. You'll have to travel a minimum of 200kms to get close to a sea. And, for a nice beach, may be, it would be 300 kms.
@robjohn Interesting, but what game is that? (If it's a secret, never mind!)
@KannappanSampath Ah, yes. But on Sunday night we only had two, but the game works for two as well. We have recently been playing a lot of these train games, and Ticket to Ride and its spinoffs as well. Both teach a lot of geography.
@KannappanSampath It might be, but recently all I hear about are MMORPG :-)
Well. I am going to finish this and head to the office where I can print freely all sort of things I might need to learn the proofs I need to cover. Bye for now.
@AsafKaragila Listen: strange women lying in ponds distributing swords is no basis for a system of government. Supreme executive power derives from a mandate from the masses, not from some farcical aquatic ceremony.
Can someone help me understand the notation (free $R$-module): $$F(S) := \oplus_{s \in S} R \cdot s$$ $S$ apparently can be *any* set. So if $S$ is not a subset of my $R$-module $M$ then what meaning is $\cdot$ supposed to have? Is it like $R \times \{i\}$ for $i$ in some index set? Or what?
It would make more sense if $S$ was a subset of $M$, my basis of $M$.
The elements of $F(S)$ formal linear combinations of the elements of $S$. Here $S$ is any set, completely unrelated to the ring. You can identify $F(S)$ with the finitely supported functions $S \to R$ (that is: only finitely many elements of $S$ are sent to non-zero elements of $r$). Think of $S$ as an abstract basis for $F(S)$.
(for some people it helps to write $F(S) = \bigoplus_{s \in S} R \cdot e_s$ so that $\{e_s\}_{s \in S}$ looks more like a basis)
@MattN This looks good. But if you write $\sum_{s \in S} \lambda_s \Box \in F(S)$ what do you put in place of the $\Box$? Writing another $e_s$ or the like, quickly leads to index overkill, and writing $\sum_{s \in S} \lambda_s \cdot s$ never leads to ambiguities.
@DavidWallace I fail to understand what sort of transformation is he talking about? And, why does he feel they are easy to do?
As in, if you want to transform a random variable in that way, I don't know of an immediate method. Even if they exist, I am sure if they will be easier than the elementary tricks.
No, I don't understand what his/her problem is either. It seems that you have answered the question; and he/she wants a completely different answer. Some people are just never satisfied!
@MattN One instance: The group ring $R[G]$: Suppose $S = G$ has a group structure. Then you can turn $F(S)$ into a ring $R[G]$ by declaring $(\sum_g a_g g) \cdot (\sum_h b_h h) = \sum_k c_k k$ where $c_k = \sum_{k = gh} a_g b_h$ (convolution product). It is simply more convenient to have $k = gh$ as a relation in $R[G]$ than $e_k = e_g \cdot e_h$.
When I see $G$ I automatically replace it with $Z_p$ or something like that. Then reading maps of finite support is funny. Thanks for pointing this out. I have a feeling this is not the last time I'll see group rings. : )
@tb The $\frac{\sin(x)}{x}$ answer has been slowly gathering votes, but not nearly as fast as the rotating cube. I think that would be my guess for the next to hit 25
Besides the fact that I still enjoy just watching the cube :-)
@Mariano: It may be unrelated, but yesterday when I tried to log into LATimes.com, they tried to use Google, but I got the message "Session Timed Out" I tried several different ways and I still got that message. This makes me afraid to log out of SE.
@JonasTeuwen if you observe it inverting, it collapses and becomes singular. :-D
@AsafKaragila @BrianMScott @HennoBrandsma Thank you so much!! You're lovely. I scored a $5.5$ out of $6$. I don't care much about the grade but when I said I was worried sick that I might've failed she laughed and said "Yours was one of the best exams." : D
