# Recent content by n_bourbaki

1. ### Journal papers in a foreign language

You learn 'enough' Russian (or whatever language) by using a dictionary. Fortunately mathematical papers often read the same irrespective of language, and indeed the words are often the same. The only difficulty with Russian is the alphabet. It is often (always?) a requirement of an American...
2. ### Is this logical reasoning correct?

This proof assumes that f and g have the same domain. It fails if they have different domains. But then by definition the functions are not equal, so there was no need to go any further.
3. ### Is it possible to find function that describes irrational things?

You have so many undefined terms it is impossible to know where to begin. What does 'the change from prime to prime' even mean? As has been pointed out before, functions can be defined that describe *anything*, the only question is whether that function has a 'nice' form, or can be calculated...
4. ### Natural isomorphism of Left adjoints

You have (natural) isos (F?,?)-->(?,G?)-->(H?,?) is the composition of (natural) isos an iso?
5. ### Doing proofs all alone

So you don't think that every implication needs to be justified? Note that I did not specify what was required to make that justification rigorous. That only comes with experience, and depends on whom you are writing the proof for. For one example, if I could assume a lot then I would not even...
6. ### R is complete?

The set S seems explicitly described. x is in S if and only if x is less than or equal to infinitely many of the a_i.
7. ### Sum of random number of random variables

E[X] is just a number, so you have to work out the variance of a constant times N. That is standard: Var(kY)=k^2.Var(Y) for k constant and Y an r.v. Don't forget that Var (Y)= E(Y^2) - E(Y)^2 as well, when you're doing things like this. So if U and V are independent Var(UV)= E(U^2V^2) -...
8. ### Doing proofs all alone

As it's your first time writing proofs, what is almost surely true is that you're not putting in enough details. A proof is not just a string of numbers and mathematical squiggles. First, make sure that your proof makes sense as a written piece of English. You can do this by giving the proof...
9. ### Structure constants of su(2) and so(3)

There is no contradiction there: SO(3) isn't simply connected. It's fundamental group is C_2 (group with 2 elements), and has SU(2) as its simply connected cover. These facts are illustrated quite nicely with 'the soup bowl trick' and quarternions.
10. ### Structure constants of su(2) and so(3)

The usual 'equal' and 'isomorphic' misunderstanding. Clearly, they are not equal, since thet are different, but equally clearly they are isomorphic.
11. ### Infinite cyclic groups isomorphic to Z

This is so utterly trivial that I cannot believe it needs a discussion. G is infinite and cyclic and G=<a>, i.e. G is the set of all integer powers of a if written multiplicatively. So in what way is the map a^k ---> k not a clear isomorphism? All powers of a are distinct.
12. ### Vector spaces

That just isn't the way to say it: a subspace of a vector space V is by definition taken to be over the same field as V.
13. ### Formal construction

I would have said the formal sum would be abaca+bbac, and an element in the group algebra. Again, we use the word formal simply because addition is not a naturally defined operation on strings.
14. ### Type of Induction

In this sense of the word, there is essentially only one form of induction. Whilst we may use the terms 'induction' and 'strong induction' for two different things they are entirely equivalent notions (i.e. each one implies the other), and your notion of induction in post 1 falls between the...
15. ### Formal construction

A word is just a word. It is called formal purely because a priori X has no structure that allows us to construct words from the letters. And that is all that is going on. We just think of these things as if they made sense, when there is no innate structure, and then show that it makes sense.