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...
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.
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...
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...
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) -...
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...
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.
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.
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.
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...
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.