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comment Could I do this to an infinite series?
Typically, we define $\sum_{k=1}^{\infty} x_k$ as $\lim_{n \rightarrow \infty} \sum_{k=1}^n x_k$ if and only if the limit exists, so this doesn't quite work. It would be correct to say that the partial sums are equal to the resulting expression, i.e. $\sum_{k=1}^n k + \sum_{k=1}^n k = \sum_{k=1}^n 2k$, but neither of these sums converge so we can't really talk about their limits.
comment Finding at least 2 elements in a set that satisfys an equation
For each $n$, temporarily consider the additive group modulo $n$, that is $\mathbb{Z}_n$. By "$-1$", we merely mean the inverse of the element $1$. In $\mathbb{Z}_5$, we have $-1 = 4$, because $1 + 4 = 0$ (and in general, $-1 = n-1$). Now, try out squaring this additive $-1$ in the multiplicative group $U(n)$ (first verify that it's always in there), and show that it works.
comment Finding at least 2 elements in a set that satisfys an equation
Namely, $1$ and $-1$ are the two elements in $\mathbb{Z}$ under usual multiplication that square to give the identity. So, what are the analogues of $1$ and $-1$ in the finite group $U(n)$?
comment Finding at least 2 elements in a set that satisfys an equation
Okay, so the group operation is multiplication mod $n$. Then actually $U(n) = (\mathbb{Z}_n)^\times$ as a group. Got it.
comment Finding at least 2 elements in a set that satisfys an equation
What group structure are you giving $U(n)$ when we determine if there are two elements $x,y \in U(n)$ that satisfy $x^2 = y^2 = 1$? If we're working in the group $\mathbb{Z}$, the only element that satisfies $x^2 = 1$ is 1.
comment Using Pumping Lemma to show a language is not regular
This is correct, you could probably be a little more concise on the ending. Just say that the fact that $xyyz$ is not in the language means the language does not satisfy the pumping lemma. So our assumption that the language is regular must be incorrect. It also couldn't hurt to write out $xyyz$ to be more explicit, that is, $xyyz = 0^{p+k}1^p 1^p$ for some $k \geq 1$, noting that $2(p+k) > 2p$.
comment Solutions of $a^{2} - 2b^{2} \equiv 0$ mod $p$
Thank you very much! The terminology really helps when researching this!
comment Why is proof of the [topological] closed graph theorem incorrect?
Thanks; I see now there's nothing guaranteeing that $N$ can be be written as $U' \times V'$ with $U'$ definitely a subset of $U$. Can I fix this approach or am I off altogether? I could use some hints going forward because I'm pretty stumped.
comment How can I better understand manipulating “operators” in mathematical relations?
Thanks. Okay, I understand that you can do this algebra of functions, but if I just have some operator or map $A : U \rightarrow V$ and I have an element $u \in U$, then when I write $Au$ I mean the result of applying operator $A$ with input $u$? Is writing the operator next to a member of its input type "implied application"?
comment What “is” a matrix in the context of a vector space?
I think I understand. Since the space of ALL possible linear maps from V to W is isomorphic with the set of ALL m x n matrices, we can work with matrices and perform arithmetic on them and interpret our results in terms of linear mappings because of the isomorphism. So would it be reasonable then to view the "multiplication" of a vector by a matrix just an expression of the matrix's corresponsing map being applied to the vector? I think I'm splitting hairs at this point but I do feel more comfortable thanks to your post.
comment Two questions about Euler's number $e$
If I may recommend a book, check out "e: The story of a number" by Eli Maor. It's an excellent read that talks all about the history of the number e and all of its implications. You can find it here (or of course look for it at a library at school or otherwise):
comment Are these 2 graphs isomorphic?
I would be interested in seeing the code, if you don't mind.
comment Notation for factorial-type pattern with a skip/step of two instead of one?
Well... that is indeed what I was looking for. Didn't know that notation existed!