Any kind of answer. I have read your answer and i want to answer in the same way as how you did. Can you tell me how to answer any kind of questions which follows the policies of MSE? @SohamSaha
One thing.. I shouldn't copy the answer from GPT and paste here right? YOu must have seen my previous answer, where i didn't copy the answer from any AI yet others say i did. What should i do? @SohamSaha
Others can say and do whatever they want. Being true to yourself should be enough. This is the current MSE stance on Gen-AI content: math.stackexchange.com/help/gen-ai-policy
I just use webs like GPT, Grammarly, edurev and so on to get the reference for my answer and to also check if i did in the correct method so that the asker won't follow the wrong path.. Is this wrong? I won't copy the content from that web, i just frame it in my own words.
but that leads to my bad reps right.. They think i copy and vote mine down.
Mention which parts are genAI content, and ensure that the core elements of your post are not AI generated. I for myself am strongly against AI content, as they are really error prone in mathematical context
But that’s just my opinion
Even reworded content should be referenced to the appropriate sources
@vigilantvino02 You should be ok. Just don't post text generated by AI. However, some recent grammar checking tools can use GenAI to rewrite your text. It's probably not a good idea to use such text, even if the program isn't actually changing your ideas.
Such tools make you sound like GenAI. And people may not believe you if you say, "Oh, it's just a translation / grammar tool!".
You can put a note in your answer like: "I used ChatGPT while doing research, but this answer was written by me, and cleaned up using a non-AI version of Grammarly".
@SohamSaha Ah, the good old Unit Cell. There's now a more efficient version of that. But the original was mind-blowing when it first burst onto the scene.
"Every closed, oriented 3-manifold $M$ contains a link $L\subset M$ such that $M-L$ is homeomorphic to a finite sheeted covering space of the Whitehead link complement"
Just a few years a new infinite growth pattern emerged from a random soup survey. The Conwaylife forum people have been doing a distributed exploration of 16×16 random starting patterns. Of course, it will take eons to completely investigate all 2^256 starting patterns. :D
Consider the definition of the exponential function as a power series, namely $$E(z)=\sum_{n=0}^\infty\frac{z^n}{n!}.\tag1$$One can derive $$E(z)E(w)=E(z+w),\quad E(z)E(-z)=E(0)=1,\tag2$$and then Rudin claims that $E(z)\neq 0$ for all $z$. How does this follow from $(1)$ and $(2)$?
Ah wait. After some thinking, I guess it is because if $E(z)=0$ for some $z$, then $0=E(z)E(-z)=1$, contradiction.
fun fact: If $M$ is a closed oriented surface with a constant negative curvature w.r.t a metric $g$, then $L_{X_1}g = L_{X_2}g$ for vector fields $X_1,X_2$ on $M$ implies $X_1 = X_2$.
I have a linear transformation $L: R^n \to R^n, y = Lx $ such that $y_i = x_i \forall i \ne h,k$ while $y_h = x_h+c x_k, c \ne 0$ I have a rectangular closed cuboid $I = [a_1,b_1]\times ... \times [a_n,b_n]$ and I gotta prove that $m(L(I)) = m(I)$
If I have an exact functor F: Sp \to D, where Sp is the \infty-category of spectra, and D is stable, complete and cocomplete, there exists a regular cardinal \kappa such that Sp is \kappa-accessible, F: Sp_{\kappa} \to D is k-accessible, where Sp_{\kappa} denotes the \kappa-compact objects, and taking Ind-completion recovers F? This seems not true in this generality but I have seen (from my understanding) it claimed somewhere
Like if F is not exact why should it be \kappa-continuous for any \kappa? Here \kappa-continuous means that it commutes with \kappa-filtered colimits
In Rudin's PMA, he says that there exists a unique $t\in[0,2\pi]$ such that $e^{it}=z$, where $|z|=1$. Then, when proving the algebraic completeness of the complex field, he claims there exists real $\theta$ such that $e^{ik\theta}b_k=-|b_k|$, where $b_k$ is a nonzero complex coefficient of a certain polynomial. I simply do not understand why such a $\theta$ exists?
$k$ could be any natural number between $1$ and $n$, for some natural $n$.
@Jakobian I never realized that I've always seen the $K_\sigma$ notation exclusively in descriptive set theory books/papers and not really in topology ones. Funny
