Is it correct to say that $\sum_{n=1}^\infty \frac{n}{n^2+1}\left(\frac{x-2}{x+2}\right)^n$, with $x \ge 0$, converges uniformly in each $I_a =(0,a]$ with $a>0$ because $\sup_{x \in (0,a]} \left|\frac{x-2}{x+2}\right|^n = \left|\frac{a-2}{a+2}\right|^n$ and the series $\sum_{n=1}^\infty \frac{n}{n^2+1}\left(\frac{a-2}{a+2}\right)^n$ converges absolutely?
Consider a compact matrix Lie group. In general, if $\exp(\Sigma_i t_i \Lambda_i)$ generates a Lie group, does $\Pi_i \exp(l_i \Lambda_i)$ also generate the Lie group?
Maybe Fulton's algebraic curve is one good introductory textbook on algebraic curves? Any other textbook you people would recommend? (as a supplementary for example)
My prof. said that for finite groups, induction and compact induction are the same, i.e., if $H<G$ as a subgroup of finite group $G$ and $R$ is a commutative unital ring, then two functors $RG\otimes_{RH}-$ and $\mathrm{Hom}_{RH}(RG,-)$ from $RH$-module cat. to $RG$-module cat. are isomorphic. Any reason why this is true?
Maybe you should try the 'fake it till you make it' strategy. You see, there are so many great math teachers on youtube, who are very good at expressing their thoughts. Try to imitate how they are doing their jobs, like how they talk, how they explain things, how they response when someone asks a question, etc. Try to imitate their body languages as well. You have the knowledge I believe, just some lack of confidence is holding you back I guess.
This, I hope, is not a duplicate; I am exercising my critical thinking here and I want to understand what going on, and the available content I have found online on this so far has not helped.
I'm getting conflicting information regarding whether $1=0.\overline{9}$ (i.e., "$0$ point $9$ recurring"...
Hi @noballpointpen They're going well, I guess. Thank you for asking. I'm behind, but I'm applying for a retroactive Leave of Absence to make up for it. (I had a lot of time off for my health.)
This uses the soft-question tag because there might be more than one valid answer, and it's a matter of guesswork to some extent; but there is a right answer (in theory).
Thoughts and Motivation:
As a follow-up to
If a hom. $\phi:G\to H$ of diagonalisable linear algebraic groups is injective, the...
To Shaun: Good, but I do not spend enough time on my studies, so my progress is slow these days. I am not in school, it's self-studying. Read few pages of Mendelson's logic yesterday and finished two problems, spent 3 hours overall. Do you have knowledge in logic? Of course, besides stuff that every mathematician knows.
@noballpointpen Three hours is good going, especially for self-study. I used to. I read some notes by Simpson and I'm interested in topos theory, but I wouldn't say I'm knowledgeable; just an enthusiastic onlooker.
I have a question on my problem set that asks "Find Taylor series representation of $f(x) = x \ln(2 + x) $ about $x = 2$." I can find the terms by hand and I have found the first 6 terms but I can't see a pattern to these terms so that I can show it with sigma notation. Would it be fine to leave it with ellipsis in the end?
@Shaun may I ask? Are you insecure -- or how would I say it, not a native speaker, sorry; maybe, "feel guilty" -- about duplicates? It is not the first time I see you mentioning that your question is not a duplicate. And you spend lots of time commenting on other posts about duplicates.
@noballpointpen I'm English. In my culture, people apologise for things a lot. Besides, I don't want to waste other people's time with a question I could have answered by searching harder.
@Shaun ah, then I understand. Indeed. Sorry, if my question was rude :) I mean, people usually don't mention their question is not a duplicate. If it is, or it is a bad question, these sayings do not help.
I would like to ask people here. Do you just post links in the chat to this site and they automagically get formatted like this link to a comment above? So it does require high reputation, like image uploading? Occasionally I wanted to post a link to MSE here but I don't know how do you format it, the help\faq pages are useless.
. . . then click on it and the link option should appear. Click on that, and it will take you to a different screen (if you're on a phone). Copy & paste that link into the chat.
https://chat.stackexchange.com/transcript/message/63727454#63727454 It doesn't work like this, Shaun. I guess, I need high reputation for the chat to format it for me.
You need high rep for an "upload" button to appear near "send" to upload images. This is not in privileges list, too. So there are minor privileges that are not documented... at least easily available.
First of all, I'm not yet familiar with more general concepts like "compactness" and this is a question from the first chapters of my real analysis course.
We can regard the reals as a union of the following sets: $\bigcup ([n,n+1]$$\cap$$\mathbb{R}$) with $n \in \mathbb{Z}$. If each of the set...
I have a question: How do you construct a regular holomorphic foliation of $\Bbb C^2$ by the class $S=\Bbb C -\lbrace0 \rbrace $ where all leaves are holomorphic to $S?$ Due to the rigidity of the foliations (because of the holomorphic condition) I am lead to believe there does not exist such a construction. In the smooth setting, it's much different because there's more leeway topologically to deform the foliation achieving the goal
Is there a straightforward argument to invalidate a construction in the holomorphic case?
@Jakobian No I was more thinking about the operation division. Addition, subtraction, and maybe multiplication are fine, but division, I really don't like it.
Hi @robjohn you could probably help me with this explanation. I was given a sequence $x_k$ defined recursively as $x_{k+1} = f(x_k)$. Also the function $f$ is a contraction mapping. We used this sequence to prove the contraction mapping principle. I was asked to show $\|x_k - x\| \leq \frac{c^k}{1-c} \|x_1 - x_0\|$. I did all this, but it was commented that this gives an estimate on how fast the sequence converges to the fixed point.
What I'm confused about is what is giving the "speed" of convergence? I know it is the term $ \frac{c^k}{1-c}$ but how should I interpret "speed" in this sort of scenario?
Oh...I get that it is a geometric series and such. I'm just trying to comprehend what is meant as "speed" in this scenario. So would I say something like "the sequence is converging at the rate of a geometric series"?
You would say that the rate of convergence is exponentially fast. Because, fixing initial seed $x_0$, $\|x_k - x\| = O(c^k)$ for some fixed constant $c < 1$.
I suppose people use "geometric" and "exponential" interchangeably. I forgot $\{c^k\}$ is called the geometric progression, I had geometric series in mind.
Assume $f:X\rightarrow Y$ is a continuous function between metric spaces such that for all small $r>0$, and all $x\in X$, $B_{f(x)}(r)\subseteq f(B_x(r))$. Assume $K$ is compact and $f|_{K}$ is injective, does it follow that $f|^{-1}{K}$ is Lipschitz?
Let $f : X \to Y$ be a measurable function. I guess we could assume that $X$ and $Y$ are (standard) Borel spaces. How does one prove that $graph(f) := \{(x,f(x)) : x \in X\}$ is a (Borel?) measurable subset of $X \times Y$?
the different notions of size, topological vs measure theoretical vs cardinal seems to be a recurring issue (what is small/negligible for one is not the case for the other:)
