A general philosophy question: I am reading through Stacks Project Chapter 10 on Commutative Algebra but feel somewhat unmotivated with just the algebraic machinery in hand. Should I do some parallel reading for others chapters as well? Are there any geometry that I can apply these algebra on? Thank you.
Hi, Anyone can help me on this very basic question will be welcome and appreicated. What is all the assumption and condition required to making sure that T has continuous partial derivatives in Two variables when applying changing of variables formula?
What is all the assumption and condition required to making sure that T has continuous partial derivatives in Two variables when applying changing of variables formula?
Reference link: http://sites.millersville.edu/bikenaga/calculus/change-of-variables/change-of-variables.html
@WilliamSun i think itβs very hard to learn from stacks project and would recommend you learn commutative algebra through other sources instead. The typical recommendation is atiyah macdonald which i is good but i personally prefer ones with a geometric flavour
I have had a first course in Atiyah from all chapters except for the dimension theory/valuation rings.
The reason why I learn stacks project is because it has more categorical constructions/homological techniques.
I feel comfortable with the algebraic materials but I am trying to apply it to, say some modules over the ring $C^{\infty}(M)$. Maybe I should just examine the examples one by one...
Heya @Jasper !! Welcome to the dark side with the rest of us :P
@Victor This will guarantee that the function you have to integrate when you change variables is still continuous. There are stronger theorems, but you don't see them in a calculus class.
I'm trying to solve this exercise: "If |G| is odd and has a normal C_17 subgroup, then it is central". I know and understand the proof of the fact that "If $H$ be a normal subgroup of prime order $p$ in a finite group $G$ and if $p$ is the smallest prime that divides the order of $G$, then $H \leq Z(G)$" (see here). But I'm not sure how to approach this exercise.
Anonymous
Say, $|G| = 17.3 = 51$. Then any conjugacy class within the subgroup $C_{17}$ must divide $3$ I think. So any conjugacy class within $C_{17}$ can either have size 3 or size 1. So we can't directly conclude that $C_{17}$ necessarily lies in the center of $G$. Could someone help me a bit with this?
@TedShifrin Are there aplenty or is there vast majority of cases that look simple like the one took in Calculus courses gnerate by computer ramdomly yet it is very challenging to verifing it by hand? What are their features? Appreicate in advance
Edited for Typos @TedShifrin Are there aplenty or is there vast majority of cases that look simple just like the one took in Calculus courses randomly generate by computer yet it is very challenging to verifing it by hand? What are their features? Appreicate in advance
Let $X = \ell^2(\Bbb R)$ and let $K := \lbrace e^{(n)} : n\in \Bbb N\rbrace \subset X$ with $e_k^{(n)} = \delta_{kn}$. I'm trying to show that $K$ is closed and bounded but not compact (think this might be a standard example of Heine-Borel failing). Obviously $K$ is bounded because $\lVert e^{(n)}\rVert = 1$ in the $\ell^2$-norm. My lecturer then says "Since $\lVert e^{(i)} - e^{(n)}\rVert = \sqrt{2}$ for all $i \neq n$, all convergent sequences in $K$ become constant after finitely many terms"
I'm being a mong and not understanding the last statement lol
or maybe I'm not being stupid but it's just so obvious that I think it's not as obvious as it is
It's what you think it is; if a sequence in $K$ converges then it'd be Cauchy. But $\|e^{(i)} - e^{(j)}\| = \sqrt{2}$ for $i \neq j$ implies that's never possible!
Unless you are eventually constant that is.
If a sequence $\{e^{(i_k)}\}_k$ was Cauchy you'd of course be able to find $k_1, k_2 \in \Bbb N$ such that $\|e^{i_{k_1}} - e^{i_{k_2}}\| < \sqrt{2}$.
I tend to avoid inequalities etc. I'd just write it as, the open ball $B_{\sqrt{2}}(e^{(n)}) \cap K = \{e^{(n)}\}$ is a singleton, so every element of $K$ is isolated in subspace topology aka $K$ is a discrete space but compact discrete spaces are finite.
@Pig in fact all weβve done in the seminars so far is to prove Galois theory of infinite extensions and then define projective limits (which seemed like a weird order to do it in lol)
Also yeah that paper is a refinement of a book by Iwasawa and iwasawa theory is kinda big in Heidelberg number theory department
@MikeMiller Not much indeed, but they remark that they assume connectedness of the groups whose lifting they want to give, which means that their representation will be in U(P) instead of all of Aut(P). But then again, I don't know if not-connected groups are ever considered in this context.
Well there is about 11 people in this room for now.
Use the Weierstrass M-Test to prove that each of the following series is uniformly convergent on the domain given. $\sum x^n(1-x)/n$ on the domain $[0,1]$. $M_n=1/2n^2$ using differentiation and $A$.
The $A$ I mentioned states that: Let $a_n=(1+1/n)^n$. $a_n<a_{n+1}$. $2\le a_n$, for all n.
@StupidKid there are two ways. Using the derivative, find the maximum of $x^n(1-x)/n$ is at $x=\frac{n}{n+1}$. Then we are looking at $\left(\frac{n}{n+1}\right)^n\frac1{(n+1)n}\le\frac1{2n(n+1)}$. Apply the M-Test.
@StupidKid The other way is to notice that $\sum\limits_{k=0}^\infty x^k(1-x)=1$ for all $x$. Then $\sum\limits_{k=n}^\infty x^k(1-x)/k\le\frac1n$ for all $x$.
@Astyx The sum of all the terms without the $k$ in the denominator is $1$, so the sum of the terms with $k\ge n$ in the denominator has to be at most $\frac1n$
First, add the bookmark. Replace the URL with that large amount of text from the bottom of the web page, save the bookmark. From this room, use the bookmark as if you were going to another page.
@Alessandro usually you only use "an" when the next word begins with a vowel, but for whatever reason you also use "an" for words beginning with an "h"
@robjohn Ok lemme tell. I changed url with that code. Then there is no save button so I touch and it is gone. And bookmark was not saved. I made in another in folder. It was there but unusable.
And I again touch the star then it's url is changed.
Are there any god who can make video of instruction lmao.
I don't use Chrome, so I can't give an exact step by step explanation of installing the bookmark. Perhaps someone else who has installed it on Chrome could help.
@robjohn Nope! But I think he should be able to figure out himself after a while. There is also the difference between the version on a computer and the version on a phone. I don't use either regularly, so I don't know about either.
@MikeMiller Good point. I wanted to have the sphere bundle naturally sitting inside the vector bundle, but even then maybe only a fiberwise norm suffices.
I am lazy dog . I mistakenly evaluate derivative wrong and then forgot to make it in alternative form then evaluate it. I my brain was so laggy that I didn't even check if I wrote it wrong lol.
@Alessandro Doubting myself again: if $S \subset C^0(K)$ (continuous functions on a metric space $K$) then $S$ is bounded, because $S \subset C^0(K) \subset B(K)$ (bounded functions on $K$) and for $f \in S$ you have $\lVert f \rVert_{\sup} = \sup_{x \in K} \lvert f(x) \rvert < \infty$
@Balarka do you want to think about something ugly some beautiful metric topology? I don't understand if I'm trying to prove something false or if I don't know basic calculus lol
A coarse equivalence between two metric spaces $X$ and $Y$ is a function $f$ such that there exist two functions $g_1,g_2:[0,\infty)\to [0,\infty)$ going to infinity, with $g_1(d_X(x,y))\leq d_Y(f(x),f(y))\leq g_2(d_X(x,y))$ for all $x,y\in X$
(so for example quasi-isometry is the special case in which $g_1,g_2$ are linear)
Now I have two metrics $d$ and $d'$ on the same $X$ such that the identity is a coarse equivalence. Are $(K(X),d_H)$ and $(K(X),d'_H)$ also coarsely equivalent? ($K(X)$ are the compact subspaces of $X$ and $d_H$,$d'_H$ the Hausdorff distances associated to $d$ and $d'$)
What is all the assumption and condition required to making sure that T has continuous partial derivatives in Two variables when applying changing of variables formula?
Reference link: http://sites.millersville.edu/bikenaga/calculus/change-of-variables/change-of-variables.html
