I think that is it.....I think the issue was me trying to wrap my head around why it would be $n-1$ well now typing this out I see why that is the case.......manifolds....
Yeah, $S^k$ is the $k$-dimensional sphere or $k$-sphere.
The guy teaching the multivariable math course at UGA this year just emailed me to complain about my sloppiness. They need to know that the composition of continuous functions $[a,b]\to\Omega\to\Bbb R$ is continuous, but I've only proved it explicitly in the book with open domains. Bad Ted.
@LeakyNun So you first find $E/\Bbb Q$ whose galois group is $S_5$ then finding index $24$ subgroup and the corresponding subfield has degree $24$ extension over $\Bbb Q$ and primitive element theorem? Does it show $\Bbb Q[x]$ has degree $24$ polynomial whose galois group is $S_5$?
I wonder how many little things like that I've screwed up — probably dozens. I told him it would be a good exercise for them to prove the general result using sequences (and I gave the sequential characterization in the text).
Yeah, I think there are lots more interesting things to worry about, and there are plenty of challenging exercises as it is, but he wants to be complete.
He was also upset that I defined a connected subset of $\Bbb R^n$ to be path connected (rather than giving the general definition). This doesn't upset me at all for a freshman/sophomore course. Shrug.
the general definition of connectedness is confusing I don't remember it completely at the moment, but the fact it is defined by intersections and lack there of makes it more abstract. at least path connected is tangible.
Yeah, authors have to make controversial pedagogical choices. I've made them.
@dc3rd The general definition isn't that bad, but useless for calculus. You can't write the space as the union of two (non-empty) disjoint open subsets. But then we get back to subspace topology issues. Duh.
i love how rudin introduces all of this metric space topology in his chapter 2 and never uses it. he also calls the chapter something like 'basic topology' and never defines the word 'topology.'
I was more so saying what topology is. i.e the "study of spaces", but nobody ever begins by telling you that a metric is a layer to the whole concept of spaces
"stretching and bending BUT NO TEARING" what is somebody supposed to do with that? also a lot of topological intuition is blended with geometric intuition, which mathematically is 'extra' structure but intuitively maybe is not.
as a side note it's not obvious that dimension is well defined. or even that an open ball in R^n cannot be homeomorphic to an open ball of some other R^m.
i guess it's well defined if you say it's the minimum over all potential n's that might work as coordinate charts.
@TedShifrin I have no idea. If it's a usual DDoS attack, it is probably from a large number of virus infected machines, so it is hard to track down where it is actually sourced.
for more on that issue, see the lawsuit townes v. bevmo, inc. it is pending in the california supreme court. should set precedent about what "dry" means and does not mean
on my site, lesliecoin.biz.ru. there are some formalities such as needing your social security number and several credit cards. it's just so we have them on file for regulatory reasons.
i last read "the time machine did it" by john swartzwelder, which is a parody of hard boiled detective fiction by a former writer for the simpsons, "last call" by elon green, a non exploitative true crime book about a serial killer who operated in gay bars in early 1990s NYC, and "empire of pain" by patrick radden keefe about the family behind the opoid epidemic.
i recommend all of them but they are very different books
math.GM is where attempts at famous problems tend to go. there are also sometimes interesting survey papers in that category. i don't know that many people read them.
it's not my area. i see a few unsupported assertions in there, reference to numerical values (to multiple decimal points) without explanation of where they came from or the error estimates involved. the style is idiosyncratic. on the last page there's a statement "= 0 > 0." they try to explain it, but that's not a good way to get people to read your paper.
this paper reminded me of a style point that even good authors ignore. the word "summation" can almost always be replaced with "sum." in my aesthetic view it's just better that way.
when i was in grad school i worked a bit on understanding louis de branges's attempts to prove the riemann hypothesis because he was using tools i was familiar with. a lot of his lemmas were true, but expressed in very idiosyncratic language, it was very hard to follow. he'd call things fields or ideals when they weren't (he didn't need them to be, but it was distracting). i gave up eventually.
if a mathematician proves a famous result but nobody can understand it, does the falling tree make a sound?
Okay. I have a problem for which I have a solution I'd like to outline, because I want to make sure the logic is right.
First, recall that $D_n = \langle x,y | x^2 = y^n = 1, xy = y^{n-1}x \rangle$ and from this presentation one can argue that $|D_{n}| = 2n$. I am asked to argue that $D_n$ is isomorphic to $G := \langle s_1 , s_2 \mid s_1^2 = s_2^2 = 1, (s_1 s_2)^n=1\rangle$.
So, first I defined a surjection from $G$ to $D_n$ by first arguing that $s_1$ and $s_1s_2$ generate $G$ and that they satisfy the same two relations as $x$ and $y$, meaning that $s_1 \mapsto x$ and $s_1s_2 \mapsto y$ defines a well-defined surjection. Then, because $G$ is generated by $s_1$ and $s_1 s_2$ subject to the two relations as $x$ and $y$ (and possibly more), we know that $|G| \le 2n$.
But, since $f$ is a surjection from $G$ to $D_n$ and $|D_n| =2n$, $f$ is in fact a bijection and hence an isomorphism.
Did anyone know that the Witten genus of a 4k dimensional compact oriented smooth spin manifold with vanishing first Pontryagin class is a modular form of weight 2k, with integral Fourier coefficients?
@user193319 "Then, because $G$ is generated by $s_1$ and $s_1 s_2$ subject to the two relations as $x$ and $y$ (and possibly more), we know that $|G| \le 2n$". Shouldn't it be $|G|\ge 2n$?
@Koro I don't think so, because there could be more relations on $s_1$ and $s_1 s_2$ which means that, possibly, some words in them could just reduce to the identity.
Like, $D_{\infty} = \langle x,y \mid x^2 = y^2 \rangle$ is the infinite dihedral group. If you throw in the extra relation $(xy)^2 = 1$, then you get the Klein four group which is finite.
So, adding in relations turns a lot of elements into the identity element.
Presentations are defined in terms of quotients of free groups by normal subgroups. If you add in more relations, you are "quotienting" out by a normal subgroup which is larger than the one you started with (or possibly equal to it), so the quotient group should be less than or equal to the quotient group you started with...if that makes sense.
How do I count number of elements in center of G={$e,x,x^2,x^3,y,xy,x^2y,x^3y$} with o(x)=4, o(y)=2 and $xy=yx^3$? Solution: G is a group of order $2^3$ so it has a non-trivial center. It means that $Z(G)$ can possibly have order 2, 4 or 8. Since $x$ and $y$ don't commute, G is not abelian so $Z(G)$ can't have order 8. If $Z(G)$ has order 4 then $G/Z(G)$ is cyclic hence G should be abelian which is not possible. It follows that Z(G) must be of order 2.
The idea is roughly the same. As you do more math you will see this process repeatedly happening: 1. we define a mathematical structure (e.g. vector spaces or groups), 2. we define a notion of "structure-preserving" function between two such mathematical structures of the same type (e.g. between two vector spaces, or between two groups), 3. we study the mathematical structure via these structure preserving functions.
@Prithubiswas so here, a group homomorphism is a function which preserves (up to some loss of information, contained in the kernel!) the group structure.
Group homomorphisms with zero loss of information (i.e. trivial kernel) are called "isomorphisms". This is just like how linear maps with trivial kernel are called isomorphisms.
And isomorphisms denote "equivalence" of the structures.
e.g. "up to re-naming" you have the same group, vector space, etc.
I used to be "mad". now i just ignore most of things and am on medication like a good boy so i dont get mad anymore :=)
I am learning about exponentials of matrices. I understand intuitively, why this $e^{BAB^{-1}} = Be^AB^{-1}$ needs to be true. But i am having issues proving this in mathematical language. writing out the definition $e^{BAB^{-1}} = lim_{k\rightarrow \infty} [Id+\frac{BAB^{-1}}{1!}+...+\frac{(BAB^{-1})^k}{k!} ]$
How do i continue?
(Obviously for $A,B \in END(\mathbb{R^n}), B$ is invertable)
My intution is, that we are prfeforming a linear transformation, and we want to consider it from a different coordinate system " hence B, B^-1$ so it doesnt matter if we change into that system and preform the transformation, or preform the transformation and change into that system.
(BAB^(-1))^k = B A^k B^{-1} so you can pull out a B on the left and a B^{-1} on the right of the whole sum. some basic facts about convergence i guess are used to justify the manipulation.
@robjohn In an old answer I integrated by parts to put an integral of the form $\int_{\mathbb{R}^{2}} f(x) g(t) \cos(xt) \, \mathrm dt \, \mathrm dx$ into a form that I could apply Fubini's theorem. I didn't know at the time that Plancherel's theorem could be used.
In a recent answer to a different question, I used Plancherel's theorem instead, although it could be made nicer if we indeed only need $f(x)$ and $g(t) \sin(xt)$ to be in $L^{2}(\mathbb{R})$ for Plancherel's theorem for the sine Fourier transform to be true.
@RandomVariable to work with this, $\sin(xt)$ is small for small $t$, but as $x$ gets larger, $g(t)\sin(xt)$ gets larger also, but then, $f(x)$ is small and the oscillation of $\sin(xt)$ is more. This looks like a splitting of the domain to analyze the integral might be useful.
@geocalc33 Not sure what you are talking about, but it might be you are talking about looking at the Fourier Transforms in "Fourier Space" and the function itself in "normal space".
Earlier on, algebra is easier to teach effectively because it's a lot more methodical ("just apply the definitions!"), which leads one to an impression that algebra is a lot more organized/structured than analysis which early on requires comparatively a lot more problem solving.
On the other hand, later, algebra becomes a very complex web of results and definitions that can be very hard to keep track of, while analysis maintains a steady balance of problem solving skills and intuitive definitions.
asymptotically, they are the same difficulty, though
What do you mean by "that diverges"? I am guessing you are asking "does every integral transform satisfying <something to do with diverging> admit an inverse kernel?"
oh, i dunno. they made my wife the chair of her department and i hear halves of phone calls and teleconferences that usually center on professors needing to do something administrative and refusing to do it, or doing it only when cajoled and bothered about it repeatedly.
like many schools they rely on adjuncts for a lot of labor, and that is its own kettle of fish. tenured profs are worse.
i wasn't, but it is probably similar. if that deals with questions of funding in things like lab sciences, it might be a slightly different situation. faculty in those areas basically have to function like middle managers or CEOs sometimes. faculty in the social sciences don't, and it shows.
one phrase i like a lot is a "goat rope," meaning a confusing situation often complicated by large numbers of people working at cross purposes.
presumably roping a goat is a difficult and counterproductive task. i used it once and was asked to explain it and i realized i didn't know where it came from.
koro: how can they do that? does indian law forbid importation of books printed elsewhere and bought at lower prices?
american law doesn't thanks to a supreme court case, kirtsaeng v. wiley, involving a math graduate student who had his friends import 'international editions' resold for cheap.
i wonder if stuff like this is a reaction to kirtsaeng, actually. i was worried when the decision came out that it might price people outside the US out of the market. price discrimination based on geography makes a good deal of sense, as annoying as it is for people in the US.
@Koro Did you figure it out? There are 4 conjugate subgroups, hence the normalizer has order 24/4 = 6. You should see what that subgroup is in a very obvious (even geometric!) way.
questions @TedShifrin, recall the question in your text where we have to consider the curve: $f(x,y) = 4y^3-3y-x = 0$? I had a question on the second part of it and the third part. I'll get to the question on the second part after since it is just algebra. But the third part went like this ( I included my rough solution, but I'm not sure about the end...I know something is missing.
Checking that the value $\phi'(1)$ agrees with the implicit function theorem lemma (implicit differentiation formula) where $\phi(x)$ was defined as
$$ \phi(x) = \begin{cases} \phi_2(x), & x \in (-1,1) \\ 1, & x = 1 \\ \phi_1(x), & x \in (1,\infty)\\ \end{cases} $$
where $f(x,y) = 4y^3-3y-x = 0$
$\phi_1(x) =$ (all that stuff)
$\phi_2(x) = \cos(\frac{1}{3}\arccos(x)),\ x \in (-1,1)$
I don't even know what $\partial \phi/\partial x (1)$ means. First, you mean $\phi'(x)$. So you have to prove that the function $\phi$ is differentiable at $1$. The implicit differentiation only works on the open intervals where you have $\phi_1$ and $\phi_2$. So what should $\phi'(1)$ equal?
What is the meaning of the phrase herding the cats? I've found one description on Wikipedia but it is not clear enough.
