@MikeMiller do you think it is interesting: If $X$ is connected and every point in $X$ is a cut point, and $Y\supseteq X$ is a compact space with $\overline X=Y$ and $Y\setminus X$ is connected, then $Y$ has no cut points.
If one responds to every message they get, it can take up a heap of time. If one responds to say they are busy, it may lead to pleading etc. Noone owes anyone a response, and I don't think it's rude.
@Miksu I really like Artin because it goes all the way from linear algebra to group theory, then covers a bunch of Lie theory and representation theory that I never learnt, and then finally to field and Galois theory (which includes a short detour on algebraic geometry and Riemann surfaces too)
I think you'll like it because it's not just an intro to algebra, but an intro to mathematics
Wow. I just checked university library and there's no copy of that book there... I live in quite small city and we have only one university where you can't study for math major.....
I should probably put it the same place I have put my various other notes, but it seems like I may not be able to update them there (I was not able to update the algebra exercises to correct the errors found at least)
I am trying to understand a proof for a cauchy sequence. It would be really great if someone could help me a bit. How does the author exactly come from the third to the fourth line here? (Yes, I see it is the geometric series)
Makes sense, yeah, the main connection I've seen between that and Sylow so far is a theorem I proved as part of the "applications" bit of my paper that a group of order $p^n$ is isomorphic to a subgroup of $UT(n,p)$
And even that just amounts to saying that $UT(n,p)$ is a Sylow p-subgroup of $GL(n,p)$ by counting, at which point you're just like okay, p-subgroups are contained in Sylow p-subgroups, and they're conjugate
I first learned what I know from Herstein and a few papers by Keith Conrad, which I've found to be quite fun but not comprehensive enough. Artin is supposed to be good for seeing algebra as part of a bigger picture but apparently introduces the topics strangely. DF drags much too long but is understandable and has a lot
I've heard some people praise Aluffi, which tries to integrate a categorical mindset. It's possible that without having seen algebraic topology (which gives a motivation for functors and the like because you actually need to juggle structures), you may find it a bit boring. Hungerford is supposed to be dry but more compact than DF. Maclane a bit longer but less dry
@Miksu that's about all I can say, hopefully that'll help with search? Or maybe I've made the choice harder, who knows?
Lang is actually a nice read once you've understood a lot of the concepts, because he does a great job of tying in algebra to other fields of math and giving a glimpse ahead. On the other hand, his second (and final) example of a monoid in section 1 of chapter 1 is the homeomorphism classes of compact connected surfaces. This is indeed interesting, but not great for people starting out.
@ForeverMozart I just sort of pop in and out. You would have a much better idea if that’s interesting than me - I don’t have much intuition for those sort of spaces like you
I was reading a paper and it introduced a Gelfand triple $H_0^1 \subset L^2 \subset H^{-1}$. I don't really get what's going on, or why we need to introduce $L^2$. I've seen this in a few different situations but I don't get what exactly is happening
Can we say the following about the following lemma? So here conformality means that the map preserve the "figure essentially" it doesn't distort it in anyway. So, we expect for the elementary domain to be transformed into elementary domain. As elementary domain we can think of it as simply connected domain, i.e things without holes. Let D be subset of C be an elementary domain and $\phi : D \rightarrow D^*$ be a globally conformal mapping of D onto the domain $D^*$. We suppose its derivative is analytic. Then $D^*$ is also elementary domain.
Let $R$ be a PID. An ideal $P$ in $R$ is said to be primary if $ab \in P$ and $a \notin P$ implies $b^n \in P$ for some $n \in \Bbb{N}$. Show that $P$ is primary if and only if $P = (p^n)$ for some $n \in \Bbb{N}$ and some prime element $p \in P$.
Here is my attempt:
Assume that $P = (p...
The assumption that phi is a conformal mapping means phi is analytic, infective, and has nonzero derivative; the assumption you make after that that its derivative is analytic is superfluous
@MikeMiller It is a domain (i.e connected open set) such that for any analytic $h : D \rightarrow C$ we have that it has a primitive i.e $F$ such that $d(F) = h$.
I do believe this is true. If $\gamma$ hits a leaf $L$ at $p$, comes back and hits again at $q$, you can chop everything off till the path from $p$ to $q$, do a waterfall construction to get it to hit $L$ only once
I think something like that should give you a transversal which globally intersects every leaf once
@AnubhavMukherjee It will hit all the leaf, and it will hit $L$ once. I guess I don't see the global thing, and Mike is skeptical which is not a good sign usually
A codimension one foliation $\cal F$ on a smooth manifold $M$ is taut if every leaf of $\cal F$ meets a closed transversal (i.e., a simple closed curve that is everywhere transversal to the leaves of the foliation). Is it true that a taut foliation admits a closed transversal $\gamma$ that meets ...
The reason it's true is because you're stating some holomorphic criterion and a conformal map is the same thing as a holomorphic map with holomorphic inverse
@Anubhav It's saying if every leaf admits a transversal to it, then the whole foliation admits a transversal. Now the thing would be to see if the construction preserves intersection number of the original transversal in contrast to the total transversal
Say $\mathcal{F}$ is the foliation on my compact fold $M$. To each leaf $L$ there is a transversal $\gamma_L$ through it. Because $M$ is compact, you can choose a finite collection of transversal $\gamma_k$'s s.t. every leaf is transverse to one of them.
So say $M_k$ is the saturated subset of leaves transverse to $\gamma_k$
These exhaust $M$, i.e., $M = \bigcup M_k$
If $L_{ij}$ is a common leaf between $M_i$ and $M_j$, and $\gamma_i$ and $\gamma_j$ hits it at $p$ and $q$, do the waterfall construction to graft them togather
Suppose $\gamma_1$ and $\gamma_2$ are two paths with $\gamma_1(0) = p \in L$ and $\gamma_2(1) = q \in L$, coming from two different sides of $L$ ("sides" make sense because $L$ disconnects it's tubular nbhd)
Consider a path $\gamma_3$ from $q$ to $p$ lying fully on $L$. So this gives me a path $\gamma_2 * \gamma_3 * \gamma_1$, which is tangential to $L$ but transverse everywhere else
So, I talked to my algebra prof, since we're set to finish the Sylow proof this Friday and thus we'll have 4 classes open to just do stuff with
He said he had in mind some subset of either talking about finite subgroups of $SO(3)$, mobius transformations, group presentations, or simple groups of small order, but that he'd be open to requests
Given that I've already done the proof that the smallest non-abelian simple group is $A_5$ (and it's the only one of order 60), I may lean away somewhat from the last one, but which of these topics strikes you as particularly interesting?
Or do you have other things you think would be worthwhile to request?
Let $A$ be a commutative ring where every element has a multiplicative norm $N$.
So if $a,b,c$ are in $A$, such that $ab = c$ Then $ N(a) N(b) = N(c)$ ( So $N$ is a multiplicative homomorphism )
Let $B$ be a commutative ring.
Usually we take $N$ to be integers.
But When $N$ is defined as ele...
> All the shapes on the left panel follow a rule. All the shapes on the righthand panel break that rule. Question: What is the rule?
While it's a very easy question to solve, current algorithms for computer vision completely fail at that task.
And yet it might be necessary to recognize certain objects.
The implication seems to be that, if we want to build better computer vision algorithms, we need an architecture that can solve the above sort of problem.
(When I wrote "current algorithms", I meant the state-of-the-art, such as deep neural nets)
Let $R$ be a PID. An ideal $P$ in $R$ is said to be primary if $ab \in P$ and $a \notin P$ implies $b^n \in P$ for some $n \in \Bbb{N}$. Show that $P$ is primary if and only if $P = (p^n)$ for some $n \in \Bbb{N}$ and some prime element $p \in P$.
Here is my attempt:
Assume that $P = (p...
Let $A$ be a commutative ring where every element has a multiplicative norm $N$.
So if $a,b,c$ are in $A$, such that $ab = c$ Then $ N(a) N(b) = N(c)$ ( So $N$ is a multiplicative homomorphism )
Let $B$ be a commutative ring.
Usually we take $N$ to be integers.
But When $N$ is defined as ele...
