I recall @leslietownes you answered my question about the definition of closed above coinciding with the topological definition of closed. I am wondering: the definition above of compact seems to imply that we are justified to deal with $G$ as being a subset of Euclidean space. This makes me think that the author means to deal with the metric space definition of closed as well. However, I am having trouble seeing why we can induce this additional metric space structure that does not come with,
and cannot even be induced in a natural way (it seems) on a general matrix Lie group
by "metric space definition"s I mean Rudin's definitions in his PoMA
silly it looks like there they are making those definitions for 'matrix lie groups' which do sit inside some set of matrices. there are a bunch of results about how some things often defined using metrics (such as compactness) depend only on the topology and not on the particular choice of metric.
this probably does get subtler for abstract lie groups but that book does not seem to be dealing with them there.
in particular, i would not assume that just because in some case an author uses characterizations like the heine borel theorem that the author somehow 'needs' the metric or that the result would somehow break down if you choose a different metric, or don't choose a metric.
rudin doesn't get into this too much, because he doesn't say what a topology is. he might have an exercise or two or something about circumstances under which different metrics define the same collection of open sets.
oh yeah i do not doubt that the listed lie groups are indeed compact. im just trying to write up a justification for being able to use a metric (because a priori it seems like an arbitrary decision which metric to use)
unless somebody is doing something very weird, the notion of 'open set' in GL(n,C) that they get from putting it inside R^{n^2} and using balls from the euclidean metric will coincide with some more 'intrinsic' definition.
i'm not sure i understand the setting. who are you trying to justify this for?
just for some notes. but the notes im writing up present the point-set topology definitions from Rudin before going into the matrix Lie theory from this book; but i am wondering if i should not present the definitions from RUdin and instead learn more general topologcy definitions
you can certainly topologize GL(n,C) in ways that don't give it the 'usual' topology coming from the metric. you might not get a lie group that way, or you might get some goofy lie group, i don't know.
but if you want to study lie groups, i don't know why you would do that, or have to justify why you weren't doing that.
i guess my initial impression is, the intro to a subject can often involve a handful of choices that seem arbitrary but maybe aren't, but the details of understanding why they aren't arbitrary often come later. after you understand concrete examples that arise when you make those arbitrary choices.
some people study matrix lie groups without even realizing that they are lie groups. they are classical objects and i don't think you have to justify why you might care about them, even if they are only a special case of something more general
someone asked a question last night about the envelope of a family of curves, with screenshots from a book that looked like it had been written in the 19th century. i presume self study, if only because i struggle to imagine the situation where someone would be dropped into that on purpose.
Oh, that was this morning. That was @Thomas Finley. I actually wrote an explanation about envelopes on his posted question and commented that the text looked like it was 19th century ... or written by physicists.
i do kinda dislike the trend of MSE questions where it's basically 'i encountered this completely at random on the internet, explain to me what they had in mind when they wrote it.' maybe with exceptions if the source is actually cited and is a book other people might use.
but yeah, i think we have a generation of people where "some random doc on the internet" is suddenly comparable to [insert major textbook here] in terms of what's reasonable to ask about it.
@leslietownes If I were a doctor, that would be amusing. However, your proposition that you are chatGPT is false because I am chatGPT. On the internet, many things are claimed and most should be investigated for veracity.
I am just starting to watch a playlist on time series analysis, and it's pretty clear from both notation and terminology (i.e., "characteristic equation") that they have some deep connection to recurrence relations. I feel like I have a pretty good handle on differential equations/recurrence relations. Can someone explain what the connection is to time series analysis at a high level, so that I can possibly transfer some of the intuition over?
i have personally found that notes are mostly good for deeper dives into something one already knows about, but never good as resources to learn a subject directly from scratch
but idk i also feel this way about lecture notes :P they always lack too much information compared to a textbook
unless of course the lecture notes essentially become a textbook
yeah, other people's notes can be really useful learning tools, but you can't use them as a textbook. in particular, it isn't helpful to interrogate them like "what did the author have in mind when they did __ instead of __"
as an example, the notes with that howler of a typo up above state, on page 1: "there should always be provisions for errors and typos while this material is being used"
you tend not to find disclaimers quite that drastic in published books
this is my own disclaimer :P "I promise not to give a precise exposition of the representation and Lie theoretic topics in these notes. Rather, I hope to introduce you to some of their physically essential concepts and show you how supreme the power of these theories are in structuring Quantum Mechanics."
Does this make sense: When simplifying the expression $\sin(\arctan(x))$, you aren't considering the values of $x$ directly but rather you're considering the values of the angle whose tangent is $x$, given by $\arctan(x)$. The function $\arctan(x)$ ranges from $-\pi/2$ to $\pi/2$, which are all in the first and fourth quadrants where $\sin$ is nonnegative. Thus, the negative values of $x$ are effectively considered, but they result in nonnegative $\sin$ values because of the range of $ \arctan(x)$.
@TedShifrin Thank you! I liked that explanation, soon, I will accept it! But , recently, I came accross a strange absurdity, and if it interests you, you might want to check it out. It was while, finding envelopes of a family of curves. But when we apply different methods, the answer comes out different, and the fun fact is, I find no explanation about that.
Find the singular solution of the differential equation $$4xp^2=(3x-1)^2,$$ where $p=\frac{dy}{dx}.$
As we know the singular solution, of a first order differential equation, is represented by the envelope of the family of curves, represented by the general solution of the differential equation. ...
@TedShifrin I see, the thing is, I have not found any book/materials that can be used to solve "singular solutions" of differential equations of 1st order. Can you suggest me ? I mean that would've been really helpful! Thanks!
@Koro It is not true that $X \times \Bbb R$ is homeomorphic to $Y \times \Bbb R$ (you can prove it; don't need anything fancy).
@onepotatotwopotato Elaborate? Trying to find a solution to a Nash embedding theorem is a PDE problem, you're trying to solve for $f$ in $f^* g_\mathrm{Euc} = g$, where $g$ is given.
But I wouldn't call that "using Nash embedding theorem to construct PDEs"
@BalarkaSen I thought geometrically: there should not be any problem in moulding Y cross R to X cross R. But it seems that there are some issues in that.
Take an embedding of an open disk in a torus, then remove its image.
i.e., take the complement
Usually you do this by consider $T^2$ as the usual subspace of $\Bbb R^3$, then picking a point $p$, an open ball $B$ around $p$ of sufficiently small radius in $\Bbb R^3$, and then consider $B \cap T^2$ which is an embedded open disk in $T^2$. The space is $T^2 \setminus (B \cap T^2)$.
Find the $p-discriminant $ of the differential equation $F(x,y,p)\equiv 4xp^2-(3x-1)^2=0.$(Note: $p=\frac{dy}{dx}$)
We proceed to calculate the p-discriminant of $F(x,y,p)=0,$ where $F$ is the function representating the differential equation in the question. So, $F(x,y,p)=4xp^2-(3x-1)^2=0\implie...
I used to use okular, which I thought was amazing, but its a complete pain in the *** to install on mac. And it uses an X11 window (or whatever its called) instead of the usual cocoa mac thing (not sure if these words are correct)
Find the $p-discriminant $ of the differential equation $F(x,y,p)\equiv 4xp^2-(3x-1)^2=0.$(Note: $p=\frac{dy}{dx}$)
We proceed to calculate the p-discriminant of $F(x,y,p)=0,$ where $F$ is the function representating the differential equation in the question. So, $F(x,y,p)=4xp^2-(3x-1)^2=0\implie...
@AlessandroCodenotti while working on this further math.stackexchange.com/questions/4715236/… I discovered that not all spaces $Y$ such that for any non-empty proper open $V\subseteq Y$ and cozero $U\subseteq X$ there is continuous $g:X\to Y$ with $g^{-1}[V] = U$ must be path-connected. The example, in fact, is a continuum. This is a property stronger than connectedness but weaker than path-connected
Perhaps this property can be studied further in continuum theory?
In universities or large institutions, where there are a lot of teachers, one teaches a subject on which he has some level of expertise. But let's say, in an institutions, there are only 4-5 teachers. They have to take subjects on which they have relatively less idea. e.g. A teacher with PhD in comm. algebra teaching PDE or complex analysis teacher teaching cryptography etc. I'm talking about extreme cases. But I wonder what do they do?
probably the best they can, not much else to do in that case
maybe ask for advice from connections
i think it would be fine if a teacher told me: let me get back at you, studied the question, and gave an answer the next class if they didn't know it on the spot
Do they study the course before the semester starts? Or do they study and teach parallelly? I guess you'd be having problems in the later situations. P.S. I am not a teacher and I don't have to teach someone in near future. It's just out of my curiosity.
I want to show that $\widetilde M$ defines a sheaf by checking the glueing conditions. Consider two basic opens $D_+f$ and $D_+g$, which corresponds to $\operatorname{Spec}B_{(f)}$ and $\operatorname{Spec}B_{(g)}$ respectively. If we consider the intersection $D_+f\cap D_+g$ under this correspondence as a subset of $\operatorname{Spec}B_{(f)}$ and $\operatorname{Spec}B_{(g)}$, then we are localising $B_{(g)}$ at $\frac{f^{\deg g}}{g^{\deg f}}$ and $B_{(f)}$ at $\frac{g^{\deg f}}{f^{\deg g}}$ respectively. To exhibit an isomorphism $\widetilde{M_{(f)}}\vert_{D_+(fg)}\to \widetilde{M_{(g)}}\v…
hm, I think I have an idea how to make a slightly cleaner arguement
by considering $\widetilde{M_{(fg)}}$ as well
and then show that $\widetilde{M_{(f)}}\vert_{D_+(fg)}\cong\widetilde {M_{(fg)}}$
In that case I need to exhibit $$ M_{(f)}\to M_{(fg)} $$ which is straightforward
and then show that multiplication by $\frac{f^{\deg g}}{g^{\deg f}}$ in $M_{(fg)}$ is an isomorphism
I'm a bit confused how to formally argue what the scalar in $B_{(fg)}$ would be, (I mean I'm guessing $\frac{f^{\deg g}f^{\deg f}}{g^{\deg f}f^{\deg f}}$, but I have to see how that follows from the identifications
A complete closed uni-leaf foliation on $\Bbb B^2= \lbrace (x,y)\in \Bbb R^2 | x^2+y^2<1 \rbrace $ can easily be constructed as follows. Begin with the standard foliation on $\mathcal H$ on $\Bbb B^2$ defined by $dy=0.$ Then obviously all leaves of $\mathcal H$ are diffeomorphic to the real line and closed in $\Bbb B^2.$ Let $h$ be a diffeomorphism of $\Bbb B^2$ defined by $h(r,\theta)=\bigg (r,\theta + \tan \frac{\pi r^2}{2}\bigg),$ where $(r,\theta)$ are the polar coordinates.
Then, $h$ sends any leaf $l$ of $\mathcal H$ to a complete curve $\mathcal H(l)$ in $\Bbb B^2,$ because each end of $h(l)$ spirals asymptotically on $\partial \Bbb B^2.$ Hence $h(\mathcal H)$ is a foliation we have desired. Note that, since $h$ is real analytic $(C^{\omega}),$ so is $h(\mathcal H).$
I understand the first sentence. The second sentence I understand to mean that this standard foliation, $\mathcal H$ is partitioning the ball by open line segments with zero slope. I understand that all leaves of $\mathcal H$ are diffeomorphic to the real line, but I'm not completely sure why they are closed in $\Bbb B^2.$ Lastly, in the fifth sentence, I'd like to get some intuition for this statement.
When they say "each end of $h(l)$ spirals asymtotically on $\partial \Bbb B^2$" can I think of this as the two ends of the open line segments being transformed so that they spiral around the ball getting closer and closer to the boundary? Why does this create a complete curve $\mathcal H(l)$ in $\Bbb B^2?$
I think there is none, yes. Because if there would one, then we would have $\phi(i^2)=\phi(-1)=-1$ but also $\phi(i^2)=\phi(i)\phi(i)$ which is impossible because no two integer gives -1 when multiplied.
I actually used $\phi(-r) = -\phi(r)$ above, I have to prove that.
OK, it's easy for at least -1: We have $\phi(-1) = \phi(-1+1-1) = \phi(-1)+\phi(1)+\phi(-1) = \phi(-1)+1+\phi(-1)$ which gives $\phi(-1) = -1$.
I don't know how to feel about the fact that there is no ring homomorphism from Z[i] to Z, though. When we are studying groups in the course, homomorphisms were generally abundant. Here I guess it's hard to find.
oneofvalts waits for the semester to end to start studying properly. :)
Let the side of the large square be $1$ and the side of the small square be $x$. Then $x^2+(1+x)^2=2$. The angle is $\frac\pi4-\tan^{-1}\left(\frac{x}{1+x}\right)$
The actual citizens of Israel have been rallying against Bibi for some time
But it is hard to imagine that the conflict will be easily resolved without international intervention. It's telling that the West sanctions Russia but not Israel for the same (or arguably worse) crimes, just because that the latter is an ally
Hopefully the dynamics in Middle East would change now that Iran is also a nuclear power.
@PlaceReporter99 I don't know if @robjohn has gotten back to you in the meantime (I am not caught up on chat yet), but I am currently not moderating. I would suggest that you use the "Contact Us" link at the bottom of any non-chat page to get in touch with SE directly.
@TedShifrin It isn't even so much that the Republicans support Israel without question---it is the small, but very vocal, evangelical community which "supports" Israel (as part of their end-of-the-world mythology), and the Republicans are, for some reason, beholden to that obnoxious group. I blame Reagan.
But I will say though that the Republican vs Democrat foreign policies regarding Israel have not changed so much over the years, regardless of the former being way more vocal about their support.
@BalarkaSen I disagree. Until about 20 years ago (I see 9/11 as kind of turning point), both Democrats and Republicans were pretty on-board to support Israel. I think that the left saw it as a kind of "reparation" for WWII, and would claim to actually support Jewish people.
However, in the last couple of decades, those on the fringes of the left have become increasingly suspicious of Israel, vis-a-vis the Israel / Palestine conflict.
And what was once an extremely fringe BDS movement has become somewhat more mainstream (though I still think that folk on the left, but closer to the center continue to support Israel).
@XanderHenderson But what are you disagreeing to? I'm talking about policies that has been enacted by the US government to discourage Israeli occupation of Palestine.
@BalarkaSen The Dems used to uncritically support Israel. In the last 20 years, that support has waned, and BDS have moved from fringe to (almost) mainstream.
@XanderHenderson I don't really see this in action; the US continues to heavily fund Israel military equipments that are used to massacre the people of Gaza and West Bank in response to what are essentially fireworks from Hamas. Also, is there really a unanimous "Democratic view" regarding Middle East? Sure, there are social democrats in the Democratic party who are vocal about it. But the Democratic party seems very disoriented and divided to me.
Most of the social democratic policies do not actually go through the senate.
@冥王Hades The left has always had an uncomfortable position with Islam. On the one hand, INCLUSIVITY! DIVERSITY! RAH RAH RAH! On the other hand, theocratic states gonna theocrat, and for example, women are often not treated well in those states.
@BalarkaSen I mean, if you are talking about political action, sure. Republicans and centrist Democrats are a super-majority in government.
@XanderHenderson Bingo. Don’t even get me started on how vile and evil islam is especially towards women and minors. I’ve lived a few years in Muslim countries, I know how rotten it is to its core
But the Democrats have been moving away from uncritical support of Israel over the last couple of decades, and I would not be surprised if, in the next 20 years, we see a different relation develop.
@冥王Hades One could use the same adjectives with respect to Christianity, or Hinduism, or any other ethno-religious group.
I don't think that Islam is the problem (any more than I think that the Mormon church is the problem in much of the American west). It is the political structures which are propped up by a state-imposed religion.
I think if US interference in Middle East slackens it will only be because it will no longer be profitable for US to maintain a military control over Middle East.
that's what i thought, ted, then i found this book called "Life the Lesliecoin Way," it's not like other religions at all, and there's this cool currency that goes with it
Disregarding ethics for a bit, an interesting thing I have learnt recently is that the reason Israel so ruthlessly massacres West Bank after every Hamas attack is because the missile interceptors for Iron Dome are quite expensive. And the Hamas knows this -- their missiles do not really do any damage but if they can crash the Israeli economy, that's a win for them.
@XanderHenderson One can use the same adjectives for almost every religion, which are really just obsolete ideas conceived by men of by-gone eras. Though I will contend that Islam is the worst offender out of them
@TedShifrin I think it’s evil because the theory itself is evil. It’s certainly true that religious people abuse it. But an ideology that calls for slaughtering nonbelievers is evil, pure evil
@SineoftheTime doesn’t make it any better. Also, “ According to the Islamic tradition, Aisha was six or seven years old when she was married off by her father to Muhammad.” en.wikipedia.org/wiki/Aisha
@SineoftheTime You’re welcome to disprove it and find a more reliable source. Not being being 9 instead of 6 makes a difference, it’s still utterly disgusting
That is why We ordained for the Children of Israel that whoever takes a life—unless as a punishment for murder or mischief in the land—it will be as if they killed all of humanity; and whoever saves a life, it will be as if they saved all of humanity.1 ˹Although˺ Our messengers already came to them with clear proofs, many of them still transgressed afterwards through the land. Quran chapter 5 verse 32
I think Balarka and I turned this into the religion room. Probably we shouldn't get too carried away, although it's an absolutely valid debate/discussion.
it's all up to interpretation, i don't see why you would think a fundamentalist interpretation of the quran is any better than a fundamentalist interpretation of the bible
