Ah okay, ye it feels more like I've completely cut it out of my life, without ever looking back it at from a rear-view mirror x'd (Hence my confusion for a second)
Oh, wanna hear something that surprised me as a TA ;x
So I'm a TA for a first-year prob thy course, and ever since we've been dealing with (continuous) random variables, I have a weekly fear that my students will ask me how the things we're working with are properly, so I spend a lot of time refreshing my measure-thy skills, and while I'm enjoying that, I have yet to receive such a question (or remark) from a student x'D
i never had to teach probability theory 'for real' although it was sometimes an adjunct to applied calculus courses. where the instinct was not to 'peek under the hood' but just grasp the language of it.
i should have guessed that.
that's funny except it gets less accurate as the cat grows older. maybe there's a string explanation for that.
i have no sense whatsoever of string theory. a friend of mine tried to explain it once, it didn't stick. he was patient and i think the explanation was probably very good. my brain just didn't light up in the right places.
@feynhat I was a math grad student at Princeton from 1981-1985 and Ed Witten was being talked about in the math department and I heard a lot about him from a physics grad student I knew. I don't know if he was there at the time, I think he went to Harvard after getting his PhD at Princeton.
Howdy @robjohn. You started at Princeton as I started my 34-year stint at UGA. Would you believe I actually have never been back to visit Princeton since my college days? No one in the department would have seen fit to invite me (and quite appropriately so!).
there's actually a restraining order against ted. he can't go back. i have several of these too. whatever, i've been banned from better universities than berkeley.
@Leslie Last time I tried to go into Evans when I was visiting Berkeley, it was a holiday and the building was locked. I sent a message on Facebook complaining to Ken Ribet (whom I've known for years).
Are you talking about Georgia, USA, or Georgia, Russia?
I was working some examples of interior points vs frontier points, before getting back to the question I was working on (which is driving me nuts), In particular I examined the set of integers $\mathbb{Z} \subset \mathbb{R}$. From applying the definitions, it seems like $int(\mathbb{Z}) = \emptyset$ and the frontier($\mathbb{Z}) = \emptyset$. Is this a correct assesment?
my dad was supposed to go on some kind of friendship mission to georgia, the former soviet republic, and then like two weeks before the trip a coup broke out. that was the end of that.
dc3rd that seems right to me, bearing in mind that all of this is relative to a choice of topology.
but for the frontier to be $\mathbb{Z}$ at least working with the definition and explanation @copper.hat had given me would mean that I can "approach" the frontier point from points in the set and from points outside the set
@Leslie I am no fan of southern cooking, despite having lived there for 34 years. A few things I do like, but overcooked vegetables and fried everything — nope.
@dc3rd for nice sets, consider the frontier to be the boundary as you might imagine it the 'real world'. given that, what do you think the boundary of the integers would be?
@copper.hat Well there is one of two ways I'm thinking about it, I'm thinking of the integers being a fixed distance apart, but any nbhd of an integer will also contain the reals.,.......so I'm going to say the frontier of the integers is the integers themself.
@dc3rd From Wikipedia: "The boundary of a set is empty if and only if the set is both closed and open", though no reference is given, I believe it to be true.
@robjohn: I ordinarily say boundary, too, but I explained to copper that I specifically chose frontier here because I have found that students get super-confused when we get to manifolds with boundary. Typically, the whole manifold (and boundary) consists of boundary points as it sits in $\Bbb R^n$.
okay so this is what is confusing me in this example: the definition of frontier point of a set $S$ is "every nbhd of $a$ contains both points in $S$ and points not in $S$. "
So in my mind I am picturing a number line, with the points as you would expect for the integers. I can take nbhds that don't contain any points of the integers except for the point $a$ itself. So are you saying the point $a$ itself is sufficient to satisfy the definition?
that's where my head was for a minute. if it wasn't confusing, then mathematicians couldn't earn the megabucks teaching this stuff. and oh, the royalties on the textbooks.
ted can pilot his solid gold yacht to long beach and we can all go to disneyland. i hear they have a version of "it's a small world" where it's just mathematicians from different countries and they teach math to you.
i went to disneyworld once. i became violently ill almost immediately and spent the day in their infirmary. the cots were sub par. it was frankly a disappointment. and mickey didn't visit, or anything.
You said that the integers are a frontier for itself. So let's take an integer, $a$ and draw a nhbd around it small enough so that the only integer in the nbhd is the centre of the nbhd. Does this then make it a frontier point?
i get why this is potentially confusing because often in limits you don't consider the point at which you are taking the limit, at all, and you insist on only using other points. but in evaluating the boundary, that's not what you do.
Actually, I did not define limit points in my text. That's going too far into topology. Of course, I defined closedness in terms of limits of sequences, so the notion is there.
wife to daughter: "do you want to go outside for a bit?" daughter to wife: "my mom said it's freezing out there." wife to daughter: "uh, i am your mom."
@dc3rd: Now that you're feeling better, did you understand why with $S=(a,b)$ the interior of $\bar S$ is just $S$? You kept claiming to prove otherwise. Do we have a proper example of an open set where a point of the interior of $\bar S$ is not in $S$?
@leslie Is your daughter already at the third-person stage for parents? How precocious.
we never did them at school, and whenever I get into a situation where I need to maximize a linear form subject to linear constraints, I feel like it should be the easiest thing ever, but I end up filling a page and getting nowhere
@dc3rd: I'll offer one hint. $\Bbb Z$ has a property that's quite different from the interval $(a,b)$. One is that it's "discrete" — each point has a neighborhood consisting just of itself. But what's another big difference?
Well when we talked yesterday, I went back and just tried to fiddle with some other sets. And then I said to myself "self" find the frontier and interior of $\mathbb{Z}$.
Ah, OK. It's perhaps the most obvious example for 8b, but what's another example? This is an important notion for later. Say in two dimensions. Not just a discrete set of points.
I was wondering when I would have to go into higher dimensions.....not just a discrete set of points....well there goes my idea of discrete coordinates...
The direction I was trying to lead you for 9a doesn't require any more dimensions. Working in $\Bbb R$ is fine, but for general purposes I wanted you to give me a (non-discrete) set in $\Bbb R^2$ every point of which is a frontier point.
diff eq is y''-2xy'+(mu)y=0 where mu is some fixed parameter >0. I got the recurrence relation a_(n+2)={ (a_0 (mu-2n)) / (n+1)(n+2) } and im not sure if its correct. I used the frobenius method
THis just came in my head, but if I restrict my set to being just the $x,y$ axes in $\mathbb{R}^{2}$ I could imagine that being a set where every point is a frontier point
Ride well, @copper. I was feeling very nostalgic for Berkeley today, as it was foggy as I went north to the Farmers Market. Very nostalgic.
Why this level of abstraction, @StudySmarterNotHarder? Why do we just not work with cardinality of $A$ and the cardinality of $A^n$ and say we have the appropriate $\Bbb Z^N$?
I'm not coming up with anything special about the other property Ted, the only things that came to mind were each point is an equal distance from each other. The other thing was the integers are not dense, but we're not in that realm of thinking and it is not the clearest idea for me yet.
then I don't understand the discussion of this paper arxiv.org/pdf/2011.11708.pdf in Appendix E, page 87-88. They have a function p(p0) = I \sqrt{(m-p0)(p0+m)} and they say that they can consider two different analytic continuations. here the complex variable is p0
I was thinking to apply the Schwartz reflection principle to the function $-I p(p0)$, since one of the hypothesis is that the function must be real on a subdomain of the real axis
the function $-I p(p0)$ is analytic in the whole upper-half complex plane, and it is real for $|p0|<m$. Therefore, I thought the analytic continuation is analytic in $\Bbb C$ minus the branch cuts
I'm not looking at the paper, and the notation is disgusting, so I won't. I'm saying that this is not about Schwarz reflection. It's about the point that you have make a branch cut in the plane to define $\sqrt z$ (e.g., you remove the negative real axis) and there are two different branches — i.e., two different holomorphic functions on the domain.
When you do something like $\sqrt{1-z^2}$ it's the same idea, but the branch cut either goes from $-1$ to $1$ or else has two rays from $\pm 1$ to $\infty$.
Regardless, there are two branches of the square root function.
That's what they're talking about with "two different analytic continuations."
There's a gap in the set, isn't there? It's not a matter of a bumpy ride. It's a matter of having to leap to get from one point to the next. You have to leave the set.
There are still two holomorphic functions. You have to choose one of two values at one particular point, and then the rest of the values are determined. I don't consider this an analytic continuation issue. It's a branch issue.
I kinda think I see what you're hinting at, but let me attempt to unravel it more here. You had also mentioned that the nbhd's of the points of the integers are the points themselves. I also know that the frontier points of the integers are the integers themselves.
So to bring everything in focus for me again. I am trying to obtain a counter example to the claim that all interior points of $\bar{S}$ are points of $S$....so using this suggestion of focusing on open intervals....
most recently we talked about the integers and its frontier. Also its differences from $\mathbb{R}$, which are its discrete and now also leaping in and out of the set, which make sit different from an interval
