@BenSteffan $\mathcal{A}$ is a barycentric refinement of $\mathcal{B}$ if for each $x\in X$ there exists $B\in\mathcal{B}$ with $\text{St}(x, \mathcal{A}) = \bigcup\{A\in\mathcal{A}:x\in A\}\subseteq B$. If every open cover of $X$ has open barycentric refinement and $\mathcal{U} = \{U_x :x\in X\}$ is an open cover with $x\in U_x$, can we find barycentric refinement $\mathcal{V}$ of $\mathcal{U}$ with $\text{St}(x, \mathcal{V})\subseteq U_x$?
I think this should be true if one takes barycentric refinement of $\mathcal{U}$, and then again, and maybe few more times
I'm not sure how they got that $U^2\subseteq W$, looks wrong
We are given that there exist neighbourhoods of the diagonal $W_n$ such that $W_{n+1}\circ W_{n+1}\subseteq W_n$ and $(W_1(x))_{x\in X}$ refines $(W(x))_{x\in X}$. Why would that imply, say, $W_2\circ W_2\subseteq W$?
@BenSteffan I'm thinking of whether I should take Physik or Informatik as Nebenfach now.. Have you heard of anyone taking either; what would you say is more mathematical at Bonn?
@ILikeMathematics Physics will have you do more "mathematics," in the sense that much of it is inherently computational. But you will have to accept the fact that physicists don't really care all that much for mathematical rigor, and the way they approach things could often be characterized, from a mathematician's perspective, as haphazard.
Computer science will have less mathematics, and I believe that most mathematics you could learn there you could also learn "in-house" by taking courses at the institute for discrete mathematics (but this could also be a benefit: you get to learn about some of the things you would learn in a course there without having to actually take one, which is generally a grueling experience)
You should also be aware that physics is by far the Nebenfach with the highest workload
whether it's worth taking depends strongly on your personal interests. if you really like things like analysis and differential geometry, it will probably level-up your computational skills and your intuition. if you're going into, say, algebra I wouldn't bother.
@BenSteffan Ah, when looking over some of the CS exercise sheets, a lot of it looked mathematical (i.e. there was some sum or function and you need to find the optimal value for some parameter etc.). Theoretische Physik I and II are both 9LP, then there is still 6LP missing which would be covered by Physik I, but that's experimental and I assume not so mathematical.
I guess one could instead go for Theoretische Physik III which is 11LP and would be overkill
Maybe theoretische Physik even has Physik I as requirement so that wouldn't work out
the bars both of entry and of doing well grade-wise in it are extremely low
hell, the exams are entirely reproductive: here's a question about the material of the course, you have 90min to vomit everything from the course related to it onto the sheet of paper in front of your
if you can recall more than, say, 50% of the material even fairly roughly you will walk out with a 1.0
presumably because our department thinks learning about rules of inference that are presumed known in the first week of math lectures for half a year ought not qualify you for additional CP
but I wouldn't have taken it anyways
having to deal with people struggling with even a basic understanding Russell's account of the mathematical structure of natural language in a seminar was enough for me
@Thorgott Thanks for the help. Unfortunately, I haven't been able to locate a definition of what the ring of regular functions on an affine open set is. If $X\subseteq\mathbb A^n$ is an affine algebraic set, then I know that $k[X]$ may be defined as the quotient of $k[x_1,\dots,x_n]$ by $I(X)$, i.e. as the ring of polynomial functions on $X$.
If $U$ is open in $X$, then I believe that to say that $f:U\to k$ is regular on $U$ means that for each $p\in U$, there is an open set $V\subseteq U$ and $g,h\in k[X]$ such that $p\in V$, $h(p)\neq0$, and $f(x)=g(x)/h(x)$ for all $x\in V$. So, since in this case we have $X'$ open in $X$, does $k[X']$ just denote the ring of regular functions $f:X'\to k$?
@Thorgott: Okay. One thing I find confusing about the definition of $f:U\to k$ being regular is that it seems to depend on the choice of the set $X$. Is this actually not the case?
@BenSteffan I was thinking about where to post about this, and since this is from an article, and related to a "known" result, I've decided to ask it on mathoverflow
even though, I'm not quite sure if it fits there. But maybe it does
I know K. P. Hart included this result, that paracompact Hausdorff spaces are strongly collectionwise normal, in one of his articles, so certainly, it seems to me they either must know or didn't check that article in detail
it would be interesting if I were to reveal some kind of inconsistency
Who is the target audience for the book series Lecture Notes in Mathematics (from springer)? Is it suitable for undergraduates? Given its name, it seems like it might be good for self-learners. However, I've rarely seen anyone recommend its books for undergraduates-or anyone, for that matter-even though the series includes a vast number of books.
it's also maybe mistaken to interpret the applicability of a descriptor like "suitable for undergraduates" across an entire range of books, even if a publisher classifies them similarly
but what ben said is true, that series is generally for people who are already active in some research area and not undergraduates
it's also maybe mistaken to attach much meaning to the term "undergraduate." What is and isn't "suitable for undergraduates" depends critically on where you are, and Springer does not market to the US alone.
but generally i wouldn't give much significance to the prose label that a publisher uses to describe a book series. and people do occasionally base things like summer programs for new researchers around texts like that
yeah pie if there's a common feature of a number of your recent questions, it is seeking to find, like, uniformity and structure across something (whether masters programs or series of textbooks) that really doesn't exist
although at a high enough level there's enough similarity from place to place that i can assume "undergraduates" (whatever that means) probably aren't actively engaged in math research
most people i know in math never paid any attention to whether books where regarded by others (let alone a book's publisher!) as 'suitable' for them before checking them out. but they also often personally concluded that books weren't suitable for them (many self studiers cling to books, whether out of a sunk costs thing because they paid for it, or because they believe they'll get the best knowledge out of whatever someone else said was the best book despite any evidence to the contrary)
it's also possible to learn things out of a book that isn't suitable for you
