Mathematics - 2024-11-26

Thorgott

00:03

@Frusciante I think your postulation also makes sense as long as we consider the factorization to be part of the datum, but I think this is intuition in terms of lifts moreso than in terms of being initial. That said, I believe the condition I gave is, in general, weaker than the full condition of being a relative colimit, hmm.

anyway, good luck tomorrow!

@X4J yes, good observation

ModularMindset

00:43

chat.stackexchange.com/transcript/message/66686577#66686577 this is not possible they would give ellipses.

Sine of the Time

01:03

Xander Henderson

01:31

@SineoftheTime Less was know, so it was easier to know most of what was known.

Jakobian

@Thorgott I read the proof of the sum of open ANR's is an ANR theorem. I could verify its true, but I don't have intuition for the proof. Which is my least favourite type of proof to read

It looks like a bunch of sets is being defined, in fact, I think there was around 10 to 15 different sets defined in that proof. Those have properties that eventually give us the desired result. Why we defined all of those? What was the intuition for it? I guess I won't know

Perhaps the person who came up with the proof, arrived at it by the brute force approach

Corollary: $\mathbb{R}^n$ and $S^n$ are ANR's for normal spaces

Proof: If $X$ is normal and $F\subseteq X$ is closed and $f:F\to \mathbb{R}^n$ then extend $f_i:F\to \mathbb{R}$ to $X$, then $f$ extends to $X$, so $\mathbb{R}^n$ is AE for normal spaces. Since $S^n$ is union of two open copies of $\mathbb{R}^n$, its an ANE for normal spaces.

sanya

01:56

Hi can someone guide me as to how to use baye's theorem here to solve this sum successfully

A bag contains 6 Balls of unknown colour. 3 Balls are drawn at random from the bag and are found to be all black. Find the probability that no black ball is left in the bag now. Initially assume all number of black Balls in a bag to be equally likely

There are 7 case either 0 black ball or 1 black or 2 black or 3 or 4 or 5 or 6 so probability of each is 1/7 ?

Jakobian

@sanya what is Baye's theorem

do you mean Bayes' theorem

also, what does it mean to "solve a sum"?

Sine of the Time

@Jakobian commutative property is not always valid :D

Jakobian

moreover, where is this "sum"?

sanya

@Jakobian yes

@Jakobian The problem that I wrote above?

Jakobian

@sanya the amount of black balls isn't random

so the probabilities in this case depend on a parameter, which is a number of black balls

sanya

02:06

Ok

IThinkHighlyOfEiligh

02:45

@SineoftheTime lol

ModularMindset

03:04

anyone here?

pie

I've been awake for over 30 hours, and strange ideas are starting to pop into my head. One of them is that Biden looks oddly similar to Gauss.

maybe I need to sleep IDK

copper.hat

03:43

old people's faces converge

4

IThinkHighlyOfEiligh

05:12

@pie that's bad for your organs to stay awake that long

Biden is a Gauss' great great great... great grandson

nickbros123

06:45

its topologin' time

psie

09:58

Exercise 3.26. If $\lambda$ and $\mu$ are positive, mutually singular Borel measures on $\mathbb R^n$ and $\lambda+\mu$ is regular, then so are $\lambda$ and $\mu$.

Definition A regular measure $\nu$ on $\mathbb R^n$ is i) finite on compact sets and ii) $\nu(E)=\inf\{\nu(U):U\text{ open},E\subset U\}$.

Attempt: If $\lambda +\mu$ is finite on compact sets, then it has to be the case that $\lambda$ and $\mu$ are finite on compact sets, so condition (i) holds. For condition (ii), since $\mu\perp\lambda$, suppose $N$ is $\mu$-null and $N^c$ is $\lambda$-null.

Since $\lambda+\mu$ is regular, for any $E\subset N$ that is Borel measurable, there exists an open set $U$ such that $$\lambda(U)<\lambda(U)+\mu(U)<\lambda(E)+\mu(E)+\epsilon<\lambda(E)+\epsilon$$ for any $\epsilon>0$, therefore $\lambda$ is regular (the first inequality above is simply due to monotinicity). A similar argument holds for $\mu$.

Question: Is this proof OK? I'm suspicious if it is assumed $E$ to be bounded above? I can't tell myself.

psie

10:15

Maybe @Jakobian has something useful to add here :)

Jakobian

11:22

@psie $E\subseteq N$?

psie

@Jakobian yes, we first want to show $\lambda$ is regular, but it is null on $N^c$, so we can just look at subsets of $N$, right?

Jakobian

I don't really like that jump. I'd apply the definition to $E\cap N$ instead

The inequalities can be just changed from strict ones to weak ones so its not a problem with infinity

After those changed I'd say it'd be okay

psie

Ok, thanks!

Ryder Rude

12:28

what r some interesting places in mathematics where the algebra $[A,B]=i$ shows up

it is a Lie bracket

hi

Soumik Mukherjee

14:19

What is the motivation behind the definition of coherent sheaves?

Xander Henderson

15:08

@SoumikMukherjee Well, you want to be able to distinguish them from incoherent sheaves.

Mats Granvik

15:51

If you really believe in the idea of tariffs, why not put tariffs between every country, every city, every neighbourhood, every street, every house and every human being.

Xander Henderson

@MatsGranvik In the US, that would violate the constitution.

But it certainly wouldn't be unprecedented in history.

Commerce Clause

The Commerce Clause describes an enumerated power listed in the United States Constitution (Article I, Section 8, Clause 3). The clause states that the United States Congress shall have power "to regulate Commerce with foreign Nations, and among the several States, and with the Indian Tribes". Courts and commentators have tended to discuss each of these three areas of commerce as a separate power granted to Congress. It is common to see the individual components of the Commerce Clause referred to under specific terms: the Foreign Commerce Clause, the Interstate Commerce Clause, and the Indi...

I suppose that congress could enact interstate tariffs, but... why? It would be super unpopular...

Thorgott

@Jakobian did you check out the relevant Figure in Sakai? the details are finicky, but I think the idea of the proof is not too strange

@Jakobian same line of thinking then shows any compact manifold is an ANR for normal spaces, interesting

@SoumikMukherjee they generalize finitely presented modules (in a very precise sense if you're working over well-behaved schemes)

Joe

16:24

In the definition of "local homeomorphism" $f:X\to Y$, we require that for all $x\in X$, there is an open neighbourhood $U$ of $x$ such that $f(U)$ is open and $f:U\to f(U)$ is a homeomorphism. Is there any particular motivation behind wanting $f(U)$ to be open in this definition?

Jakobian

$f(U)$ isn't open automatically, if it weren't then I wouldn't really say that $f$ really preserves local properties of $X$

the idea, to me, is that we want to preserve local properties of $X$ to local properties of $Y$

and demanding that $f(U)$ is open, is a condition so that we have something happening locally in $Y$

Thorgott

yeah, the condition basically ensures that $X$ and $Y$ look germinally the same

Ben Steffan

16:47

accidentally answered a homework question from the course I'm tutoring on the main site because I didn't look at the new exercise sheet 💀

Thorgott

LOL

Ben Steffan

the best thing is, another tutor apparently did the same thing, on a different homework question, a few weeks earlier

the asker is so sneaky too
"so I was reading these K-theory notes" no you weren't stop lying 💀

it's fine, nobody really cares. these things are not grade relevant and asking about them online is not a crime or anything

if anything they are sabotaging themselves by asking online a day after the new exercise sheet is published instead of spending more time trying to work the questions out themselves, etc.

Ryder Rude

17:04

@Jakobian hi. I messaged u on the physics chat

Thorgott

must be extra funny for them, too, given you're not posting pseudonymously

Ben Steffan

indeed lol

Thorgott

I remember one funny instance a couple years ago where somebody posted an exam question and Emily Riehl herself wrote an answer saying "meet me in my office"

4

Ben Steffan

LOL

Thorgott

17

A: Universal properties of $\mathbb Z$

I'd be happy to discuss this in person either during or after your oral exam. Come find me in my office.

Ben Steffan

17:10

that is absolutely hilarious

Xander Henderson

And completely off-topic. That answer has prevented the question from being deleted for years.

Jakobian

@RyderRude hi

Ben Steffan

sic transit...

Jakobian

I did saw you saying that I might try learning GR if I want something in physics that is just mathematics at its core

I don't know though, I think it was a weird whim for me to try and search for something like that, and I'll just continue on learning topology

Ryder Rude

@Jakobian oh. u can check it out whenever u r interested

Jakobian

17:22

@Thorgott yeah. I wonder how could one include manifolds that are finite unions of open copies of $\mathbb{R}^n$, to weaken the compactness hypothesis

Ryder Rude

but it is mostly metric and topology doesn't play a big role

Thorgott

well, finite unions are always included

Jakobian

yeah. I mean what condition could we impose on a manifold that is weaker than compactness so that this still holds

Ryder Rude

there is something called Topological Quantum Field theory tho, which I've read is mostly pure math

i think it is just physics inspired techniques to compute topological invariants

little to do with real world physics

Thorgott

@Jakobian a condition that ensures it's a finite union of open Rns, you mean?

Jakobian

17:25

yeah

Ryder Rude

Topological quantum field theory

In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. While TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for mathematical work related to topological field theory. In condensed matter physics, topological quantum...

Ben Steffan

ah, Jakobian would have to do a different kind of topology for that :)

Ryder Rude

@BenSteffan why is it different

i think it deals with manifolds while Jakobian doesn't work with manifolds mainly

Ben Steffan

Jakobian does point-set topology, but TQFTs belong much more closely to algebraic topology and homotopy theory

Ryder Rude

oh

Ben Steffan

17:36

maybe more geometric topology, in fact, but same conclusion

Ryder Rude

it looks far too removed from physics

Ben Steffan

really? you should ask Urs Schreiber how he feels about that :^)

(don't)

Ryder Rude

but it says it has an action, which is close to physics QFT

Jakobian

@BenSteffan the direction I am going seems awfully close to set-theoretic topology too

Ryder Rude

@BenSteffan i think i once came across Urs saying that this stuff would be useful for quantum gravity

Ben Steffan

17:40

Urs says a lot of stuff when the day is long

Jakobian

well... maybe not entirely, a lot of it is about zero sets and the like

Ryder Rude

@BenSteffan yes

Jakobian

but zero sets for nice spaces don't matter since they just become closed sets and so on if your space is nice enough so... the most interesting parts are in set theory examples

Ryder Rude

i kind of bought it back then. but now I've learned to be skeptical of Urs

and of the nlab crowd in general :P

Ben Steffan

yeah, approach with caution

Ryder Rude

17:42

lol

Jakobian

but yeah I would say that the experience I have is entirely different from the experience of someone who is an algebraic topologist

Ben Steffan

yeah, it's not really the same field at all

Jakobian

I believe that the things I am learning will eventually take me to something useful though

and by useful I mean, for something outside of general topology

Thorgott

@Jakobian actually, it's always true for connected manifolds

and disjoint unions of ANRs are ANRs

Jakobian

@Thorgott how do you show that?

Thorgott

17:49

3

A: A connected manifold can be covered by finite open subsets

Good question. The answer is positive. First of all, as in my answer here, there exist $n+1$ families ${\mathcal F}_1,..., {\mathcal F}_{n+1}$ of subsets in $M$ such that: Each ${\mathcal F}_i$ is a union of closed (and tame) pairwise disjoint $n$-balls $B_{ij}, j\in J_i$, where each $J_i$ is (a...

Jakobian

thanks

so you just need to demand finite amount of connected components then

nickbros123

For $S \subseteq Y \subseteq X$, this seems like a neat little result $\operatorname{int}_Y(S)=\operatorname{int}_X(S \cup Y^C) \cap Y$

Jakobian

looks like someone took complements in the expression $\text{cl}_Y S = Y\cap \text{cl}_X S $

Thorgott

hmm, I was gonna say that arbitrary disjoint unions of ANRs are ANRs, but I suppose I don't know if this is true in your setting

it's true if ANR means ANR for metric spaces

Jakobian

Are you sure its true for metric spaces?

Because the proof that I saw in Hanner's article seems to work for countable unions of open subspaces, and the one for separable metric spaces uses that the separable metric spaces are Lindelof

Thorgott

17:59

yeah, if you want to extend $f\colon A\rightarrow\coprod_{i\in I}X_i$ from a closed subset $A\subseteq Y$, you extend $f_i=f\vert_{f^{-1}(X_i)}$ to an open subset $U_i$ of $Y$ and then all you need is to find a collection of open $V_i\subseteq U_i$ s.t. the $V_i,\,i\in I$ are pairwise disjoint and still intersect $A$ in $f^{-1}(X_i)$. this is possible if $Y$ is metric, but I don't know if it is possible if $Y$ is normal

Jakobian

but we are talking about disjoint unions, I suppose

sounds like something that might be true for a collectionwise normal space

Thorgott

oh yeah, that would do the job

Soumik Mukherjee

18:19

@Thorgott I see, thanks

The Empty String Photographer

18:47

I made something in Desmos, it maps complex numbers to other complex numbers with lines, based on a function. It creates nice patterns for certain functions. Here is the graph: desmos.com/calculator/sgqkew1npa

FabrizzioMuzz

19:36

Anybody have any insights with regard to this question? math.stackexchange.com/questions/5002192/…

@Thorgott How do I format a link like this?

Sine of the Time

just send the link of the question without text

2

Q: Counting the number of unique color combinations on a grid?

I have an 8x8 grid with a pattern involving 5 colors that can look like this: I am not sure how to describe the pattern in mathematical language, but they are supposed to create continuous borders/boundaries such that each color must be connected to least one of the same color How many unique co...

combinatorics

FabrizzioMuzz

@SineoftheTime got it, thanks

copper.hat

@TheEmptyStringPhotographer looks like the top of my head

The Empty String Photographer

Huh

copper.hat

the desmos plot you mentioned

Sine of the Time

19:40

are you balding? :(

FabrizzioMuzz

so we can probably simulate this

using your plot

copper.hat

@SineoftheTime no, but i get my hair cut to a #2 (fairly short).

i used to shave/#0 but people treat you quite differently when you have little hair

the purpose being to minimise hair care time :-) i used to do a lot more outdoor stuff

Sine of the Time

yeah, hair care is annoying

I'm not going to a barber since 2021

Jakobian

19:56

@Thorgott That every point-finite open cover of a normal space has a shrinking is a theorem of Lefschetz, apparently

from his book about algebraic topology

at least that's what Nagami seems to imply

Thorgott

that's cute, I didn't know who it was due to

copper.hat

20:25

when i was younger i used to play with iron filings and magnets trying to get patterns like the above, but it never worked

ModularMindset

20:42

1

Q: Where should I begin to understand $\mathcal I_{\Gamma}$ in complex space?

Motivation: What is the largest surface of revolution with constant positive Gaussian curvature that can be embedded in $X^3=[0,1]^3$ with a pair of antipodal corners as cone points? It can be shown that there is a unique surface solution to this question. Therefore, replicating this surface thre...

differential-geometry reference-request riemannian-geometry complex-geometry vector-bundles

Any advice for me on how to pursue in complex space?

Also I had the thought that maybe I shouldn't deal with an embedding at all and do things intrinsically

ModularMindset

21:06

It's neat that in real space the Gamma set gives rise to an augmented multigraph structure comprised of the six quartics - orthogonal pairs lying on coordinate planes. And the restriction of the normal bundles to these quartics gives a nice line bundle which vanishes at six points corresponding to the vertices of the graph 🤔

Ben Steffan

21:23

I dislike how Grothendieck writes

why name anything when you can just refer to everything as a "canonical map" in some capacity

maybe his parents fed him a bowl of lambda calculus for breakfast every day when he was young

Thorgott

@BenSteffan I do that too tbh

Ben Steffan

boo :(

Thorgott

my average commutative diagram has at least half the arrows unlabeled cause they are canonical

Ben Steffan

Ever since attending a Schwede seminar I am scared of the word canonical

and of the word "technical" I guess

copper.hat

just blast the word canonical

Ben Steffan

21:36

on an unrelated note, Waldhausen has passed away

Jakobian

Suppose that $\delta$ is a real-valued, non-negative function such that $\delta(x, z)\leq A\max(\delta(x, y), \delta(y, z))$. Can we prove that $$\delta(x, y) \leq A\delta(x, x_1)+A\delta(x_n, y) + 2A\sum_{k=1}^{n-1} \delta(x_k, x_{k+1})?$$

Xander Henderson

21:52

@copper.hat You need a canonical cannon for that...

copper.hat

the canon being the need for a canonical cannon

Xander Henderson

Canonically.

copper.hat

shoot, you're right

@Jakobian i would imagine that you would need some powers of $A$ in there?

Xander Henderson

Look at me! I'm Grothendieck! Canon canon canon!

copper.hat

he was homotypical

Ben Steffan

21:57

Grothendieck? not so much, but he was always scheming

copper.hat

albeit i have no idea what homothetical means

Jakobian

@copper.hat no, and the inequality seems to hold

Sine of the Time

what symbol do you guys use for the Lebesgue measure? $\mu$ or $m_n$?

Jakobian

$\lambda$

Xander Henderson

@SineoftheTime $\lambda$.

copper.hat

22:00

$m, \lambda$

Xander Henderson

Or $\lambda^n$ for $n$-dimensional Lebesgue measure.

Sometimes $m$ if I am feeling metalic.

Sine of the Time

thanks

Jakobian

I was staring at this but it makes sense, $D(x, y)\leq 2D(x_1, y)$ and the final inequality makes sense because of induction

NewViewsMath

A number of posts/answer I've made, some from many months ago, some closed, just got an enormous amount of downvotes out of nowhere.

Does SE have the problem with people downvoting a ton of other things because they disapprove of one thing you've said, or is this just a strange coincidence?

copper.hat

they seem to do periodic sweeps for pattern voting.

Xander Henderson

22:14

@copper.hat No, not really.

Ben Steffan

@NewViewsMath I don't see that reflected in your profile

Xander Henderson

Generally, we need a flag.

Jakobian

@copper.hat doesn't seem to be in this case

Xander Henderson

@NewViewsMath I don't see anything actionable, but if it continues, please raise a flag.

"Enormous" also seems to be a bit of an exaggeration.

Jakobian

the posts that got downvoted seem to also have a lot of votes for closure

NewViewsMath

22:15

https://math.stackexchange.com/questions/4851750/determining-which-characters-split-or-fuse/4904474#4904474

This has gotten three downvotes in the last day, which is strange. It also has a lot of votes for closure, but is closed

If I'm not contributing well, I understand the downvoting, but this has been closed for months

Xander Henderson

@NewViewsMath Like I said, I don't see anything actionable. It looks, to me, like someone saw the post and voted to close it. When this happens, the post is bumped into the close votes review queue, which brings it extra attention, which can lead to downvotes and votes to close.

Jakobian

@NewViewsMath This post in particular contains no work of your own, it is a PSQ, a problem-statement question

Xander Henderson

@NewViewsMath I don't know what you mean by "this has been closed for months". It hasn't. That's why it is capable of getting close votes now.

NewViewsMath

I marked it as having an accepted answer

Xander Henderson

And you have called extra attention to the question by asking about it here. Which may attract more downvotes.

NewViewsMath

22:18

The one I provided in the hopes it would help future people struggling with the same issue

Xander Henderson

@NewViewsMath That is very different from being closed...

NewViewsMath

I agree many of these posts as PSQ, usually a "where does one learn this" Is there a better way/place to ask those sort of questions?

It's rather difficult to include my own work trying to determine what a definition means

Jakobian

Perhaps at university

NewViewsMath

lol, my advisor has told me multiple times "I don

Jakobian

Online, I don't know of any place which would do that, at which you can get sufficient expertise on the topic

NewViewsMath

22:22

't know, it's your job to figure it out"

Jakobian

you just need to make more effort towards your questions

here (as in, in this chatroom) you can ask questions, but you can't expect answers
our interests rarely intersect

NewViewsMath

oh certainly, that's why I've answered a number of my own questions, once I determine it, hopefully it helps somebody else. Other times, I've been able to get solutions. (One recently was very straightforward, and I hadn't realized)

I struggle to see a better way to say "these three papers say XX is a commonly known thing, but no text of the subject seems to describe XX, where do I go learn?" Should I just not direct such questions here?

Jakobian

You can certainly post them here if you put the work into your questions to include such things as background etc.

for example this is my own question, we can analyze what I did here

I've defined something and mentioned this is common in the literature, then I proceeded to define something that I actually wanted

then I defined another thing common in literature

I gave some examples

and lastly I asked question and partial solutions to it

so I was clear about the concepts, I've provided how it works, and when it works, that I know of

this is bullet-proof question to me

no one would close that, I think

NewViewsMath

alright

could I send one and we see what I might do to improve similar things in the future?

Jakobian

You just have to dig into what you can add, some of it is creativity, some of it is a learned skill

but you need to indicate, that this is not just another low-effort question somehow

you need to work on the contents of it

I don't know how else to help you, good luck

NewViewsMath

22:38

ok

thank you

leslie townes

23:14

some of the clarifying comments that you added to your own question in response to people asking you about it in february, are the kind of thing that it would normally make sense to edit into the question and not leave in comments. the comments from january from people are offering you examples of things that were missing from the OP and are still missing because you didn't edit them in

it also helps to broaden the audience for the question as much as possible, e.g. if this isn't specifically a question about ATLAS but is actually just a question about what you can and cannot deduce from a character table, it would be helpful to phrase the question like that in the first instance, as doing otherwise (1) makes it seem like people have to do their own research to figure out what the question is, and (2) makes it seem like the question is only about some specific reference

so if possible in this example i would edit the question to explain what "case G.2" is (what in the world would make anybody think that this was standard terminology? why would you want to narrow your audience to only those people who are going to click through and solve the mystery about that it is) and offer citations to ATLAS only as an example. quotations of ATLAS would also be preferable to citations for this purpose

it isn't obvious at first glance that "ATLAS" isn't some software package, which might also have pissed people off. it isn't, but how would a prospective voter know that. questions about software packages are very routinely downvoted

Claudio

23:29

if I need to study uniform conv of a sequence of functions on $I = [0,c]$ and the sequence is defined separately on $I_1 = [0,a), I_2 = [a,b), I_3 = [b,c], \bigcup_{i=1}^{3}I_i = I$

I can then study uniform convergence separately on each $I_i$

Sine of the Time

@Claudio is it a question or a statement?

Claudio

23:46

a question hahah

can I then study...?

Ben Steffan

you can study anything you want :)

Claudio

anyways I got the right answer so I guess the answer is yes

Sine of the Time

You can study the UC over I by studying the UC over I_j

Claudio

@BenSteffan hahah, thanks I needed to hear that

@SineoftheTime perfect

thanks

Sine of the Time

:)

Xander Henderson

23:49

@SineoftheTime What is it that you are suggesting I do at UC Irvine?

Claudio

@SineoftheTime we started the brief measure theory part in the last lecture on friday

I understood maybe a good 10%

Sine of the Time

I've never studied measure theory

@XanderHenderson study UC?

Xander Henderson

Seems tautological...

Claudio

@SineoftheTime Oh, I thought you did my bad :p anyways I might to have to lower to 8% now that I'm rereading my notes

Sine of the Time

I've only "studied" Lebesgue measure

Ben Steffan

23:52

@Claudio at least the subset of things you understood is not a null set

Jakobian

I've finally understood why fully normal is equivalent to paracompact + normal, by showing that a normal open cover (i.e. it admits a sequence of star-refinements) is numerable (i.e. it admits a subordinated partition of unity). My proof is that a normal open cover induces a certain continuous pseudometric, and since metric spaces are paracompact Hausdorff, this then induces a subordinated partition by taking Kolmogorov quotient.

Claudio

@SineoftheTime yeah that's what we're gonna focus on as well

Ben Steffan

did you know that lebesgue is french for "the besgue"

IThinkHighlyOfEiligh

the best guy

:)

Claudio

@BenSteffan lol

I might have to use that next time

Sine of the Time

23:56

Bro invented a whole theory on integration in his thesis

IThinkHighlyOfEiligh

Without lebesgue measures we'd all be running around like chickens

Claudio

@SineoftheTime Nah, I'mma do my own thing, Spiderman

IThinkHighlyOfEiligh

Turing did some stuff too, man, like crack the Nazi's encryption

So did Emmy Noether, the list goes on...

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$Mathematics$ Mathematics