Mathematics - 2022-11-16

copper.hat

00:34

@DarwinZou I think the answer is probably generational. I have tried for decades (starting in the'90s) to go 'paperless' but whenever I do mathematics I fall back to pencil & paper.

leslie townes

ted transitioned from vellum to paper and said, one technological change is enough.

incidentally i'm running out of old materials for ted to use in these jokes. we've done clay tablets, papyrus, and "scrolls" of unknown composition.

and now vellum.

Ted Shifrin

Don’t forget coal on tablet.

leslie townes

so there will be one coming up where equations are smudged onto cave walls next to illustrations of the day's hunt. at that point maybe i'll have to find a new joke.

D.C. the III

I mean the stable LeslieCoin is probably the talk of the town with the crypto-world finally beginning to go up in flames.....

leslie townes

oh yes, as other markets crash, lesliecoin reigns supreme.

PM 2Ring

00:50

I prefer BuffyCoin, which was founded on the principle that you can take the cryptocurrency out of the crypt, but you can't take the crypt out of the cryptocurrency. Of course, it's based on proof of stake.

D.C. the III

@PM2Ring Violation of principles if it is called BuffyCoin and you are referring to the same Crips I think you are referring to....

copper.hat

02:32

none compare to bogcoin.

one potato two potato

03:27

0

Q: Computing $H_*(T^2,T\vee T)$

I'm currently trying to compute the homology of $T*T$ where $T = S^1\times S^1$ and $*$ is a join operation. One known is that $X*Y\simeq S(X\wedge Y) = S(X\times Y/X\vee Y)$ where $SX$ is a (unreduced) suspension of $X$. To do this first note that $$\tilde{H}_n(X*Y)\simeq\tilde{H}_n(S(X\wedge Y)...

Fresh AT question

copper.hat

03:49

best to leave questions age a little.

2

Koro

04:31

@Thorgott I had discussion on this in another room and I have understood this now.

Koro

05:08

How to show that $\mathbb Q$ is not locally compact?

Suppose on the contrary, that $\mathbb Q$ is locally compact. Take any $x\in \mathbb Q$, there exists a compact space $C\subset \mathbb Q\implies C$ is compact in $\mathbb R$ and that $C$ contains an open set $U\subset C$ of $\mathbb Q$ that contains $x$.

I don't see how this gives a contradiction.

$U=V\cap \mathbb Q$ for some open set V in $\mathbb R$.

copper.hat

maybe show that every neighbourhood has a cauchy sequence that does not converge?

Koro

x is in U and so is in V. There is a d>0 such that $(x-d,x+d)\subset V\implies (x-d,x+d)\cap \mathbb Q\subset U$. Now take an irrational number $i$ in the interval, then by density of the rationals there is a sequence of rationals in the interval converging to i. The sequence must have a limit point in C. But clearly that doesn't happen.

here I have used the fact that $\mathbb R$ (and therefore $\mathbb Q$) is a metric space (this is to ensure that a sequence in C converges in C).

one potato two potato

Btw the question I asked in the above post, if $\iota_*$ is injective, is true. Because the generators are 2-cells in CW complex

So I'm gonna delete the post after an hour.

copper.hat

05:24

my knowledge of manifolds is not manifold.

one potato two potato

Computing a product of two tori using cellular homology was helpful instead of using Kunneth formula.

copper.hat

i cannot disagree.

Cotton Headed Ninnymuggins

06:31

How do I earn reputation the fastest?

Not Tfue

Stop playing video games and using social media...

leslie townes

find the biggest, toughest looking dude in your neighborhood, walk up to him, say "hey man, do you---" then sucker punch him.

6

Cotton Headed Ninnymuggins

Lol, street rep

I'm just saying reputation don't come unless I make petty edits to people's posts. My answers are almost never accepted, etc.

Not Tfue

You can make bot account then upvote your own post.

Cotton Headed Ninnymuggins

I actually tried that years ago, those accounts get removed real quick

leslie townes

06:41

it's a difficult question to answer, because reputation and site policies are essentially structured to make reputation difficult to scale up. if there were a strategy for pressing the gas on reputation, bad actors would exploit it until the rules changed or the accounts got kicked off.

Not Tfue

Lol for real?

Cotton Headed Ninnymuggins

and then the reputation literally gets taken back

yes lol

Not Tfue

Didn't even thought about it until you mentioned it.

leslie townes

i do think it's hard to earn rep by posting interesting answers. a lot of the huge reps you see are from people who got a lot of upvotes long ago when it was easier.

posting an interesting question is another way of getting upvotes, and you don't even have to know the answer! :D but you do have to post an interesting question, which is harder than it sounds.

Not Tfue

I wish I could remove all of my dumb math question :(

Cotton Headed Ninnymuggins

06:44

And then I had a question recently about a sequence converging and someone assumed I meant series convergence and decided to single-handedly completely change my question which was ridiculous

I don't know what the intention was with that. They must've assumed I'm a complete idiot not knowing what question I even want answered.

leslie townes

sorry to hear that, that kind of edit is discouraged i think. it is common enough for questions to raise the reaction of "wait, do they mean __?" but that can be done in a comment asking the OP if they meant __ or to otherwise clarify something.

its very common for questions to be significantly improved by clarifying edits, but it should generally be the OP that does that.

Not Tfue

star for good joke

Cotton Headed Ninnymuggins

Someone asked if I meant series even though I had 0 indication that there was a sum happening and my rationale and definitions had nothing to do with a series. Within a few hours someone else changed it, very strange.

leslie townes

07:06

yes, that's an example of an edit that should have been rolled back. i see that you did roll it back.

one potato two potato

07:23

@copper.hat But I usually use Kunneth formula first and next compute cellular homology to be sure.

Koro

Suppose that $X_\alpha$ is locally compact for every $\alpha$ and compact for all but finitely many $\alpha$, then the product space $\Pi_\alpha X_\alpha$ is locally compact.

How to prove this statement?

Proof of the statement requires Tychonoff theorem but I don't see how that applies here as we are given X_a compact for all but finitely many.

Infact, I don't even understand why Tychonoff is required here at all.

one potato two potato

07:53

Qualifying exam problems are really a treasure. So many interesting problems I've never seen before.

one potato two potato

08:12

I can't see how $N$ slides over to $f(N)$. So the sliding is taken along the surface I guess. The disk is stretched so that it contains $N$ and $f(N)$ and the sliding is taken inside of that stretched disk? It would be helpful if you draw some pictures in your mind. — one potato two potato 22 hours ago

Can't understand how sliding of $N$ to $f(N)$ occurs.

one potato two potato

12:41

0

Q: Chainging the hypergeometric series $F(\alpha,\beta,\gamma;z)$ to an integral form

This is problem 6.9 in Stein complex analysis. Let $$F(\alpha,\beta,\gamma;z) = 1+\sum_{n=1}^\infty{\alpha(\alpha+1)\cdots(\alpha+n-1)\beta(\beta+1)\cdots(\beta+n-1)\over n!\gamma(\gamma+1)\cdots(\gamma+n-1)}z^n$$ be the hypergeometric series for $\alpha,\beta\in\Bbb C$ and $\gamma\neq 0,-1,-2,\...

integration complex-analysis gamma-function beta-function

A minor chaos

Thorgott

@Koro nice

@Koro try proving it without Tychonoff

Jakobian

13:01

@Koro well for $x\in \prod_\alpha X_\alpha$ grab open sets $x_\alpha\in U_\alpha$ with compact closure for each $X_\alpha$ which was assumed to be locally compact. This is your open neighbourhood with compact closure

The Tychonoff is used to prove that this closure is indeed compact - it's homeomorphic to product of $\overline{U_\alpha}$ for $X_\alpha$ locally compact, and $X_\alpha$ otherwise, that is we have a product of compact spaces

Now try to prove the reverse direction.

@CottonHeadedNinnymuggins just answer honestly. Sometimes the reputation just comes in time

Difficult things go unnoticed the most, but sometimes they too find a way.

That said, to earn reputation faster, try answering a bunch of easy questions, like on the calculus level

people like when it's nice and easy, then they go and upvote you

because they can understand what the hell you are saying in the first place

Sha Vuklia

13:41

I'm reading about localisation, and in my book (Neukirch - Alg Numb Thy) the following remark is made:

I don't understand why if $X$ is finite or omits only finitely many prime ideals, then only the prime ideals from $X$ survive in $A(X)$. For the finite case, consider the localisation of a domain $D$ at a non-zero prime $P$. Then $0$ will still be prime ideal of $D$ and $D_P$, while $0$ doesn't belong to $X=\{P\}$. So I don't think I'm interpreting correctly what they mean by "only the prime ideals from $X$ survive in A(X)".

Should I read it as "only the prime ideals $Q$ that are contained in some $P\in X$ survive in $A(X)$"? Then the finite case makes sense at least.

Sha Vuklia

14:12

Or I can see it in the special case of a Dedekind domain and non-zero primes

never mind, it seems to be a mistake (someone else had pointed it out too)

Jakobian

I don't find the appeal of this part of algebra. Too dry

@AlessandroCodenotti what does $H(M, X) <\varepsilon$ means where $M$ is a subcontinuum of $X$?

ILikeMathematics

If the derivative exists, then the difference quotient $\frac{f(x) - f(a)}{x-a}$ strives against a limit for $x \to a$.
Is there a case where the difference quotient doesn't strive against a limit for $x \to a$?

There is $f(x) = |x|$, then the difference quotient strives against $1$ or $-1$ depending on $x \to 0^{\text{+}}$ or $x \to 0^{\text{-}}$ for $a = 0$, but that means it's striving against two limits, not "no limit", right?

s.harp

14:39

Suppose I have a complex polynomial in n variables. Is there a name for the property $\sum_{i=1}^n (\partial_i P) ^2 = 0$?

Thorgott

14:52

@ILikeMathematics sure, consider $x\sin(1/x)$. this extends to a continuous function with value $0$ at $0$, but the difference quotient at $0$ diverges by oscillating.

Xander Henderson

15:28

@ILikeMathematics I don't understand the phrase "strives against" in this context. What do you mean?

Koro

@Thorgott et @Jakobian I'll get back to you soon.

Suppose that V is a real vector space of dimension n. Then the dimension of the space of all k tensors on V is $n^k$.

Ted Shifrin

17:15

@s.harp property? It rarely holds.

user532965

17:26

hello may I ask algebraic geometry question?

BAYMAX

Can we write an expression for all eigenvalues of a matrix to be less than one in absolute value?

Thorgott

feel free to ask any question, answers aren't guaranteed of course

BAYMAX

10

Q: Proving an implication of two dimensional matrix.

If $A = \begin{bmatrix} x & 1\\ y & 0\end{bmatrix}, B = \begin{bmatrix} z & 1\\ w & 0\end{bmatrix}$, for $x,y,z,w \in \Bbb{R}$. I have observed by considering many examples of $x,y,z,w$ that: If all the eigen values of $A^2B$ and $AB^2$ are less than one in absolute value $\implies$ $\det(AB+A+I)<...

linear-algebra matrices determinant

monoidaltransform

Can finite dimensional CW complexes always be embedded in some $\mathbb{R}^n$?

Kabir Munjal

Sir how to generalize Powers of Matrices in summation notation??
A^n in Sigma Notation??
This question was asked by my sir when we were studying basic maths for physics and Linear Algebra(Vectors and Matrices at JEE level) chapter came..
Edit (A is a square matrix of order m)

Sha Vuklia

17:44

@Thorgott Mind if I ask a question about orders anyways?

Xander Henderson

@NewGuy Per the room description, don't ask to ask. Just ask.

Koro

Are modules used in algebraic topology also?

Ted Shifrin

@monoidaltransform with finitely many cells?

Lots of presentations of $\Bbb Z$-modules, yes

Koro

Ohh okay. Lot of abelian groups then

Ted Shifrin

And more generally …

But the stuff you learn in working with modules is essential

Koro

17:59

I see. Thanks for the info.

Sha Vuklia

@Thorgott I figured it out (though I must say I only realised the final step while I was trying to formulate my question in the chat, so that did help me)

Alessandro Codenotti

Whenever you have a (co)homology theory with coefficents in any ring different from $\Bbb Z$ you are working with modules

@Jakobian No clue without context, what is H? Maybe Hausdorff distance?

Jakobian

18:15

@AlessandroCodenotti do you know something called a shore?

Hausdorff distance doesn't quite make sense

$M$ is a subcontinuum of $X$

ah wait. Maybe it is Hausdorff distance

ah yes. It is in fact Hausdorff distance. Thanks!

also see here sciencedirect.com/science/article/pii/S0166864113004422

Thorgott

@monoidaltransform finite ones can, yeah

@ShaVuklia well, that's great too

additional to cohomology theories with arbitrary coefficients, modules also pop up in other places, e.g. by looking at modules over the cohomology ring of a space or by working with modules over the group ring of the fundamental group

Ted Shifrin

But that’s well beyond first-year AT :)

user123234

18:40

If I have a stopping time $T$ which is finite a.s. and I want to compute $P_x(K_x\circ \theta_T|T<\infty)$ how can I rewrite this? So I'm a bit confused since we condition on $T<\infty$ and not on a sigma algebra

$K_x\circ theta_t$ is not so important. I only want to understand the conditioning part

Goku

Is there a chat I could speak to a moderator with? I've got a minor issue I'd like to deal with

Ted Shifrin

Maybe CURED chatroom? Or wait and ask Xander.

Koro

multilinear algebra is confusing. So many symbols! Am I the only one having this feeling?

Goku

Well I've raised the issue in the Math's Mods office chat, hopefully the moderators will see that

Koro

Finding basis of space of alternating k tensors on V

Ted Shifrin

18:55

What’s the big deal?

@Koro

Jakobian

multilinear algebra is easy

Thorgott

the trick to doing multilinear algebra is to realize it's just linear algebra on different spaces

Jakobian

for alternating $k$-tensors on $V$, just take $\mathrm{d}e_{f(1)}\wedge ... \wedge \mathrm{d}e_{f(k)}$ where $f:\{1, ..., k\}\to \{1, ..., \dim V\}$ is some increasing function

Koro

So if V is of dimension n, then I understand that if k>n, then the alternating k tensor space is the zero space (any k tuple of basis elements will have repetition so alternating maps on it will give 0). Now if k is atmost n, then Munkres says that -We show first that an alternating tensor f is entirely determined by its values on k-tuples of basis elements
whose indices are in ascending order. Then we show that the value of f on such k-tuples may be specified arbitrarily.

I don't see how coming with 'indices in ascending order' intuitive.

i.e., for someone doing this for the first time.

Thorgott

the idea is that if you know the value on some $k$-tuple, you also know the value on any permutation of that tuple by virtue of being alternating

Koro

19:00

So it feels like memorizing at times.

Jakobian

well what do we know about alternating tensors
there can't be repeats, and the order has to be fixed because otherwise we can just shuffle them

Koro

@TedShifrin :(

Curio

Does uniform convergence imply absolute convergence?

Thorgott

sorting tuples by ascending order is just one way of fixing a system of representatives for all $k$-tuples up to permutation, which is something we want to do because of the previous remark

Jakobian

@Curio of a series? No

Curio

19:01

Function series

Thanks

Jakobian

Take any convergent series of numbers which doesn't converge absolutely and let functions be constant

Ted Shifrin

You can try my lectures, Koro, although I define things without using tensor product. Perhaps it’ll improve your intuition.

Jakobian

By properties of wedge product itself, we know we have to have at most such vectors, and it turns out they suffice as well. Is my intuition

Mary Star

Hello! How are you?

I want to show that it holds that $P_sG_k=\tilde{G}_kP_s, \ s\geq k$, where we get $\tilde{G}_k$ from $G_k$ by exchanging two entries $\ell_{j_1,k}$ and $\ell_{j_2,k}$ with $j_1,j_2\neq k$ ($j_1\neq j_2$), that are exchanged also at the permutation $P_s$.

Could you give me a hint how to show that? I don't really have an idea.

Goku

I saw a question right now that was a duplicate of my own from a while ago and I flagged it, I'm hoping that was the right call?

Mary Star

19:13

I have posted the question in the main : https://math.stackexchange.com/questions/4578124/matrix-exchanging-elements

I would be glad if someone could help me because I got stuck.

Ted Shifrin

Is yours answered, Goku?

Koro

I understand the proof now. Thanks all.

Ted Shifrin

@MaryStar You need to establish your notation to start with.

Entries of what? Permutations of what? Are these all matrices? Very unclear.

Alessandro Codenotti

@Jakobian hmm is this the thing in a dendroid where a point is a shore if there are two open sets such that every arc from the first to the second goes through the point? Or am I mixing it up with something else?

Jakobian

@AlessandroCodenotti hmm... it might be related. From what I understand it's just a property stronger than being noncut

Goku

19:31

@TedShifrin yes, by yours truly in fact

Mary Star

19:42

@TedShifrin It is not given more information. $G$ and $P$ are matrices, especially $P$ is a permutation matrix and $\tilde{G}_k$ is a new matrix which is $G_k$ when we exchange two entries $\ell_{j_1,k}$ and $\ell_{j_2,k}$ with $j_1,j_2\neq k$ ($j_1\neq j_2$).

I suppose that $G_k$ is the matrix of the steps of the LR-decomposition. @TedShifrin

So LR decomposition with pivot where P is the corresponding permutation. @TedShifrin

Mary Star

20:03

Do you have an idea? @TedShifrin

Alessandro Codenotti

@Jakobian ahh that's the thing with arbitrarily big continua missing the point

Yes it's related but I forgot the details. I got interested for a question I was trying to solve but I couldn't make much progress and I moved on to other projects

Ted Shifrin

@Goku Then you should vote to close because there’s a duplicate with an accepted answer.

@MaryStar Think about what conjugating by $P$ does ($PAP^{-1}$).

Jakobian

20:20

Do you ever go into the General Topology chat btw?

Yeah, I think I got interested a bit in continua again :)

do you know of any ways to find references for the things you're interested in? Is there a method, or do you just look around and see

articles, say

I want to stop just reading books and try and see if I can find something on my own

at some point at least

schn

20:35

I have been given the following function $$h(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} \Re\{a(k)e^{i(kx-\omega t)}\}\mathrm{d}k.$$ I have also been given $a(k)$, but I think it need not be specified. I want to determine the Fourier transform $\hat{h}(k,t)$, in particular, $\hat{h}(k,t=0)$.

If the real part in the integral was removed, so that we would have $$f(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} a(k)e^{i(kx-\omega t)} \mathrm{d}k.$$ Then $a(k)$ would just be the Fourier transform of $f(x,t=0)$. However, unfortunately I can't just remove the real part and so I am stuck.

robjohn

In which variable is the Fourier Transform to be taken?

schn

From $x$ to $k$.

robjohn

Is $a$ real valued?

schn

Yes.

robjohn

At first glance, it looks like the even part of $a$

But I’m sitting in a waiting room on my phone, so take that with a grain of salt

schn

20:46

OK :)

Ted Shifrin

@robjohn I hope all is well.

robjohn

This is for a friend. My wife has an appt tomorrow, and I’m waiting on insurance to see if I can get the next test I need.

It’s fun getting old.

Ted Shifrin

I am way ahead of you. Two major heart surgeries and a cancer surgery. But goofing off now. All fondest wishes to you all!

schn

I would be very grateful if you could explain the even part of a function. Suppose $a(k)=2\pi \delta(k)+C/((k-1)^2+2)$, where $\delta$ is the Dirac delta and $C$ is a constant.

Sha Vuklia

@Thorgott I think now I do have a question that I am unable to solve.

I'm trying to show that $\mathfrak p\supset\mathfrak a$ is the only prime ideal containing $\tilde{\mathfrak a_{\mathfrak p}}$. It's clear to me that $\mathfrak p$ does contain it. However, I was unable to use the local prime correspondence, and it was pointed on in a stack post that this doesn't work indeed. Instead, looking at primary decompositions should do the trick.

I figured if I can show that $\tilde{\mathfrak a_{\mathfrak p}}$ is primary, then the claim is indeed true. However, I was unable to do this. Do you have any ideas?

Btw, here $\mathcal O$ is a 1-dim Noetherian domain

robjohn

21:02

@schn the even part of $f$ is $x\mapsto\frac{f(x)+f(-x)}2$

Alessandro Codenotti

@Jakobian I just google around or ask my advisor, she knows a lot of continuum theory

schn

@robjohn Ok, thanks

Alessandro Codenotti

@Jakobian No, I barely even read this chat lately

Sha Vuklia

@TedShifrin O dang Ted, I don't know if those are past or future treatments, but that is heavy, hope you're doing decently alright

robjohn

Dirac Delta is even, so it is its own even part

Sha Vuklia

21:06

Or maybe I should try to understand lohll's answer in math.stackexchange.com/questions/3074990/…

Alessandro Codenotti

Also the book hyperspaces (which also does a lot of continuum theory) has plenty of references to papers and open questions, it's a great starting point @Jakobian

Sha Vuklia

oh, I understand that answer now

@Thorgott I guess never mind again. Sorry for pinging you constantly

Ted Shifrin

@ShaVuklia Way past. Cancer gone almost 11 years.

Sha Vuklia

Oh, good to hear:)

Goku

@TedShifrin Ted-1 Cancer-0

Jakobian

21:17

@AlessandroCodenotti ah alright. Thanks

Thorgott

@ShaVuklia no worries, glad you figured it out

copper.hat

21:36

@robjohn hope it works out.

monoidaltransform

22:39

0

Q: terminology gromov hausdorff uniform convergence term

I know that if $X$ is a compact metric space and $X_n$ are such that $X_n\rightarrow_{GH} X$ then that notation means $X_n$ converges to $X$. However, we does it mean for one to say that if $X$ is compact, then we can find $X_n$ that approximates $X$ uniformly? Does it mean that there exists $C\g...

metric-spaces riemannian-geometry

Mary Star

23:08

@TedShifrin isn't this about the steps of the gaussian elimination, i.e. Changing rows / columns due to pivoting?

Goku

Does a generalization of syntheic methods exist when it comes to "Langley's" problems? I've read that such papers have been published but surprisingly I found none so far

leslie townes

23:33

wikipedia references somebody's unpublished result for a recent example solution of such a problem. i don't know if it's 'synthetic,' but a web archive of it (in japanese) is linked on that page.

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$Mathematics$ Mathematics