Mathematics - 2024-10-27

Debug

01:00

Need a topology expert

0

Q: What is a/the definition (if it even exists in literature) of a piecewise continuous function in general topology?

I can't find any information on the web about this. Definition. I have a function $f : X \to Y$ between two metric spaces such that there exists a sequence $X_{i}, i \geq 1$ of closed subsets of $X$ such that $\bigcup X_i = X$. For each $i \geq 1$, there is a continuous function $f_i : X \to Y$ ...

Binky

09:56

Hello

@SineoftheTime What level of English do you have?

PM 2Ring

@copper.hat Thanks, that's sweet. A couple of years ago, I posted some Japanese blues here, eg

May 12, 2022 at 4:38, by PM 2Ring

Also see I Can't Quit You Baby by the Shoka Okubo Blues Project. Juna on bass is an awesome slap bass funk player. Shoka also plays drums.

Juna now plays with the amazing Jazz Avengers. Here's an example of their work, although one of the sax players is a temporary substitute for a lady who's currently on maternity leave.

Jazz Avengers in HK 15/5/2024 -No 10 Why not?

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Recently, their guitaist fell ill, and Shoka filled in for her. Now Shoka is a blues player, not a jazz guitarist, but she did a fine job here. The interplay towards the end between her and Ami on soprano sax is spectacular.

THE JAZZ AVENGERS - Sphere (Live) /五所川原 FOREST BLUE

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@copper.hat Religious people have been using GPT since the early days, even using it to write sermons. It didn't take long before someone created JesusGPT...

@copper.hat The latest LLMs are more impressive. They're less likely to make errors. OTOH, that's mostly superficial apoearance, since they still don't know what they're talking about. It's not like Wolfram Alpha, which actually has rules of algebra & calculus explicitly encoded into its structure.

My new name for these bots is ChatGTA: Grand Theft Autocomplete.

Here's an earlier funk tune from The Jazz Avengers, Pick Up the Pieces, written by a Scottish group.

🎷THE JAZZ AVENGERS🎷Pick Up the Pieces

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Sine of the Time

11:04

@Binky basic

nickbros123

11:58

finite union of nowhere dense sets is also nowhere dense. is this statement only true for metric space?

nickbros123

12:11

ok i think this is generally true

Soumik Mukherjee

12:40

@nickbros123 yes

nickbros123

@SoumikMukherjee whenever im starting a problem in metric spaces I gamble on going down two routes: epsilon balls, limit points/ derived set arguments, etc- the distancy route, or neighbourhoods, maximal open set, minimal closed set arguments etc- the topology route.

Ben Steffan

be happy you have all of those approaches :)

nickbros123

yes :)

idk much of topology- except metric spaces (which also I donno much of)- but it "feels" like home. Ill probably do a reading under a prof this winter in topology

Soumik Mukherjee

Nice

nickbros123

when im doing algebra I just dont get the same feel, like whenever im doing smething like calculating upper central series of D_8 or something, im like, "why am i doing this?". such questions dont come when im doing topology. ofcourse its not good for me to close myself so early- and I wont- but its just a feeling

Ben Steffan

12:54

haha

I feel you

Jakobian

14:15

@nickbros123 The knowledge in general topology makes me understand why things are the way they are for metric spaces a lot better

So it feels more at home to talk about topological spaces, or Tychonoff spaces, in general

Sine of the Time

Bet you feel at home when you study Polish spaces

nickbros123

14:56

lol

Debug

Bounty 🤑

3

Q: Conjecture: the integer cosets of ideals are Noetherian

I have strong evidence that this fact will help prove continuity of $\pi_2(n)$ the twin prime average counter on the naturals. Continuity with respect to the Furstenberg topology that is. See my recent post history under continuity for more details. Let $U_1 \subset U_2 \subset \dots$. This s...

abstract-algebra elementary-number-theory ideals conjectures noetherian

Debug

15:20

$500 bounty (mse money)

Pizza

16:04

Hi

Sine of the Time

@Pizza sup

Soumik Mukherjee

Hi

Pizza

@SineoftheTime inf

Sine of the Time

:)

Pizza

16:22

Sine of the Time

@Pizza doesn't look hard

Ben Steffan

but that's impossible to do... oh wait

Binky

@Pizza its 0

Pizza

:)

where an x can be inserted to make the derivative more complicated

Sine of the Time

@Pizza subsitute every number with an $x$ :)

Pizza

16:45

@SineoftheTime How long did it take you to prepare for the analysis 3 exam?

Sine of the Time

Hard to tell

I've remember I've heard it was hard, so I focus from the first lessons

so it's hard to estimate how long did it take

Are you studying methods?

Pizza

I was just looking at an exercise now, but I haven't done much yet

@SineoftheTime Did you find it difficult?

Sine of the Time

yes

@Pizza which topic?

but analysis 3 is different from your exam

Pizza

@SineoftheTime calculate residues in the poles

@SineoftheTime yes I know

Sine of the Time

if you know how to differentiate, you should not have problems

Pizza

16:54

only the profs called it a kind of analysis 3 that's why

Sine of the Time

analysis 3 and complex analysis

Binky

I need advice, I need an analysis book 1 but I don't know which one to buy, advice?

Sine of the Time

you need it to do what?

Binky

to study the analysis 1 topics that are normally studied

Sine of the Time

@Binky it's hard to tell

Binky

17:00

Because I read that there is a need for one, online notes are not enough

Sine of the Time

so do you want to use it to prepare analysis 1? Maybe it's better if you ask your professor

Binky

no, but I need them for tests

That is, I am studying without books and I have some deficiencies

Sine of the Time

are you studying mathematics?

Binky

no, I am not enrolled in any faculty yet

Sine of the Time

so which tests are you preparing?

It's not clear what are you searching for

without more infos, no one can help you

Binky

17:04

I need an analysis book 1 to study things in advance

because I know that these are topics that are already studied in high school

Sine of the Time

are you Italian?

Binky

yes

Sine of the Time

ok, I can tell you the standard books used

Binky

Ok

Sine of the Time

but I don't think it's that useful, your professor should tell you. Moreover, if you won't do math, I don't think if you'll use them

The standard books are: Giusti, Marcellini Sbordone, Pagani Salsa, Giacomelli Dal Passo, Bramanti

Binky

17:10

You can also tell me if you know any books in English

Pizza

But sometimes you wrote exercises of analysis 2, what did you need them for?

Sine of the Time

Spivak, Apostol and Rudin

@Pizza It's not clear what Binky is doing actually :D

Binky

@Pizza Yes, I took them from internet, it was to practice, but I saw that I can't do them very well

Sine of the Time

@Binky was the exam on differential equations high school stuff?

Binky

It wasn't an exam

it was like a form to fill out to see how much you knew about that topic

Sine of the Time

17:17

never heard about that

Pizza

@SineoftheTime From what I understand he wants to see these things in advance that he will study later at university

Sine of the Time

still it feels strange

he jumps from computing residues to evaluating double integrals and solving DEs

Pizza

yes, and now he starts from analysis 1, I don't understand...

However, I don't think it's a good way to study to do all these subjects together

Sine of the Time

if he wants to study rigorously, it's a bad idea

if he only need to learn how to compute things, then it may be acceptable

Binky

Yes, I was only interested in computing

Almanzoris

17:28

Greetings.

Sine of the Time

hi

Almanzoris

How's everything going?

Sine of the Time

normal

what about you?

Almanzoris

Going fine. I've got some work to do, but yesterday I watched the UFC fights. There were a few really good ones.

Especially the stellar one.

Lucky Chouhan

@Almanzoris I miss Broke Lesnar :')

F5

Almanzoris

17:46

@LuckyChouhan I have never watched the WWE at all. But yeah..., those things have to be enjoyed in their "present". Sooner or later, it reaches its end.

copper.hat

18:35

@PM2Ring not a huge fan of the hype surrounding "ai" at the moment. i liken it to a smart, lazy, drunk, unprincipled assistant. can be helpful, but unreliable.

anak

19:43

Feels kind of stupid, but suppose I have a non-vanishing normal vector field to a hypersurface $H\subset M$. I want to define a function $\rho\colon N\to\mathbb R$ for a neighbourhood $N\supset H$ such that $\rho^{-1}(0) = H$ and $d\rho_p\neq 0$ for each $p\in H$. This is easy with the implicit function theorem around some point in $H$, and it's easy to come up with functions without the $d\rho$ constraint using tubular neighbourhoods, but I can't get anything satisfying everything.

With the tubular neighbourhood idea, I would want to say that since regular points are generic, we can perturb our function $C^\infty$-small (?) to get a function, but this isn't to clean. I am also wondering if the normal vector field condition is even necessary?

Xander Henderson

Tubular Bells > tubular neighborhods.

Thorgott

do it in charts and use a partition of unity to glue?

Xander Henderson

Tubular Bells (Pt. I)

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Ben Steffan

ah yes tubular, everyone's favourite SMW level

Xander Henderson

@BenSteffan *shudders*

That is, like, PTSD inducing.

Ben Steffan

19:55

don't you like the p-balloon xander

isn't it the most fun of all the powerups

Xander Henderson

@BenSteffan That is false. Goomba shoe for the win.

Ben Steffan

that's not an smw powerup

Xander Henderson

@BenSteffan Don't care.

The goomba shoe is still better.

Ben Steffan

O. K. 👍

Sebastiano

@Joe Thank you very much for your comment: math.stackexchange.com/questions/4990092/…

@Joe I have given a comment. My best regards.

anak

20:02

@Thorgott I have trouble showing that 0 is a regular value in this case. Maybe tthe normal vector field can be used for this, but I guess I will have to think about it longer.

It would have to guarantee that all my local defining functions $\rho_\alpha$ have the same sign at $d\rho_\alpha|_p$ for $p\in H$, I guess?

I guess you can do that by making sure that I guess the normal vector field defines an "in" and "out direction, and so as long as you pick $\rho_\alpha$ so that they agree on sign on the in an out sides, then you get that?

Seems doable.

But now I wonder what's wrong/missing with my previous suggestion via tubular neighbourhoods? Am I implicitly appealing to this normal vector field?

Thorgott

oh yeah, I suggested nonsense

your assumption implies the normal bundle is trivial, so the tubular neighborhood trivializes and we can just take the projection $N\cong H\times\mathbb{R}\rightarrow\mathbb{R}$

anak

20:17

Oh it's implicitly through taking the neighbourhood as trivialized!

Thanks!

For some reason I was taking it for granted that $N\cong (-\epsilon,\epsilon)\times H$

copper.hat

20:47

i think u2 took inspiration from mike oldfield when it came to repetitive grind themes

dfnu

Hello! I am totally new to the chat. I have been posting questions and answers for some time, but now I have only a curiosity for which I think a question would be too much, and hoped for some help here

Is there anyone around that is experienced in real analysis?

copper.hat

why don't you just ask the question?

Ben Steffan

"Just ask; don't ask to ask." - Chat Description :)

dfnu

:) ok, I was just wondering what that meant.

copper.hat

ask forgiveness, not permission

my personal favourite is given a Lipschitz continuous curve $c:[0,1] \to \mathbb{R}^n$, can it be reparameterised as a differentiable curve?

Jakobian

20:56

@dfnu lots

dfnu

Ok. I was studying the functions for which one can differentiate asymptotic equivalences. So that f'~g' whenever f~g, (Sort of the converse of L'Hopital Rule) and found out that this is related to functions that preserve asymptotic equivalence

Xander Henderson

Real analysis is dumb. Do $p$-adic analysis.

copper.hat

real analysis is the mother of all sirens.

dfnu

so I was wondering if this is something known and presented in some book

Ben Steffan

@XanderHenderson shudders

dfnu

20:57

I don't even know what a p-adic number is...

Xander Henderson

@copper.hat So it's some kind of klaxon?

@dfnu It is an element of the Cauchy completion of the rational numbers with respect to the $p$-adic absolute value.

Ben Steffan

Really, I thought it was an element of the field of fractions of the $(p)$-adic completion of the integers...

copper.hat

@XanderHenderson i was thinking more along the lines of a different kind of noise, as in Θελξιέπεια, for example

Xander Henderson

The $p$-adic absolute value is a measure of how "divisible" a number is by a prime number $p$. Fix a prime number $p$, and observe that any rational number $x$ can be written as $x = p^n (a/b)$, where $a$ and $b$ are not divisible by $p$. The $p$-adic absolute value of $x$ is $p^{-n}$.

Jakobian

@dfnu don't worry, no one has focused on what you're asking yet

Xander Henderson

20:59

Roughly speaking, the $p$-adic absolute value measures the $p$-ness of a number.

Jakobian

or, do worry?

dfnu

ahahha ok

Xander Henderson

*ba-dum-tish*

Ben Steffan

@XanderHenderson ...

who was it that said this in a lecture again? Deligne?

copper.hat

@dfnu what is your question?

dfnu

21:00

in the meantime I ll get to learn smth about p-adic numbers

Xander Henderson

@BenSteffan Lots of people.

'tis an old joke.

dfnu

@copper.hat Do I find anywhere (books, papers...) something about the connection between functions that preserve asymptotic equivalences and functions for which one can differentiate asymptotic equivalences?

psie

I'm reading Theorem 2.44 in Folland, the change of variables formula for linear transformations. He proves $$\int f(x)\,dx=|\det T|\int f\circ T(x)\,dx$$ for Borel measurable functions and then says that the result for Lebesgue measurable functions follows as in an earlier theorem, but it's not very clear how it follows from that theorem. How does one obtain the formula for Lebesgue measurable functions if it is already true for Borel measurable functions?

Moreover, will $f\circ T$ also be Lebesgue measurable if $f$ is Lebesgue measurable (here $T$ is a linear operator from and to $\mathbb R^n$). I recall Folland warning about compositions of Lebesgue measurable functions not being Lebesgue measurable.

@psie I guess I need to start with indicator functions, then advance to simple functions, etc. The usual drill.

Xander Henderson

@psie If it is Lebesgue measurable, how far from being Borel measurable is it? (at worst)

psie

@XanderHenderson a null set away...always :)

Thorgott

21:11

@XanderHenderson I'm trying to sit in on a non-archimedean geometry lectures this semester, surely close enough

Xander Henderson

@Thorgott Good for you. The archimedean property is a straight jacket for the mind. Free yourself!

Ultrametric spaces for the win!

Ryder Rude

hi

Xander Henderson

21:32

@RyderRude High.

Ryder Rude

@XanderHenderson high ;)

Xander Henderson

@RyderRude High.

Hoe.

Hee.

Haw.

Hay.

Who?

Ryder Rude

highway

Ben Steffan

height

Ben Steffan

22:05

morphisms in Segal's category can both be active and inert at the same time

it will then also be semi-inert

it can in addition also be null, but then it will be unique

copper.hat

22:41

@dfnu i do not, sorry.

@psie perhaps look at math.stackexchange.com/a/1206264/27978

psie

@copper.hat Ok. In this case, I know $f\circ T$ is Lebesgue measurable even though $f$ is Lebesgue measurable and $T$ is Borel measurable...the feared combo

It follows because Borel null sets are invariant under $T$ and $T^{-1}$

Hence by monotinicity, Lebesgue null sets are also invariant under $T$ and $T^{-1}$

I didn't give a lot of details in my message above, but it would have been too long otherwise.

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