@copper.hat Wait... there was a quarter in which Advanced Hacky Sack wasn't being offered at UCSC? Man... standards there have fallen...

@XanderHenderson :-)

:-)

Asaf Karagila's knowledge on set theory is impeccable

His knowledge of whisky seems to be pretty good too.

polymath

He's an extremely concise communicator, who was once a regular in this room.

of course, real whiskey has an e...

yes

🥃

vague irony in the fact that i am not a whiskey fan myself

nor I

my da used to work for Irish Distillers.

The gypsy king has some Irish heritage in him, no?

Fury?

yup

my understanding is that he is English of Irish Traveller backgroud, so yes.

One more looong month of waiting...

Usyk shaved his head

:-). when i was a kid, my dad & myself (well, my dad) would bet on some big matches. Joe Frasier was our favourite.

cool

Ali was incredible, but his taunting didn't sit well with our then notion of sportsmanship, so we never bet on him.

Everybody loved to hate Ali.

My respect for Ali grew independent of his boxing.

But he was mean to Frasier.

Fury tries to imitate the rope a dope.

The heavyweights get the attention, but i prefer watching lighter weights, albeit i find Mayweather incredibly boring to watch

I was quietly hoping that Paul would get his ass (or ear) handed to him. Seemed like more of an arranged match unfortunately

my tolerance for violence wanes as i age :-)

good night!

cya

@Thorgott I see. Thanks

If a function $f(x,y)$ is jointly continuous in $x$ and $y$...does it mean it is continuous?

I'm asking because it is claimed $$A_rf(x)=\frac1{m(B(r,x))}\int_{B(r,x)}f(y)\,dy$$is jointly continuous in $r$ and $x$.

It's so strange, because it is claimed that $(A_r f)^{-1}((a,\infty))$ is open, so somehow joint continuity must imply continuity in this case.

I think jointly continuous in $r$ and $x$ means that the function is continuous if we fix either $r$ or $x$.

@psie yes

@psie no

psie that latter notion (which is generally distinct from continuity in the product topology) is more frequently called 'separate' continuity

presumably there is some kind of integrability condition on f to ensure that that function is even defined

Reading stuff on set theory is like, it's a bit dry for me, yet feel like it's nonetheless important, a la, stuff on AC, zorns lemma, transfinite induction / recursion etc. I think I need to take a tutorial from one of the professors or PhD students on this cuz I am not able to motivate myself to self study this stuff

Guys, $\widetilde{SL(2,\mathbb{R})}$ geometry is beautiful.

Thurston's

@leslietownes since we mentioned topology

yes, there are topologies appropriate to discuss separate continuity instead of joint continuity

what is the categorical interpretation of all of this

:)

I guess we need to ask @BenSteffan

@nickbros123 as cool as it is i think we non set theorists could all skip all of it and be entirely fine

or like 95% fine. i have run into maybe two or three examples of something i thought was true being false and the counterexample was like 'well the first uncountable ordinal blah blah'

Was it by chance the S omega ? :)

@leslietownes The first uncountable ordinal is very important

I think I know what you mean, and those counter-examples probably relate to convergence of sequences and how they don't determine the topology

that, or you mean something like Dieudonne measure

first uncountable ordinal may also be used to write down Borel sets explicitly

the thing i had in mind was the bottom of page 5 in cms.dm.uba.ar/academico/materias/2docuat2021/…

i had the honor of recommending the rejection of a paper premised on exactly that mistake

@leslietownes they're defining Dieudonne measure there as far as I see

anyway we had that example from noted set weirdo bob solovay

no they're not doing that, I just browsed it very briefly , I'm wrong here

I see what they're doing

constructing a set which sections are Borel but isn't Borel

there's a guy who is still publishing papers in operator theory just ignoring all of that

its interesting, I never saw such construction

@leslietownes how does it relate to operator theory?

the example of a Borel set I mean

i can't answer without outing someone who still works in academia but sometimes you need or at least hope that certain constructions are measurable and absolutely all of the difficulty is in proving that they are measurable

and if you ignore stuff like that everything is a whole lot easier

well... :P

@leslietownes even if you just explain the relationships of how in the field you connect operator theory to measure theory?

i am not going to provide much in the way of clarity on this point, but L^2 spaces connect operator theory and measure theory

i shouldn't have said anything

those notes themselves point in a couple of directions, their author was an operator theorist

my cat is wandering around the house acting like i haven't fed her but i have

on another note, I am searching through papers of Blair, trying to see where he proved the characterization of $z$-embedded set $S$ in terms of $S$-separated sets

@leslietownes maybe expecting to get more

absolutely trying to get more

she usually does this with my wife, she can make one of us believe that the other didn't feed her and then get two meals

but my wife is on a work trip so it's just me and i know what she's been fed

maybe she doesn't realize that you can't do that to one person twice

cats aren't that smart, I guess?

yeah i think there's a limit of feline intelligence and we are reaching it

within her mental boundaries she is very sharp

I wonder how exaggerated are limits of human intelligence :P

within our mental boundaries we are very sharp

eh... debatable

if problem-solving is one of the skills you need to be good at to be called "intelligent" then that feels like a very stupid ability to measure intelligence by, indeed

i am more interested in problem creation

I suppose that's what a cat does when she asks for two meals

@leslietownes if you're not part of the problem, what are you doing with your life?

by the way does anyone know of example of $A\subseteq \beta \mathbb{N}$ that is not separable?

I found a paper which implies that every dense subset is separable

@copper.hat I'm sorry to bother you again, but how does $x_n\to x_0,r_n\to r_0$ and $\|y-x_0\|<r_0$ imply $\|y-x_n\|<r_n$ (for sufficiently large $n$) and what does this have to do with continuity? I can not work it out: $$\|y-x_n\|\leq \|y-x_0\|+\|x_0-x_n\|<r_0+\epsilon,$$for sufficiently large $n$, but how do I bound further to get $\|y-x_n\|<r_n$?

apparently if $X$ is hereditarily separable regular space then $|X| \leq 2^{\aleph_1}$

but this would imply that under $2^\mathfrak{\aleph_1} < 2^\mathfrak{c}$ we have that $\beta\mathbb{N}$ contains a subset that isn't separable

which is not to say we can do that in ZFC but is good enough for me, I guess

I think this inequality is consistent with set theory

yeah its consistent with set theory

I have this business idea called Problem purchasing ltd. The idea would be to open an office and have people come and sell their problems to me. For a really good problem I would pay several thousands up to a million. For easy problems I would only pay a few dollars. How does that sound for a business idea?

might not be the correct place to ask this but ill ask anyways: does anyone know why latexworkshop ( texlive whatever) on VS code renders low quality pdf? render as in the final pdf on its own is fine, but the preview on VS code itself is like i can count the pixels

Why is it $2\pi \sqrt{\varphi(\tau)}$ in that last sentence?

@pie That book is published in the Springer "Developments in Mathematics" series. The intended audience of that series is researchers, advanced graduate students, and postdocs. Do you fall into that audience? If not, it might be a bit over your head.

Of course, even if you are not in any of those categories, it is possible that you have developed sufficient expertise in the field, and that you might find the text useful.

And note that the text is published under an open access model, so you can just download it and check it out. If it goes over your head, nothing lost.

@XanderHenderson Thank you for the clarification! I had thought that the Graduate Texts in Mathematics series was the only one aimed at researchers, advanced graduate students, and postdocs. I am far away from being one of those categories.

The graduate texts series is aimed at a less advanced audience than the developments series.

yeah the GTMs are basically textbooks for anybody. developments in mathematics is, welcome to my nightmare world

@MatsGranvik great idea, please let me know when you start the office:)

@XanderHenderson I consider myself one of those, at least in spirit

if anyone disagrees then I'd like to know - would be nice to hear if I'm wrong

if anyone agreed then that'd be good to hear too

@copper.hat I worked it out. You don't have to reply anymore. Enjoy your Saturday, go to church tomorrow and work hard on Monday! :)

@leslietownes "yeah the GTMs are basically textbooks for anybody. " so for undergrads ?

@pie a lot of them read those books, yes

WOAH! can you give me an example of a GTM book that is appropriate for undergrad?

@pie Most GTMs are appropriate for an advanced undergraduate.

But not just any advanced undergraduate---the point of saying that something is aimed at an audience of graduate students is to emphasize the sort of prerequisites that a text might have.

@XanderHenderson I didn't expect that at all. but what does "advanced undergraduate" even mean?

Lee's Introduction to Smooth Manifolds, for example, is in the GTM series. But anyone with a basic background in calculus (including multivariable calculus), real analysis, and topology should be fine.

@pie Well, typically, "advanced undergraduate" (in the US system) means someone who is in their last year or two of undergraduate study. This is the kind of student who has already completed their basic classes, and is ready to start specializing a bit.

@XanderHenderson Could you please elaborate on what kind of basic topics an advanced undergraduate student is typically expected to have completed? As someone studying on my own, I’d appreciate understanding what foundational knowledge is considered essential before specializing further.

@pie Again, "advanced undergraduate" literally means an undergraduate student who is in the advanced stages of their degree, i.e. one in the last year or two of their program.

I think that you are trying to over-interpret the phrase.

At most US institutions, an "advanced undergraduate" in mathematics is going to be a student who has completed a year of calculus, some kind of introduction to higher mathematics course (often a discrete mathematics course, but it could be anything), probably linear algebra and an introduction to diffy Qs. They should also have taken a few more specialized classes, e.g. analysis, algebra, etc.

May I ask: Is discrete mathematics course important? I thought that Id I studied number theory and set theory that will cover the discrete mathematics course, but maybe I am wrong.

@pie Read the whole sentence.

I said "some kind of introduction to higher mathematics course".

Some kind of very elementary discrete math course is often used for this purpose in the US, but any kind of class could serve this purpose. It is not so much the content of the course that matters. Rather, the course is used to introduce students to the idea of how to prove things.

@pie it wouldn't cover discrete mathematics, no

For me, the course that filled that transitional role to higher math was real analysis out of Folland's undergraduate text, as well as another book by some authors who I've forgotten. Dangello, maybe?

Yeah, Dangello. amazon.com/Introductory-Real-Analysis-Frank-Dangello/dp/…

nobody is going to give you permission to read a book. alternatively, i give you permission to read any book

who gives a crap who anybody says a book is for

i'd cosign everything xander said btw. your random person at a college is probably going to be out of place in a GTM. every college has students who would not be out of place in a GTM

if a book is confusing you, read something simpler

if a book is not confusing you, read something more complex

@leslietownes This. Like I said above, the particular GTM volume which started the conversation is open access. You can just download it an read it. If it is too hard, try another book.

I have a book on my shelf that I paid \$60 for, and which I have attempted to read multiple times. I make a little more progress every time, but it is a difficult text. Someday, I might finish it...

every book should frustrate the reader a little bit

what's the point of reading something that you already know

what are necessary and sufficient conditions on f so that $f(x^3)$ is smooth?

can someone pls help

$f: \mathbb{R} \to \mathbb{R}$

In order for $f(x^3)$ to be smooth, it is both necessary and sufficient that $f(x^3)$ be smooth.

smooth=$C^{\infty}$ ?

@SineoftheTime yes

@XanderHenderson how to prove that what you say is necessary?

and what about f(x) = x^{1/3}

@smth ???

in order for $P$ its necessary and sufficient that $P$

@smth Do you know what a "tautology" is?

it depends on what is is

@copper.hat Thanks, Mr Clinton.

@psie glad to hear it, sorry about the slow response, my church is a fire trail in a nearby park :-)

haha :) ok, cool

@Jakobian I agree

I am confused about this: for f(x) = x^{1/3}, f is not smooth, but $f(x^3) = x$ is smooth

@XanderHenderson would not let him near my daughter, but i think he was one of the most effective recent presidents.

You have to be careful with those exponents

@copper.hat Oh, I agree 100%.

@smth why is that confusing? $x^3$ 'smooths out' the curve at 0

I am disappointed with his leadership of the party since then---I think that the Clintons are very much an important factor as to why there don't seem to be any young, viable democratic candidates in national politics.

@XanderHenderson sorry, I haven't read your answer well, but I still have question, how I know that this condition is not too general, how to know that there is no better one?

But he did actually manage to get a lot of his agenda through. Much more so than Bush or Obama.

@smth Did I make any such claim?

the irony was that many of his policies were 'Republican' in nature

I said that, in order for $f(x^3)$ to be smooth, it is both necessary and sufficient for $f(x^3)$ to be smooth.

@XanderHenderson but why is that necessary?

@copper.hat Indeed. Like most Dems since the 80s, he is/was, at best, center right.

how I know that?

yep, cultivation of replacements is not a strong point

@smth How can $f(x^3)$ be smooth if $f(x^3)$ is not smooth? Clearly, this is a necessary condition for the smoothness of $f(x^3)$.

Again, do you know what a tautology is?

@XanderHenderson yes, alright

@smth he is not adding any infomation, it is like saying a car is white iff a car is white

The statement I made is completely empty. In general, in order for A to hold, it is both necessary and sufficient for A to hold.

@smth with any invertible function $f$, smooth or not, you have $f(f^{-1})(x)) = x$.

It is a kind of joke, which is meant, to a degree, to point out that your question is kind of vague. What kind of condition are you looking for?

I get that it is tautology alright, but I am sure that that answer is not what I need

@XanderHenderson sorry, I see

Let me think

@smth It is not clear what you do need. What is the context in which you are asking this question?

To be honest, I have no idea, so the whole problem is a bit more, I am given two atlases on $\mathbb{R}$, and the first one is $(R, id)$, and second is $(R, \psi)$, where $\psi(x) = x^3$. And the question is What are necessary and sufficient conditions for f:R \to R to be smooth, when domain is with first atlas and codomain with second

and when I write down coordinate representation

I get x \to f(x^3)

and I what that to be smooth?

by smooth I mean C^\infty(R)

another question is same but f is now a map between R as manifold (R, psi) and R as manifold (R, id)

here, I need (f(x))^{1/3} to be smooth

you need to be careful what metric you are using on domain/range

metric?

why I need metric here

@copper.hat @XanderHenderson

@smth what do you mean when you write $(\mathbb{R}, \psi)$?

only one chart, map psi: R \to R, psi is what I have mentioned before

$\psi^{-1}$ is not smooth..

i thnk i am missing the point of your question

smoothness is relative to a set of coordinate charts, which is presumably where we're getting

you can just ask the question

@c

@copper.hat why we need that to be smooth?

and also, isn't it smooth between (R, \psi) and (R, id)

if you want a clear answer, you need to ask a clear question

can you ask me what is not clear?

because I am stuck here

Presumably, you are attempting to answer a question from some set of lecture notes or from a text of some kind. Perhaps start by (a) telling us the source of the problem and (b) carefully copying the precise statement of the question into chat.

This feels very much like an XY problem to me....

xyproblem.info

@Jakobian What? so should I read a book on it?

@pie what for

I just like to learn math, I don't know what to study or what not to study, is the course important for future courses? and if not the stuff that number theory or set theory won't cover it where can I find them (other that discrete math )

@pie What is your end goal?

"I like to study math" isn't really a goal.

Also, at what level are you?

@XanderHenderson at the moment I don't have a clear goal for studying math, I just study math because I like it (although it is not my major ), I want to be a mathematician and get into academia. I read baby rudin, hoffman and kunze and parts of Ahlfors and ODE book

@pie So you are an undergrad?

yep

Okay, so, as an undergraduate, there should be a degree program (or degree programs) in mathematics. There should be an outline of courses you need to take in order to obtain a degree in mathematics, or to complete a minor in the field.

Take those classes. Talk to an advisor in the mathematics faculty, and see what advice they have.

@XanderHenderson I appreciate your advice, but unfortunately, I can’t take a minor in my country because our higher education system doesn’t allow it. Additionally, the quality of higher education here, especially in mathematics, is quite poor. That’s one of the main reasons I didn’t choose to major in math. It’s also why I don’t follow the courses offered here or seek guidance from advisors in the mathematics faculty.

@leslietownes confidence building

pie i would take all of xander's advice as applicable to the system in whatever country you want to get into academia in

its harder to see what people are doing in universities than it used to be because everything is increasingly behind course management software

but it is still sorta possible

the new castles

are they also upon tyne?

once