Mathematics - 2021-10-19 (page 2 of 2)

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7:01 AM

@Koro $[-1,0)$ and $(0,1]$ are open sets in $S$.

$S$ is disconnected if there exists $U,V$ non empty disjoint open such that $S \subset U \cup V$ and $S$ intersects both.

Note that $S $ is contained in, not equal to the union of two open sets.

Clearly $\{-1,1\}$ is not connected in the reals, and equally clearly not the union of two open sets.

@love_sodam: yeah but how does that help me proving that S is disconnected in R?

good night folks!

gnite

@copper.hat it seems intuitively correct but never saw this definition before.

good night @copper :)

7:05 AM

Well, clearly the union of two open sets is open, but there are non open sets that are disconnected, so the definition you are using cannot fly.

@copper.hat hmm, that's my confusion. How does the open set definition hold in such cases ? :(

it doesn't

i don't know where you are getting that from.

here for example: math.hws.edu/eck/metric-spaces/connectedness.html

there is a difference between open in the ambient space and open in a subspace.

yeah, i understand that and that's why I asked about connectedness of S in R.

and not in S.

7:08 AM

either you are working entirely in $S$ or you are working as a subspace. take your pick.

For this question, I'm working in $\mathbb R$, the set of reals.

read definition 6.2 carefully

ok

If $S$ is as you have above, then $[-1,0), (0,1]$ are open (in $S$).

ah, I got it @copper. Thank you so much :) :)

I was using the definition wrongly.

7:21 AM

any way to think about the fact that $\lim \limits_{x \to 0} \frac{1}{x} - \frac{2}{\sin 2x}$ converges to 0? sketching a table of results using $x = \frac{1}{n}$ with $n \in \Bbb N$ does not make this obvious at all and makes it look like it converges, but using l'hopital's rule we get the convergence to 0

or $x = \frac{1}{-n}$, with $n \in \Bbb N$

oh, i was thinking of sine as cos

terrible mistake

nevermind that question

2 hours later…

9:41 AM

It seems that arXiv is down: isitdownrightnow.com/arxiv.org.html downforeveryoneorjustme.com/arxiv.org

1 hour later…

10:49 AM

59351734: I'd like to edit large parts of this message and the ones that followed it, @robjohn.
That the Taylor expansion occurs uniformly in the integration variable $u$ presumably means that, for arbitrarily small enough bandwidths $h$, the remainder $g$ can get arbitrarily close to $0$ independently of the value of $u$. However, it seems like $g$ can still be a function of $u$ despite this fact, if it was understood correctly. Hence estimating the integral in which it appears in by simply moving it outside the integral is maybe not possible, i.e. $$F(h)=\int_\mathbb{R} k(u) g(h,u)\mathrm…

@MartinSleziak it's up for me now, but maybe you know that already.

This and the two that were sent shortly after.

@schn which message?

@robjohn Yes, arXiv seems to be working again.

@robjohn I used the "reply to this message"-function incorrectly in the message above, however, the link in the following message seems to work.

11:08 AM

@schn The condition you need is $\lim\limits_{h\to0}\sup\limits_{u\in\mathbb{R}}\frac{g(h,u)}{h^m}=0$.

3 hours later…

1:50 PM

@robjohn Please undelete this post, which was wrongly deleted by a moderator.

2:35 PM

@copper.hat. You are like that on measure theory!!

I did not know it's that huge and big topic in math. Thanks anyway.

@copper.hat. Can you please recommend video lectures/book other than the one in Stanford! That one is too rough.

Alessandro Codenotti

3:19 PM

Terence Tao has an introductory book to measure theory that is very well written

3:39 PM

i like bartle's 'integration'

4:03 PM

of course, Bogachev

too new for me

Never hoyd of that.

Alessandro Codenotti

@Thorgott Bogachev is missing too many important results, Fremlin is a much better choice

per google, it came out in 2006. just after i stopped caring about measure theory

measurable cardinals are not important results :P

4:07 PM

1100 pages ... good for reading in one night

Alessandro Codenotti

(Bogachev is seriously good, but I would most definitely not recommend it for a first exposure)

I was going to say that it looks both advanced and sophisticated.

Alessandro Codenotti

@TedShifrin Fremlin is 5 or 6 volumes of crazy technical and advanced measure theory

it's probably the single most densest and technical math text I know

also, it will take you a lifetime to do all the exercises in Bogachev

Well, this was never a subject that appealed much to me, anyhow.

4:08 PM

there's a ton of them, and most really hard

Alessandro Codenotti

It contains everything you need at a level of generality way higher than you need, together with a billion things you don't need to know

sounds great

As I said last night, the grad course I took was from Segal & Kunze, so the Daniell integral. I read Halmos and Royden when I was studying for my analysis qualifying exam first quarter at Berkeley.

volume 12,000 of Answers To Questions Nobody Asked, a series of books

Alessandro Codenotti

@Thorgott just today I was looking at the proof of analytic determinacy from a measurable with some other phd students

4:10 PM

royden is OK but not popular enough for it to be reprinted often enough. the version i got for a class, bought new, had glue and sets of pages falling out of it

you're a bad example

alessandro, stop doing that

Modern book production. Both my linear algebra book and my multivariable math book fell apart on numerous students. I actually made the publisher replace them, but they would only do it for students who'd bought them new.

i remember thinking 'this would be a classic text if i could find pages 130-150 for my next homework assignment'

I raised a huge stink. Freeman finally threw away hundreds of books and made their printer do a redo. I think the main issue is cheap glue and not enough used.

4:15 PM

Why can we use concepts of set theory when studying logic

in law school i ordered a book at a substantial discount from its author, i think i even put cash in the mail, because water had invaded where the books were stored and there was a run where all the copies had minor damage

@Math Huh?

I mean it’s just a question since logic is supposed to be the foundation and come before set theory

yet most logic books use the notion of a set

@Math math.stackexchange.com/questions/1681857/…

the above is an example of how you can extend a first-order deductive system to be able to prove set-theoretic statements

Sure but it doesn’t answer my question

4:18 PM

Set theory is part of logic.

logic books are written at varying levels of specificity. i'd expect a good one to be clear about what it is assuming, even if what it is assuming is set theory.

Ooo Hellowww

@Math I wouldn't necessarily say that logic is supposed to come before set theory. If you have no sets, how do you make sense of quantifiers?

Huh? What does that mean?

I mean predicative logic, dealing with the quantifiers $\forall$ and $\exists$

if you have no set to quantify over, it's hard to use them

I'd say a combination of first-order logic and set theory is the foundation of math

4:26 PM

And where do predicates live if not in a set

Salut @Astyx!

of course,you can do propositional calculus without set theory, but as soon as you get to first order logic, you need some set theory

Salut!

Comment ça va ?

Of course, at the naive level (e.g., teaching the typical US intro to higher math course), basics of sets (inclusion) run parallel to basics of logic (implication). I found it helpful to the students to make this stuff all clear.

Ça va, plus ou moins, @Astyx, et toi? Il nous faut des crêpes, n'est-ce pas? :D

@TedShifrin yes, when I TA'ed freshman LA dealing with sets and logic at the beginning, I also stressed the connection between logic and set theory like impication and inclusion, intersection and cojunction etc.

4:31 PM

Apparently when the Bourbaki influence was at its paroxysm, kids did set theory in kindergarten

@LukasHeger Observation: Say you have a sheaf on the site of relatively open subsets of a topological space $X$, given a filtration $\{X_i\}$ of $X$, as we discussed earlier. Then we could (1) restrict it to each "grade" $X_i$ or (2) restrict it to each "associated graded" $X_i \setminus X_{i-1}$. The former produces a sheaf on a similar site, the latter produces a genuine sheaf on a topological space.

@Astyx my parents did some set theory in elementary school

@TedShifrin Moi ça va :) à quelle occasion ?

Hi balarka!

When you state the axioms of ZFC you don’t quantify over a set.

hi @Astyx

4:33 PM

@Math you need a universe of discourse

im doing topos theory as you can see

I keep forgetting what a topos is

i dont know what a topos is lol

its the category of sheaves on a site or something

yeah, that's a fine definition

I know 2 out of those three words !

4:35 PM

of course there are theorems characterizing which categories arise as the categories of sheaves on a site, which can be taken as alternative definitions

@Astyx pourquoi pas? pour refêter notre réunion il y a plusieurs ans.

@BalarkaSen on a camping site?

what a site
said the blind algebraists

badum tss

@TedShifrin Je ne dis jamais non à des crêpes

Tu es sur Paris ?

Je ne voyage point à ce point-ci :(

(But sorta cute pun.)

you can take stalks of sheaves on a site right?

4:40 PM

i don't know what's happening, but i want a crepe

just direct limit

@BalarkaSen yes

excellent stuff

but stalks won't give you as much information if your topos doesn't have enough points

yeah but shouldnt be an issue for my topos

4:42 PM

true

i think my stalks are pretty much direct sum of stalks if you approach a point from many different directions in the filtration

seems true

i cant believe im doing this

I can't believe you've done this

then stop doing it

4:46 PM

can't, i need to use this language to prove something i want to

it's not too late

@Balarka has indeed gone to the other side.

sigh

no, gromov uses sheaves of simplicial sets on topological spaces

i am just doing it on a site instead

it cant be helped

i will donate to a fund to support people who have to deal with this in real life, in your memory

4:50 PM

lol

slight change of subject, but this morning my daughter saw my cat sniffing around a plate where we'd had some cat grass, and she immediately said "LIVVY NEEDS MORE CAT GRASS." she's fully internalized the concept that we are here to serve the cat

yes, dogs have owners, cats have servants

5:14 PM

My cat, having damaged her various claws after surgery, now gets a second week of head in cone and drugs. Ugh. Yup, servant.

@leslietownes The earlier, the better. It prevents a lot of anxiety when serving cats later.

@robjohn Is your pup feeling better?

@Ted sorry to hear it! i hope she isn't using the drugs recreationally.

@TedShifrin there are small improvements, probably due to some of the meds, but we still have not heard the results of the tests that were done last week, so we don't have anything better to do for her. She is hanging in there, though.

@TedShifrin I am not only a cat slave (three masters), but I am also feeding, medicating, and taking the sick pup out to the backyard every 2 hours.

Crazy. The test results were due a week ago. I dunno what you do other than throw a tantrum.

5:23 PM

The tests were done a week ago.

It feels like longer.

It was last Tuesday. If I don't hear back by this afternoon, I will call and inquire.

Ah, my bad.

But you're right; it does seem like longer. I am busy most of the day caring for her at this point.

preparing the food and meds and feeding her takes about 45 minutes of each 2 hour period. I only feed her 6 times a day, so that gives some rest at night these days.

Hi guys , good morning !

5:28 PM

Hi, BKJA.

Oops, its actually Rover , I just recharged to Rover, but it seems it didn't worked :/

the hawk is back. second day in a row. perches on a nearby tree and does the trademark hawk call.

some profile updates need a little time to go into effect across all instances of your identity.

Hm ok

it's rover now. at least that's what i'm seeing.

Ok

5:33 PM

I see Rover

i saw BKJA1 until shortly after my remark regarding the red-tailed hawk.

i love that hawk sound. keeeeeeeaaaaaaarrrrrrr. they use it in movies to stand in for every predatory bird, even where the red tailed hawk is not native.

Yes, I saw rover this time, it took time

birding can detract from your enjoyment of movies because of shoddy sound work. you'll be watching something, and then wonder "they better have a scene that explains why the northern mockingbird is apparently outside this building in europe."

tears you out of the moment.

@Leslie: I have never seen a vulture

koro, you need to visit california. we've got 'em.

5:35 PM

:)

I used to think that they went extinct

they're ugly birds, and kind of scary up close. they're much larger than they look.

the california condor is vulture-like in appearance and also terrifying up close. they are enormous birds.

they are an amazing story of conservation. at one point there were fewer than 30 of them.

enormous. wingspan is something like 10 feet.

@leslietownes woww !

wow, didn't like me linking that image.

Why can't a person have an irrational height?

i dunno, why?

5:40 PM

I dunno either. Never mind that question please. It just came to my mind :)

i mean, people don't have rational or irrational heights. those are properties of numbers, which require a choice of units.

California condor

The California condor (Gymnogyps californianus) is a New World vulture and the largest North American land bird. It became extinct in the wild in 1987 when all remaining wild individuals were captured, but has since been reintroduced to northern Arizona and southern Utah (including the Grand Canyon area and Zion National Park), the coastal mountains of central and southern California, and northern Baja California in Mexico. Although four other fossil members are known, it is the only surviving member of the genus Gymnogyps. The species is listed by the IUCN as Critically Endangered. The plumage...

photos don't do it justice

those things are huge.

Yuge, u mean?

yes. a lot of people are saying this.

no I meant to consider unit as centimetres and since height of a person increases and then after certain age becomes constant (let's call it limiting height). So could that limiting height be an irrational number (in cm)? That's what I meant to ask but never mind that question please.

i dunno if height is well-defined enough to admit to rational vs. irrational level of analysis.

a friend of mine in law school asked how tall i was, i said six feet, which is true. she didn't believe me, she thought i was taller. it turned out that a lot of men on dating apps round up to 'six feet' because they use inches and not centimeters and the roundness of that number matters to them. i think she broke up with a guy because of this.

i honestly don't think anyone cares about how tall anyone is, which makes it sad to lie about it.

my daughter is going to be tall. her height is 99th percentile for her age.

5:48 PM

Although it may not matter now for a person to have irrational height or rational height but who knows. As far as the question is concerned, as you said Leslie, terms are to be defined first. So I take my question back for now :)

the thing to keep in mind is that rationality vs. irrationality is an infinite precision distinction. we don't have that in real life. the width of a skin cell could throw it off.

if some length in real life can have an irrational length, maybe height is the same?

but most likely all real life lengths are rational

if atoms have a size

@shintuku how about length of diagonal of a square of unit length?

hm nevermind that wouldn't follow, we could have a sequence of atoms of irrational length

i think that irrational numbers are very real!

5:52 PM

yeah you're right, i don't see why height couldn't be irrational

if you could mesure it at arbitrary precision

hmm, we may not have equipment (as on date probably) to measure it though

but likely real life lengths are continuously changing a very slight bit

just contact with a surface probably modifies true size

yeah the question needs re-phrasing/modifications/more clarity but the spirit of the question is clear, I think.

yeah, you'd want a single point in time, about which even a second later would probably change the answer

Building a home with all doors/windows etc. with irrational dimensions will be cool, I think.

5:57 PM

rolls $\pi^e + \root4\of 5$ eyes

most likely the best we'll ever be able to do is to say a length is within some specified $\epsilon$ of an irrational number

an open neighborhood ofc

@Koro If you have fixed units, the probability of having any rational dimension is lose to zero given manufacturing uncertainties.

i'm OK as long as 'being within some positive epsilon of an irrational' is an interesting property. it is, isn't it?? isn't it?! answer me

I disagree. Anything manufactured is measurably rational.

@leslietownes not of any irrational, a given irrational

6:02 PM

haha :) @prof. Ted.: $\pi^e$ reminds of a joke (rather a mnemonic) about why $e^\pi\gt \pi^e$. it goes like this: $\pi$ looks like a bed and $e^\pi$ is a person that has lifted $\pi$ (bed) on its head and $\pi^e$ is sitting on $\pi$ (bed) so the former has more strength and so the inequality.

Can I manufacture an exact diagonal of an exact square? Nah.

Too complicated for me!

that's what i'm saying, we need to be speaking of bounds

i'm gonna patent this

Leslie will sue.

$\pi^e$ is very important in number theory

@copper.hat yeah, I think. We can't measure anything with 100% accuracy

6:04 PM

because $(\pi^e)=(p)$

not even rational numbers.

at Lukas; what's $(p)$?

@LukasHeger Ha ha.

the ideal generated by the prime $p$

primary decomposition.

6:05 PM

@shintuku I was joking, $\pi$ and $e$ have a different meaning in the context that $(\pi^e)=(p)$ holds

Xander Henderson

@LukasHeger and that is equation to $(np)$, right?

which implies $p = np$, of course

Xander Henderson

@shintuku Up to isomophism.

p=np, what is this branch of mathematics?

is it in number theory?

alphabetics

6:09 PM

it is a bathroom decision i face many times at night

@Koro computational complexity i think

the wonders of aging.

Xander Henderson

@copper.hat Just $p$. It prevents you from waking up in the middle of the night.

@Koro en.wikipedia.org/wiki/P_versus_NP_problem

@shintuku ah, I think that's part of computer science?

6:11 PM

@XanderHenderson i suppose it should be Shakespeare's to pee or not to pee

i think it has some aspects of math too

@shintuku @shin: I have heard about it but since this came up in the context of $\pi^e$ above that's why I was wondering :(

no that's all a joke at our expense

:)

there's a computational-complexity tag on mathse, so i guess it is math too

6:15 PM

Anyone knows about Computational engineering?

not getting it's course structure online

there are also tags on MSE for physics, chemistry, crystallography and cartography so I guess those are math too?

this must be some sort of conspiracy

it may be risky to infer relevance from tags. sometimes people who object to closure/deletion cite tags. i do not follow this closely.

working in fuzzy logic atm

good luck.

7:08 PM

A high energy physicist working in lorentzian geometry and quantum gravity decided to propose to they're girlfriend. When she saw the ring they said - will you marry me? She said it's beautiful and I will marry you. Okay said they - just so you know - the ring represents a causal diamond. And I work for the government

7:41 PM

How should I think of a cyclic module geometrically? I know modules are like generalized vector bundles over a space, but I can’t really imagine an interesting situation in which a module is cyclic (maybe something like attaching a 1-dimensional tangent space to a single point, but where does this occur).

Xander Henderson

7:52 PM

@LukasHeger The existence of a tag only indicates that some user added that tag to some question in the past, and that the question has not been deleted. If you look into a tag more deeply, you can get a better idea about what is relevant and what is not.

In genera, the mere existence of a tag does not indicate that everything which could be tagged as such is on-topic.

And there are several tags which I think are generally off-topic (there are history tags, for example, which I think are better asked on the history of math and science site).

I have this condition $$\lim\limits_{t\to0}\dfrac{f(x,t)}{t\phi(x,t)}=0\,\text{uniformly}\, x\in \mathbb{R}^N$$

can i deduce by using the definition of the limit that

$$\forall \varepsilon>0, \exists \delta>0, \forall t\in\mathbb{R}, |t|<\delta\Rightarrow |f(x,t)|\leq \varepsilon |t||\phi(x,t)|,\, \forall x\in\mathbb{R}^N$$

hello

sorry i forget to say it

@XanderHenderson yeah actually I wanted to show that the argument "computional-complexity has a tag on MSE, therefore it's math" by @shintuku is invalid

not saying that computional complexity isn't math

Xander Henderson

@LukasHeger Sorry---I should have read farther up. Consider this me backing you up. :D

And there is certainly an intersection between computational complexity and mathematics.

But neither is a subset of the other.

8:08 PM

what i write is correct ?

3 hours later…

11:35 PM

@copper.hat. Do you have applied book recommendations about convex optimization please other than Stephen!

@Avra I have never found a satisfactory applied convex optimization book. Why do you not like B&V?

We may have different ideas about what applied means...

How's your knowledge on basic manifold concepts Copper?...I'm trying to wrap my head around some aspects of them

or anybody else in the ether presently here...

Xander Henderson

@dc3rd It is generally best to simply ask, rather than ask to ask.

i don't do curvy spacy thingys

:-)

but Ted is a man of many manifolds...

11:53 PM

@XanderHenderson so you're asking @dc3rd to ask, while asking him not to ask to ask?

Xander Henderson

@LukasHeger I wasn't asking them anything. I was giving an explanation. They can do whatever they like. :P

anyway @dc3rd if you have some questions on manifolds, just ask them, maybe someone will chime in

speaking only for myself, i am reluctant to go 'on the record' as knowing X before i know what the question about X will be. this may be a practical concern about asking to ask.

fair enough....... So let me see if I can verbalize the ideas properly through an example:

Say we have a hypersurface: $F(x,y,z) = x^{2} + y^{2} - a^{2}$ (in reality I know it is a cylinder but just for the sake of example assume it is something wild)

since it is a hypersurface it is just a collection of points that satisfy the specified conditions. Assuming that around some point $a$ the conditions to be considered a manifold are satisfied.

Is what the manifold doing is providing us with a "visual representation" of our surface within the neighbourhood of our point $a$? Is that what is …

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