12:23 AM
Is there a trick to checking whether or not I can isolate $y$ from an equation? For example, I have something like
$$y + 2\ln |y+1| = x + 5\ln |x-3| + C$$ Aside from spending 30 mins playing around with my algebra, is there anything I can look for?

12:35 AM
@AkivaWeinberger Hello Akiva
@AkivaWeinberger Did you read Rudin book ? analysis?

@CookieToast I wouldn’t hope for an explicit solution for $y(x)$ in that case, no
It’s not always obvious though

@Semiclassical is there a general method of telling that a function has no closed form solution without having to mess around with algebra for a while?

General method, probably not
But if you’ve got both powers and exponentially of y, that doesn’t bode well
For instance, there’s no elementary way to solve $x=ye^y$ for $y$

so can we just get wild embeddings of any surface

yup

12:49 AM
by cutting the horns off of alexander's horned sphere, cutting out a disk from the desired surface, and gluing the horn-cap in?

take connected sum with alexander horned sphere

Gotcha @Semiclassical

sweet

Lol the guy across the street from my house has gotten locked out of his car with the car horn stuck

Sweet

12:57 AM
meant to say *exponentials not exponentially above
What's going on here right now is snow and a good load of it

How does one operates with Big O notation
it is very comfusing soemtimes

@Jacksoja I would if I didn't have so much homework. Our first exam is in a week and a half, and our prof hasn't even started lecturing :O

it's up to about 9 inches now

@Semiclassical do you live somewhere that snows a lot, or is this a new thing?
@Jacksoja Diff. Eq. :P

1:00 AM
Minnesota

@CookieToast no way your prof will run thru everything in 1 week

so it's not unusual to have a storm like this in the winter, but it's definitely not what we get on a day-to-day basis.

Ik, I don't know what he's thinking @Jacksoja. Luckily I got ahead over winter break.
@Semiclassical My roommate is from MN and she says it can get pretty hairy there

yeah
it's not cold right now at least
by which I mean it's 30 F outside
so it's freezing, but it's not too cold :P

1:16 AM
An irrational number is a number not expressible by a ratio of integers, which means that its decimal expansion goes on forever, and never repeats in any sort of pattern. Does that mean that it is impossible to theoretically come up with a function $f(n)$ that will return the $n$th digit of an irrational number?

The Bailey–Borwein–Plouffe formula (BBP formula) is a spigot algorithm for computing the nth binary digit of pi (symbol: π) using base 16 math. The formula can directly calculate the value of any given digit of π without calculating the preceding digits. The BBP is a summation-style formula that was discovered in 1995 by Simon Plouffe and was named after the authors of the paper in which the formula was published, David H. Bailey, Peter Borwein, and Simon Plouffe. Before that paper, it had been published by Plouffe on his own site. The formula is π = ...

For example, $f_{\pi}(1) = 1, f_{\pi}(2) = 4, f_{\pi}(n) = digit(\pi, n)$

So at least in binary, there actually is an algorithm

Does this concept extend potentially to all irrationals?

No clue. I'd guess not.

1:18 AM
Also, what's the difference between an algorithm and a function?

the number of manipulations involved, really
or, well

I ask because I'm curious about the function $f:\mathbb{R} \to \mathbb{R}$ where $f(x)$ returns the average of the digits of $x$. I was thinking for irrationals you could just take the limit of the Bailey-Borwein-Plouffe-esque formula for each irrational as $n \to \infty$.

the function is the mapping (in this case, from n to the nth digit)
the algorithm is the process which cranks out this mapping.

Ah, so an algorithm is basically a function involving "steps".
And even if not, $f:\mathbb{Q} \to \mathbb{Q}$ would be interesting

There's a remark to the following effect: "D. J. Broadhurst provides a generalization of the BBP algorithm that may be used to compute a number of other constants...Explicit results are given for Catalan's constant, $\pi ^{3}$, $\log ^{3}2$, Apéry's constant $\zeta (3)$ (where $\zeta (x)$ is the Riemann zeta function), $\pi ^{4}$, $\log ^{4}2$, $\log ^{5}2$, $\zeta (5)$, and various products of powers of $\pi$ and $\log 2$."
So there's some crazy stuff out there
That said, there's a big difference between pi and an arbitrary irrational number.
We know a looot of formulas for pi.

1:25 AM
$\pi$ is stupid, $\tau$ is life!

I prefer to eat at least $3$ pi, thank you very much :P

I am confused... what do you mean by "average number of digits" of an irrational number?

I'm not completely sure, but I was thinking that if a function exists to calculate the nth digit of an irrational, then maybe you could take the $\lim_{n \to \infty} \frac{f(n)}{n}$

That will go to zero
since $0 \le f(n) \le 9$

@xander I by no means have a well thought out cohesive definition here :P
Yeah, whoops. It should be something like $\lim_{n \to \infty} \frac{\sum_{k=0}^{n}f(k)}{n}$

1:31 AM
If $a_n$ represents the $n$-th digit of the decimal expansion of a number, it might be meaningful to ask about $$\lim_{n\to \infty} \frac{1}{n} \sum_{j=1}^{n} a_j,$$ but I suspect that this would be $\frac{1}{2}$ for almost every real number

where I mean "almost every" in the sense that the set of numbers such that the above limit is not $\frac{1}{2}$ is of Lebesgue measure zero

Would you define lebesgue measure for me real quick?

The set of natural numbers which aren't even primes is Lebesgue null, now that I think about it

Though I still understand your overall meaning

1:32 AM
and by $\frac{1}{2}$, I actually mean $4.5$

:thinking:

or no
that's not what I mean
what do I mean?
yes! 4.5...
@CookieToast Define Lebesgue measure real quick... uh... okay
um...

Just wave your hands, make some vague references, say "lebesgue" a few times, and say "let that sink in for a few hours"

1:34 AM
basically, you want to be able to measure the "sizes" of arbitrary sets of numbers (or whatever)
and you want your tool for measuring these sets to give the "expected" values to easily understood sets

>arbitrary

for example, the "measure" of an interval $[a,b]$ should be $b-a$
@Daminark I'm getting to that!

>tfw no AoC
Lolol yeah fair I'm just messing :P

It turns out that you can't measure completely arbitrary sets in this way, but you can come close-ish
The tool that does the job is called Lebesgue measure

1:36 AM
It's a pastime of mine to just interject inconveniently with technical stuff

So its an aproximation of sorts?

it isn't an approximation

$\longleftarrow$ knows nothing about set theory :P

Lebesgue measure can be though of as a function from a subset of $\mathscr{P}(\mathbb{R})$ to $\mathbb{R}_+$ that assigns a "size" to sets
and this isn't set theory, it is measure theory

1:37 AM
You define the length of a set to be the inf of the lengths of coverings by intervals. For some sets, you try that but it turns out it just doesn't make sense, so you cop out and say they aren't measurable

or you talk about the Hausdorff outer measure ;)
in dimension 1
which is the same as Lebesgue outer measure on the real line
BUT BETTER!

Tru
Actually we should define the Lebesgue measure as the Haar measure on R

That is the typical approach, honestly
except without the word "Haar" thrown about, 'cause topological groups scare baby grad students

Lol I feel like I'm in the womb rn

but I prefer to think of Lebesgue measure on $\mathbb{R}^n$ as the $n$-dimensional Hausdorff measure, normalized so that the unit cube has measure 1
because I am an analyst, not an filthy algebraist

1:40 AM
You're still an analyst. Disgusting

But yeah usually I just thought you do either Caratheodory directly, or if you're a hipster you use Riesz representation

Hey, if I have $\frac{dy}{dx} = \frac{f(x)}{f(y)}$ that must satisfy $y(b) = b$, then I end up with $\int f(y) dy = \int f(x) dx +C$ and $C$ just equals zero right?

you need Caratheodory to get a complete measure, and one typically gets Hausdorff measure by applying the Caratheodory construction to the Hausdorff measurable sets

Ugh geometers, go back to those jumping jacks you do whenever you explain something

I think that using the RRT would be circular somewhere along the way, but I can't put my finger on why

1:42 AM
Waving hands so much you borderline achieve lift

@Balarka

I am not a geometer

then what ARE you?!

@XanderHenderson I actually think it's been done, you can define this Daniell integral or something and it just works out

1:43 AM
I am a topologist. I wave my hands so much I fly into the outer space

ha!
@Daminark There is probably a way of doing it, I'm just not at all comfortable with it

I mean after that Lie bracket stuff you were doing earlier I think you're starting to change. Denial is the first stage

ha!
axiomatizing integration seems like a cop out
Also, I worry that if we abstract integration too much, we'll end up doing category theory.
and that sounds painful

"ears perk up*
Actually one of the problems we had in functional last week was to show that if you have some normed space X and a closed subspace Y such that Y and X/Y are reflexive, then X is reflexive

oh, the wife and kid are home; time go make dinner
laters

1:47 AM
The grad students found this functorial way to do it cleanly I think, it was quite painful directly
See you!

bye @XanderHenderson!

@Daminark Dami my man :D
@Daminark I kinda need help with some analysis

I can try

@Daminark can I send yoyu email ?

@Daminark did you read the book of Rudin ?
because the question is from there

1:55 AM
@Jack sure, what's yours?
@Kasmir I have gone through a reasonable amount of it some time back

Hmm nice
@Daminark you got same email ? i Think sending this is better than typing here

Sure

@Daminark Done :)

2:10 AM
hi Demonark and Kasmir

I got nothing
Hey @Ted!

@TedShifrin Ted :D
@TedShifrin am doing real analysis and I kinda need some help :D
mosly on how would one succeed in such course

Having already done algebra proofs, it's reasonable to try analysis, Kasmir. But it's hard.
A good portion of my videos is (multivariable) analysis stuff.

nice,so I got this question about closure
E (bar) = intersection of E_alpha

Um?

2:17 AM
where E_alpha denotes the family of all closed sets containing E
I don't quite get that ´><

Oh, you should have said that.

haha

So the closure is the smallest closed set containing $E$.
That definition is given in my course, yes.

looking it up right now :D

2:20 AM
well let me see if I got it right
We have E subset of X
We define something called the closure of E to be the smallest closed Set containing E
Hmm many questions arises
well first can this intersection be empty ?

Not if $E$ is nonempty.
If $E$ is empty, the answer is, yes, the empty set.

I dont have a feeling for this, and why is the empty set both open nd closed

well a set is closed iff its complement is open

Well, the complement of the empty set is $X$, which, of course, is open.
(and closed, as well)

2:24 AM
:D
its very odd

Nah.

I mean do we have use of them being defiend both as open and closed?

No, you can check from the definition that $X$ is open (as a subset of $X$) and that the empty set is open (vacuously), so each is closed.

from the titlles of lectures nothing hints about containing this material ><
okay thanks Ted =p one last comment
in defining closure of a subset E, why do we need the intersection ?

Because that's how you get the smallest closed set containing $E$.

2:27 AM
can you give me some intuition behind it , sorta of a Picture to understand it

Just throw in barely enough points to make $E$ be closed. Those are called limit points.

okay thanks :D
Ill keep Reading and see if it makes more sense now :)
I have few homework problems for later, if you still here Ill ask ya :D
You are the best Ted :D

Look for the lecture on closed sets (after the one on open sets).

let me see if I can find it
oh open sets in R^n
but no closed sets
lecture 14,15 I assume

2:31 AM
you had in 14 , balls in R^n
I assume that one has something to do with this :)
ill check them out :D

Should be after that.

yepp :)
watching now :)

@Kasmir: I just finally got in on a different browser. Lecture 16. Sequences and Closed Sets. Grr. You should have figured that out.

@TedShifrin Yes but accually it starts at the end of 14
so i need to Watch end of 14 ,15 and 16
I did figure it out :D

Well, for all the open set stuff too.
OK, have fun. :0
BTW, my definition of closed set is different, but I prove that it's equivalent to your definition.

2:35 AM
haha yeah I wanted clear understanding so I ll start from there =p I have my lecture in 9 hours , so I got plenty of time to be prepared =p
Roger that ! we also use different definitions in class
and part of HW to prove the equiv of them
@TedShifrin thanks again as allways :)

2:51 AM
what is the correct way to write a domain of a function is R but not 2 specific numbers ?

1 hour later…
4:08 AM
Morning chat

4:27 AM
lies
it is evening chat
because 'merka!

wakes up after sleeping since like 4pm
it is 11:30pm
Good morning
@Faust Where are you, anyway? Australia?

I lives in the Canada its near the USA

Thanks, I did not know where Canada was

>morning

:thonk:

@AkivaWeinberger its really big actually, i live on a little tiny small island but its apparently huge.

4:36 AM
Hi handsome folks
Morning Faust

@KasmirKhaan !

I could not figure out your time zone

hows the math going?
its currently 8:37pm

you said morning in each 6 different 4 hours period of the day

i am as far west as you can get before it becomes east.

4:37 AM
okay so not morning -_-
am taking 4 Courses faust
it is hell
wbu ?

lol i did 5 last semester
only taking 4 this semester :)

how did you do it?
Dami stop starring messages

93.5% average but i did hw 80hrs+ a week

Wut

that cant be true
:D

4:39 AM
managed to get 5 A+'s

11 hours a day ?

how do I find the minimum of $\dfrac{3\sqrt3}{\cos \theta}+ \dfrac{1}{\sin \theta}$ using AM-GM.
attempt:

12hrs a day on weekdays 10 hrs on each weekend day

well done man :D
I also study alot but i take breaks in the middle

@Faust Right, so you're on Vancouver Island.

4:40 AM
so the net would be 8 hours
sup akiva

I'm on an island as well but it's Long

long Island ?

im Autistic so its hard to remember to take a break, by time i need a break i havent moved in probally 8hrs+

$\theta \in (0, \dfrac{\pi}{2}) \implies \dfrac{3\sqrt3}{\cos \theta}+ \dfrac{1}{\sin \theta} \ge 2 \sqrt{\dfrac{6\sqrt 3}{\sin 2\theta}}$

Faust you continue to surprise me

4:41 AM
actually fell on the floor yesterday because of it

why ?

@AkivaWeinberger yeah =)

how are you now?

@KasmirKhaan Yeah

for minimum of LHS,
$\sin 2\theta = 1 \implies \theta = \dfrac{\pi}{4}$

4:42 AM
i forgot to move for about 8 hrs and i was in an awkward position when i tried to move my back was locked up, so when i tried to move i just fell on the floor

Sorry to hear that

Though I literally mean the island
I'm in Brooklyn

actually laid there for about an hour

Oh wow

before i could get up but my backs better now

4:42 AM
Brooklyn best borough tbh

take some 30 mins walks each day man

@AkivaWeinberger can you send me some pizza i hear its good there?
2

its not healthy to sit alot
fax it

i cant walk when i thinking about math
i realize hours later that i am really lost with no idea how i got where i am

Faust tbh you did not seem so dedicated before
its very good to hear that you taking things serious

4:44 AM
being in school costs alot of money if i do well people give me money so i should do well ^^

neat :D

i also like math

did you convince mathein to do that thign on summer?

anything i like is easy to do.

he is your best help imo :D

4:45 AM
yeah

exactly

he said he would

neat :D

i half want him to do it german

But if you could include me in it would be great too :D
I want soemthing to do on summer ( math related )

4:46 AM
sure summer is a ways off he seems nice so should be ok =)

:D
anyways
Kasmir got some analysis thing to do
So see ya later yall :)

@KasmirKhaan i am supposed to do a research project on some wierd non-commutative geometry and c* algebra
you can do it with me if you want.

algebra is nice
5
am taking rep theory now

lucky!

if i do well there, id be able to help on algebra
haha why lucky

4:48 AM

it is not

and i want to take it

i just got in because not many took it
also like 7 more undergrad with me

the rest are master or more

4:49 AM
run away!

@XanderHenderson you are the only person here I want to hit :D

@XanderHenderson we didnt even bring pitchforks

anyway Peace handsome folks

aw :(

gnight

4:50 AM
night

Night

@Xander you're outnumbered!
We will attack with... idk homework help or smth

i hope everyone had a good day

heh
I held office hours today and NO ONE SHOWED UP
it was a good day

i just goto OH and ask random crap i find intresting

4:55 AM
that is what you are supposed to do

i asked my linear algerbra professor about why i needed the axiom of choice for a topology problem and some random question about set theory

as long as you give priority to those that actually have questions

so far havent asked a question related to what we have covered in class but meh.

but if you are thinking about asking for letters, then it is good if the faculty know you, think you know what you are doing, and don't find you annoying

lol

4:56 AM
you laugh, but it's true
DON'T BE ANNOYING!

i have no idea what i am doing

undergrads seem to have a real problem with that one

what means annoying?

Along similar lines, how would one know if they annoy someone? Like, I don't think too many people are openly pissed but I can't say if any of them just roll their eyes completely despite seeming rather happy

do they make eye contact?
or wait... no... they're mathematicians
hrm...
By the way, do you know how to tell if a mathematician is an extrovert?

4:59 AM
eye contact hard for me i always forget when thinking

I mean I think they do for the most part?

They look at your shoes while talking to you, instead of their own.

why?

It's a joke

Okay that is gold

5:01 AM
i no get it

More seriously, if people seem like they are engaged with you and enjoying your company and/or questions, they are not annoyed.
But if they start checking their email, or watching the clock, or playing tetris on their cleverphone, you should leave

hmm
sounds like more of a having to deal with first years asking silly things
i had one ask today, what if i let my set be an element of itself and then i do .... blah blah
so the universe always contained the universe as a set element
i stopped listening after that but he kept asking the prof

I don't think that's particularly prevalent here, first years are likely to be a bit intimidated for that

There's a bunch of set theories that allow $x\in x$

Man, back when I was an anthropology major, I took a sociolinguistics class.
One of the other students in this class was an English major.

5:09 AM
In fact, you can have multiple objects whose only elements are themselves.

He just couldn't get enough of the word "plosive".

(Despite the fact that it seems like they'd all be $\{\{\{\dotsb\}\}\}$ and thus equal)

Which wasn't even relevant. I feel like that is the equivalent of "But Russell's paradox is a thing I read about on the eenterwebs!"
@AkivaWeinberger Yes, like the (removed comment).

For some reason I thought you said sociology and not sociolinguistics

5:10 AM
My sister likes the word "fricative"

k well im exhusted so m e sleeeeep now

mmm... fffffffffricatives!

Ffffffffφφapproximants

that was a cool class---it was taught by one of Chomsky's students

I like how the words begin with the sounds they describe. Plosives. Fricatives. Jaffricates.

5:12 AM
heh
okay, I gots to sleeps
laters

Nasals. Sibilants. Laterals.
Rhotics.

5:29 AM
Why is that people answer here on Math.SE much more quickly than Physics.SE or Chemistry.SE?

5:46 AM