Mathematics - 2016-01-28 (page 1 of 2)

Quality

00:26

Can anyone help to conform an answer about volume of revolution? I have been trying to get answer for two weeks now math.stackexchange.com/questions/1612615/…

Antonio Vargas

00:38

@Evinda yeah, $\sum |x_{mj}|^2 \to \sum |x_j|^2$, so the inequality will remain.

(assuming $n$ is fixed)

I'm an artist

01:16

Late here again, but the results are comeing one after another. Very awesome days here!

coming

Secret

02:48

Is there are term for the most random thing possible?

Suppose we have a bunch of lights A,B,C,D,E,F

A blinks every 2 seconds
B blinks when A blinks 2 times
C dims and brightens according to the function sin^2(wt)
D blinks randomly in a way such that on average the probability of it is light follows a gaussian distribution
E blinks in a pattern that resembles a random walk
F blinks in a way that depends chaotically to A and B

Can there exists a lamp G such that it blinks so randomly that even when one tries to

Joshua A

03:05

... is this a real question @Secret

Secret

Put it in another way, I often heard that when people mentioned about something is random and statistical, they often form normal distributions

I am trying to look for something that is the extreme version of random such that it cannot be described by any probability distribution, and I am wondering what's the math concept that describes that

Or in short, what is the mathematical concept that can give the effects of true randomness?

Secret

03:17

that it is so unpredictable that even if you measure it multiple times, the probability that it happens cannot be determined even in principle, other than it is nonzero

Joshua A

You'll need to look into superposition

Secret

03:36

superposition of what?

usukidoll

03:59

hey guys do you know if the cartesian product is a countable finite set...

Partly Putrid Pile of Pus

04:11

Is the disjoint union of $\{1,2,3\}, \{2,4,6\}, \{1,2,4\}$ equal to $\{(1,0),(2,0),(3,0),(2,1),(4,1),(6,1),(1,2),(2,2),(4,2)\}$

usukidoll

?

Partly Putrid Pile of Pus

If we take the disjoint union of $n$ sets of cardinalities $s_i$, is the cardinality of the disjoint union equal to $\sum_{i=1}^n s_i$

usukidoll

cardinality is the size of the set right?

Partly Putrid Pile of Pus

Yep

Just trying to make sure I 'get' the disjoint union

Oh I do get it, yay

Benjamin R

Sooooo.... there is an accepted answer to a question that states: "Atan2 usually takes (x,y) not (y,x)" which is completely bogus.

3

Q: Find the angle between the 2 points (50.573,-210.265) and (117.833,-80.550)

I am attempting to find the angle between the 2 points (50.573,-210.265) and (117.833,-80.550). Is my calculation correct because a program is giving me a different answer? It says the angle is 27'24'27.27 DMS dx = x2 - x1; dy = y2 - y1; angle = Atan2(dy,dx) * 180 / PI; angle = 62.59242 My re...

trigonometry

The entire answer is basically a comment which is wrong.

How does one edit that?

Quality

04:25

Please help me someone with a question I have been on two weeks

0

Q: Volume of revolution help

I have a question, part of which is more theoretic and one is an application of it which I am also looking to know if it works. For example, say we wanted to calculate the volume of the solid of revolution of the region bound by $y=x^{2}-2 , y=0$ , about $y=-1$ ( but only consider the part abov...

calculus proof-verification definite-integrals

usukidoll

DMS ? As in Degrees Minutes Seconds? wow I hardly see any math problems with that

Benjamin R

especially when comparing it with degrees

Peter Woolfitt

@BenjaminR what's the problem here?

Partly Putrid Pile of Pus

@BenjaminR His comment on angle between points is entirely correct, so it isn't basically a comment which is wrong

Benjamin R

@PeterWoolfitt It's saying that Atan2 takes (x, y) this is completely false. It takes

true

what I said was an overreaction

shall I just remove the misstatement and add a link to Wikipedia for atan2 instead?

Partly Putrid Pile of Pus

04:29

As for the (x,y) thing, I think you may be right

Peter Woolfitt

Oh I see, okay

Benjamin R

I am 100% right about that, but he 100% right about the lines.

ABcDexter

hello everyone ^_^

Semiclassical

bearing in mind that it's from a well-known and repped user, it might be worth checking things

Benjamin R

@Semiclassical that's why I didn't just hack and slash :-)

ABcDexter

04:33

@Quality what's with the current answer?

Semiclassical

but it looks like you're right. atan2(y,x) is the standard based on Wikipedia

it'd be nice if they said which program they used, though :/

Benjamin R

It's the standard for every programming language I have ever used, too.

Partly Putrid Pile of Pus

I can't seem to get either result, that they obtained lol

Benjamin R

Technically it's not exclusively a trigonometry question, let alone a pure trig question, right?

I think leaving a comment is probably best given the answerer's rep he will surely respond.

Semiclassical

the way that problem is phrased is weird

is the 'program result' referring to that from the pseudocode, or something else entirely?

Benjamin R

04:49

I don't think that's pseudocode. Looks like standard C or C++ to me: http://www.cplusplus.com/reference/cmath/atan2/
The whole thing is weird, I am leaving a comment with questions, seems like the answerer understood the context. If so, he can edit the question and include tags etc.

Can you guys refresh and read my comment and see if that is sufficient?

Semiclassical

seems fair. point is that the (x,y) issue seems like it should be attributed to the asker's unnamed 'program' rather than the stated code

Benjamin R

yeah

okay I am glad I asked here rather than just stomped my feet

thank you all!

Semiclassical

np

Partly Putrid Pile of Pus

05:32

What's a type A Lie algebra?

user174558

06:58

Hey @robjohn happy new year!

Balarka Sen

07:22

Someone claims that Jordan-Schoenflies implies given an embedded circle in $\Bbb R^2$, among all the paths which connects an exterior point to an interior point, there must be one which hits the circle only once.

I don't see why that should be true. It seems a very fractal-like curve can be whipped up so that any such path hits the curve in infinitely many places. What am I missing?

Mithlesh Upadhyay

Hello everyone! Would you like to check this integral question.

https://math.stackexchange.com/questions/1579690/the-value-of-double-integral-int-01-int-0-frac1x-fracx1y2-dx

Balarka Sen

Ok, I am thinking of smooth paths, in which case I am pretty sure this is false. But I still don't know why it should be true.

Balarka Sen

07:45

Nevermind.

usukidoll

08:01

$ z_i \in Z \backslash {o}$ what's this mean?

Balarka Sen

hi @iwriteonbananas

usukidoll

@BalarkaSen does what I type make sense or did I screw this up... that's supposed to be z_i in z \ {0} btw

Balarka Sen

I don't know what that means. Maybe $\Bbb Z - \{0\}$.

usukidoll

yeah the complement defintiion

but does it look right?

Balarka Sen

I can't answer that without more context. But I am thinking about something else right now.

usukidoll

08:08

this -> Let $P_{n}$ be the set of all polynomials of degree $n$ with integer coefficients. Prove that $P_{n}$ is countable.

but I think I've bombed
Suppose we have the following polynomial with integer coefficients

$z_{0},z_{1},z_{2}...z_{n} \in Z$

for $z_{0}x^{n}+z_{1}x^{n-1}+z_{2}x^{n-2}...+z_{n-1}x$ $z_{i} \in (Z \backslash \{0 \}) $

So by the theorem is Z is countable and $(Z \backslash \{0 \})$ is countable than the Cartesian product is countable... and I messed up..oh well I tried

iwriteonbananas

@BalarkaSen Good morning

Balarka Sen

I am not actually reading that, so you should ask someone else. Sorry. I am thinking about another problem.

iwriteonbananas

Are you still awake, @MikeMiller?

usukidoll

k :/

Balarka Sen

@iwriteonbananas This guy is weird. I wonder if the Antoine's horned sphere works, but that's something I can't visualize efficiently

Idle001

08:14

wb @skullpetrol

iwriteonbananas

@BalarkaSen Yeah, weird question. Took me a minute to parse

The Substitute

08:27

Off topic. What is the purpose of having squares in the definition of variance? Why not just use $|X-EX|$ instead?

Huy

09:01

@TheSubstitute: 1. The absolute value function isn't differentiable everywhere. 2. You want to make large deviations have more weight than small deviations.

Balarka Sen

09:12

@iwriteonbananas lots of hard questions today: math.stackexchange.com/questions/1630257/…

Lembik

09:43

Can anyone help me with a notation problem? I want to say the "number of different vectors that Mx can take when consider over all vector $x \in \{-1,1\}$"

how do you write that properly?

$M$ is a matrix

Huy

$|\{Mx: x \in \{-1, 1\}\}|$?

Lembik

@Huy Looks good! Is that in some sense the size of the range of $M$?

Huy

in some sense, I guess

Lembik

what would you call $\{Mx : x \in \{-1,1\}\}$?

Huy

not very interesting set

Partly Putrid Pile of Pus

09:55

This^

Lembik

@Huy I think it's a very interesting set!

Huy

it contains at the most two elements

doesn't seem too exciting to me

Lembik

@Huy $M$ is $m$ by $n$. The max number of elements is $2^n$

oh oops :)

Huy

not the way you write it

Lembik

$x \in \{-1,1\}^n$

thanks!

Huy

09:58

that's better

Lembik

it's not at all clear to me what properties of $M$ determine the answer

@Huy do you have any idea?

Huy

@Lembik: not right now, maybe determine it for a few matrices you're interested in and see if there's a pattern?

Lembik

I did that

it's utterly mysterious to me

there is some loose relationship to $det(MM^T)$

but it's quite loose

user174558

@Huy I just watched Solace, great movie.

Mary Star

10:36

Suppose we have the cylinder $R$: $x^2+y^2\leq 1$, $0\leq z\leq 1$. How can we calculate the integral $\int_R x^2dV$ ?

user174558

10:54

@MaryStar Are you taking multivariable calculus now?

Mary Star

Yes. @Jasper

robjohn

@Jasper ...to you, too.

user174558

@robjohn I hope you are not affected by the winter storm. 50 people died in US and 100 in East Asia.

robjohn

@Jasper No. We are still waiting for the rain that we were told was coming. We are still in a drought.

user174558

@robjohn Get the best witch in the country to perform a ritual.

Mary Star

10:59

Which are the limits of the integral for $x$ and $y$?

Partly Putrid Pile of Pus

What actually is a type $A$ Lie algebra?

user174558

@MaryStar You need to express y in terms of x. It's a circle on the xy-plane.

Tobias Kildetoft

@PartlyPutridPileofPus basically $\mathfrak{sl}_n$

Partly Putrid Pile of Pus

Basically?

Tobias Kildetoft

Well, I might also call a product of such type $A$

also, over the reals, you get some others I think

More precisely, it is a semisimple Lie algebra whose root system is of type $A$

Mary Star

11:02

So do we get $-\sqrt{x^2-1}\leq y\leq \sqrt{x^2-1}$ ? And which are the limits of $x$? $[-1,1]$ ? @Jasper

robjohn

@MaryStar That integral is the same for each $\mathrm{d}z$ slice...

Partly Putrid Pile of Pus

@TobiasKildetoft Okay, thanks, these words will give me a good starting point!

Mary Star

@robjohn What do you mean?

robjohn

just what I said

Partly Putrid Pile of Pus

@TobiasKildetoft So type $A$ is referring to type $A_n$ where this has dynkin diagram consisting of $n$ connected nodes

robjohn

11:05

$$\int_0^1\int_0^1\int_0^{2\pi}r^2\cos^2(\theta) \,r\,\mathrm{d}\theta\,\mathrm{d}r\,\mathrm{d}z =\frac\pi4$$

Tobias Kildetoft

@PartlyPutridPileofPus Right (and with no double or triple edges)

Partly Putrid Pile of Pus

I think the term is simply laced

Tobias Kildetoft

@PartlyPutridPileofPus Not quite. Type $D$ is also simply laced

Partly Putrid Pile of Pus

Okay good

Oh okay

Oh I see

Tobias Kildetoft

simply laced (as far as I recall) just means no double or triple edges

Partly Putrid Pile of Pus

11:07

Where can I learn about associated compact groups?

Mary Star

@robjohn So, do we use cylindrical coordinates to calculate the integral?

Tobias Kildetoft

@PartlyPutridPileofPus Not sure what a good resource is, as I don't really know the Lie group stuff related to this

robjohn

@MaryStar I just used polar coordinates to compute the integral for each $z$

Mary Star

Ah... having calculated then the integral for each z, how do we use then the limits of z? @robjohn I got stuck right now...

robjohn

@MaryStar it is the same for each $\mathrm{d}z$ slice

the same for each $z$. $\int_0^1A\,\mathrm{d}z=A$

Mary Star

11:12

Ahh... I got it!! Thanks a lot!! :-) @robjohn

Huy

@Jasper I agree, but unfortunately the following Bond movies aren't remotely as good.

(unless you were referring to a different movie)

Tobias Kildetoft

@Huy Did you get your exam results yet?

Huy

@TobiasKildetoft: Yep, I passed. He probably realized the questions he asked weren't really what most students had prepared for.

Tobias Kildetoft

@Huy Ahh, nice

Balarka Sen

The ideal questions for an oral Lie algebra exam is of course "prove that Lie theory is internal to $(\infty, n)$-topoi".

Mary Star

11:20

I have also an other question...
I want to calculate the surface integral of $F(x,y,z)=(0,0,z)$ on the unit sphere with parametrization
$x=\sin u \cos v, y=\sin u \sin v , z=\cos u$
$0\leq u\leq \pi, 0\leq v\leq 2\pi$
with positive direction the direction of $T_u\times T_v$.

Could you give some hints how to calculate this? @robjohn

Partly Putrid Pile of Pus

@TobiasKildetoft Do two different Lie algebras ever have the same Dynkin diagram? Does the Dynkin diagram distinctly classify a Lie algebra?

Tobias Kildetoft

@PartlyPutridPileofPus Over $\mathbb{C}$ the (semisimple) Lie algebras are completely classified by their Dynkin diagrams (i.e. by their root systems)

@PartlyPutridPileofPus I wrote a short note on it at pure.au.dk//portal/files/56984898/classification.pdf and also as an answer here at math.stackexchange.com/questions/427135/…

I am not really familiar with how this changes over other fields

Partly Putrid Pile of Pus

I shall read this, appreciated

Tobias Kildetoft

They are identical, so you can just read one of them

robjohn

@MaryStar Look at this answer.

Note that the area between two parallels of latitude is $2\pi r$ times the distance between the parallels.

Mary Star

11:30

I haven't really understood the fact that the positive direction the direction of $T_u\times T_v$... @robjohn

Partly Putrid Pile of Pus

What area does your research belong to @TobiasKildetoft?

Tobias Kildetoft

@PartlyPutridPileofPus algebraic representation theory

I'm an artist

@robjohn hey, I got amazing results this night!!!

Tobias Kildetoft

mainly algebraic groups, but I have also done some things with applications of 2-representation theory for representations of Lie algebras, and recently some study of positively based algebras

Huy

what's a totally geodesic boundary?

Partly Putrid Pile of Pus

11:33

I see you submitted a paper 2 days ago

Tobias Kildetoft

@PartlyPutridPileofPus Yeah

Partly Putrid Pile of Pus

For one to understand this paper, would I need to read MM1-6? (Among otherthings, I mean necessary not sufficient)

Tobias Kildetoft

@PartlyPutridPileofPus No, the present paper is fairly elementary. The main "deep" results used are the Perron-Frobenius theorem and the Schauder fixed point theorem (which is just used a single place)

I also think we have made it self-contained for the most part

(though I suppose some of the examples and results do require knowledge of Kazhdan-Lusztig theory)

Mary Star

At the question you linked it is asked for the volum, right? Why is this surface integral the volume? @robjohn

Balarka Sen

11:49

This guy is a nontrivial guy. He really knows his stuff.

4

Huy

is that your second account?

Balarka Sen

i wish

Huy

if I had a free wish I wouldn't wish for a particular MSE account

Balarka Sen

if I had a free wish I'd ask for 2 more free wishes. and so on and so forth.

Huy

then you'd die before actually getting something useful out of it

Balarka Sen

11:53

well, I could stop whenever I want. and I'd get exponentially many more wishes at each step. it's all p-adics.

Huy

no, you can't. you just said "and so on and so forth"

it's too late now

RIP.

Balarka Sen

well, I don't have a free wish.

Huy

@BalarkaSen: I hereby offer you a free wish. RIP.

Balarka Sen

not a very reliable offer.

Huy

you're not supposed to answer anymore.

go back to your while-loop.

Balarka Sen

11:58

my wishes are looping, not me.

Huy

you're wrong

Balarka Sen

"whatever floats your boat"

you planning to lock yourself up in this chat now that your exams are over?

Huy

my exams aren't over

Balarka Sen

ah?

robjohn

12:25

@I'manartist Great! what kind?

@MaryStar Did you notice that the surface area was computed first?

I'm an artist

@robjohn series with harmonic numbers but very advanced ones

Tobias Kildetoft

@PartlyPutridPileofPus Do you plan to read it, or were you just curious btw?

Partly Putrid Pile of Pus

@TobiasKildetoft To be honest, I have never read a research paper. I think since you seem active here it would be beneficial for me to read it and ask you questions, if that isn't a problem

Tobias Kildetoft

@PartlyPutridPileofPus That would be fine

@PartlyPutridPileofPus Though as I said, some of it probably requires knowledge of KL theory

So the main things should be readable, but the application to Hecke algebras are probably harder without that knowledge

privetDruzia

Hi @TobiasKildetoft, @PartlyPutridPileofPus, @I'manartist, @robjohn, @BalarkaSen, @Huy

:)

Balarka Sen

12:35

hi

privetDruzia

found a question online, where the first part got me quite curious

1

Q: Understanding derivative notation in those equations

I am given the following set of equations from a physics course, which is about longitudinal waves in rods. My questions are: On the second line you have $ (\frac{\partial \Delta}{\partial x})dx $ If you are already specifying you are doing a partial differentiation with respect to the x-dir...

derivatives physics partial-derivative

the first question actually

not the two other ones

@BalarkaSen, any idea?

Balarka Sen

I am not reading the question, sorry. I have other things to do right now.

Huy

you mean you are the same user but since that user got banned in chat yesterday you're using this account to ask again? @privetDruzia

privetDruzia

i know the owner of the other account, we study at the same university

and are in the same class

he s not a close friend

and his first question is one I have as well

lol didn't know he got banned

@Huy

privetDruzia

12:56

@Huy, any idea?

MphLee

hi sorry if i disturb, i'd like to know how to write text outlined. someone can help me

privetDruzia

on SE?

Huy

@privetDruzia: I won't look at questions from that person, let alone answer them

privetDruzia

well just answer my via chat, is fine enough

:p

Huy

please don't ping me anymore for that question

privetDruzia

12:57

@MphLee just add space, I guess. It works when u want tou outline a list of different

points

MphLee

I need to delete a some parts of my question because I've discovered an error.

add space before? i mean i want a line overthe words

privetDruzia

I am not sure whether I udnerstand u correctly

r u familiar with latex?

MphLee

I don't want it inside a formula

privetDruzia

just try $\vec{this is my text}$

MphLee

ok I try it

privetDruzia

13:00

oooh yes u worte OVER the words

Balarka Sen

"0.0. What’s this about, and why should I care?

0.0.1. Is this one of those manuscripts that I need to know 25 alternative definitions of n-Category and 16 generalisations of the étale site in order to get started?"

Quill

I heard there's magical ninjas here who can read magical ninja syntax

privetDruzia

this will put a line ABOVE the words

idk in that case

Quill

in The 2nd Monitor, 3 mins ago, by Dan Pantry

((a … → b) … → [a] → *) → (a …, Int, [a] → b) … → [a] → *)

wtf is that

MphLee

okok thx

Balarka Sen

13:01

"0.0.2. But I know 26 definitions of n-Category and 17 generalisations of the étale site, are you saying this isn’t for me?"

MphLee

really there are 26 alternative definitions?

o.o

Balarka Sen

lolwut "0.0.4. That’s all very clever, but I’m a topologist, I hate French mathematics, and I can mimic that sort of thing using $K(\Gamma, 1)$"

Huy

when did you start using "lolwut"

I haven't seen that expression since 2010

Balarka Sen

"0.0.8. I’m not buying that. Fibre products in 2-categories are only well defined up to equivalence, so the group axioms are devoid of sense in a 2-category, and this whole 2-group thing is a lot of sad rubbish"

@Huy I use that occasionally.

Michael Albanese

@BalarkaSen What are you reading?

Balarka Sen

13:08

This thing. The title is mildly interesting, because I have once wondered whether there is an analogue of higher homotopy groups in the algebraic setting, but unlikely I understand much of the math there.

Section 0.0 is a fun read nonetheless.

Huy

@BalarkaSen: 5 times since 1 January 2015, so approximately every 78th day.

I'm looking forward to April 15th @BalarkaSen.

Michael Albanese

That's nice. I thought those were genuine comments about someone else's work, in which case, that would not be nice.

Balarka Sen

Oh, gee, that'd indeed be un-nice.

MphLee

hey balarka do u know a way to put a line over my text, (not on TeX) like I did an error

Michael Albanese

@MphLee Try this <strike>text</strike>

It produces <strike>text</strike>.

That didn't work. It works on the site though.

MphLee

13:14

omg thank u so much

Michael Albanese

No worries.

More generally, MSE supports HTML.

Huy

what happens if I use </body>

Dan Pantry

You can use a triple hyphen for strikethroughs in the chat and I would imagine on the main site, too.

~~like so~~

@Huy SO will escape it to >/body< ;-)

Huy

too bad

I'm an artist

13:50

@privetDruzia hi

Pretty busy these hours with some research.

BBL

Mike Miller

@iwriteonbananas Just ask the questions... I'll see them eventually.

Evinda

Hi @DanielFischer !!! Suppose that we have $Y=\{ x \in \ell^2(\mathbb{N}) | \exists n \in \mathbb{N} \text{ such that } x_j=0 \forall j>n \}$.
Why does it hold that $Y$ is not closed?
Suppose that we pick a sequence $(\overline{x}_m)$ in Y such that $\overline{x}_m \to x$.

$(\overline{x}_m=(x_{m1}, x_{m2}, \dots, x_{mn}, 0, \dots, 0, \dots)$.

Will x not be of the form $x=(x_1, x_2, \dots, x_n,0, \dots, 0, \dots)$

Michael Albanese

@Evinda: How can you choose a sequence $(\overline{x}_m)$ converging to $x$? You haven't said what $x$ is.

Can you think of a sequence of sequences in $Y$, such that they tend to a sequence not in $Y$?

Evinda

@MichaelAlbanese $x \in \ell^2(\mathbb{N})$.

No, that's why I ask... So the set is closed, isn't it?

Michael Albanese

14:07

No, it isn't.

A general element of $\ell^2(\mathbb{N})$ is of the form $(x_1, x_2, x_3, \dots)$. Can you think of a sequence of sequences in $Y$ that approximate $x$?

Evinda

@MichaelAlbanese Y is dense in $\ell^2(\mathbb{N})$ so for all $x \in \ell^2(\mathbb{N})$ there is a sequence $(y_n)$ in $Y$ such that $||y_n-x|| \to 0$.
Or am I wrong?

Michael Albanese

That's true. If you can find a sequence $(y_n)$ in $Y$ converging to $x$, then you would actually prove this.

If you know $Y$ is dense, then if it were closed $Y = \overline{Y} = \ell^2(\mathbb{N})$ which is false, so $Y$ is not closed.

However, I think it is easier to write down a sequence $(y_n)$ in $Y$ that converges to $x$.

user153330

@I'manartist i couldn't, did you send it then delete it?

Mike Miller

@iwriteonbananas: I think I told you earlier flatness wasn't the same as path-independence. :) But as a response to your comment, no, the projective plane does NOT support a flat metric.

Evinda

@MichaelAlbanese Could you explain to me why $\overline{Y}= \ell^2(\mathbb{N})$ ?

Michael Albanese

14:18

That follows from the density of $Y$.

user153330

:27159055 well that's trivial to me haha

I'm an artist

@user153330 what do you mean by that's trivial to me?

Evinda

@MichaelAlbanese You mean that we pick a sequence in $Y$ , let $(x_m)$ such that $x_m \to x \in X$ and show that $x \notin Y$ ?

@MichaelAlbanese So do we use a theorem?

Michael Albanese

Yes. You do not use a theorem.

You can do it by hand.

user153330

@I'manartist (anyone who saw your performance and how it improved over time will come to that conclusion)

Semiclassical

14:20

morning chat

Evinda

@MichaelAlbanese I mean that what you said: That follows from the density of $Y$.

user153330

@Semiclassical bmorning

Michael Albanese

Density of $Y$ in $\ell^2(\mathbb{N})$ implies $\overline{Y} = \ell^2(\mathbb{N})$ does not require a theorem, it follows quite quickly from the definition of density.

I'm an artist

@user153330 hehe, thanks.

user153330

@all do you remember there was an account who had a challenge of finishing many textbooks over 2 years?

i didn't see him for so long

I'm an artist

14:22

@user153330 I discovered a new family of amazing series, and their difficulty is insane after a certain point. However, last night I finished the generalization.

user153330

@I'manartist well i love hardcorish results

Evinda

@MichaelAlbanese A ok... If we have shown that the distance $d(x,Y)=\inf \{ ||x-y||_2: y \in K \}$ cannot be attained, doesn't this imply that Y is not closed? Or am I wrong?

I'm an artist

@user153330 hehe, I like them a lot. They give a sense to my life.

Michael Albanese

Because $Y$ is dense, that distance is zero and it is attained by $x$ if and only if $x \in Y$.

I think you're overthinking this.

Please allow me to suggest a strategy.

Evinda

@MichaelAlbanese Ok

Semiclassical

14:26

oh, btw, i meant to mention that i get what you mean about how nice the closed form is for that integral we'd been discussing @I'manartist

I'm an artist

@Semiclassical niceeeeeeeeee :-)

Semiclassical

the one indexed by $n$, not the $t$-dependent generating function

Michael Albanese

Consider the sequence $x = (1, \frac{1}{2}, \frac{1}{3}, \dots ) \in \ell^2(\mathbb{N})$. Note, $x \not\in Y$.

I'm an artist

@Semiclassical Yeah (the stuff privately discussed)

@Semiclassical So you like that result?

Michael Albanese

Can you find a sequence of sequences in $Y$ which converge to $x$?

Semiclassical

14:27

yeah, though I think the generating function makes it crystal clear why you end up with a sum of zetas for the $n$th case

I'm an artist

@Semiclassical Yes, indeed

Semiclassical

namely, because $H_t$ itself generates zetas and dividing by the denominator resums them

it makes me wonder if there's a well-known integral representation of $H_t$ that i don't know about

Evinda

@MichaelAlbanese $\left( 1, \frac{1}{2}, \frac{1}{3}, \dots, \frac{1}{n}, 0, \dots \right)$ while $n \to +\infty$ @MichaelAlbanese

Michael Albanese

Yes!

user153330

@all do you remember there was an account who had a challenge of finishing many textbooks over 2 years?
i didn't see him for so long

I'm an artist

14:30

@Semiclassical $$-\int_0^1 t x^{t-1} \log (1-x) \, dx$$

Michael Albanese

Now you have a sequence $y_n = (1, \frac{1}{2}, \frac{1}{3}, \dots, \frac{1}{n}, 0, \dots)$ in $Y$ which converges to $x$ in $\ell^2(\mathbb{N})$, but $x \not\in Y$. So $Y$ is not closed.

Semiclassical

that'd do it

I'm an artist

@Semiclassical :D

@Semiclassical I sent it to a journal with a simple solution (I mean my version).

Semiclassical

and an integration by parts lets one trade the first term for $x^t$

right

I'm an artist

yeah

Evinda

14:33

@MichaelAlbanese I got it... Thanks a lot!!! :)

Michael Albanese

No worries.

Semiclassical

so i guess $\int_0^1 \frac{x^t}{1-x}\,dx$ also works, which is pretty obvious once one expands that denominator

I'm an artist

Yeah, it should.

I'll be back a bit later. I need to finish now something (in a hurry).

BBL

Semiclassical

mmkay

iwriteonbananas

@MikeMiller Sorry, I don't know why I missed your remark about flatness not being the same as path-independence

Gah, I feel like crap today

Balarka Sen

14:37

That's me everyday.

iwriteonbananas

Not getting anything done.

Semiclassical

@I'manartist actually, scratch that. integrating by parts doesn't work directly since the boundary term would diverge.

might be able to fix it by taking the integration to be on $[0,1-\epsilon]$ and letting $\epsilon\to 0^+$ at the end

user153330

@all do you remember there was an account who had a challenge of finishing many textbooks over 2 years and he had a wordpress where he recorded his updates?
i didn't see him for so long, his name was commiting to a challenge i guess

@Jasper ^^

@BalarkaSen ^^

Balarka Sen

I haven't seen him for some time, no.

But he's alive: math.stackexchange.com/users/233746/alex-clark

Or at least, he was two days ago.

user153330

@BalarkaSen that's not him for sure

Balarka Sen

14:40

Alex Clark is committing, I assure you.

It's just another account he made up.

user153330

oh, i'm really confused now, latest time he updated his wordpress he said he still didn't finish axler LADR and rudin, and now he's working through Lee's?

Balarka Sen

That was at least a year ago, if not two. But he has progressed a lot through the years indeed.

user153330

a lot indeed, chapeau to him

GBeau

I just got accepted into graduate school :D :D :D :D

8

Mike Miller

Nice. Where?

GBeau

14:52

Stony Brook

Top 15 for my subfield, super excited

Mike Miller

Cool. Say hi to @MichaelA for me.

What do you want to study?

GBeau

high energy particle experiment

Mike Miller

I see.

Balarka Sen

@GBeau Congrats.

GBeau

thanks

user153330

15:09

@GBeau nice news! congrats man!

Semiclassical

15:26

@GBeau Any particular subfield/researcher within that you're after? (The UMN has ties to nuetrino physics, for example)

GBeau

@Semiclassical accelerator physics, unless my mind gets changed in my first year before I join a research group (which is possible)

@Semiclassical my undergraduate research is with ATLAS at the LHC

Semiclassical

mmkay

so you might end up being one of those people who periodically cycles across to LHC for research duties

GBeau

yeah

Evinda

16:36

Let $(X, \rho)$ be a metric space and $x \in X, A \subset X (A \neq \varnothing)$.
We have $x \in \overline{A}$ iff $d(x,A)=0$.

We suppose that $d(x,A)=0$ .

We want to show that $x \in \overline{A}$.

Suppose that $x \in X \setminus{ \overline{A}}$.

Since $X \setminus{ \overline{A}}$ is open, there is an $\epsilon>0$ such that $B_{\rho}(x, \epsilon) \subset X \setminus{ \overline{A}} \Rightarrow \{ y \in X: \rho(x,y)< \epsilon \} \subset X \setminus{\overline{A}}$.

How can we find a contradiction? @DanielFischer

Martin Sleziak

17:37

@ThomasAndrews I would opt for closing as duplicate rather than deleting, since the links in the comments might be useful for other users. And, in addition to that, duplicates can be useful. But it's definitely no big deal. As it is your post, it is your call what to do with it. (Basically the main reason I responded was that you pinged me before the deletion.)

@Evinda Just in case it is any help, you can have a look at this question and other related posts.

But if you want to try finish your proof, from $B(x,\varepsilon) \subset X\setminus A$ you shold be able to get that $d(x,A)\ge\varepsilon$.

Evinda

18:47

@MartinSleziak What can I do so that I get this? I am confused right now...

Balarka Sen

Hi @PedroTamaroff.

Martin Sleziak

@Evinda If you know that $d(x,a)\ge\varepsilon$ for each $a\in A$, then $d(x,A)= \inf_{a\in A} d(x,a)\ge\varepsilon$.

Evinda

@MartinSleziak How do we know that $d(x,a)\ge\varepsilon$ for each $a\in A$?

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