Mathematics - 2019-06-04 (page 1 of 2)

Leaky Nun

00:17

Watching Ellipticman lecture on illuminating curves, very silver

Ultradark

00:27

Ellipticising elliptic elliptic on elliptic elliptics, very elliptic

BananaCats Category Theory App

01:01

:math.stackexchange.com/questions/3249094/…

Please upvote

I'm coding that stated algorithm now

using Python

it will be posted as one file

Also please comment if you know about discrete optimization and groupoids

Or if you have any questions

Mathphile

02:20

can we prove that there are no integer solutions to $y=\frac{x^x}{x!}$ other than (1,1) and (2,2)?

N. Maneesh

3

Q: property of a two subset of the space $X=C[0,1]$ with its usual 'sup-norm' topology

Consider the space $X=C[0,1]$ with its usual 'sup-norm' topology.Let $S$={$f \in X:\int_0^1{f(t)dt \ne 0}$} $S_1$={$f \in X:\int_0^1{f(t)dt = 0}$} $1$.Then $S$ is (a) open , (b) dense in X , (c) connected-which property holds for $S$. $2$.Then $S_1$ is (a) closed, (b) connected, (c) com...

general-topology

How do I prove $S$ is dense?

Consider $\psi(f)=\int_0^1f$. Kernal of $\psi$ is a hyperplane. co dimension 1. So, $S_1$ is nowhere dense. Hence $X\setminus S_1=S$ dense.

Am I correct?

Buddhini Angelika

05:22

When considering $Cay(\mathbb{Z}_{p} \times \mathbb{Z}_{p},\:S)$ where $S=\{a,\:b\}$ with $\vert a \vert= \vert b \vert=p$, $q$ copies of $p \times p$ grid structures connected along a $q$ cycle will be present. And the $p \times p$ grid structure $p$ cycles connected like $p$ rows and columns will be present (which are formed due to the two elements of order $p$ in the generating set), right?

Thanks a lot in advance

Balarka Sen

09:10

@BananaCatsCategoryTheoryApp yeah boi

their new album was ok tho

didn't hype me out that much

skullpatrol

11:49

\o @nitsua60

UserX

12:18

I have a problem with a particular type of exercises. I gotta show that $H=<[2]_4, [4]_6>$ is normal to Z4 X Z6, and I can by showing it is a subgroup and Z4 X Z6 is abelian and all subgroups of abelian groups are normal, but then I gotta find to what known group is the quotient Z4 X Z6/H isomophic to.

In the past on these exercises I dealt with quotient groups where the "nominator"(cant translate that, you get what I mean, the group on top of the quotient) is a direct sum of cyclic groups of coprime order thus cyclic, then it's easy showing that the quotient group of cyclic group is cyclic and there is only one cyclic group per order up to isomorphism.

Now that "nominator" is not cyclic I'm stuck, please tell me I don't have to write out the whole quotient group to see what it's isomorphic to. What do I do?

nitsua60

12:31

@skullpatrol o7

skullpatrol

12:44

quite the shocker in the world of heavy boxing @nitsua60

can't wait for the rematch

Anthony Joshua vs. Andy Ruiz Jr.

Anthony Joshua vs. Andy Ruiz Jr. was a heavyweight professional boxing match contested between Anthony Joshua and Andy Ruiz Jr. The event took place on June 1, 2019, at Madison Square Garden in New York City, with Joshua's WBA (Super), IBF, WBO and IBO heavyweight titles on the line. Joshua was originally scheduled to face undefeated challenger Jarrell Miller, who was replaced by Ruiz after Miller failed three drug tests. Ruiz won the match via technical knockout in the seventh round. == Background == Joshua has repeatedly spoken of his desire to fight the undefeated WBC champion Wilder for all...

Tanvir

13:15

Hi, Can anyone help me to figure out this problem?math.stackexchange.com/questions/3245142/…

Ajay Mishra

14:04

@Tanvir what is the real problem?

I mean, what you are supposed to find?

Tanvir

14:22

@AjayMishra I suppose to find the area of a spherical cap that is crossing the intersection(if it cross)

nbro

14:50

Are the following derivatives correct?

Balarka Sen

@UserX What's the order of the quotient? What are the possibilities?

user193319

Let $F(\Bbb{N})$ be the collection of all finite subsets of $\Bbb{N}$ (think of each set as ordered), and let $\{p_1,p_2,...\}$ be an enumeration of the primes. I believe the map $\{n_1,...,n_\ell\} \mapsto p_1^{n_1} ... p_{\ell}^{n_\ell}$ is a well-defined injection...does that seem right?

anakhro

Good morning (afternoon to you?), @BalarkaSen

Balarka Sen

Yup, afternoon. Morning, @anakhro

Did you ever figure out the ball thing

user193319

If so, then this proves that $F(\Bbb{N})$ is countable.

Wait, maybe $\{n_1,...,n_\ell\} \mapsto p_1 ... p_\ell$ is easier to think about.

anakhro

14:55

@BalarkaSen no, but I feel like you don't even need the ball.

Ajay Mishra

@Tanvir I've solved the problem but I don't know where do draw the drawing. Where do you draw? I tried Geogebra, Windows Paint, google draw.

anakhro

Just put the symplectic coordinates on $\xi$ and try a similar argument.

Balarka Sen

Darboux-like thing?

anakhro

Yeah, Darboux on $\xi$ then extend it.

Balarka Sen

Gotchu

Ryan Unger

15:05

@BalarkaSen can I repurpose $\Bbb P^n$ for PDE

the parabolic n-space

Balarka Sen

no

Ryan Unger

@BalarkaSen it's $\Bbb R^n\times\Bbb R$

makes perfect sense

anakhro

@RyanUnger that would be $\mathbb P^\text{ainful}$

Alessandro Codenotti

15:47

@user193319 That works

$[X]^{<\omega}$ has the same cardinality as $X$ for every set $X$ (assuming choice)

(This is actually equivalent to choice I'd guess)

Yes it is definitely equivalent, it's easy to give a group structure (even a ring structure) on the set of finite subsets of $X$, which can be transported through this bijection to $X$ and "every set has admits a group structure" is equivalent to choice

anakhro

16:14

A cute cardinality problem: If $X$ is an infinite set, does there exist an uncountable family of infinite subsets of $X$ such that the pairwise intersection is finite?

Once you know the answer, it's "obvious", but it's cute because of how creative some solutions can get.

Alessandro Codenotti

I've heard that problem and solutions a few times before so I won't spoil it, it's really nice

nbro

@AlessandroCodenotti Can you please see my question above?

2 hours ago, by nbro

Are the following derivatives correct?

Alessandro Codenotti

No, I'm very bad with that kind of things

Hi @Ted

Balarka Sen

Hi @Ted

nbro

Oh, ok

Ted Shifrin

16:21

Hi, demonic @Alessandro, a @Balarka.

nbro

Maybe Mr. Ted can check that :P

Ted Shifrin

It makes no sense to me.

I have no idea what $\dfrac{dy}{d(x+\Delta x)}$ is supposed to mean.

And with partial derivatives things only get worse.

anakhro

Hello Ted.

Ryan Unger

@Semiclassical trying to picture that neighborhood if I take the graph to be the devil's staircase function o.O

Ted Shifrin

hi @anakhro

anakhro

16:30

@TedShifrin are you fond of music?

Ted Shifrin

Depends on the music, of course.

anakhro

What music are you fond of?

Ted Shifrin

Classical, folk.

anakhro

Is Jethro Tull included in "folk"? :P

Ted Shifrin

I listened to them in high school.

Balarka Sen

16:35

I have been listening to Chopin often

Ted Shifrin

Piano or other, a @Balarka?

Alessandro Codenotti

Jethro Tull are awesome, I heard them live twice

Balarka Sen

@TedShifrin Yeah, went through the Nocturnes some time ago

Ted Shifrin

Very cool. Back later ... French Open back live.

Alessandro Codenotti

@AlessandroCodenotti Well they more like Ian Anderson concerts to be fair

anakhro

16:40

I would have pinned Ted for being more of a megadeth/slayer kind of guy, but

shrugs

Alessandro Codenotti

Jethro Tull and heavy metal are not mutually exclusive, I can prove that :P

anakhro

@AlessandroCodenotti what kind of stuff do you like other than Jethro Tull?

Balarka Sen

@anakhro Lol

On that note, Judas Priest is pretty good

Especially their last album; it kicks ass

Semiclassical

Yo

Alessandro Codenotti

@anakhro I'd say my favourite genre is prog rock, both classical stuff from the late 60s and 70s to modern prog rock, but I generally appreciate all technical things and experimentations regardless of genre. For an extreme example on the other end of the musical spectrum I quite like some mathcore and technical death metal groups

I also listen to a lot of post rock

Balarka Sen

16:45

Hi @Semiclassical

Semiclassical

@AlessandroCodenotti Explosions in the Sky?

MatheinBoulomenos

Hi @Balarka @Alessandro @Semiclassical @anakhro

Balarka Sen

I listened to their first album a few days; so good

"How Strange, Innocence"

Hi @Mathein

Alessandro Codenotti

@Semiclassical Sure, they're one of the big names in post rock

Hi @Mathei

MatheinBoulomenos

I like tool a lot

Balarka Sen

16:47

GY!BE is my favorite post rock band

nbro

@TedShifrin Hi Ted. $dC/d (x + \Delta x)$ means the derivative of function C with respect to function $g(x) = x + \Delta x$

Alessandro Codenotti

My profile picture here is the cover of Departure Songs by We Lost the Sea, probably my favourite post rock album

Semiclassical

Got to see them perform once. Pretty awesome

Balarka Sen

F#A#$\infty$ man

Alessandro Codenotti

@MatheinBoulomenos agreed, they're awesome

@BalarkaSen That and lift your skinny fists are masterpieces

Balarka Sen

16:48

@MatheinBoulomenos Have you listened to Porcupine Tree by any chance

Semiclassical

@BalarkaSen yeah. I didn’t really like their latest album though. “The Wilderness” really didn’t have a cohesion to it

MatheinBoulomenos

@BalarkaSen I have, and I like it

Ryan Unger

@MatheinBoulomenos long time no see

Balarka Sen

@Semiclassical Ah I didn't listen to that one

Semiclassical

There were parts I enjoyed but eh, didn’t appreciate it much as a whole

Abdullah UYU

16:49

Hi, is there anybody here that watched this playlist youtube.com/playlist?list=PLOHFP8vrRmSHNsQ2AMS7_3l0Xj8gEmhTz ?

ÍgjøgnumMeg

y0

Balarka Sen

@Mathein Cool. Probably one of my favorite modern prog rock bands

MatheinBoulomenos

@RyanUnger long time no see indeed

Alessandro Codenotti

@AbdullahUYU No but if he's teaching out of his book it's probably good

Ryan Unger

@Semiclassical I forgot to ask, did you graduate?

Alessandro Codenotti

16:50

Porcupine Tree are awesome! Wilson's solo stuff is riskier, some is very good, some is meh at best

Ryan Unger

@MatheinBoulomenos are you still at Hberg?

MatheinBoulomenos

yes

taking a break semester atm

Semiclassical

@RyanUnger yeah

Balarka Sen

@Alessandro The Raven That Refused To Sing is probably the only album by him which I like (and like a lot)

anakhro

@nbro what is the part of it you are unsure of?

Hi @MatheinBoulomenos !

Ryan Unger

16:51

@Semiclassical so what are you doing now

Alessandro Codenotti

@BalarkaSen That's definitely his best solo album if you ask me as well

Insurgentes isn't bad either

Semiclassical

Been doing some non-thesis related research lately. Quantum foundations stuff

Ryan Unger

post doc?

nbro

@anakhro I am actually quite sure it is correct, but I would like someone to confirm it

Alessandro Codenotti

I don't like Hand. Cannot. Erase. and To the Bone though @Balarka

Semiclassical

16:52

No. Sorta stuck between things for now

Balarka Sen

They're bad. And he keeps saying pretentious things about them lol

Steven Wilson has a massive ego (not unjustified, but) in reality

Semiclassical

might get a postdoc out of this eventually but not immediate

Got a conference talk out of it, though, which was neat

anakhro

@AlessandroCodenotti if you like experimental rock, The Physics House Band is pretty good.

Abdullah UYU

@AlessandroCodenotti I suppose it's like so. Nevertheless I liked them.

Ryan Unger

that's nice

Alessandro Codenotti

16:54

@BalarkaSen That's true

@anakhro Never heard of them, I'll check them out! Thanks for the suggestion

nbro

@anakhro What do you think about it? Have you had a look at it?

anakhro

@nbro you just apply the chain rule, linearity of derivative, a rule of the derivative, and distributivity of multiplication over addition.

Balarka Sen

I came to stumble upon PT from Opeth which I also like

Alessandro Codenotti

I'm listening to Calypso right now, pretty cool so far

Semiclassical

Math thing: I’m trying to tease apart two different vector space constructions in probability theory

anakhro

16:56

Calypso is a great track.

That's the first one I listened to, as well. :P

@Semiclassical which ones?

Alessandro Codenotti

Do they only make instrumental pieces?

nbro

@anakhro Yeah, I kinda did that

Semiclassical

Or, well, two different inner products on such

nbro

I am only asking for a confirmation of the correctness

anakhro

@nbro ya u gud, d00d

@AlessandroCodenotti as far as I have heard, yes.

Semiclassical

16:57

Start with a probability space and consider the set of random variables on this space with finite variance

The idea is that each rv is a vector in a vector space

nbro

@anakhro :P

Alessandro Codenotti

Nice, I quite like instrumental groups, most experimental stuff tends to be instrumental

But sometimes experimenting with voice leads to really cool stuff too, gentle giant being the best example I'd say

nbro

@AlessandroCodenotti Check out Chinese Man then ;)

anakhro

Do you like The Mars Volta at all? They are pretty experimental (imo), but they have vocals.

Alessandro Codenotti

Indeed I do

Semiclassical

16:59

But I’m seeing two notions of inner product on this set. One is just (X,Y)=E[XY], while the other is (X,Y)=E[XY]-E[X]E[Y]

nbro

Chinese Man - Maläd

►

Simone

Must continuous functions be total?

Alessandro Codenotti

I have nothing against vocals, it's just a tendency I've noticed for experimental groups to have instrumental pieces (apart from very heavy genres with harsh vocal styles)

Balarka Sen

Kraftwerk is also pretty good

Semiclassical

Those two inner products agree on r.v.s with zero mean

anakhro

17:00

@Semiclassical is the latter like a "normalized" version in some sense?

nbro

Chinese Man are instrumental hip hop

Semiclassical

@anakhro yeah

Balarka Sen

@anakhro They agree when you quotient by the space of degenerate random variables, yep

Simone

Is the square root function continuous for example?

Alessandro Codenotti

The Mountain by Haken is another example of a great prog (metal) album with interesting vocals

Balarka Sen

17:01

That shit is amazing

Semiclassical

I think it amounts to the following (which goes in hand with what Balarka is saying I think)

Balarka Sen

I had Haken's Affinity on repeat a few days ago

Alessandro Codenotti

@BalarkaSen The whole album is amazing, but the song The Cockroach King is at another level

anakhro

@Simone a function is single valued and total.

@Simone so by the very definition of "function", all continuous functions are total.

Balarka Sen

@AlessandroCodenotti The Cockroach King is my best song from their new album

anakhro

17:02

Continuity doesn't play a part.

Semiclassical

In the first case, your vectors aren’t really rv’s but equivalence classes of such, where X~Y if X=Y a.e.

Alessandro Codenotti

@BalarkaSen That's not from the new one, it's from The Mountain (which was before Affinity)

Balarka Sen

What? It's from Vector, no?

Simone

thanks

Alessandro Codenotti

The new one is Vector, it's also very solid, a bit more on the metal side than The Mountain

Semiclassical

17:03

Whereas in the second case you weaken that to X-Y=c a.e. for some real c

Balarka Sen

Oh it isn't

Weird

Alessandro Codenotti

@BalarkaSen Nope, it's from The Mountain

Balarka Sen

Damn

Semiclassical

So I guess that’s what relates/differentiates them

The second case is by construction “coarser” since two rvs can be equivalent under the latter without being so for the former

nbro

I ordered a frappucino :P

Who remembers this?

Semiclassical

17:08

And if you do think of the vectors in that way, then the definition (X,Y)=E[XY] makes sense so long as you choose X,Y to be zero-mean representatives of their equivalence classes

Probably a better way to say that tho

MatheinBoulomenos

@Semiclassical to me it sounds like the two vector spaces differ by the constant functions, i.e. you can obtain the coarser one from the "finer" one by quotienting out constant functions, so that there is a SES $0 \to \Bbb C \to V_1 \to V_2 \to 0$

Balarka Sen

That's what I said!

MatheinBoulomenos

oh sorry, didn't read it

Balarka Sen

It's ok :)

I mean once you quotient by the degenerate rv's, your inner product becomes the more natural L^2 inner product, so why not

Semiclassical

@BalarkaSen elaborate on that?

Tanvir

17:12

@AjayMishra you can write the answer in my post. for drawing I use vision so you can use that as well. But I actually need the analytical solution.

Semiclassical

For reference, when people talk about L2 on probability spaces, they seem to consistently mean the “finer” case

Balarka Sen

Any random variable can be thought as a measurable function on a probability measure space. So $X : (\Omega, \mathcal{A}, \Bbb P) \to \Bbb R$ is your random variable; the connection with the distribution function is that $F_X(x) = \Bbb P(X^{-1}(-\infty, x])$ - this is really the pushfroward measure $\Bbb P^* X$ on $\Bbb R$ (you just need to know the values of the measure on left-infinite right-closed intervals to determine the whole measure on the real line)

Semiclassical

The bit that caught my eye in that statement, to be clear, was the “more natural” statement

Balarka Sen

In that case, $\Bbb E[X]$ is nothing but $\int_\Omega X(\omega) \Bbb P(\omega)$, integral over the measure space. The connection with the usual definition for absolutely continuous random variables is that if $X$ is absolutely continuous with pdf $f_X$, then this integral is the same as $\int_{\Bbb R} x f_X(x) dx$ by a change of variables argument, and using the fact that by Radon-Nikodym, $X^* \Bbb P = f_X d\lambda$

Semiclassical

i mean, for the “finer” case the inner product is E[XY]. That seems pretty L2 to me

Balarka Sen

17:17

Then $\Bbb E[XY]$ is $\int_\Omega X(\omega) Y(\omega) \Bbb P(\omega)$ for any square-integrable pair of random variables $X$ and $Y$ (i.e., of finite variance).

That's the L^2 inner product

MatheinBoulomenos

If $V$ is a Banach space and $W$ is a closed subspace, then $V/W$ is Banach via the quotient norm $\|v+W\|= \inf_{w \in v+W} \|w\|$. If $V$ is Hilbert, then this satisfies the parallelogram identity, so that $V/W$ is again a Hilbert space

Balarka Sen

(In the first paragraph I meant $X^* \Bbb P$ instead of whatever the fuck typo I made, and in both paragraphs I meant $X_* \Bbb P$. That's the notation for pushforward of a measure)

Semiclassical

right. But you seemed to be saying that it was L2 once you quotient by degenerate r.v’s, whereas it seems like quotienting by equality a.e. seems to be L2 just as well

Tanvir

@AjayMishra If you want, you can draw it by hand and post the picture as well.

Balarka Sen

@Semiclassical I mean, Cov(X, Y) is not the L2 inner product before you quotient.

It is after you do that

MatheinBoulomenos

17:20

@Semiclassical it seems to me that if you do what I described with L2 and the subspace of constant functions, you get the inner product for the coarser space

Semiclassical

@MatheinBoulomenos that’s what I was thinking as well

But it seems strange to call the inner product in the latter case as “more naturally” L2

Balarka Sen

Right, because E[X] is the orthogonal projection of X to the subspace of constant functions in L2

Semiclassical

Though it seems like splitting hairs at this point

For purposes of definition, the finer version is probably the better one since you can get the coarser version from it

Balarka Sen

I haven't actually seen a place where this formalism is actively useful in probability honestly. It's a nice formalism, but is it useful?

Semiclassical

Well, the coarse version shows up (rather implicitly) in a 1937 paper I’ve got on the brain

anakhro

17:30

>1937

Semiclassical

Whereas the finer version shows up pretty commonly in the literature

There’s a whole book on Hilbert space methods in prob/stats, apparently

Balarka Sen

Interesting. I'd like to read it.

Ryan Unger

Balarka is now an analst

Semiclassical

I’ve only seen it via google books, mind

Ryan Unger

how the hecc are we suppoed to keep up with him

Rithaniel

17:32

Balarka is a speedy math boi

Ryan Unger

here I am trying to show that a set is connected

Balarka Sen

^ dont believe him

Ryan Unger

Balarka is doing Aleph17 stratified probabilistic hodge theory

Balarka Sen

it a subset of a sobolev space

Semiclassical

Anyways. The nice thing about the coarser version is that the direction cosine between two random variables w/r/t this inner product is their (Pearson) correlation coefficient

Ryan Unger

17:34

@BalarkaSen don't believe me?

Balarka Sen

@RyanUnger I wrote a smallish exposition on the holonomic approximation theorem just for fun. Want to read it?

Ryan Unger

sure

actually I am trying to show that a connected set can only have on time

Rithaniel

I'm over here in the position of needing to apply for PhD programs in the Fall of 2020.

Ryan Unger

no sobolev space

@Rithaniel that's a long time away

Rithaniel

That's like, one year away and I don't even know where I would like to apply.

Semiclassical

17:36

Which means that analytic statements about the pairwise correlations amongst a set of random variables, turn into geometric claims about the associated vector space

Rithaniel

Currently the plan is to stay on at my current university for the spring semester as a grad student, though. For income and not missing any semesters.

Semiclassical

So that’s one way the formalism can be insightful

Ryan Unger

tfw connected =/= path connected

well I guess I'll just die then

Semiclassical

@RyanUnger eww

Ryan Unger

@Semiclassical so there's a function that's continuous in nice situations, I found out it's not continuous in mine, so now I'm on an epic journey to find out how discontinuous it is

and for awful reasons this is leading me to all the counterexamples in analysis

Balarka Sen

17:47

@Ryan sent via discord

Semiclassical

Oof

Ryan Unger

I got it

does it replace parts of EM

or supplement

anakhro

Is there a good EM book for mathematicians?

Griffiths is a pain

Balarka Sen

in the first two pages i go over formalisms about jets, in the next few pages i talk about a "counterexample" which is not really a counterexample, and in the last few i handwave the proof

Ryan Unger

Jackson should be fine right

Balarka Sen

17:49

there are a few pictures :3

Ryan Unger

Ok I remember EM being impossible in some places

Balarka Sen

@anakhro Electromagnetic duality for children

That's the best book

Ryan Unger

and I went on an MO hunt which lead me to some papers Mike told me about

and then I gave up

Balarka Sen

EM is ok the material is just confusing

they do their best

u dont want to try Gromov's PDR

the original source lmao

anakhro

@BalarkaSen I was about to accuse you of making up the name of a book as a joke

then I googled it :o

Ryan Unger

17:51

I can probably avoid that material with a prudent career choice

Balarka Sen

Likely yeah

Ryan Unger

unless someone invents an h principle for minimal hypersurfaces

Balarka Sen

@anakhro 1) the book exists 2) despite the title, it's not for children lmao

quite the opposite

anakhro

Yeah, I can see that.

I was meaning more of an intro to EM, but I guess this gives me something to save and hide in a folder I will never touch again.

skullpatrol

Are you going to start learning to read German or French soon @RyanUnger?

Ryan Unger

17:53

No

skullpatrol

Russian?

Ryan Unger

Don't want to be accused of collusion

anakhro

@skullpatrol he did however start learning Afrikaans.

skullpatrol

:-)

\o @TobiasKildetoft

Tobias Kildetoft

@skullpatrol Hi

skullpatrol

17:58

How's life?

Tobias Kildetoft

good

skullpatrol

\o @TedShifrin

Ted Shifrin

howdy @skull

heya @Tobias

Balarka Sen

rehi @Ted

Ted Shifrin

rehi, a @Balarka

Tobias Kildetoft

17:59

Hi @TedShifrin

Balarka Sen

Did you ever explain why you put the "a"? :P

Mathphile

16 hours ago, by Mathphile

can we prove that there are no integer solutions to $y=\frac{x^x}{x!}$ other than (1,1) and (2,2)?

some help?

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$Mathematics$ Mathematics