Mathematics - 2017-07-09 (page 4 of 7)

user84215

12:06

How can we type x^y^z^a ?

Astyx

x^{y^{z^a}}

user84215

Thanks

Balarka Sen

Hi @Alessandro

parvin

hi
what does this mean?

σ-algebra,on a set X is a collection Σ of subsets of X that includes the empty subset, is closed under complement, and is closed under countable unions and countable intersections. The pair (X, Σ) is called a measurable space.

Secret

12:23

Last night dream: A weird species of bird with a meager sponge like beak

user84215

Which part is ambiguous for you?

user84215

Why does \left{\frac{\sqrt x}{y^3}\right} not work?

Astyx

You need \{ instead of { after \left and \right

$\left\{{3\over 2}\right\}$

({ and } are special symbols)

user84215

Ok. Thanks.

Akiva Weinberger

@Secret This is horrifying

@parvin What part of it is not clear to you? (I don't know how much you already know)

Secret

12:38

eh? Probably because my drawing sucks so much to be able to illustrate it properly, or because of the sponge like beak

But basically, this bird is kinda like a cross between a hen and a peafowl, and as large as a cassowary

Leaky Nun

@Secret nice to hear that you're no longer dreaming about mathematics

Balarka Sen

the menger sponge is mathematics, though

Secret

Well, movie dreams are kinda rare. Also I actually have different types of dreams, but I tend to share themes relevant to a chat room for some given dream (unless it is cool enough)

which is why you guys so far read of my dreams as maths

Akiva Weinberger

@nequit Probably

(André Nicolas's answer is probably useful)

Secret

The only place where dreams tend to be shared in full is h bar, because the theme of h bar is so beyond physics most of the time

thus making it don't need to be physics

Anyway, the story about this bird as depicted in the dream is that the small chick you saw is one I took cared of (as "scripted" by the dream's own history) for some time and I visited it with an old man at some stacks of biohabitat tanks. The chick can teleport from tank to tank using transient portals

As for the adults, they walk around in the theme park which is where the biohabitat tanks are

I also theorised while within the dream that their sponge looking beak might be a suction component similar to those of an octopus

One possible guess on why portals are involved might be because Johnrennie and Kaumudi are playing Portal and they discuss about the levels

Akiva Weinberger

12:47

One of the first levels in Portal has large glass rooms with portals between them

Not that this necessarily has anything to do with your weird dreamworld

(Portal 2 has a similar opening level as well, though with a timing element removed)

Secret

So in some sense, I had a hypothesis that my dreams can be put into a broader mathematical context, that my dreams might be an encryption (plus adding extra obfuscations) of what I have experienced in the waking life in the past:

Let $S$ be the class that contains all content that made up my dreams. There exists an inverse map $f$ such that $f(S)$ recovers a subclass of my sensory experience at some time periods before the date of the dream

(though I strongly doubt my dreams can form a proper class, cause nothing in real life is known to be countably infinite )

Akiva Weinberger

The infinite is sometimes just a neat device for switching around quantifiers

I'll come up with a concrete example later

Balarka Sen

@Secret you sure you don't do drugs man?

Akiva Weinberger

I don't think he needs to

Balarka Sen

I am not even sure how to put myself that high to have those dreams

Akiva Weinberger

12:55

@Daminark @KasmirKhaan @TedShifrin Heh, just found this coincidentally

Secret

@BalarkaSen Absolutely no drugs. The only medication I am currently taking are cyclosporine and prednisolone, both are for my kidney disease

I don't even drink alcohol nor coffee

I, however, tend to have vivid dreams only when I slept long and good enough, and the previous night doing something mentally intensive such as sorting out my chemistry calculations and thinking about ordinals

If my sleep is too short or too poor or both, I don't even be able to recall my dreams in sufficient detail

Here are what I think to be the 7 basis mechanisms that are responsible for my dreams, based on analysing my 6 years of dream log:

1. Desire and conscious antiselection: Things I am conciously aware of and/or desires have a low probability to be selected as dream elements
2. Shock factor: Mechanism 1 is overriden if said event is too emotionally shocking
3. 3 hour rule: Things I experience and did 3 hours before going to sleep are highly likely to be selected as dream elements, provided mechanism 1 is not triggered
4. Proximity rule: If something familar is close by, the probability of being selected by dream element increases. This result in a spike in frequency of appearance

2

Leaky Nun

@Secret no tetris effect?

Akiva Weinberger

Tetris is hated by his subconscious and consistently takes hostile roles

Secret

Tetris effect tend to manifest via mechanism 5, but sometimes because of conflict with mechanism 1, it is not 100% effective

Akiva Weinberger

Tetris is fairly repetitive, though

Secret

13:04

Mechanism 1 tend to dominate most mechanisms as far I knew, which is why I cannot incubate my dreams

Akiva Weinberger

So I can't say, like, "Secret, have a dream on topological actions of $S_n$ on $\Bbb R^2$"

(Actually, does that even exist? Idk)

Secret

yeah, unless somehow I (or my unconscious self) found it interesting enough to trigger mechanism 2, which is what I guess what happened when Brown Ninja asked me that majorisation question on the other day and I spent half a day trying to solve it

Balarka Sen

@AkivaWeinberger You can have a dumb action

gx = x for all g

Sn does act interestingly on R^n by permuting the coordinates, though

Secret

O and one more thing: Like all dreams, my dream cannot produe content that I never experienced before. Evidence for that in my dreams include I never dreamt about ordinals before I learning them properly, I never had 4D dreams before I know about the tesseract and so on

Akiva Weinberger

I think I had $B_n$ in mind at first and then got confused, tbh

That acts by a weird twisting thing

What's "majorization"?

(Also my phone just tried to autocorrect 'w' to 'without boundary' I think)

Secret

13:09

Majorization

In mathematics, majorization is a preorder on vectors of real numbers. For a vector a ∈ R d {\displaystyle \mathbf {a} \in \mathbb {R} ^{d}} , we denote by a ↓ ∈ R d {\displaystyle \mathbf {a} ^{\downarrow }\in \mathbb {R} ^{d}} the...

user84215

What is the importance of invisible parentheses?

Secret

It is a preorder on the components of two vectors, such that the sum of subvectors of one is always $\geq$ than the other

Akiva Weinberger

@aminliverpool I don't know

What are invisible parentheses

Secret

Even if you guys managed to show there exists an algoritm that can consistently trigger mechanism 2 so that you can incubate my dreams indirectly, the result is not guarenteed to be useful. An analysis of my dreams from 2015-2017 found my dreams are in general poor at solving existing problems (because it often invent its own system of knowledge that make no sense in the real life context, but make perfect sense under the dream's setting) but good at creating new structures and ideas

Balarka Sen

@AkivaW yo you should learn riemannian geometry with me

user84215

13:14

@AkivaWeinberger I mean using \. in formula typing

Secret

braces in mathjax (and latex as well) are indications where arguments are specified, putting a \ before them escapes them so they are treated as symbols rather than part of a command

For example:
{ab} gives ${ab}$

but \{ab\} gives $\{ab\}$

user84215

@Secret Did you answer to my question?

Secret

yup

user84215

But my question is about invisible parentheses.

Secret

$\left( \right.$?

user84215

13:23

I mean the command \.

Secret

$\.$

Give me an example where the invisible bracket is in action, cause as seen above, just typing it does not seemed to be recognised by mathjax

user84215

try \left.

user84215

for example: \left. x \right.

user685272

what does that have to do with invisible parentheses?

( )

^parentheses

Secret

$\left. \right.$ are used when you only want one of the \left or right perenthesis. For example

Abcd

13:29

How to solve this question?

I know how to compare electron affinity between atoms but not ions.

Secret

$\left(\begin{matrix}blah \\ blah \\ blah\end{matrix}\right.$

user84215

you can do it if you do not insert one of the parentheses. It does not need to use invisible parentheses.

Secret

$\left(\begin{matrix}blah \\ blah \\ blah\end{matrix}$

will not render, \left must pair with \right

Here's a more practical example on how it is used:
$$f(x)=\left\{\begin{matrix}0,x\neq q \\ 1, x = q\end{matrix}\right.$$

user84215

Thanks.

Secret

$\Bigg($
https://tex.stackexchange.com/questions/38868/big-parenthesis-in-an-equation

pilko

13:35

Has something changed, LateX not working anymore in my chat ?

user685272

it works for me

pilko

arf found the reason, http changed to https

user685272

$( \big( \Big( \bigg( \Bigg($

parvin

@AkivaWeinberger closure under countable unions and closure under countable intersections, measurable spaces, countable spaces !

I don't know about these; i searched but all i got was some unfamiliar math-talks!

user84215

Is it necessary to adjust the size of parentheses manually?

Secret

13:42

A brief summary on AI impacts. For mathematicians, they might want to have a quick read about it and use this as a guide on thinking about how it will affect exploration of mathematical structures and searching algorithms

pilko

I have a doubt about an answer I made : math.stackexchange.com/questions/2352152/… , I find it very simple but got 0 vote while the other two keep being upvoted for hours. Did I wrote something that terrible ?

user84215

The command \left( does that for you automatically.

SteamyRoot

It doesn't necessarily do it like you want it to, especially if you're working in a multi-line situation.

Secret

and it is also quite poor for exponential tower expressions

pilko

yep, I often fall back to $\exp$ in many cases stuff getting too small.

Secret

13:51

For me, I invented a new symbol for complicated exponential towers, though currently only math chat people know about it and it is probably not very useful outside of investigations regarding tetration

pilko

it's a popular subject though.

Secret

Jun 7 at 14:11, by Secret

The next section will require some slight change of notations here as nesting brackets get a bit annoying when there are $n$ of them. Let $\mathop{\Large{E_L}}_{i=1}^n a_i$ to denote an exponential tower of terms $a_i$ all left associative, and let $\mathop{\Large{E_R}}_{i=1}^n a_i$ for the right associative case. Tetration is then given by $\mathop{\Large{E_R}}_{i=1}^n a={}^n a$. Now

Fargle

Hello chat.

Balarka Sen

Hello individual.

Fargle

grumbles

Alessandro Codenotti

14:30

Hi chat dwellers

Fargle

Hello, @Alessandro, how goes it?

Alessandro Codenotti

14:43

Meh

I'm studying for a probability exam, which is not a subject I'm fond of

Fargle

14:58

Probability's never been my favorite either.

I couldn't tell you why--I don't have a similar disdain for analysis as much as I like to joke in that direction.

Secret

My issue with probability and combinitorics is that I often count the wrong things or forgot to count some things

Akiva Weinberger

Combinatorics is the math that counts

@parvin Do you know what "countable" means?

Fargle

@BalarkaSen Huh, interestingly indeed. Highly nontrivial but not nearly nice enough to be transitive--the origin is in its own orbit.

Balarka Sen

right.

the "diagonal line" is preserved

Fargle

Still faithful, though, unless I'm misremembering the definition of faithful.

Balarka Sen

15:09

well faithful means gx = x iff g = id

that's not true by what you said

Fargle

I thought it meant that no element acts trivially on the whole set except the identity.

Astyx

Hi people

Fargle

Hello @Astyx

Astyx

What's up ?

Fargle

Not much. Reading books like a nerd. You?

Astyx

15:16

Just finished my oral exams for Polytechnique

Fargle

Nice! How do you think you did?

Astyx

Bad for physics, but the rest was good

I try not to think about it until I have the results :p

Fargle

That's wise.

Astyx

Since it's not constructive to do so

In any case I have other exams to distract me from thinking about it until then

(yay)

What books are you reading ?

Fargle

I've been trying at Rudin for a long time but I haven't been able to make much progress at it. I blame a lack of mathematical grit.

Astyx

15:19

Oh, so mathematical books :p

(What was I thinking ?)

Fargle

I could have been reading a selection from @Balarka's library.

I prefer to stay sane, however. ;P

Astyx

Wise decision

Fargle

I kind of want to find an analysis book that's more conducive to self-study, but Rudin seems so complete that it would be a shame to stop.

Balarka Sen

I have some excellent suggestions for you, Fargle [unhinged grin]

Typhon

@Secret That is interesting indeed.

Astyx

15:22

I probably am not the best person to advise you on this

Fargle

@BalarkaSen It'd better be real analysis. leers

Not, ahem, surreal analysis.

Secret

arxiv.org/pdf/1307.7392.pdf

Fargle

You found that almost too quickly.

Vrouvrou

Hello, someone know how to write $\sup_{a,b}$ such that b under a ?

Secret

$\sup_{a_{b}}$

Vrouvrou

15:34

no not like this

for example a is ||u||\leq 1

and b is u\in X

i tried a\\b but this don't work

@Secret

SteamyRoot

$\displaystyle \sup_{\substack{a\\b}}$ is probably prettier, though.

so yeah, use substack

Vrouvrou

@SteamyRoot thank you

Secret

Testing
$$\atop{a\\b}$$

It works. Now to...

$${}^0_0\atop{0\\0\\0}_0^0$$

bleh, not working as I expected

PVAL-inactive

PoMA is probably fine for self-study.

There's better books than RaCA though

I've heard good things about FA but never used it.

Fargle

15:50

I'm sure there are. I'm just having trouble with the former because I feel like pretty early on he just starts throwing proofs at you without much in the way of elaboration.

FA?

PVAL-inactive

I didn't know what Rudin book you are talking about.

So I listed all 3

Fargle

I meant PoMA, i.e. baby Rudin.

Ted Shifrin

Hi @Fargle, PVAL

Fargle

Hi @Ted

Ted Shifrin

Rudin is the master of the masterful proof ... not the greatest pedagogue.

Fargle

15:54

I'll be back in 5-10 minutes, gotta run an errand real quick.

braces for Ted to criticize conflation of an adjective with an adverb

Ted Shifrin

happy erranding

Fargle

Always.

Balarka Sen

errandeous

Ted Shifrin

well, he's supposed to be errandeous

but perhaps it's a myth'

Fargle

I'm back.

Astyx

16:03

Me too

Fargle

I guess I should just stick to Rudin. I'll get more out of it. I just feel like I'm moving slowly.

Ted Shifrin

oh hell, @Astyx is here too?

Astyx

I can leave if I'm bugging you

Ted Shifrin

LOL, so obliging :P

Balarka Sen

@Ted learning the sectional curvature

Astyx

16:08

I try my best

Ted Shifrin

@Balarka': We talked about that once before (I told you it's Gaussian curvature of exp of the subspace).

Balarka Sen

Yeah, I know

that was a nice intuition

Ted Shifrin

I'd give you the moving frames exercise to prove that, but ... :D

Balarka Sen

I have accepted that moving frames is not my thing and have moved on :)

@AkivaW was supposed to learn differential forms with me, but he didn't

Ted Shifrin

rolls eleven eyes

well, prove it some other way, then

But don't make the mistake of thinking that $\exp(\Lambda)$ is a totally-geodesic submanifold.

Balarka Sen

16:13

Hm, interesting. I don't think I have a counterexample off the top of my head.

Ted Shifrin

a counterexample to whom?

oh

well, don't think about constant curvature spaces ...

Balarka Sen

An example of a Riemannian manifold such that there's a 2-plane which does not exponentially project to a totally-geodesic submanifold (locally)

Ah, ok, let's see

Fargle

@Ted Not to distract you, but: if $y^n > x$, why is it that $\frac{y^n - x}{ny^{n-1}} < y$?

$x,y > 0$ as well.

Ted Shifrin

Distract me from what? I'm not thinking about Balarka's issue.

Fargle

lol, fair enough

Ted Shifrin

16:25

Yuk.

Where did you get that?

Fargle

It's part of Rudin's proof that positive numbers have a positive nth root.

Although, now that I ask it, it seems really trivial.

Ted Shifrin

So $n\ge 1$?

Fargle

Because clearly $(n-1)y^n > -x$, and then yeah.

Yep.

Well, $n > 1$. $n = 1$ is easily handled.

Wait, no, he never has to assert that, I don't know why I'm being overly careful.

Ted Shifrin

I don't know why you're staying bogged down in chapter 1 ...

I'm pretty sure he's talking about positive integers $n$.

Fargle

I've made it to chapter 3. I'm just trying to make sure I understand every part of every proof.

Yes, he is, I double-checked. My bad.

That was just a detail of that particular proof that wasn't obvious to me, but for some reason I find things become obvious once you ask someone else, before they even answer.

Ted Shifrin

16:30

That's why working with someone else is a good way to learn math.

Fargle

Hey @Balarka, do baby Rudin with me. :P

Ted Shifrin

Balarka' only answers if you put the apostrophe at the end.

Balarka Sen

@Fargle I should be learning too many things unfortunately.

Actually I might do a few chapters with you, I need some analysis for my admission tests

Ted Shifrin

I'll send you guys the takehome final I wrote for the guys who did a Rudin reading course with me years ago.

Balarka Sen

Excellent!

Fargle

16:36

I was about to say you'd already sent it to me, but I think that was an Artin-based exam instead. I'd be glad to have it.

Ted Shifrin

Actually, the algebra I sent you wasn't Artin-based. All the stuff from when I taught out of Artin in 1982-3 is long gone.

Fargle

Oh, I see.

Ted Shifrin

Sad to have thrown away so much of my professional life ...

Balarka Sen

Thanks, @Ted

Fargle

Yeah, I appreciate it as well.

Ted Shifrin

16:40

You won't say that after you get stuck :P

Fargle

If mathematics were easy, it wouldn't be so fun.

Or so they keep telling me...

Balarka Sen

@TedShifrin I still can't come up with a satisfyingly neat example. I'm thinking of giving $S^3$ some weird metric.

Ted Shifrin

Hmm, I wouldn't know how to compute with that.

Semiclassical

hi chat

Ted Shifrin

heya Semiclassic

Semiclassical

16:50

any good math today?

Astyx

Nothing but hexapoles for me

Ted Shifrin

Balarka's trying to think of an interesting Riemannian geometry example.

Semiclassical

Hexapoles?

Ted Shifrin

they're more complicated than tadpoles

Semiclassical

I know mono-di-quadro-octo-etc poles.

Ted Shifrin

16:52

heya EricS

Balarka Sen

The thing is most standard > 2 dimensional non-constant-curvature manifolds I know have rather tedious geodesic descriptions

Eric Silva

hi hi

Ted Shifrin

You can think about the geometry challenge I gave Balarka', too :)

Astyx

@Semiclassical This

Semiclassical

I think a configuration of six charges with no monopole or dipole moment would necessarily have a quadropole moment

Ted Shifrin

16:53

well, you could try Lie groups with the bi-invariant metric, Balarka' ... but that wasn't my first idea.

Astyx

Used to absord neutrons

Semiclassical

I wondered if that's what configuration you meant

Balarka Sen

@Ted Should they not have constant curvature?

Eric Silva

what is the challenge

Ted Shifrin

say what, Balarka'?

Eric: Give me an $\exp(\Lambda)$ ($\Lambda$ a 2-plane) that is not totally geodesic.

Semiclassical

16:54

I think that configuration indeed doesn't have a net charge or a dipole moment. But it should still have a quadropole moment.

@TedShifrin I don't even know what that means :(

Balarka Sen

I thought Lie groups with the biinvariant metric has to have constant curvature; can't you g-translate the curvature operator to various points on the Lie group?

If that makes sense

Ted Shifrin

You know what exp means, @semiclassic — follow geodesics.

Big error, @Balarka'.

Semiclassical

Though my brain did go to Lie groups once I saw $\exp$ so I'm happy enough with taht.

Ted Shifrin

Surely $SO(n)$ is not a sphere.

Doesn't need that kind of exp, Semiclassic.

Fargle

Oh God Lie groups

Astyx

16:57

Oh godly groups

Semiclassical

I don't recognize what the $\exp$ of a 2-plane is supposed to mean, though. Best I can guess is something like "image of $\Lambda$ under $(x,y)\to(e^x,e^y)"$ and I imagine that's not even close.

Balarka Sen

@Ted Huh. Interesting. So homogeneous spaces need not be constant curvature.

Ted Shifrin

No, no, @Semiclassic: It means (in a general Riemannian manifold) — follow the geodesics with tangent vectors in that given $2$-plane $\Lambda\subset T_pM$.

Indeed, @Balarka. Very not constant curvature. All Grassmannians, flag spaces, etc.

Semiclassical

Hm.

Balarka Sen

Is the opposite direction true? Need constant curvature spaces be homogeneous?

Ted Shifrin

16:59

Great question.

Semiclassical

So for M=S^2 and p=north pole, would that be lines of longitude?

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$Mathematics$ Mathematics