Mathematics - 2018-01-22 (page 4 of 6)

Abhas Kumar Sinha

05:00

@AkivaWeinberger can you solve that ?qph.ec.quoracdn.net/… is the question wrong?

Faust

@AkivaWeinberger that wasa good read

Akiva Weinberger

@AbhasKumarSinha Skew-symmetric means $M^\top=-M$ and $N^\top=-N$, right?

Mt=-M, Nt=-N

Abhas Kumar Sinha

okay

Akiva Weinberger

Also, it turns out that if $M$ and $N$ commute (that is, if $MN=NM$), then $M$ and $N^{-1}$ commute (that is, $MN^{-1}=N^{-1}M$). Similarly, $M^{-1}$ commutes with $N$ and $M^{-1}$ commutes with $N^{-1}$. (You should prove this, it's not too hard)

Abhas Kumar Sinha

@AkivaWeinberger also this one - qph.ec.quoracdn.net/…

Akiva Weinberger

05:02

In other words, everything commutes with everything.

Abhas Kumar Sinha

qph.ec.quoracdn.net/… is difficult to get done, as I've tried it with 3 different methods and failed

both are same

Akiva Weinberger

That one looks significantly harder

I'm bad at geometry

Abhas Kumar Sinha

okay

its from iit, so its hard

@AkivaWeinberger you know a good way to get iit done without coaching?

Akiva Weinberger

What's IIT?

Are the circle and line in that problem tangent?

Abhas Kumar Sinha

its a entrace exam, with having world's hardest question

its iit

even mit teachers can't do for them

@AkivaWeinberger no, i don't think

Akiva Weinberger

05:06

Hm, looks like they are tangent, and they meet at $(\sqrt2,1+\sqrt2)$ if I'm not mistaken

Abhas Kumar Sinha

hmmm

Akiva Weinberger

Wait no

That makes no sense

Abhas Kumar Sinha

yea, that's what i want to say, it makes no sense

Akiva Weinberger

I cheated and graphed it. They are tangent and they meet at $(1,2)$.

desmos.com/calculator/k0yjs7tygd

Abhas Kumar Sinha

good tool, but those are not available in exam

also, this question is made to be solved in 1 min

there are timing factors that sucks in exam

Akiva Weinberger

05:08

Oh good lord

Well I'm dead then

Abhas Kumar Sinha

hehehe

Akiva Weinberger

I don't even remember how eccentricity is defined, to be honest

Abhas Kumar Sinha

@AkivaWeinberger how much of maths you can do from this year question paper - jeemain.nic.in/webinfo/PDF/Paper01_RBS_English_Hindi_SetA.pdf

@AkivaWeinberger me too, i too haven't remembered till now

I love wiki and its the best book of all

heh XD

Akiva Weinberger

I've also forgotten about the moment of inertia

Abhas Kumar Sinha

that's easy

just wiki those, also there's a mnemonics to remember those stuff

Akiva Weinberger

05:11

7. is weird. I thought gravity changed linearly inside the planet?

Abhas Kumar Sinha

*Mnemonic

no

not everytime

that's physics and maths

Akiva Weinberger

Oh, wait, that is an option

so it's 7(d)

Abhas Kumar Sinha

i didn't have solved it yet

I've not studied that chapter properly

Akiva Weinberger

I also know nothing about how electricity works

or what the parts of a circuit diagram mean

Abhas Kumar Sinha

I tried to give a mock test for that chapter, but I scored 30 marks, (a lot of -ve marks for wrong answer) and time sucks

Its difficult exam in the whole planet

only in india

@AkivaWeinberger you are phd or something like that?

Secret

05:14

Why are you Indians have such difficult exams and yet having the worst textbooks in existence?

Akiva Weinberger

Right, part B, chemistry... I took the Chemistry AP test like two years ago so maybe if I studied again I'd do decently on that bit

@AbhasKumarSinha High school

Part C, math, now that should be easier

Abhas Kumar Sinha

you are in US?

@AkivaWeinberger

Akiva Weinberger

Yes

AP is Advanced Placement

Abhas Kumar Sinha

so, when you were of 10th grade, then you should have completed this exam

and cracked those

its difficult for professors and students have to do those

lol

Secret

High schoolers nowadays knew a lot lot more than back in my generation wow

Akiva Weinberger

05:16

62 looks hard and interesting

Oh, wait, there's a symmetry...

...or is there...

Abhas Kumar Sinha

no idea for 62....

dumb

Secret

Akicva: I thought you are a postgrad given your level in set theory, number theory and some geometry

Akiva Weinberger

Me neither, I don't think my idea works anymore

Secret

Akiva: I thought you are a postgrad given your level in set theory, number theory and some geometry

Akiva Weinberger

@Secret I'm like a tiny bit older than Balarka

Secret

05:18

Wow it does seems like I am the oldest and most immature of the people in the chat lol. Everyone easily have more than 20 years mental age older than me

Abhas Kumar Sinha

@Secret @Secret You've not seen NCERT Offical books till now? those are one of the best books, made by 108 researchers of the community

Secret

Never heard of them until now. Should check later

Abhas Kumar Sinha

@Secret those are even better than MIT

Akiva Weinberger

@Secret Are you forgetting Ted?

And I think me and Balarka are the only high schoolers on here

Oh, and Meow

Secret

Ted is old but a lot mature

Abhas Kumar Sinha

05:19

@Secret i'm 14

lol

I think I'm the youngest here

Akiva Weinberger

Oh, I misunderstood

Secret

I am postgrad age but I have a very childlike personality

Akiva Weinberger

You're 14 and learning matrices?

Abhas Kumar Sinha

yep

Akiva Weinberger

Nice!

Abhas Kumar Sinha

05:20

@AkivaWeinberger you?

Secret

as evidenced by most of my incomprehensible rambles

Abhas Kumar Sinha

@AkivaWeinberger your age?

Akiva Weinberger

18

Abhas Kumar Sinha

that's great

Akiva Weinberger

I think 14-year-old me was dimly aware of how matrix multiplication worked and had a distaste for the idea of vectors

Secret

05:21

26

Akiva Weinberger

(I've since changed)

Abhas Kumar Sinha

@AkivaWeinberger I've leaned matrices doesn't mean i'm that perfect in that

I'm still a beginner in it.

@Secret so you'r collage or high school?

Secret

College chemistry phd

Abhas Kumar Sinha

phd?

that's awesome

Secret

I do kinda behave like a 13 year old though

Abhas Kumar Sinha

05:23

hehehe

lol

me too

stay hungry, stay foolish

@Secret you too try chemistry section of previous year paper - jeemain.nic.in/webinfo/PDF/Paper01_RBS_English_Hindi_SetA.pdf

Secret

Since 8 I knew the promise of growing up is a sugar coated lie, thus I fought hard to keep my dreams

and stay young

Akiva Weinberger

@AbhasKumarSinha For linear algebra, I highly recommend this ten-episode series on it

youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

Abhas Kumar Sinha

@Secret yea

@AkivaWeinberger will check

Akiva Weinberger

Er, 11 actually

Secret

Will check when off mobile

Akiva Weinberger

05:25

It's the 3Blue1Brown one @Secret

I thought you already had seen it

Abhas Kumar Sinha

freehomedelivery.net/class-12-ncert-book-mathematics-part-1

freehomedelivery.net/class-12-ncert-book-mathematics-part-2

for NCERT @AkivaWeinberger

Akiva Weinberger

(The same channel also has a series on calculus, as well as other videos on miscellaneous topics)

Abhas Kumar Sinha

for main section - freehomedelivery.net/ncert-books-class-12-pdf-download-free

Akiva Weinberger

Achievement get: spell "miscellaneous" correctly on the first try

Abhas Kumar Sinha

@AkivaWeinberger Should I go to coaching or prepare myself at home?

Akiva Weinberger

05:27

(Though I think I get points taken off for initially misspelling "achievement")

@AbhasKumarSinha I'm not sure, but maybe coaching because then you always have someone to ask if you have questions

The most important thing for a test like this, I think, is to do a lot of timed practice tests.

If available.

Abhas Kumar Sinha

Shouldn't I ask my doubts here?

Akiva Weinberger

You can, but there's no guarantee that someone will know the answer

Abhas Kumar Sinha

okay

@AkivaWeinberger that's a great channel

Akiva Weinberger

(Also, it's interesting how Indian English uses the word "doubt" where most other varieties of English would use the word "question")

Abhas Kumar Sinha

i love the way

@AkivaWeinberger yep, i see

Akiva Weinberger

05:30

Spanish uses the word duda the same way

Just a random linguistics fact

Abhas Kumar Sinha

duda

Akiva Weinberger

Doubt

Abhas Kumar Sinha

World it too big, someone might knew the answer

perhaps who made the question

Akiva Weinberger

Yeah, there's nothing stopping you from asking questions on here, but I feel like face-to-face might be better

But I'm not 100% sure at all

And it's your decision

Abhas Kumar Sinha

hmmm

Akiva Weinberger

05:41

Cool picture:

Showing how $(\sum n)^2=\sum n^3$

$(1+2+3+4+5)^2=1^3+2^3+3^3+4^3+5^3$ and so on for general $n$

Abhas Kumar Sinha

that's the formula to get golden ratio

Scott flansburg's show did that but i've no idea except construction

@AkivaWeinberger from where you get those cool stuff?

Akiva Weinberger

reddit.com/r/visualizedmath

Sorted by "top"

Abhas Kumar Sinha

hmmm

seems to be cool

mathematics is just a set of pattern which are random at times and are not there many times .

Akiva Weinberger

mathworld.wolfram.com/JohnsonsTheorem.html

I think I see why that's true ^

though it took a lot of staring

(Demonstration: reddit.com/r/visualizedmath/comments/7nyq6l/johnsons_theorem, the point being that the red circle is the same size as the black ones)

Abhas Kumar Sinha

okay

but that theorem is still difficult for me

I was trying to derive a formula to get sine values of all angles (in degrees)

Akiva Weinberger

05:53

The theorem I posted just now?

Abhas Kumar Sinha

later found that same already happened in S. L. Loney's book

no

I was trying to make that

It was already invented

Akiva Weinberger

Oh, I see

Like, writing $\sin(3^\circ)$ explicitly

Abhas Kumar Sinha

similar

art no. 77 in sl sloney's trig book

lol - scontent.fbbi1-1.fna.fbcdn.net/v/t1.0-0/cp0/e15/q65/s320x320/…

Akiva Weinberger

Oh, that reminds me of this

Abhas Kumar Sinha

its difficult, but i did that

just take 3 variables and you'll get a 3d graph line solution

Akiva Weinberger

05:57

Positive whole values?

Abhas Kumar Sinha

for those

Akiva Weinberger

Because it turns out that the smallest set of integers that works is:

a=154476802108746166441951315019919837485664325669565431700026634898253202035277999,
b=36875131794129999827197811565225474825492979968971970996283137471637224634055579,
c=4373612677928697257861252602371390152816537558161613618621437993378423467772036

No, it's not a computer error, and yes, that is the smallest solution

Abhas Kumar Sinha

alon amit's book has that question

Akiva Weinberger

He has a book?!

Abhas Kumar Sinha

yep

i saw his name

i don't know him

who's he?

Akiva Weinberger

06:00

He posts answers to math questions to Quora a lot

and they're often quite good

I took the above solution from one of his posts, in fact:

quora.com/…

Abhas Kumar Sinha

okay

you'r on quora?

Akiva Weinberger

I look at it on occasion

Here's another cool thing:

bl.ocks.org/travisdoesmath/c0d08396a0bf8d5de8519ded00bb7866

40 double pendulums, starting in almost exactly the same position (so it looks like just 1 double pendulum)

Abhas Kumar Sinha

would you mind if i'll see your quora answers?

Akiva Weinberger

but then they start to diverge

I haven't been writing answers, I've just been looking at stuff posted there

Abhas Kumar Sinha

@AkivaWeinberger great explanation

Akiva Weinberger

06:02

Here: quora.com/profile/Akiva-Weinberger

Abhas Kumar Sinha

followed u

Akiva Weinberger

Thanks

Abhas Kumar Sinha

great :D

for those who think that 0.9999...... = 1, there's a problem

i think

Why Every Proof that .999... = 1 is Wrong

►

Not every proof really proves that 0.999999........... = 1

Akiva Weinberger

> Published on Apr 1, 2012

Abhas Kumar Sinha

yup

I'm watching this now

Akiva Weinberger

06:17

I'm going to bed now

Good night

and April Fools'

Abhas Kumar Sinha

great

hehehe

XD good night

Secret

08:25

@AkivaWeinberger O I am responding to Abhas about that chemistry past paper

Balarka Sen

09:08

@LeakyNun ?? It's a group. It's the group of unit quaternions

(Which makes it into a Lie group in fact)

Alessandro Codenotti

Hi @Balarka

Balarka Sen

Maybe you meant it is not per-se a group

Hi @Alessandro

Brown Ninja

12:10

Hi,
$XY = Z$ where all $X$, $Y$, and $Z$ are unitaries. Is it possible to find an $X$ for any $Y$ and $Z$?

if there are some conditions, can you please tell me what they might be?

Maybe $X = Y^\dagger Z$ solves the problem :D

Silent

If $a,b,r,s$ are integers, then $ra+sb=1$ implies $\gcd (a,b)=1$. How?

Krijn

@Silent What happens if another number $k$ divides $a$ and $b$

Silent

12:26

If another number is not 1,0,-1, then gcd(a,b)>1. @Krijn.

Oh! k also divides ra+sb

hence k divides 1 , got it

thanks

user8469759

13:35

hi guys, quick question let $X$ be a random vector in $\mathbb{R}^m$ and let $f : \mathbb{R}^m \rightarrow \mathbb{R}^n$ how can I derive the distribution function of $Y = f(X)$?
I'm sure this is pretty standard, at least as procedure
but I can't manage to find any reference

Semiclassical

13:52

@user846975 Look up change/transformation of random variables

The main trickiness is that $f$ need not not be one-to-one. For instance, if $f(x)=x^2$ then $f(X)$ will only have support on the positive reals

user8469759

can you point out a specific link?

or is it the very first one?

Semiclassical

use your google-fu

2

user8469759

that's what I'm doing

too many links

and I can't see the case I want

Semiclassical

Well, what’s the case you want?

Just R^n to R^m?

user8469759

well yea, it's most general case

Semiclassical

14:00

If that’s all you know, you’ll have a hard time finding anything

Balarka Sen

@Semiclassical Hey so remember the mechanics question I asked you last year about center of gravity of consecutive blocks stacked over another?

user8469759

how about n = 3 and m = 2?

and as more specific case

f is an homeomorphism

Balarka Sen

So we concluded that the $n$-th block from the top should stick out $\frac{1}{2n}$ amount from it's center (of gravity) for it to be balanced properly (all the blocks are unit length, say)

Semiclassical

Without more info, you know basically nil about $f^{-1}$ beyond it not being injective

Balarka Sen

And the paradox was that $\sum 1/(2n)$ diverges, so the tower could technically stick out arbitrarily long in the horizontal direction

Semiclassical

14:03

@BalarkaSen sure

Balarka Sen

There's a new video Vsauce posted where he constructed a tower which sticks out ~3 meters from the center of the bottom-most block

The Leaning Tower of Lire

►

Semiclassical

@user8469759 Is it actually possible for f to be a homeomorphism if the domain and codomain have different dimensions?

user8469759

it is

take the definition of surface

tipically given

Semiclassical

What?

user8469759

I specified from $R^m -> R^n$

R^2 -> R^3

Semiclassical

14:07

The case of a surface is a 2D subset of R^3

user8469759

I don't get your question

basically I have a function from R^2 to R^3

homeomorphism

Mitch

@AkivaWeinberger Thanks, that makes sense. I kind of get it (I mean I totally get it, but I'm just trying to figure out the non-obvious difficulties with -not- using jargon). I'm pretty sure it was Voevodsky who had the rejoinder to Feynman, but I just can't find it or remember right.

user8469759

and if I pick (u,v) random, I'd like to know what's the distribution of x,y,z

(u,v) random or taken from some given distribution

about the existance I took the definintion of surface given by my differential geometry book

Semiclassical

That doesn’t make sense. Take $(x,y)\mapsto (x,y,x^2+y^2)$

That function is one-to-one but not onto

user8469759

it is, if you take the range

not the whole R^3

Semiclassical

14:11

But then your codomain isn’t R^3

user8469759

ok I see what you mean

what I meant is a subset of R^3 that allows you to define an homeomorphism

like a surface

Semiclassical

hmm...

Probably simplest to take m=2, n=1 here

user8469759

but isn't there any approach to the case I proposed?

the distribution of the function is usually defined in terms of differentials or jacobians

there most be something I can use in my case

Mitch

@AkivaWeinberger I had a lot of trouble with that. It was the pineapple that threw me

Semiclassical

the R^n to R^n case is discussed here: ocw.mit.edu/courses/electrical-engineering-and-computer-science/…

I suspect a model example of what you want is the map $\theta\to (\cos\theta,\sin\theta)$

That’s a map $f:\mathbb{R}\to S^1\subset \mathbb{R}^2$

Main complication I foresee is that there’s an infinity of different $\theta$’s which map to the same point on the unit circle

All of which should contribute to the density of $f(\Theta)$

user8469759

14:33

what I want to do is the following

1. Take a surface patch (very small) of a given surface

2. assume (u,v) are random

3. given this what's the distribution of the normals given by the gauss map

at the end of the day I guess this is equivalent to find the distrubution of the polar angles given u,v

so it is actually a map from R^2 to R^2

because it is an homeomorphism, the distribution of the normals given the polar coordiantes is deterministic

so we have something like $p(n | \theta, \phi) = \delta(n - n(\theta,\phi) )$

Alessandro Codenotti

15:07

Sanity check: pointwise limit of nondecreasing functions is nondecreasing?

PVAL-inactive

@AlessandroCodenotti If this were true you could probably prove it.

anakhronizein

@PVAL-inactive any idea what a linearisation of a vector field is at a singular point?

I was thinking just the pushforward of $X\colon M\to TM$ at 0.

well p such that X(p)=0

But is $T_0TM \cong T_pM$?

Alessandro Codenotti

@PVAL-inactive I think I have a proof, but I'm unsure

Balarka Sen

TM is 2n-dimensional, so how can that isomorphism hold...?

PVAL-inactive

@anakhronizein I know all these words but you are using them strangely.

oh

you want to compute dv for a vector field v:X \to TX?

anakhronizein

15:21

Well it's just that I am reading a paper (Giroux) and he uses "linearisation" of a vector field (notably, a contact vector field). I am trying to figure out what he means so that I can try to figure out another thing he mentions.

PVAL-inactive

Which paper?

anakhronizein

Effectively though, yeah I am asking if I can view dv at a singular point x of V as a map $T_xX \to T_xX$.

It's on convexity in contact topology. Can you read french? There is also an english version I can dig up for you.

PVAL-inactive

Convex surfaces?

anakhronizein

I'd say yes, but I have not gotten to the convexity part in the paper yet. ;)

PVAL-inactive

I theoretically should know the material which is almost like actually knowing it.

anakhronizein

15:24

Heh

gdz.sub.uni-goettingen.de/pdfcache/PPN358147735_0066/…

en francais

PVAL-inactive

show me where you are looking

anakhronizein

few.vu.nl/~pasquott/Giroux-convexity.pdf in english

Ummm let's see

First mention of the linearisation is at example 2.6, the proof of it

pg 6 of the french pdf

also pg. 6 of the english one

But I am needing to understand the usage of linearisation to try to figure out what is meant by a later comment, remark (ii) on the same page(s) just a bit after about the trace of the linearisation and the divergence.

PVAL-inactive

they mean the derivative of the time zero flow.

that's the only thing I can think of.

anakhronizein

$d(\phi_0)$?

PVAL-inactive

yeah

anakhronizein

15:30

Ah, I see. I did not consider that.

Are you familiar with the remark about the divergence?

Or perhaps know a reference that might have such a remark?

Maneesh Narayanan

in Room for Maneesh Narayanan and BAYMAX, 23 mins ago, by Maneesh Narayanan

@BAYMAX https://math.stackexchange.com/questions/2616224/what-is-the-guarentee-of-exista‌nce-of-the-function-t-phit-in-the-proof

please help me.

really I am not spamming.

PVAL-inactive

I'm really bad with traces.

anakhronizein

Well thanks for your help anyways. I will pursue this derivative of the flow and see if it tells me anything about the trace!

PVAL-inactive

There are lots of expository sources on this material which are probably easier to read.

anakhronizein

Do you have any suggestions?

PVAL-inactive

15:35

e.g. Geiges, Ozbagci-Stipsicz

anakhronizein

Oh, geiges I find difficult.

PVAL-inactive

There's some nice notes by Etnyre on convex surfaces.

anakhronizein

One of my open questions on stackexchange is about a proof of his which I do not follow.

Yeah, Etnyer basically uses Giroux as the main outline.

And doesn't elabourate on the divergence, from what I saw.

PVAL-inactive

I'm really sleep deprived now.

I'll try looking at your question and this when I'm less so.

anakhronizein

It's fine, don't worry about it!

Though looking at Ozbagci-Stipsicz, they do only use the term "linearization" in terms of the flow at a generic singularity.

So that's good news since that means you are probably right.

But they do not have the divergence comment.

Transcript for

$Mathematics$ Mathematics