Mathematics - 2017-04-16 (page 1 of 6)

Tim The Enchanter

00:52

Hey chaps

@TedShifrin Incidentally if you had to choose between vectors and complex numbers for geometry in $\mathbb R^2$, what would it be?

DHMO

how is |G:H| defined for groups?

Balarka Sen

01:09

@DHMO Number of cosets of H in G.

DHMO

number of cosets?

Balarka Sen

Yes, number of cosets.

DHMO

say G is 2Z and H is Z

or have I reversed them

Balarka Sen

You have.

DHMO

oh

so it is 2 because we have 2Z+0 and 2Z+1?

Balarka Sen

01:12

Yes.

DHMO

wow

7 hours ago, by user193319

I never knew that H v K is a group

so 2Z v 3Z is still a group?

Balarka Sen

It is defined to be a group.

2Z v 3Z is all of Z in Z.

DHMO

wait what?

I thought 5 isn't in 2Z V 3Z

Balarka Sen

5 = 2 + 3

DHMO

I thought V means union

Balarka Sen

01:19

Then you thought wrong.

DHMO

what does it mean?

Balarka Sen

H v K is the subgroup completion of H \cup K. It is generated H and K.

DHMO

that's nice

Maks

Hi !

DHMO

@Maks hola como estas

Maks

01:30

Quick question, is anyone familiar with karnaugh maps ?

@DHMO Aca ando, de vacaciones en las sierras, vos ??

DHMO

@Maks ocupado :p

Maks

No hay vacaciones por alla ?

DHMO

hay examen

Maks

Ahh jaja

Que estudias ?

DHMO

muchas temas

Wheat Wizard

01:44

I have a mathematics question that I don't think would be very on topic because it is a opinion based. Can I ask it here?

Maks

Que carrera digo

Akiva Weinberger

04:17

You know how angles of a triangle always add up to 180 degrees, but when we say "But what if they didn't?" that kinda leads us to non-Euclidean geometry

Why can't we do that with more theorems

BAYMAX

PID $\implies$ UFD. So what is missing in UFD which cannot be a PID?

ok, they lack the property that every ideal in a PID is a principal ideal.

Balarka Sen

@AkivaWeinberger Wellll... I sort of think that non-Euclidean geometry comes more fundamentally from the unprovability of the fifth postulate from the other four (which is what you boil down to if you try to unwrap the proof of the 180 degree thing anyway).

So it's sort of more natural.

I think of discovery of non-Euclidean geometries as a proof of logical independence of the fifth postulate from the other four, than as something random you come up with if you change a theorem or make it not hold.

Daminark

04:34

Hey everyone!

@Akiva Well, a lot of things are like that

Thunda

is anyone here? got a really weird broken thing

Daminark

Complex numbers, and further stuff like quaternions and all

Thunda

complex numbers and exponents

excuse my lack of latex, but

e^2 i pi (x/T) => (e^(2 i pi)) ^ (x/T) => 1 ^ (x/T) => 1
integrating both sides from 0 to T w/ respect to x gives
(T/(2 i pi)) e^(2 i pi (x/T))|0, T = T
(T/(2 i pi)) (e^(2 i pi) - 1) = T
T = 0

Will Njundong

05:49

anyone currently active?

Thunda

no

Balarka Sen

06:12

Hi @Alessandro

Alessandro Codenotti

Hi @Balarka

How are you?

Balarka Sen

Not bad.

Secret

I wonder, if one did try to 3D print a menger sponge down to maybe the 19th iteration, will the thing become too fragile to hold itself...?

SBM

Wait, what?

3D Print a sponge

Secret

Menger sponge

In mathematics, the Menger sponge (also known as the Menger universal curve) is a fractal curve. It is a three-dimensional generalization of the Cantor set and Sierpinski carpet, though it is slightly different from a Sierpinski sponge. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension. The Menger sponge simultaneously exhibits an infinite surface area and zero volume. == Construction == The construction of a Menger sponge can be described as follows: Begin with a cube (first image). Divide every face of the cube into 9 squares, like a Rubik...

SBM

06:22

Cool

Daminark

Hello frenz

SBM

Hello @Daminark

Somebody on MSE was playing with 2017

3

Q: Computing $x^{2017}+\dfrac{1}{x^{2017}}$ given the value of $x+\dfrac{1}{x}$.

If $x+\dfrac{1}{x}=2017$, then the value of $x^{2017}+\dfrac{1}{x^{2017}}$ is, A) $2017^{2016(\sqrt{5}-\sqrt{3})}$ B) $2017^{2016(\sqrt{3}-\sqrt{2})}$ C) $2017^{2\sqrt{3}}$ D) $2017^{3\sqrt{2}}$ E) None of the above Progress: We could make use of the identity $$x^{m+n}+\dfrac{1}{x^{m+n}}=\...

algebra-precalculus

Daminark

Good lord...

SBM

06:41

...

Daminark

Seems like the answers given all point to none of the answer choices being true

But that definitely looked intimidating

SBM

Yes

Wait a second

I guess ... there exists a way to solve this

Daminark

Hey @Alessandro!

Alessandro Codenotti

Hi @Dami

Daminark

How's it going?

Hey @Akiva!

Akiva Weinberger

06:59

@Secret It would probably exceed the limits of your 3D printer. And if you somehow do it anyway, it would probably break records for the lightest object ever.

Hm, never mind, it would only be (20/27)^19 times its original weight, which is like 0.003

So not, like, infinitesimal

Secret

07:19

Wow, that is lighter than a graphene aerogel

2

SBM

:

Really?

Is it even possible?

CowperKettle

08:38

Russian mathematician Alexey Ivanov was fined $350 for standing on the street with this poster

Balarka Sen

lol really

why

CowperKettle

Of course I have no idea what these formulas mean

@BalarkaSen Because Putin is afraid of losing power.

So everybody gets detained and fined, and imprisoned for 15 days

Balarka Sen

i don't get it.

CowperKettle

More that a 1000 people were detained for participating in peaceful demonstrations on March 26

Got sentences, 15 days in prison, fines.. 10 to 20 thousand rubles

Balarka Sen

"The activists write that Ivanov was detained during an unsanctioned rally on March 26, when he witnessed numerous arrests, took a sheet of paper and wrote on it a mathematical “formula for chaos.” “Half a minute later, he was approached by people in uniform and, of course, without explanation taken to the police bus” — the “Movement of 14%”."

lol!

CowperKettle

08:42

Yep.. several people got detained in my city, Yekaterinburg

Balarka Sen

Yikes.

CowperKettle

I went to the demonstration, but was afraid of being detained... Got laywers' phone numbers in advance, etc.

Balarka Sen

That's really scary. What's this demonstration for?

CowperKettle

The demonstration was against corruption

After a movie about Russia's Prime Minister Medvedev who alledegly got one billion dollars in bribes

In Yekaterinburg, the demonstration was not allowed, because the authorities said "the slogan of the demonstration, Against corruption, runs contrary to the Constitution of Russia"

Always Confused

A similar thing I've seen in fb that was done with a mathematician in aeroplane. I cant remember details.

CowperKettle

08:45

Here is the movie about the Prime Minister, with good English subtitles

18 million people watched it, but not a single mention on Russian TV

Balarka Sen

@CowperKettle That's a terrible action against a peaceful demonstration. I did not know about this documentary.

Anonymous

Any method to calculate area of triangle directly without solving for the coordinates, given the equation of the three lines? Solving for the coordinates is time-consuming. Ideas, people?

Anonymous

@AlwaysConfused Please read it again...

Anonymous

It says the equation of the three lines are given

Anonymous

Your method would be even more time-consuming..

Always Confused

08:55

5

Q: Is there any math formula that can be used to describe shape of leaves?

There can be great variation in shape of leaves. Neverthless they are fairly simple shapes. How far are we in describing shape of leaves in mathematically rigorous way? Do we have a general formula that can be used to describe all shapes (or atleast reasonably big subset ) of leaves?

botany zoology

Balarka Sen

@blue google to the rescue

CowperKettle

@BalarkaSen Here's an article, with a photo that went viral in Russia

Anonymous

@BalarkaSen Thanks...I had tried searching it earlier..couldn't find it

Anonymous

Looks useful

Balarka Sen

@CowperKettle Thanks. I have started reading about the protest; I had no idea about any of this terrible situation in Russia.

Anonymous

09:01

math.stackexchange.com/a/906793/400242 The proof is pretty clever :-)

Space Otter

09:14

If they can make a graph of futerama characters I'm pretty sure they can do leaves.

VanessaBrown

WHY THERE'RE NO NUMBER THAT CAN DIVIDE 0?

DHMO

@VanessaBrown why are you shouting?

0 is a multiple of 0

VanessaBrown

Oh, I'm not shouting

I used CapsLock

Sorry!

DHMO

0 is the only multiple of 0

but 0 is a multiple of every number

so I thought every number divides 0

VanessaBrown

Oh! I understood!

Because I'm in grade 6 so I'm not very good at Math

Mathmetics

DHMO

09:19

alright

VanessaBrown

1+1

equal 10

DHMO

in binary

Space Otter

09:33

@VanessaBrown Ask siri that question

Cookie monster easter egg

BAYMAX

09:54

let us all do a transformation

DHMO

@BAYMAX ?

BAYMAX

In the $z$ plane ,

$$

$0<y<\frac{1}{2.c}$, $c>0$ under transformation $w = u + iv = \frac{1}{z}$

let's consider case-1 ,$c > \frac{1}{2}$

DHMO

what is the original question?

SBM

:

BAYMAX

We have in the $z$ plane the strip , given above , now what is the figure or curve we obtain when we do the transformation $w = 1/z$ onto $w$ plane?

Anonymous

10:01

When a function is said to be continuously differentiable in [a,b] does it mean that the function is only right differentiable at point a ? Or does the left derivative also exist ?

DHMO

@blue its left derivative cannot exist

Anonymous

@DHMO Yeah, but suppose the function exists beyond a on the left. Even then?

DHMO

@blue I don't know

Anonymous

I think we can't conclude anything...

Anonymous

not sure

Anonymous

10:03

@DHMO right

Anonymous

Thanks anyhow :-)

Anonymous

Balarka Sen

@blue What does that even mean? If your function is just defined on $[a, b]$, only the right derivative makes sense at $a$. The left derivative does not even make sense, let alone saying it exists.

Anonymous

Can we conclude anything about the first option? :/ @DHMO

Anonymous

f''(a)=0

Anonymous

10:12

@BalarkaSen I meant supposing the function is defined even on the left of a...

Anonymous

Yeah, on rethinking...it doesn't matter

Anonymous

Only the right derivative matters

BAYMAX

oh that seems nice questions Multiple choice based,what are they,any online course@blue

Balarka Sen

@blue That makes no sense! $f$ is a function on $[a, b]$! How can it exist on the left?

Anonymous

@BalarkaSen No. It is differentiable on [a,b]

Drew N

10:13

it could be differentiable on [a,b] but defined on a larger set

Anonymous

Not "defined"

DHMO

@blue it means $f$ is defined on $[a,b]$.

Anonymous

@DHMO Really? How? See Drew N's comment

DHMO

It is implied in the question

Balarka Sen

@blue @DrewN But no information is given about the function outside $[a, b]$. Suppose I give you $f(x) = x$ on $[0, 1]$. That extends to a buckload of different functions on the left; $f(x) = |x|$, $f(x) = x$. The latter is differentiable at $0$, the former isn't.

Drew N

10:15

it's bad terminology

DHMO

"continuously differentiable" is an adjective

Drew N

usually when you say "on", you mean it's defined on the set

and only that set

DHMO

remove it and you get "Let f be a function on [a,b]"

Balarka Sen

Even if it extends to a larger set, it doesn't so uniquely.

Anonymous

@BalarkaSen Yeah, I get that now. It depends on the specific function.

Drew N

10:15

in order for the question to be properly interpreted, you have to pretend it's also defined in a neighborhood of a and b

Anonymous

Anyhow, leave it

Anonymous

Can we get back to the current question please? =P

DHMO

@DrewN why? it is only twice differentiable on (a,b)

Drew N

well how else are you going to get differentiable at a and b?

you'd get at best left or right differentiability otherwise

DHMO

@BalarkaSen how is [R:S] defined for rings?

and for fields?

Anonymous

10:17

@DrewN That'll suffice

Anonymous

Only right derivative suffices

Balarka Sen

@DrewN That's how it is defined for functions on manifolds with boundaries.

You only care about one-sided derivative.

Anonymous

@BalarkaSen Yes.

Anonymous

Any idea about option A?

Anonymous

10:18

It doesn't say anything

Anonymous

About double differentiability at a

DHMO

ya

Balarka Sen

In any case, look at $f(x) = x^2$ on $[0, 1]$. $f''(0) = 2$.

DHMO

it isn't the first time we see your question f-ing up

Anonymous

@DHMO lol

Anonymous

10:19

These questions are stupid

Balarka Sen

I see you also want $f(b) = 0$. But it's easy to modify it into that.

So option A doesn't hold anyway.

Anonymous

Option B and D look okay

Anonymous

oh no

Anonymous

We don't know if it is thrice differentiable

Anonymous

So only option B?

Anonymous

10:22

@BAYMAX Na, these are past year papers from wbjee...you'll get them online...

Balarka Sen

Ya, it seems only B is correct. But why does it say multiple correct, then?

I don't get it.

DHMO

@BalarkaSen how is [R:S] defined for rings?
and for fields?

BAYMAX

any link @blue

Anonymous

@BAYMAX toppr.com

SBM

Rings? Fields?

Balarka Sen

10:24

@DHMO What does it even mean to say [R : S]? I don't know. You could look at the multiplicative groups and ask about their index.

BAYMAX

thanks

Balarka Sen

Subrings are... not a very natural thing to study.

BAYMAX

$[R:S]$

Balarka Sen

On the other hand if I is an ideal of R, then you could just define the index as the order of R/I. I am not sure if this is a very natural thing to study.

BAYMAX

well are you talking about field extension

Balarka Sen

10:27

For fields you talk about degree of an extension.

BAYMAX

yup

Balarka Sen

But it's not the analogue of "index" per se.

Anonymous

wtf

BAYMAX

actually what are R and S?

Anonymous

10:28

They saay there is no point in (a,b) where second derivative is 0

Anonymous

But before that they show that there is an x where f''(x)=0

Anonymous

wtf

Balarka Sen

They are giving an example of a function where f'' is nowhere zero. It is a modification of $f(x) = x^2$.

Anonymous

@BalarkaSen But in the answer they gave option C as correct

Anonymous

:/

Anonymous

10:31

That is clearly wrong

SBM

:(

Anonymous

And in wbjee 2015 they gave this as a mcq

SBM

What?

Balarka Sen

@blue Why does C contradict their example?

C says there is some $x$ such that $f''(x)$ is not zero

They just gave an example of a function where $f''$ is not zero anywhere.

Anonymous

@BalarkaSen I mean, C is given as an answer to the original question. The example is made up by them and not related to the question.

Balarka Sen

10:33

Ah. C is clearly incorrect, take $f(x) = 0$

For all $x$ in $[a, b]$.

SBM

:[

Anonymous

@BalarkaSen Yep...

Sid

@blue you giving wbjee?

Anonymous

@Sid Yeah...

Anonymous

You?

Balarka Sen

10:36

of course the only right answer is B. whoever says otherwise speaks garbage

Sid

we have a group on whatsapp meant just for this. i am giving jee advanced

@blue can we make a private chat?

Anonymous

I don't want to appear for it though

Anonymous

They ask crap questions

Anonymous

=P

Sid

yes but there are people on the group taking wbjee

from west bengal right?/

Anonymous

10:37

@Sid From assam...but yeah, i have relatives in wb

Astyx

Hi chat

Sid

@blue Make a private chat with me and give me ur whats app we will dicuss problems from wbjee. i might be giving

i will introduce u to others. do u go to cuaching?

Anonymous

@Sid I'm sorry....but I don't have a personal mobile....(I think I told you earlier)

SBM

Oh, all the best for JEEs

Sid

@SBM do u have jee

i mean are you taking this exam?

Anonymous

10:39

@BalarkaSen Hehe. Thank you :-)

SBM

If I appear it that's next April.

@Sid

Sid

you are in 11th?

going to 12th?

@SBM

Balarka Sen

I am going to protest march if JEE invades this chat.

Sid

wait @BalarkaSen im asking something

Anonymous

@BalarkaSen Invite Ocelovsky :D March together!

Sid

10:40

@SBM i will start a private chat

Balarka Sen

@blue I'd also like to protest march with this.

Anonymous

@BalarkaSen lol

Anonymous

what is that?

SBM

350$ for that bad

Balarka Sen

rusreality.com/2017/04/15/…

SBM

10:43

@BalarkaSen

Anonymous

@BalarkaSen I'm sorry man. I think I shouldn't have mentioned the word "JEE". It attracts too much unwanted attention :P.

Anonymous

@BalarkaSen omg =D

Balarka Sen

"formula for chaos". I like that.

SBM

yep

Astyx

Being a mathematician is a dangerous occupation

SBM

10:48

You get to destroy universes.

@Astyx mathematically

Fawad

Find the sum of the series $2^2 + 4^2 + 6^2+.....+(2n)^2.$

Anonymous

@Fawad Take 4 common out...

Anonymous

Yes. Now use the formula for sum of squares of first n terms...

Fawad

Got

Anonymous

11:34

How to find the minimum and maximum value of any second degree expression like $ax^2+by^2+2gx+2fy+2hxy+c$ ? (without calculus)

LeGrandDODOM

@BalarkaSen have you taken IIT-JEE ?

Balarka Sen

no

@blue Complete the square.

Anonymous

@BalarkaSen I know. But is it possible in every case to complete the square (coefficients are real) ?

Balarka Sen

Yes. This is called diagonalization of a quadratic form.

One diagonalizes the symmetric matrix which represents the multivariable second degree expression.

Anonymous

@BalarkaSen Which symmetric matrix? Can you link any article on this topic? What you said does seem correct to me considering that we can always find a maxima/minima in such cases using partial derivatives.

Anonymous

11:50

batty.mullikin.org/quadratic.pdf

Anonymous

Found one

Jack Lam

I am looking for a limit problem of the form:

n^a (L - ∫_0^t f_n(x) dx)

where f(n) converges to f

the integral converges to L

and a is such that the limit exists

as n goes to infinity

Balarka Sen

@blue Any quadratic form in any number of variables can be represented by a symmetric matrix. Write it as $Q(x_1, x_2, \cdots, x_n) = \sum_{i=1}^n \sum_{j = 1}^n a_{ij} x_i x_j$; then the matrix is $A = [a_{ij}]$ - notice that $Q(v) = v^\mathsf{T}Av$ where $v = (x_1, x_2, \cdots, x_n)$ is a vector.

Jack Lam

I don't know anything else about the problem I am looking for

except maybe that L is √2

or 1/√2

Balarka Sen

If you diagonalize the symmetric matrix $A$, you can see from the above that you end up with an expression $Q(x_i) = \sum_{i = 1}^n b_i x_i^2$.

$b_i$ corresponding to the diagonal entries of the new, diagonalized, matrix $A'$.

So you can google "diagonalization of a symmetric matrix", which will turn up with a bunch of ways one can systematically do this, and that would answer your question.

Anonymous

11:57

@BalarkaSen Aha, I see now. Very interesting :-)

Balarka Sen

Indeed.

I think completing the square is my favorite high-school algebra fact all-time. It's secretly diagonalization, which is so very useful in math.

Alessandro Codenotti

Even the classification of quadratic curves is mostly completing the square

Topologicalife

Hi. Is there a way to simplify $(x+1)^{n-2} \cdot x^2$?

Anonymous

simply?

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$Mathematics$ Mathematics