Mathematics - 2016-11-25 (page 4 of 6)

Vrouvrou

15:00

hello what is the norm on the dual space ? please @BalarkaSen

heather

I thought I understood what they are doing, but nevermind, that doesn't seem right

MartianCactus

yeah me too

heather

what book are you using, out of curiosity? @MartianCactus

MartianCactus

how old r u heather?

heather

8th grade

MartianCactus

15:01

oh nice

im in 10th

heather

nice

MartianCactus

so maybe i will have to wait for another guy

i will brb

heather

okay

Evinda

Hello!!! We have that $u(t,x)=v{\left( \frac{x^2}{t}\right)}$, $z=\frac{x^2}{t}, t>0$.

I want to calculate $v'(z)$.

We have that $v'(z)=\frac{\partial{u}}{\partial{t}}\frac{dt}{dz}+\frac{\partial{u}}{\partial{x}} \frac{dx}{dz}$

We have that $t=\frac{x^2}{z}$ so $\frac{dt}{dz}=-\frac{x^2}{z^2}$.

Also we have $x^2=tz$. How can we compute $\frac{dx}{dz}$ ?

MartianCactus

ww.meritnation.com/cbse/class10/textbook-solutions/math/rs_aggarwal_(2015)/1_10_‌1_1176 here it is

Astyx

15:03

w missing

MartianCactus

isnt that vectors or something

heather

is what vectors?

Evinda

Hello @Astyx
Do you maybe have an idea?

Astyx

Huh ?

Evinda

We have that $u(t,x)=v{\left( \frac{x^2}{t}\right)}$, $z=\frac{x^2}{t}, t>0$.

I want to calculate $v'(z)$.

We have that $v'(z)=\frac{\partial{u}}{\partial{t}}\frac{dt}{dz}+\frac{\partial{u}}{\partial{x}} \frac{dx}{dz}$

We have that $t=\frac{x^2}{z}$ so $\frac{dt}{dz}=-\frac{x^2}{z^2}$.

Also we have $x^2=tz$. How can we compute $\frac{dx}{dz}$ ? @Astyx

Astyx

15:08

What is $z$ ?

Evinda

$z=\frac{x^2}{t}, t>0$ @Astyx

Astyx

But is $x$ a real ?

Shin Kim

is it weird to use typewriter font \texttt to denote sets?

Evinda

Yes, it is @Astyx

Shin Kim

i wanna distinguish between sets and structures. i could use \mathfrak or \mathcal but \mathfrak is somewhat difficult to read and i use \mathcal for different purpose.

Astyx

15:12

This seems weird

to me

Evinda

Why? @Astyx

Astyx

Because you can't go from $v({x^2\over t})$ to $u(x,t)$

But I might be wrong

I'm not an expert

Evinda

@Astyx Let $n=1$ and $u(t,x)=v\left( \frac{x^2}{t}\right)$. I ahve to show that $u_t=u_{xx}$ iff $4zv''(z)+(2+z)v'(z)=0, z=\frac{x^2}{t}, t>0$.

I have shown that if $u_t=u_{xx}$ then $4zv''(z)+(2+z)v'(z)=0$.

It remains to show the other direction.

Astyx

Have you tried anything ?

Evinda

I thought to find $v'(z)$ and $v''(z)$ , but I didn't know how to calculate $\frac{dx}{dz}$

Astyx

15:18

What are $u_t$ and $u_{xx}$ ?

Evinda

$u_t=v'\left( \frac{x^2}{t}\right)\left( -\frac{x^2}{t^2}\right) $

and $u_{xx}=v''\left( \frac{x^2}{t}\right) \left( \frac{2x}{t}\right)^2+v'\left( \frac{x^2}{t}\right) \frac{2}{t}$ @Astyx

MartianCactus

can someone check out my question a couple of messages above?

heather

$\sin\cdot -\sin(x) + \cos(x)\cdot\cos(x) = -\sin^2(x)+\cos^2(x)$, right?

MartianCactus

trigno in 8th?

thats pretty cool

heather

@MartianCactus, I'm not actually taking trig, I'm just learning about it on my own =) but thanks

MartianCactus

15:29

same

i learnt trig in 9th before it was taught in 10th :P

khan academt?

heather

yeah and youtube where I didn't understand things

MartianCactus

i cant help u with the question tho..i have not studied that yet

oh nice

and you do pogramming too?

which languages?

heather

python mainly

MartianCactus

oh nice

i know JS and am still learning Java

heather

but I also know a bit of JavaScript and HTML/CSS (I first learned on Khan Academy)

oh, nice!

MartianCactus

15:32

i learned JS from KA and a little bit of HTML too :)

but didnt take the full course

of HTML

and was on 'advanced JS' midway

heather

have you tried codecademy?

has a lot more languages, though not as in-depth as KA

Mary Star

Could someone of you take a look at the edit part of my question: math.stackexchange.com/questions/2029913/… ?

MartianCactus

15:48

@heather thats why i left it

its not indepth

heather

@MartianCactus, makes sense. Where are you learning Java then?

MartianCactus

treehouse

its a subscription based thing

but they have REALLY indepth guides

i learned object oriented programming from there which i couldnt understand at KA

heather

nice

MartianCactus

but they do get a little vague after sometime tho

heather

I'll have to look at it

MartianCactus

15:52

but u have to pay

heather

depends on how much it costs, I suppose

MartianCactus

25$ for basic sub

100$ for pr sub

and 1000$ for real pro

i have the basic

its good

heather

yeah, that's probably too much

MartianCactus

btw where do u live?

heather

don't have a consistent job yet

Iowa, USA

you?

MartianCactus

15:53

wait u in 8th

job?

india

heather

i mean I don't have one, so I can't make money consistently

MartianCactus

ur in 8th grade

heather

yeah, so?

MartianCactus

isnt your dad supposed to do the money makin stuff?

heather

yeah...but he wants me to pay for stuff like this

MartianCactus

15:55

its good stuff

ur not buying games off steam

ur doing something productive

also u can do a 1 week trial and see if u like it

but they do get vague in advanced lessons

heather

i'm sure its good, and I know i'll be doing something productive (i might do the one week trial) but i still doubt i'll be able to persuade my dad to pay for it =)

MartianCactus

so u might need some help frm external sources and java docs too

oh

heather

so then its up to me to pay for it

MartianCactus

no just go up to him and tell him that programming has a bright future, tell him that you will make a great game/whatever you wanna make and give him $50 back, that might do it

making $50 from the game that is..

heather

=P i wish but there's no way i'd be able to make a game/app that would make 5, let alone 50

MartianCactus

15:58

u just need to BELIEVE it

u watch naruto?

heather

no, what's naruto?

MartianCactus

nvm then

heather

but just believing aside, i have to be realistic

i don't think i could make something that would earn me any money whatsoever

MartianCactus

no u can

like, once u get a sufficient amount of training, and if u are ambitious enough, u CAN

thats wat im trying

heather

well, maybe i'll try making an app/something without treehouse and see how that goes

MartianCactus

16:01

yeah go for it!!

once u succeed , then maybe u can get enoug for the subscription :)

and if u fail

keep on trying!

it all comes down to how bad u want it

ur in 8th grade so studies are probably heavy too

and its difficult to manage between thos

both

but i can always find a way

ok now i gtg, brb o/

heather

see you o/

Steve

Hi people, I've just been introduced to systems of differential equations which isn't making much sense right now for me. Here's my question:

Let $M$ be an arbitrary real $2\times2$ matrix.

Show that if $x(t)$ $\land$ $y(t)$ $\in$ $C^1(\mathbb{R})$ and that if $a(t)$ $\land$ $b(t)$ $\in$ $C^0(\mathbb{R})$, then the following system of differential equations is linear:

$$\begin{bmatrix}
x\,'(t)\\
y\,'(t)\\
\end{bmatrix}=\mathbf{M}\begin{bmatrix}
x(t)\\
y(t)\\
\end{bmatrix}+\begin{bmatrix}
a(t)\\

heather

16:17

what is $\int \cos^2(x) \, dx$?

would it be $\sin^2(x) + c$?

I'm not sure.

Astyx

No

Differentiate $\sin^2$ to see this is not true

And to integrate it linearise it

Steve

@heather you're doing calculus in 8th grade? That's impressive. Keep it up

heather

@Astyx, not sure what it means to "linearise" it, sorry

Astyx

Use trigonometric identities to get something without any product between trigonometric functions

For instance in your case $\cos^2x ={1+\cos 2x\over2}$

Which is easier to integrate

heather

oh...good to know

thanks

so then you take out 1/2, get int of 1+cos 2x and then you get x + -sin2x times 1/2

Astyx

16:26

Exactly

Semiclassical

more generally, a product of trig functions with frequencies $\omega_1,\omega_2$ is equivalent (by the product-to-sum formula) to a superposition of trig functions with frequencies $\omega_1+\omega_2,\omega_1-\omega_2$.

Astyx

You should learn trigonometric identites (the earlier the better)

math.univ-lyon1.fr/~frabetti/TMB/TMB-CM5-fcts-circulaires.pdf

(In french, but mostly formulae)

heather

@Astyx, okay, thank you

Steve

@Semiclassical Hi how are you :) can you help me with my problem? described above

Astyx

My pleasure @heather

Semiclassical

16:28

Not interested, sorry.

Steve

Alright

Does $x(t),y(t) \in C^1(\mathbb{R})$ denote that the elements in the functions are real? What does the power to 1 indicate?

Astyx

You should write $x,y \in C^1(\Bbb R)$

And it means that the functions $x$ and $y$ are functions whose domain/codomain is/are $\Bbb R$ (not sure, I find this notation abiguous)

And that they are differentiable once, and that their derivative is continuous

$C^2(\Bbb R)$ would mean they are differentiable twice, and their second derivative is continuous, etc.

Mary Star

16:44

Let $\rho=\sqrt[3]{\frac{1+\sqrt{5}}{2}}$.

We have that $\rho$ is a root of $f(x)=x^6-x^3-1\in \mathbb{Q}[x]$, that is irreducible over $\mathbb{Q}$.

How can we find all the roots of $f$ ?

Astyx

Solve $x^2 -x -1 =0$

This gives you two roots @MaryStar

Each of which you can take the cubic root

robjohn

$x=\phi$ and $x=-\frac1\phi$

Astyx

And multiply the result by $\omega$ a cubic root of the unit

Which gives you a total of 6 distinct roots

robjohn

$e^{\pm2\pi i/3}$

Astyx

Now $f$ is polynomial in $x$ of degree 6, thus it has at most 6 roots

So these are all the roots of $f$

Steve

16:47

Cheers. @Astyx

Astyx

My pleasure

Steve

Just noticed that there's a hint to my problem.

Hint: Show that the vector space transformation $f: C^1(\mathbb{R})\times C^1(\mathbb{R}) \rightarrow C^0(\mathbb{R})\times C^0(\mathbb{R})$ is given by $f(x) = x' - Ax$

So I should just isolate $(a(t),b(t))$?

Astyx

What is $f$ ?

Mary Star

So, do we find the y that satisfy the polynomial y^2+y+1 by the driscriminant?
We get $y_{1,2}=\frac{1\pm \sqrt{5}}{2}$. Then by setting $y=x^3$, we get the following roots:
$\sqrt[3]{\frac{1+ \sqrt{5}}{2}}=\rho, \sqrt[3]{\frac{1- \sqrt{5}}{2}}$, or not?
Why do we have to multiply each of them by $\omega$ ?

Astyx

These are two roots

You need 6

If $a$ is a root, then $a\omega$ and $a\omega^2$ also are

So you can get 6 roots from the 2 you have

(I'll let you figure out why this is true)

Ramanujan

17:00

And looks everyone is facing difficulties when DHMO is not there now

@MartianCactus did you got that now?

Astyx

Haha

Ramanujan

@heather did you got that?

MartianCactus

no is till need explanation @Ramanujan

Vrouvrou

17:18

please if $u_n\rightarrow u$ and $\phi$ is continuous can we conclude that $\phi(u_n)u_n\rightarrow \phi(u) u$ ?

Alessandro

what does $u_\rightarrow u$ mean?

Vrouvrou

(u_n) converge to u

Mary Star

Ah ok, thank you!! :-) @Astyx @robjohn

I have also an other question... We have that $f(x)=x^6-x^3-1\in \mathbb{Q}[x]$ is irreducible over $\mathbb{Q}$. Is it also irreducible over $\mathbb{Q}[\omega]$ ?

Alessandro

that's true then

look up sequential continuity for more info

Ramanujan

@MartianCactus k^2 <( 8/5)^2

meow-mix

17:22

good morning, everyone

Ramanujan

So…k<±8/5

MartianCactus

yes

ok

Alessandro

hi @meow

Ramanujan

k<8/5 and also k<-8/5

MartianCactus

true

Balarka Sen

17:22

@Alessandro Anything fresh today?

Ramanujan

So k lies between?

MartianCactus

no

how?

Ramanujan

Wait

MartianCactus

k is smaller that -8/5 and it is also smaller that 8/5

meow-mix

if k is less than -8/5 and 8/5, we can state that k < -8/5

MartianCactus

17:24

yes

meow-mix

so whats the problem

Alessandro

We starting talking about paths in my topology class today, we should start talking about fundamental groups and homotopy sometimes soon @Balarka

MartianCactus

the book says that -8/5 < k < 8/5

meow-mix

whats the problem

like, what does it ask you

Balarka Sen

@Alessandro Very cool.

Alessandro

17:25

also I had this abstract algebra exam I mentioned, writing down all the elements of a $16$ elements field and their minimal polynomial was painfully tedious, but the rest was fine

heather

@meow-mix, hello

@Ramanujan, that's what I got as the answer to the problem, yes

Balarka Sen

Glad to hear.

Ramanujan

@MartianCactus take x=k and a=8/5

meow-mix

yeah

either x < -a

or x > a

where did you guys mess up

Astyx

Hi everyone !

meow-mix

17:27

hi @astyx

Ramanujan

I too don't know the third step,long ago I I round that

MartianCactus

yeah, i dont understand how THAT happens

meow-mix

well

if (x-a)(x+a) is greater than 0

Astyx

How are you ?

meow-mix

then neither (x-a) or (x+a) may be 0

correct?

Astyx

17:29

By "greater than" you mean $\gt$ ? If so, you're right

meow-mix

yes lol

@MartianCactus correct?

MartianCactus

yeah

Astyx

Oh that wasn't a question, right ^^'

meow-mix

ok, so if (x-a) and (x+a) can't be 0

that is

well, theres two cases

x + a > 0

MartianCactus

they have to be negative or positive

meow-mix

17:30

and x - a > 0

MartianCactus

why

Alessandro

Are you still studying for exams or did you find enough time for some interesting mathematics @Balarka?

MartianCactus

if both of them are negative , then negative * negative = postivie which is greather that 0

Astyx

$\Vert$

Ramanujan

@MartianCactus when we take (-) common,lessthan changes to greater than and vice versa

meow-mix

17:32

hey, @Astyx im having trouble with a problem in my book, and i feel really stupid

MartianCactus

wait

Astyx

Tell me, I'll see if I can help

MartianCactus

i dont understand

wait i will send a pic of the problem

Astyx

@MartianCactus $a^2 \lt b^2 \iff |a|\lt |b| \iff a\in ]-|b|, +|b|[$

Does this help ?

meow-mix

Prove that $|\mathrm{Aut}(C_n)| = \phi(n)$

where $C_n$ is the cyclic group of order $n$

and $\phi(n)$ is Euler's totient function

Astyx

17:34

Do you know the typical representant of $C_n$ ?

meow-mix

what do you mean by that

Argon

If I define $\operatorname{Arg} z$ to map to $[0,2 \pi)$, then $\operatorname{Log} 1=0$. I is then the series for $\operatorname{Log}$ simply the standard logarithmic Taylor's series $\log(1+x) = \sum^\infty_{n=1} (-1)^{n+1}\frac{x^n}n$?

Astyx

Well what groups do you know that are cyclic of order $n$ ?

Ramanujan

meow-mix

$\mathbb{Z}/n\mathbb{Z}$?

Astyx

17:36

Yep

Alessandro

Let's say I have a morphism $\psi :C_n\to C_n$, you can ask me the value of $\psi (n)$ for some $n$ in $C_n$, how many do you need to ask me before you can uniquely determine what's $\psi$ @meow?

MartianCactus

why cant (a+x) and (a-x) be negative?

meow-mix

must it be a homomorphism?

Astyx

Now there's a property of generators of $\Bbb Z / n\Bbb Z$ in modular arithmetics

Alessandro

yes

Ramanujan

17:37

@MartianCactus can I know which step you not got?

MartianCactus

5th

Ramanujan

@MartianCactus because in fourth step we took that as positive

Astyx

Sorry, I'm tired, I misread your question @meow-mix

meow-mix

it's ok

Alessandro

I think what you were saying is still relevant to this problem @Astyx , but @meow should be able to work it out by himself!

Astyx

17:40

Anyway Alessandro is leading you on the right track

meow-mix

@Alessandro well there are $n!$ possible morphisms

Astyx

Sure @Alessandro

meow-mix

$(n-1)!$ morphisms that mapp the identity to the identity

Astyx

@meow-mix Try and find a sufficient condition on $\psi(n)$ for a certain $n$ for $\psi$ to be an isomorphism

That's not true @meow-mix

You're confusing permutations and morphisms

meow-mix

oh wait

im thinking of automorphisms

Jessy Cat

17:43

Hello fellow kitty.

This chat's going to the cats.

meow-mix

wait no... isomorphisms map $n$ elements uniquely to $n$ elements

Astyx

They do

Jessy Cat

Any of you lovely people want to help a cat out with a Fourier Series/Sturm-Liouville problem? math.stackexchange.com/questions/2030456/…

meow-mix

so why arent there $n!$ possible bijective morphisms?

Astyx

Can you answer that by yourself ? :)

Alessandro

17:46

because not all permutations are morphisms, they need to be bijections and be compatible with the group structure

@meow what's your definition of a cyclic group by the way? That's going to be useful here

meow-mix

my book defines it as a group isomorphic to $\mathbb{Z}/n\mathbb{Z}$

Astyx

Meh.

Alessandro

groan

user227867

@meow-mix Do you like cats?

meow-mix

yes

Alessandro

17:48

I mean it's true, but that's definitely not a great definition to work with

Astyx

^

Alessandro

Hm, let's see, what do you know about generators?

meow-mix

well i know about them, but this book hasn't discussed them yet

(im reading a new algebra book)

Astyx

What book is this ?

meow-mix

aluffi's

Alessandro

17:50

That's interesting, I wonder how you're supposed to solve this exercise without talking about generators

meow-mix

yeah i probably could solve it w/ generators but i don't want to

that is

i dont want to go against the book

Astyx

Well you can solve it with generators without being explicit about them

Especially with that definition of cyclic group

Alessandro

yeah I think that's pretty much the only approach

and one can notice it even without knowing explicitly what a generator is

user227867

Interesting that kids these days say the author of a book without saying the title. =P

Astyx

Discussing cardinality of the image of the morphisms in function of the image of a generator (without saying it's one) should be an alternative

MartianCactus

17:54

example 15

is the one i cant understand

the last step of example 15

Astyx

Did you see my answer @MartianCactus ?

meow-mix

@Astyx i'd imagine isomorphisms map generators of order $k$ to generators of order $k$

Astyx

Prove it !

MartianCactus

sry for the inactivity i was doing something else

Steve

@Astyx

Astyx

17:55

@Steve

meow-mix

@Astyx well i could prove that a homomorphism $\phi$ satisfies $\phi(a^n) = \phi(a)^n$

Steve

$f$ is the underived system of differential equations I think

meow-mix

so if $\phi(a^n) = 1$, and $\phi(a)^n = 1$, and $n$ is the minimum, then, its true?

Astyx

Yes, but that's not sufficient

I think your book wants you to prove it for $\Bbb Z/n\Bbb Z$ (and it's better to do so if you don't have generators yet)

Alessandro

Actually, with your definition, we're just discussing automorphisms of $\mathbb{Z}/n\mathbb{Z}$. If you know $\phi(a)$ and $\phi(b)$ you also know $\phi(a+b)$ right?

meow-mix

17:57

yes

wait

Astyx

And therefore my first answer was not as off-topic as I thought it was :p

@Steve what was the question again ?

meow-mix

so if i know a generator $g$, and i know it maps to $\phi(g)$, then i can find out the homomorphism?

Alessandro

prove it

meow-mix

well

Astyx

Try to rely on you intuition to prove such things, that's how you'll progress.

Don't rely on us to tell you wether you're on the right track

Sophie

17:59

what CAS can open the product $(x+a+bw+cw^2+dw^3+ew^4)(x+a+bw^2+cw^4+dw+ew^3)(x+a+bw^3+cw+dw^4+ew^2)(x+a+bw^4+‌cw^3+dw^2+ew)(x+a+b+c+d+e)$ for me and collect the $x^k$ terms? I for obvious reasons don't want to open the 625 term beast

Astyx

(not that I don't want to help you)

Transcript for

$Mathematics$ Mathematics