Mathematics - 2015-02-26 (page 3 of 4)

Sawarnik

8:02 PM

Hi @ABeautiful

A Beautiful Mind

Hello.

Sawarnik

How have you been recently? :)

Ted Shifrin

hi @robjohn, @Sawarnik, Jasper

A Beautiful Mind

@Sawarnik Very bad. I think I am at the worst point in my life.

Sawarnik

hi @TedShifrin :)

robjohn

8:07 PM

@TedShifrin Hey, Ted!

Mike Miller

morning, ted

Sawarnik

@ABeautifulMind :O :O Can I ask why?

Ted Shifrin

good night, @Mike

robjohn

@TedShifrin And here I thought that Mike was in my time zone...

A Beautiful Mind

@Sawarnik Mental problems, nothing new. But I hope to get well soon.

robjohn

8:08 PM

@ABeautifulMind are you seeing a doctor?

Mike Miller

well, it's morning somewhere

Ted Shifrin

he's in his own loony zone

Balarka Sen

@Ted!

A Beautiful Mind

@robjohn Not now.

robjohn

@MikeMiller I looked outside and it did not look as if it were night (good or not)

Ted Shifrin

8:09 PM

@robjohn: It'll be worse when I move and am in the same (loony) time zone.

hi @Balarka

Sawarnik

@ABeautifulMind Its sometimes puzzles me that people fully aware of their mental problems aren't able to help themselves.

Mike Miller

@robjohn Well, you could also easily interpret what I said as short for "I hope you had a good morning"

Ted Shifrin

it often strikes me as the barber trying to give himself a haircut, @Sawarnik

Balarka Sen

i picked up a nice idiom today.

Sawarnik

hmm.

robjohn

8:11 PM

@TedShifrin Is that why my ears are missing?

Balarka Sen

"an abbreviated piece of nothing"

Vrouvrou

Hello, if $P:\mathbb{R}^N\rightarrow\mathbb{R}$ is $\alpha-$ homogeneous i.e $P(tx)=t^{\alpha} P(x)$ what is $\frac{\partial P}{\partial x_i} P(tx)$ equal to ? please

Balarka Sen

i hope to use and abuse this

Ted Shifrin

is that all that's missing, @robjohn?

Balarka Sen

evilgrin

Pedro Tamaroff

8:11 PM

@Sawarnik That's pretty silly.

Ted Shifrin

you tend toward long-winded pieces of nothing, @Balarka :P

3

Sawarnik

@PedroTamaroff probably.

Ted Shifrin

howdy mr @Pedro

Balarka Sen

you mean the Spoderman, @Ted

Huy

@TedShifrin: Today, my course in Game Theory started.

Axoren

8:12 PM

So many people online right now

I looked away for a second.

Ted Shifrin

@Mike, @Pedro: Here's a nice conceptual calculus/geometry problem. Suppose you have a region in the plane and you radially symmetrize it about the $x$-axis. (So the arc of the region at distance $r$ is rotated to make it symmetric about the $x$-axis.) Prove that the arclength of the boundary cannot go up when you do this.

Axoren

Anybody got any insight on this? math.stackexchange.com/q/1166199/187120

Ted Shifrin

congratulations, @Huy: Do you still have a class?

Hi, @Axoren.

Sawarnik

@BalarkaSen why did you remove your age? :D

Huy

@TedShifrin: What do you mean by that?

Axoren

8:13 PM

Hey, @TedShifrin

Vrouvrou

@TedShifrin hello, please have you an idea about my question please

A Beautiful Mind

@Sawarnik Mental problems are harder than math problems.

Mike Miller

He's asking if you're still teaching.

Ted Shifrin

@Huy: My first (and second and ...) lectures often chase students away.

Axoren

@ABeautifulMind For a moment, I thought you were talking about mental math. I almost agreed with you for the wrong reason.

Huy

8:14 PM

@TedShifrin: Well now that they chose my course they have to stay. ^_^

Ted Shifrin

What question, @Vrouvrou?

Sawarnik

@Axoren hehe

@ABeautifulMind I see.

Pedro Tamaroff

@TedShifrin I don't understand the procedure. You're not reflecting it through the $x$-axis, I presume?

Vrouvrou

@TedShifrin this: Hello, if $P:\mathbb{R}^N\rightarrow\mathbb{R}$ is $\alpha-$ homogeneous i.e $P(tx)=t^{\alpha} P(x)$ what is $\frac{\partial P}{\partial x_i} P(tx)$ equal to ? please

Pedro Tamaroff

Ah.

I get it.

Ted Shifrin

8:15 PM

No, @Pedro, you're rotating the arc around the origin to as to make it symmetrically positioned with respect to the $x$-axis.

Pedro Tamaroff

I though you made a copy of it so that you get a symmetric figure.

You're moving that.

OK.

A Beautiful Mind

@robjohn I might go back to the mental hospital I went to, we'll see.

Balarka Sen

@TedShifrin $f : S^n \to S^n$ be a map that sends both of the two hemispheres to the northern hemisphere via $g : D^n \to D^n$. $f$ has degree zero. This means there exists $x, y$ such that $f(x) = x$ and $f(y) = -y$. That proves the Brouwer fixed point theorem, doesn't it?

Pedro Tamaroff

@TedShifrin Something something Cavalieri.

Ted Shifrin

Your notation makes no sense, @Vrouvrou. Do you mean $\dfrac d{dt} P(tx)$?

Balarka Sen

8:16 PM

Maybe discontinuing the meds were not so good of an idea, @ABeautifulMind

robjohn

@ABeautifulMind If things are not getting better without help, help seems indicated.

Ted Shifrin

akin to that, yes, @Pedro. So area is preserved. We want to see length of the boundary goes down.

A Beautiful Mind

@robjohn Things did not get better with help either.

infinitesimal

What makes you think @ABeautifulMind that you are at the worse point in your life?

Pedro Tamaroff

@TedShifrin You said it cannot go up, but now you're telling me it decreases.

Ted Shifrin

8:16 PM

Huh? @Balarka via $g$?

well, iIdidn't want to type so much, @Pedro. Unless it started symmetric, it will go down.

Axoren

@PedroTamaroff If something is non-increasing, it can only be decreasing or constant.

Vrouvrou

@TedShifrin $x\in \mathbb{R}^N$ i mean the partial derivative of P

A Beautiful Mind

@infinitesimal Well, it could be worse you mean?

Pedro Tamaroff

@Axoren Thanks, I didn't know that. =)

Axoren

@PedroTamaroff :P

Ted Shifrin

8:18 PM

That notation is very confusing and ambiguous, @Vrouvrou. Do you mean the $i$th partial of $P(tx)$, or do you mean, specifically, the partial with respect to $x_i$? These are different. I hate this notation.

A Beautiful Mind

@infinitesimal These days it fluctuates a lot.

infinitesimal

@ABeautifulMind you said you feel that you are at the worst point, right?

Mary Star

When we have that $\overrightarrow{a}=m \overrightarrow{b}+n \overrightarrow{c}$, does this mean that $\overrightarrow{a}$ is at the plane that $\overrightarrow{b}$ and $\overrightarrow{c}$ define??

Ted Shifrin

@Axoren @Pedro: That's not right.

Alizter

@TedShifrin That's easy Ted, $g \approx 9.8$

Balarka Sen

8:19 PM

Yeah. Having a fixed point means that there must be a fixed point while mapping the northern hemisphere to the northern hemisphere. Having an antipodal point means that there must be an antipodal point while mapping the southern hemisphere to the northern hemisphere. Precompose with another antipodal map or something like that to conclude Brouwer fixed point thm.

@Ted

Pedro Tamaroff

@TedShifrin What he meant is that "not $>$" is "$\leqslant$". I suppose.

A Beautiful Mind

@infinitesimal Yup, just a feeling.

Mike Miller

More long-winded pieces of nothing?

Ted Shifrin

Oh, for two numbers.

Axoren

@TedShifrin Of course, if it's not continuous, we can't say that.

Ted Shifrin

8:20 PM

Even if it's continuous, what you said is wrong, @Axoren.

Axoren

@TedShifrin Can you give a counter example?

Ted Shifrin

You might have some $x$'s where the function is constant and then it decreases elsewhere.

Balarka Sen

@MikeMiller What is wrong in there?

Vrouvrou

@TedShifrin the partial with respect to $x_i$

Mike Miller

Hard to say, because it's completely incomprehensible.

infinitesimal

8:21 PM

Listen to that feeling my friend @ABeautifulMind and go back to the doctor

Axoren

@TedShifrin I see. It would be locally decreasing or constant, but not necessarily globally so.

My mistake.

Right, @Axoren.

Lunchtime.

Enjoy your meal.

@Balarka: I can't figure out what you're doing.

user159870

8:22 PM

@robjohn How did you find these two antipodal points???

Balarka Sen

Well we are given a map $g : D^n \to D^n$.

Ted Shifrin

@Vrouvrou: So if you have $P(2x,2y)=f(x,y)$, what is $\partial f/\partial x$?

There's no reason you can "compactify" it into a map $S^n\to S^n$, @Balarka.

Axoren

@user159870 If you average two points, what do you get? You get the midpoint between them. If you do this for every opposite point on a circle, you'll get the origin every time.

Balarka Sen

$f : S^n \to S^n$ is defined by sending the upper and lower hems of the sphere to the lower hem of $S^n$ via $g$.

@TedShifrin I never said anything about compactifying.

Ted Shifrin

You're using $g$ somehow to define your $f$? I don't see how you glue on the equator.

Balarka Sen

8:24 PM

Equator is glued via identity.

Ted Shifrin

Oh, I see, you're using $g$ on each of the hemispheres, gluing along the boundary.

Balarka Sen

Yeah.

infinitesimal

@ABeautifulMind you don't need to go back on meds, but just mark this point in your life as "rock" bottom any change in how you feel from this point on must be for the better, right?

Balarka Sen

Claim : The degree of this fellow is zero.

Alec Teal

Hey guys, I've missed you

I was banned for a month for calling someone a tard on SO

infinitesimal

8:25 PM

Hi pal

Ted Shifrin

Oh, so if you assume neither fixed point nor antipodal point, then you contradict degree $0$?

Axoren

I'm going to leave this here: http://math.stackexchange.com/q/1166199/187120
Hopefully, one of the 14 people here will be able to think of a reason why I'm wrong.

Mary Star

When we have that $\overrightarrow{a}=m \overrightarrow{b}+n \overrightarrow{c}$, does this mean that $\overrightarrow{a}$ is at the plane that $\overrightarrow{b}$ and $\overrightarrow{c}$ define??

Do you have an idea?? @TedShifrin @robjohn

Ted Shifrin

I'm glad to know I'm not the only one you've ever infuriated, @Alec.

Yes, @MaryStar.

Balarka Sen

@TedShifrin Yeah, degree 0 implies fixed point and antipodal point.

If no fixed point, you can homotope by straightline homotopy to the antipodal map, which doesn't have degree 0.

infinitesimal

8:27 PM

Welcome back @AlecTeal :-)

Ted Shifrin

Although isn't it easier just to prove Brouwer directly?

Mary Star

Why does it stand?? @TedShifrin

Alec Teal

Oh @TedShifrin you say that but yet I read "I love you Alec, and I'm secretly jealous as you've never called me a tard"

Thanks @infinitesimal

Balarka Sen

Hatcher says do it this way @Ted

Ted Shifrin

OK, I'm about to meet with my Honors thesis student. She will have a proof of the problem I posed @Pedro above.

Oh, ok, @Balarka.

Balarka Sen

8:27 PM

Hatcher also says it's the original proof of Brouwer.

So, does my proof work, @Ted?

Axoren

@AlecTeal Why'd you call them retarded? You should find other ways to win arguments, like proving them wrong.

Alec Teal

@Axoren hold

Ted Shifrin

My favorite proof uses degree theory for smooth maps, supposedly due to M. Hirsch.

By definition of plane, @MaryStar.

I haven't thought out the details, @Balarka, but it seems right.

I need to leave now ...

Alec Teal

@Axoren imgur.com/gallery/4zhN1c8

Bye @TedShifrin

Balarka Sen

Good enough for me. I am happy with handwaves.

Vrouvrou

8:29 PM

@TedShifrin $\frac{\partial f}{\partial x}(x,y)=\frac{\partial P}{\partial x}(2x,2y)$ and here i don't know how to continu

Balarka Sen

:P

Mike Miller

I didn't know you were into physics, @Balarka

Balarka Sen

I am not into physics!

Stan Shunpike

Are Lebesgue sums analogous to Riemann sums in the sense that $\eta$ is like the value of a step function but rather than using partitions of interval we use measures?

Axoren

@AlecTeal That meme doesn't answer my question. It just shows me that you're willing to make a spectacle of your situation for social media.

Alec Teal

8:30 PM

Read the comments and description and stuff

Alizter

@StanShunpike I think so.

Mike Miller

Oh? I thought you said you were into handwaving without proof?

2

After all, that's precisely what physics is...

Alec Teal

Also yeah, @Axoren I was bored!

And I can make memes using pictures of my dogs, here's Toby:

robjohn

@user159870 I mapped $a/|a|$ and $-a/|a|$

Alizter

:|

Pedro Tamaroff

8:31 PM

@AlecTeal Please control your inner 12-year old.

Balarka Sen

@MikeMiller Nah, I just think out the ideas geometrically first.

Axoren

@AlecTeal So you just regularly call people retarded and respond sarcastically?

Alizter

@PedroTamaroff So do you enjoy the web?

Alec Teal

@Axoren programmers do.

Axoren

Please don't do that anymore.

Stan Shunpike

8:32 PM

@Alizter okay thanks.

Alec Teal

@Axoren just to be fair, here's my other dog Lila i.imgur.com/65ixdcf.png

infinitesimal

@AlecTeal you should know that you can't teach a fish to climb a tree no matter how many times you call him a retard ;-)

Alec Teal

There we go, it worked that time.

is done now

Axoren

@AlecTeal I'm a CS PhD student. You can't claim that just because you're a programmer it's okay to blatantly flame people. The site's got standards and you have to meet them maturely.

Alec Teal

@Axoren I have no respect for CS students and programming. Have you ever even used version control? The commit log is where you bitch about your collegues

Balarka Sen

8:34 PM

@Mike If you want, I can send you the solutions of 2.2 written rigorously with no handwavy argument.

Alec Teal

Start a private chat if you want to talk about it seriously.

infinitesimal

Didn't any of my last comment make sense @AlecTeal?

Balarka Sen

But for that, I'd first have to see if I can do the exercises, and then really do the exercises and then LaTeX them. Ugh.

Alec Teal

@infinitesimal I didn't really read after the word "fish"

Axoren

@AlecTeal You're just showing that you're not something worth talking to, at this point.

Alec Teal

8:35 PM

Your loss.

Pedro Tamaroff

@Axoren There's an ignore button. =)

infinitesimal

:(

Axoren

@PedroTamaroff Using it soon.

Alec Teal

I can ignore without technical help =)

But anyway, hiya everyone.

Stan Shunpike

If in Lebesgue sums $\eta$ is the analog of step functions in Riemann sums, why don't we call it a step function? Or is it called that and I just didn't know it

infinitesimal

8:36 PM

Use it it often ;-)

Pedro Tamaroff

@StanShunpike One calls them simple functions.

Stan Shunpike

@PedroTamaroff why? Does the idea of a step function not apply?

Axoren

Makes me wonder what other people from Kalamazoo are like, however.

Vrouvrou

@TedShifrin ?

Alec Teal

@StanShunpike not really, the point is that there is some partition that will actually work with a step function.

Pedro Tamaroff

8:39 PM

@StanShunpike You can take any measurable set instead of an interval.

Although you can use step functions.

Look at the Daniel scheme.

Alizter

@Pedro did you get the spiderman joke?

infinitesimal

@AlecTeal nice to have back (skullpatrol)

Pedro Tamaroff

@Alizter OH. I just got it.

Alec Teal

Hey @infinitesimal why the name change?

Pedro Tamaroff

I thought you'd joke about poor Uncle Ben.

Alec Teal

8:40 PM

Hmm

Mary Star

So that two vectors define a plane do the vectors, and therefore laso the plane, have to pass through the origin?? @TedShifrin

Stan Shunpike

@PedroTamaroff @AlecTeal Okay, thanks guys.

Alec Teal

@MaryStar that's a trivial vector space question

As in you know the page that defines a vector space, the one after it will explain basis.

infinitesimal

They will never catch the infinitesimal @AlecTeal ;-)

Alec Teal

@infinitesimal is it a temp one?

infinitesimal

8:42 PM

Yep

Mary Star

Why does this stand?? I got stuck right now... @AlecTeal

Alec Teal

@MaryStar do you know what a basis of a vector space is?

Alizter

Find a 10 digit number such that the first n digits are divisible by n and contains the digits 0-9 distinctively.

Alec Teal

1234567890

FINALLY A NUMBER THEORY QUESTION I CAN ANSWER

Balarka Sen

doesn't work.

1234567 is not divisible by 7

Axoren

8:45 PM

@AlecTeal 2 is not divisible by 3.

Mary Star

It is a subset of independent vectors that produces the space, right?? @AlecTeal

Alec Teal

@Axoren 2 is the 2nd digit

@MaryStar yes

Alizter

@AlecTeal 1234 is not divislbly by 4

Balarka Sen

@Axoren if i understand the question correctly, it asks for a 10 digit number such that chopping off til n digit gives something that is divisible by n

and i was right.

Alec Teal

Oh right,

Good luck then @Alizter

Alizter

8:46 PM

It is a challenge to you

I have already solved it

There is one such number

infinitesimal

Can you prove it is unique?

Alec Teal

@infinitesimal probably just do it by induction

*brute force even

Mary Star

And because of the fact the $(0, 0, 0)$ has to be contained in the basis, each vector and each plane has to pass through the origin?? @AlecTeal

Alec Teal

No no no @MaryStar the 0 vector is not in the basis

infinitesimal

One "no" is enough pal :-)

Mary Star

8:50 PM

Why does each vector and each plane has to pass through the origin?? I got stuck right now... @AlecTeal

infinitesimal

Recall the "fish" :P

Alec Teal

@infinitesimal don't you... grr

@MaryStar you need to brush up on your Linear Algebra

This is like... definitions

:O

@Alizter 3816547290

@BalarkaSen yup :)

8:56 PM

Fairly easy problem to code up on PARI/GP.

Alizter

@BalarkaSen I did it pen and paper

only hard bit is checking div by 7

Balarka Sen

Right.

Alizter

once thats done there are 8 cases that can be checked by hand

Balarka Sen

That's why I coded it up.

Maximilian

(1/cscΘ)^2 = sin^2Θ right?

Alizter

8:57 PM

Its easy to code up

Axoren

$N = x_1x_2x_3x_4x_5x_6x_7x_8x_9x_{10}$, $x_1$ can be any number as all of them are divisible by 1. $x_2$ must be even to be divisible by 2. $x_1 + x_2 + x_3$ must be divisible by 3. $x_3x_4$ must be divisible by 4. $x_5$ must be 0 or 5 to be divisible by 5. $x_6$ must be even and $x_4 + x_5 + x_6$ must be divisible by 3 to be divisible by 6. Good luck with divisibility by 7. $x_6x_7x_8$ must be divisible by 8.

$x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9$ must be one of the following numbers: $\{9, 18, 27, 36, 45, 54, 63, 72, 81, 90\}$. $x_{10}$ must be 0.

Alizter

Now do the same but for hexadecimal

and 16 digits

Balarka Sen

I'm leaving. Bye.

:P

infinitesimal

:-)

Balarka Sen

That was just a figure of speech.

infinitesimal

9:00 PM

You should've put quotes around the the word "leaving" ;-)

That allows you to use whatever meaning you like.

Alizter

@BalarkaSen I am too lazy to write a program for now but I will check various 'base' cases later

Balarka Sen

@Alizter How many integers n are there such that n = sum of factorials of their digits?

Hint : There aren't many of those.

user159870

I am mixed up.

I have to clarify it in my mind... we show that the image is also a circle using translation and inversion. Correct???

How do we know for which two points we calculate the average???

Why did you take the points $\frac1{1-|a|}$ and $\frac1{1+|a|}$ ???

@robjohn @Axoren

Maximilian

I think I did my math wrong.. I have to get cotΘ=-1/5 to sinΘ, Θ is in quadrant 4.. The book says (-5sqrt(26))/26, and I got sqrt(26)/5

I used the 1+cot^2Θ=csc^2Θ identity, squared -1/5 to 1/25, and then added 1 which I got 26/25 and square rooted that?

I failed somewhere

G-man

You have calculated cosecant, not sine.

Also you forgot to add minus sign

Balarka Sen

9:09 PM

Here, @Mike : If you have a map $f : D^n \to D^n$, use this to construct a map $F : S^n \to S^n$ that maps both the upper hemisphere and lower hemisphere to the lower hemisphere by $f$. This is has degree zero as it's not surjective, so it must have a fixed point $F(x) = x$. This means there must a fixed point of $F$ restricted to the lower hemisphere. That induces a fixed point on $f$.

Maximilian

Ohhhhhhh I see

Balarka Sen

Fairly easy.

Maximilian

Oops, I did 1/cscΘ to make sinΘ, but i didnt do it to the other side.. oops

I think lol

rubik

Why are you all using MathJax? The chat does not render it. Or is it my chat?

G-man

Its your chat

rubik

9:11 PM

Ah ok there are plugins, I read that now

Ahah perfect now

G-man

27 people and still silence...

26 now

rubik

xD

G-man

Time, Dr. Freeman?

Alizter

@BalarkaSen Will solve

rubik

@G-man Oh I didn't know about G-Man!

G-man

9:20 PM

@rubik wh

rubik

I looked up your quote

G-man

@rubik how'd you know him now? Did you like look up thecqu

Maximilian

When a question has tanΘ<0, it means if your answer is sinΘ or cosΘ they're both negative yes?

G-man

Sorry I am on a phone and I have big fingers, so there are some unforeseen consequences...

robjohn

@user159870 The points $\frac{a}{|a|}$ and $-\frac{a}{|a|}$ are on the line that goes through antipodal points on the unit circle. Thus, its image will intersect the destination circle perpendicularly. Since its image is the real line, the real line must pass through the center of the destination circle

G-man

9:25 PM

@Maximilian only one of them is negative

Maximilian

Ah... Ok.. damnit.. I get it now

I had cos and was finding sin and cos was positive anyways, so that would make sense.

I overlook the simplest of things:P

infinitesimal

@user159870 making a sketch may help :-)

user159870

Which is the line that goes through antipodal points on the unit circle??? The Line $at$ ??? @robjohn @infinitesimal

Balarka Sen

@user159870 Let $z$ be a point on the unit circle. What is the antipodal point?

user159870

Is it $-z$ ??? @BalarkaSen

Balarka Sen

9:32 PM

Yes. Now what is the formula for the line that joins $z$ and $-z$?

Try recalling some cartesian geometry.

user159870

Is the Line $k \cdot z$ ??? @BalarkaSen

infinitesimal

please stop using three question marks :-) @user159870 isn't one is enough?

Balarka Sen

Formula for a line is usually not an expression, @user159870

And what you wrote is just a point. $z$ is a point, $k$ is a constant so $kz$ is just a point.

user159870

Isn't it $y=kx$ ? @BalarkaSen

infinitesimal

Thank you :)

Balarka Sen

9:37 PM

What is $k$?

G-man

Does anyone have an idea as to how the weather people calculate chance of rain? I don't suppose they can use the favourable outcome to total outcome approach there, so what kind of probability do they use?

Balarka Sen

If $k$ is anything, then no. There is a particular definite value of $k$ for which the line of yours goes through a pair of antipodal points in your circle.

infinitesimal

Sounds like a good question for earth science.SE @G-man

Balarka Sen

Recall that the points you are given are $(x, y)$ and $(-x, -y)$ in cartesian coordinates, @user159870

G-man

I'm not asking about the earth science, but the math. Because with classical probability it is always 50%

user159870

9:40 PM

$k$ should be so, so that the Line rotates around the origin, or not? @BalarkaSen

Balarka Sen

That doesn't make any sense.

What is $k$?

I mean, it has a value, right? It can't be arbitrary.

Maximilian

Which fundamental identity is best for finding sinΘ when you have tanΘ? I tried 1+cot^2Θ=csc^2Θ and did not get the right answer any ways I did it:P

Balarka Sen

@user159870 OK, so tell me what's the formula for a line first.

Any arbitrary line on the cartesian plane.

infinitesimal

It is a lot more complicated than that @G-man

Maximilian

I could use tanΘ=sinΘ/cosΘ but I don't have cosΘ

robjohn

9:45 PM

@Maximilian Have you installed ChatJax?

Nope

OK.:!

What is it

@Maximilian see the LaTeX in chat link in the right sidebar

Balarka Sen

I'd have to hit the bed soon.

Maximilian

9:46 PM

Ya

robjohn

@Maximilian It allows you to see $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$ rendered properly

Maximilian

:O

G-man

Its weird right when responses time so perfectly?

robjohn

@user159870 remember I said that $\frac{a}{|a|}$ and $-\frac{a}{|a|}$ are on a diameter of the unit circle?

Maximilian

:O Thats magic

robjohn

9:50 PM

@user159870 thus the line through them intersects the unit circle perpendicularly

@user159870 and that line maps to the real line

Balarka Sen

Talk about killing a fly with a javelin

G-man

by making the javelin Pierce its eye.

user159870

@BalarkaSen A Line on the cartesian plane is: $y=\lambda x+b$. Correct?

robjohn

@user159870 since analytic functions are conformal, the real line must intersect the destination circle perpendicularly

Balarka Sen

@user159870 Right. So if your line passes through $(x_0, y_0)$, what d'you get?

If your line passes through $(-x_0, -y_0)$, what d'you get?

robjohn

9:53 PM

@user159870 which means that the real line is a diameter of the destination circle. Therefore, $\frac1{1-|a|}$ and $\frac1{1+|a|}$ are antipodal points

Balarka Sen

Since your line passes through both, what d'you get after combining the equations?

Can you tell something about the value of $b$ and $\lambda$?

I think that's enough hint for you.

Maximilian

So close

user159870

@robjohn why does the Line map to the Real Line?

Maximilian

The problem is tanθ=-sqrt(6)/2, cosθ>0

And I solved for secθ first, and got sqrt(5)/2

robjohn

@user159870 The line between those points is the line $\{at:t\in\mathbb{R}\}$

@user159870 $f(at)=\frac{at}{at-a}=\frac{t}{t-1}\in\mathbb{R}$

Maximilian

9:56 PM

then when solving for sin I got -sqrt(30)/5, and the real answer is -sqrt(15)/5

Balarka Sen

@user159870 Plus, note that you're in a unit circle. That should tell you something about $\lambda$

OK, I'm off.

Maximilian

sqrt(5)*sqrt(6)=sqrt(30)?

user159870

The line passes through $(x_0, y_0)$: $y-y_0=\lambda(x-x_0) $ (1)

The line passes through $(-x_0, -y_0)$: $y+y_0=\lambda(x+x_0) $ (2)

$(1)+(2): 2y=2\lambda x \Rightarrow y=\lambda x$

@BalarkaSen Correct?

What does $f(at)=\frac{at}{at-a}=\frac{t}{t-1}\in\mathbb{R}$ mean? What Information do we get from that? @robjohn

Maximilian

Damnit, I fail again

I see my mistake

robjohn

@user159870 That says that $f$ maps any of the points on the line $at$ to the real line

Axoren

10:24 PM

@robjohn, can you take a look at this question?: math.stackexchange.com/questions/1166199/…

I've been bashing my head over this for nearly 12 hours straight

Actually, probably more. 12 hour since I gave up and posted it on Math.SE.

Mary Star

What does it mean when 3 vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are at the same plane??

Axoren

@MaryStar It means that they lie on the same plane and that any linear combinations of them will also lie on the same plane.

Mary Star

At an exercise I saw the following:

If $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are at the same plane, then
(i) all are equal to $\overrightarrow{0}$
(ii) all are at the same line and one is not the $\overrightarrow{0}$
(iii) two are not the $\overrightarrow{0}$ are they don't belong to the same line

Why do these 3 cases hold?? @Axoren

Axoren

10:58 PM

@MaryStar If they're all $\vec 0$, they lie on EVERY plane, as every plane passes through $\vec 0$.

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