Mathematics - 2015-01-13 (page 2 of 3)

7:00 PM

@Chris'ssis I changed the variable for t=tan(x) then did polynomial division

@Chris'ssis but it seems easier your way

how do i see that $Aut_q(S^2) \cong \mathbb{Z}_2$ where $q:S^2\to \mathbb{R}P^2$ is the covering map that sends a point on the sphere to the line through the point and the origin

Chris's sis

@Mircea Just another way :-)

iwriteonbananas

it seems clear that the only homeomorphisms $\varphi:S^2 \to S^2$ with $q \circ \varphi = q$ are the identity and the map that sends $x\mapsto -x$

but how do i prove that there is no other?

robjohn

@Mircea $\frac{x^4}{1+x^2}=x^2-1+\frac1{1+x^2}$

Mircea

@Chris'ssis how's your work comming along?

@robjohn that's exactly what I did

Chris's sis

7:04 PM

@Mircea I've been working on my book and so far things are excellent! Thanks! :-)

Mircea

@Chris'ssis glad to hear that

Huy

Not much going on in the chat today. :(

robjohn

@Mircea Then $\int_0^{\sqrt3}(x^2-1)\,\mathrm{d}x=0$ and $\int_0^{\sqrt3}\frac{\mathrm{d}x}{1+x^2}=\frac\pi3$

Mircea

@Chris'ssis I got an offer from Uni of Liverpool for the Computer Science & Mathematics course :D

Chris's sis

@Mircea aaa, awesome!!! :-) Glad for you!

Mircea

7:05 PM

@robjohn yes, I solved it

@Chris'ssis Thanks! Now I'm waiting for the other unis I applied to to answer

Chris's sis

@Mircea Where would you like to study?

Mircea

@Chris'ssis I'd mostly want to get into Lancaster University

Huy

@Mircea: Are your parents rich?

Mircea

@Huy not really but I'll take a study loan

Chris's sis

@Mircea I see. Is there anything special for that you'd like to study there?

Mike Miller

7:07 PM

They're good, @robjohn, though I guess you deleted that message. Keeping busy. Teach linear algebra in a couple hours.

Huy

@MikeMiller: Remember all the cool applications we gathered for you!

Mircea

@Chris'ssis Well, it's one of the best universities in UK and I also really like the curriculum of the CS&Math course

robjohn

@MikeMiller I deleted the message since Hippalectryon made a bad joke which prompted the deletion.

Huy

@Mircea: How do you measure it being one of the best?

Chris's sis

@Mircea thecompleteuniversityguide.co.uk/league-tables/rankings

Indeed.

Mircea

7:10 PM

@Huy 3 different rankings

@Huy or actually 2

Huy

@Mircea: Which ones?

Mike Miller

@iwriteonbananas Deck transformations respect the covering map. Pick a basepoint of $S^2$; there are only two places it could go. (Why?) Now show that this choice of where the basepoint goes determines the whole automorphism (think paths and the unique homotopy lift)

Mircea

@Huy well the first would be the complete university guide and the second I have to think a bit to remember

Mike Miller

@robjohn Ah... he's got my number :)

Mircea

@Huy theguardian.com/education/ng-interactive/2014/jun/02/…

iwriteonbananas

7:12 PM

@MikeMiller oh i see

Huy

@Mircea: Ah, okay, cool. Too bad they only list UK unis. Any particular reason you want to study in the UK?

Mircea

@Huy I don't know any other language besides romanian and english; UK offers student loans that cover all of the university fees; It's not as far as USA; I was too late to apply in USA

Huy

@Mircea: Okay.

N3buchadnezzar

@Huy I got it =)

Huy

@N3buchadnezzar: Good. :)

N3buchadnezzar

7:18 PM

@Huy Finished your work?

Huy

@N3buchadnezzar: Not before 2016.

=P

heellooo

hi there

@user153330 Hello

7:23 PM

@user153330 welcome

user153330

@Chris'ssis @Mircea how're you? = )

iwriteonbananas

@MikeMiller mike, do u know where i can find old exams on this topic? (anything from point set topology to automorphism group of a covering)

Chris's sis

@user153330 Not that bad, thanks! ;)

Mircea

@user153330 Pretty good, thanks for asking!

user153330

@N3buchadnezzar may I ask you a question?

N3buchadnezzar

7:24 PM

Sure?

dustin

Would anyone be willing to share their thoughts on a good measure theory text?

user153330

@N3buchadnezzar where did you learn Tikz? from a book?

N3buchadnezzar

I just did stuff, and asked when i got stuck

Mircea

@dustin maybe this helps? youtube.com/watch?v=YgVha7Vvfes&list=PL87215166B6D41337

N3buchadnezzar

@dustin I liked Tao

dustin

7:26 PM

@Mircea I am looking for a book more than lectures online

Mircea

@dustin then idk

user153330

@dustin sure

N3buchadnezzar

@user153330 mirrors.ctan.org/macros/latex/contrib/tkz/tkz-euclide/doc/… + tex.stackexchange.com/questions/220503/… =P

Like this?

user153330

@N3buchadnezzar i recognize that one, it's from an answer by robjohn

Mike Miller

Nope @iwriteonbananas

N3buchadnezzar

7:28 PM

@user153330 I was thinking of updating that answer :p

iwriteonbananas

@MikeMiller ayte

user153330

@N3buchadnezzar key thanks = )

N3buchadnezzar

Just draw an amount of pictures with a nonzero measure, then you will be a tikz-master.

Hippalectryon

@Chris'ssis oops wasn't there could you repost that last one ?

ok thanks

Chris's sis

@Hippalectryon That one is a very nice integral. If you don't approach it properly, it will make you suffer. :-)

Vrouvrou

7:37 PM

I'm in the metric space $(\mathbb{R},d)$ where $d(x,y)=\frac{|x-y|}{1+|x-y|}$ and i have to prove that $I_n=[n,+\infty[, n\in \mathbb{N}^*$ is closed, is it right to take a convergent sequence $(x_k)\in I_n$ so $ x_k\geq n,\forall n\in \mathbb{N}^*$ then, $\lim_{k\rightarrow +\infty}x_k\geq n, \forall n\in \mathbb{N}^*$ i.e., $x\in I_n$ so $I_n$ ??

Thank you

Hippalectryon

@Chris'ssis Anyway, how goes the book ? How many pages already ? :P

Chris's sis

Also this integral is pretty precious $$\int_0^{\pi/4} (x-\tan(x))^2 \ dx$$

user153330

@N3buchadnezzar lol advice loaded

Hippalectryon

Precious ?

Chris's sis

@Hippalectryon hehe, I won't share these things now. :-)

Hippalectryon

7:41 PM

@Chris'ssis ok ok

negociations failed. next stage : spying

Chris's sis

@Hippalectryon lol :-)

Hippalectryon

:D

Mathy Person

Anybody want to give any input on this problem? math.stackexchange.com/questions/1103069/…

N3buchadnezzar

@user153330 ? :p

Mathy Person

oh

nevermind

Hippalectryon

7:45 PM

@MathyPerson Use the up arrow to edit messages

Mathy Person

$title$

aha

got it

Hippalectryon

test Confirmed, [title](banana link)

Mathy Person

title=banana

Hippalectryon

:D

Mathy Person

banana

yayy

double testing

user153330

7:47 PM

@N3buchadnezzar advice loaded = your advice is now installed in my head

N3buchadnezzar

:p goodie

Mathy Person

lol

Any advice on this? Math.SE problem

N3buchadnezzar

@MathyPerson cry

Sawarnik

hi people

Mathy Person

@N3buchadnezzar already did

user153330

7:53 PM

@Sawarnik hiiiiiii

Sawarnik

@user153330 hey :D

user153330

@Sawarnik how is it going??

Sawarnik

@user153330 well what? :D

user153330

@Sawarnik wuuut????

Chris's sis

Why Mathematica is not able to compute the value of $$\int_0^{\pi/4} \frac{\tan(x)}{x} \ dx$$?

Mircea

8:02 PM

I'm going to sleep. Good night/day, everyone!

Chris's sis

Maybe should I write them an email and asked for more details about this issue?

@Mircea Good night! Take care!

Mircea

@Chris'ssis Thanks! See ya!

Mathy Person

i have three problems left ahh

(in math)

Hippalectryon

@Chris'ssis Why does it seem strange to you ?

Chris's sis

@Hippalectryon I don't see where is the difficulty, but maybe I miss some crucial points.

Hippalectryon

8:06 PM

@Chris'ssis There isn't a single one out there which can compute it, I believe

Except Sage, maybe. Let me check.

Chris's sis

@Hippalectryon OK

@robjohn did you attend the integral above so far?

Hippalectryon

Do you have a nice form for $\int\cos(x^a)dx$ ? The usual one looks bad.

Sage is offline for me :/

sage

evinda

@DanielFischer If a set $X \subset \omega$ is bounded then $(\exists k \in \omega)(\forall y \in X) y \leq k$. If a set $X$ is not bounded, then $(\forall k \in \omega) (\exists y \in Y)y>k$.
I have I understood that $(\forall k \in \omega) (\exists y \in Y)y>k$ is the negation of the statement that has to hold when $X$ is bounded.. But could you explain me why $(\forall k \in \omega) (\exists y \in Y)y>k$ has to be true?

Chris's sis

Ups, I have done something wrong here $$\int_0^{\pi/4} \frac{\tan(x)}{x} \ dx$$ (let me reconsider my calculations)

N3buchadnezzar

8:26 PM

@Chris'ssis eazy

Chris's sis

@N3buchadnezzar Show me your way

N3buchadnezzar

@Chris'ssis I was thinking $\tan( a \arctan x)/x$

Lemme ponder

@Chris'ssis Darn. I was thinking abut $$ \int_0^{\pi/2} \frac{x}{\tan x}\,\mathrm{d}x $$

Chris's sis

@N3buchadnezzar hehe, that's a different story :-)

N3buchadnezzar

Yeah, there $x = \arctan( 1 \cdot \tan x)$ works fine ;)

Chris's sis

@N3buchadnezzar Yeap.

robjohn

8:37 PM

@Chris'ssis Are you talking about $\frac{\tan(x)}{x}$?

Chris's sis

@robjohn Yeah.

N3buchadnezzar

@Chris'ssis Well $$ \int_0^{\pi/4} \frac{\tan x}{x} + \left( \frac{\tan x}{x} \right)^{-1} \,\mathrm{d}x = ?$$ ^^

Chris's sis

@N3buchadnezzar What do you mean?

I can do that, but I need to work some more, I have the whole way in mind.

Espen Nielsen

8:55 PM

Hi everyone. Sorry if this isn't the appropriate place to ask, but I don't think it warrants its own question at Meta. What is the usual practice regarding questions like math.stackexchange.com/questions/851945 , which are easily answerable but very old, and the OP hasn't been active for 6 months?

Easily answerable?

hi all

hi rsz

@TedShifrin Maybe I misread. I thought the question was about ruler and compass constructions. Never mind.

Ted Shifrin

@Espen: I have no clue whether there's any merit to the question or not, but I certainly have never seen anything about it.

rsz

9:08 PM

i am stuck with some task which i can't figure out, i think that there is something really simple in fact but can't get to it, if some of you guys could point me to something it would be helpful,
((2^63)-1) this is the number in base 10 which i would need to convert to base 16
i am thinking of converting the number somehow so i can get something reasonable, has anyone of you some idea about it?

Espen Nielsen

If the only allowed operations are with straightedge and compass, it cannot be more expressive than the persian geometry which allows intersections of conics. It is known that this geometry can at most express 5th roots.

Ted Shifrin

@rsz: Start with $2^{63}$. Can you figure that out in base $16$? Easier, what about $2^4$, $2^5$, $2^6$, ...

Well, no, @Espen, we can construct a regular 17-gon with ruler and compass, for example.

Espen Nielsen

@TedShifrin But you are still limited to constructing field extensions of order $2^n$. Constructing a 7th root will give a field extension of order 7, right?

Ted Shifrin

no, order $6$, still not a power of $2$, though. I'm just quarreling with your statement that you can't get past 5th roots. Gauss knew all this stuff :P

Espen Nielsen

@TedShifrin When I said "past 5" I meant in the sence that 5 is the greatest prime divisor. I guess I didn't express myself clearly. :P

Ted Shifrin

9:15 PM

I'm still confused, @Espen. What about 17?

Espen Nielsen

@TedShifrin It has to do with constructible angles. The cosine of $2\pi/17$ can be expressing using iterated square roots.

Ted Shifrin

I understand all that. I don't see how it fits with your 5 stuff.

rsz

@TedShifrin: i can deduct the fact, that the result gets always multiplied by two, but that what it should do, how can this help me?

Ted Shifrin

What are the place values base 16, in terms of powers of $2$, @rsz?

Espen Nielsen

@TedShifrin Argh, it seems I was confused. The geometry of conic sections does not support 5th roots. Only roots of order divisible by 2 or 3. The point still stands, though, that given a parabola in the plane, there is no way, using only a straightedge and compass operations, or taking intersections of conics, to produce a 7th root of unity.

Ted Shifrin

9:23 PM

I don't know anything about using conics other than circles to construct points. I've never seen it discussed. Cool questions ...

Espen Nielsen

It's actually suprisingly hard to find good treatments of it.

But this is one of them, it seems. math.cornell.edu/~dwh/papers/geomsolu/geomsolu.html

Alex

if the last two digits of a number are 00 does that mean it is divisible by 4?

Ted Shifrin

well, Alex, is 100 divisible by 4? :)

Hi, btw

rsz

@TedShifrin i don't understand your question

Alex

yea 25

Ted Shifrin

9:28 PM

@Espen: Thanks. I have one of Hendersen's elementary geometry books, but I don't think it's in there.

So, then any multiple of 100 will be divisible by 4, right, Alex?

Alex

yes i think so

Ted Shifrin

@rsz: What is the base 16 representation of $2^4$?

wondering how I got stuck on algebra duty today

rsz

it's 10

Ted Shifrin

And what is $2^8$?

Espen Nielsen

@TedShifrin It's a shame. This is the kind of geometry that all 2nd year math students should learn. =)

Ted Shifrin

9:31 PM

LOL ... We have enough problems with just basic stuff, even with most math majors, @Espen.

rsz

that is 100 in base 16

Ted Shifrin

Teaching abstract algebra to 3rd and 4th year math students in the US, many of whom are going to teach high school, I discovered years ago they didn't know laws of exponents.

OK, @rsz. Now can we figure out $2^9$ or $2^{10}$?

rsz

yes those are like 200 and then 400 and the next one would be 800 and so on

Ted Shifrin

ok, so, what is $2^{63}$?

Espen Nielsen

@TedShifrin Wow! I think this is a common theme though. The curriculum moves so fast that most don't have time to become sufficiently intimate with the basics. I often find myself experiencing the same symptom...

Ted Shifrin

9:35 PM

well, basics from elementary school is a bit depressing :(

Espen Nielsen

That's true.

Ted Shifrin

I'm teaching a senior-level differential geometry class and I have to remind students of basic trigonometry and first-term calculus.

Espen Nielsen

Haha, they probably didn't think about those concepts after handing in their exams. :P

Ted Shifrin

not to mention hard stuff like $u\cdot v = \|u\|\|v\|\cos\theta$ :D

Espen Nielsen

(Full disclosure though, I conducted the geometry discussion above with my Galois theory book open...)

Ted Shifrin

9:38 PM

Well, I've taught it a few times, @Espen :P

Espen Nielsen

It's hard to compete with 35+ years in the business. :P

rsz

@TedShifrin the number is 8000000000000000

Ted Shifrin

I'm quitting soon enough, don't worry :P

Right, @rsz. Now subtract 1 :)

You'll need your letters for (base 10) 10, 11, ... , 15.

Alex

hi @TedShifrin thanks for the help the last few days btw

Espen Nielsen

Well, it was nice talking to you, Prof. Shifrin. I have to get back to writing my thesis.

Ted Shifrin

9:41 PM

Have fun, @Espen. My pleasure too. I didn't know about cube roots with parabolas. Very cool.

Most welcome, Alex.

Alex

i also learnt divisibility rules

they are so useful

Ted Shifrin

Yup

Alex

if it is divisible by 4 can i assume that it is divisible by 8 ? same with 3 and 9 ?

rsz

@TedShifrin thanks, i think i finally understood how to get to the result, all i needed to understand was the relation between the different levels i mean 2^2,2^3,2^4...

Ted Shifrin

That's why I led you that way, @rsz :P

No, Alex. You have those backwards. If a number is divisible by 8, then it's divisible by 4. But 12 is divisible by 4 and NOT by 8.

Alex

9:50 PM

this is all for military selection process. so many speed distance time questions. learning the fractions has been the hardest part but im getting it now aha !

Ted Shifrin

wow, well, it's good skills to have

Transcript for

$Mathematics$ Mathematics