Mathematics - 2014-12-04 (page 4 of 9)

Meanwhile, here's a related article of Arnold about mathematical education that I take more seriously than the article of Thurston (for reasons which would earn me another smack from Ted) pauli.uni-muenster.de/~munsteg/arnold.html

beginner

11:04 AM

@BalarkaSen what is residue for modolo arithmatic?

Anonymous

@WillHunting how does taking an online course get you a PhD?

Will Hunting

@beginner I have a Permanent Head Damage.

Venus

Why can't I tag Random Variable here in chatroom?

Will Hunting

@AshwinGokhale Well, there are many such programs around these days, not for math though.

Balarka Sen

@beginner I am not going to survey on this. Read up some few chapters of a standard elementary number theory book.

beginner

11:05 AM

@BalarkaSen can i only ask you to verify things, not to teach?

Balarka Sen

On the other hand, isn't this explained on Dummit-Foote already?

Anonymous

@WillHunting I never asked anything about what you were studying?

Will Hunting

@AshwinGokhale I did a math degree ten years ago.

Balarka Sen

@beginner You can ask me if you get stuck on any problem. I am too busy with my studies to explain elementary concepts which are very well explained in standard books.

beginner

@BalarkaSen ok i will do that only then thank you

Balarka Sen

11:07 AM

I'll advice you to pick up a book and go through it rather than picking up terms, @beginner

Will Hunting

@beginner Instead of reading random books proposed by random strangers, you should figure out a course of study first.

beginner

@WillHunting i will finish dummit and foote before i am '15 or so'

Will Hunting

@BalarkaSen Advise is the verb and advice is the noun.

Balarka Sen

@Will It's a chat dude. Do you expect everyone to spell correctly? :P

Will Hunting

@beginner I think you should finish a course in linear algebra first before reading that book.

beginner

11:09 AM

noun is a thing or a place, verb is a describing word?

Anonymous

@beginner why don't you check the schedules of tripos?

beginner

adjective is a verb noun?

@BalarkaSen did you start on dummit and foote?

Balarka Sen

I started mathematics with elementary number theory, if that's what you mean.

beginner

@BalarkaSen what was your text book order?

or survey order as you call it hehe

Balarka Sen

@beginner Not all books are surveys. The order of books are hard to recall but I think I studied calculus before number theory.

beginner

11:10 AM

oh so i am lost then :(. you were doing dummit and foote at my age

but i didnt start them yet

Anonymous

Don't compare your start with somebody else's @beginner

Balarka Sen

@beginner How about, say, beginning with elementary number theory? It's a quite self-contained subject.

beginner

balarka inspires me ashwin

ill be back later

Venus

Hi @M.N.C.E. Nice to see you here ^^

Balarka Sen

@beginner What you should do is to be less overenthusiastic. It's dangerous. Have you studied set theory?

Ugh covering spaces are hard. @DanielFischer can you help me with something?

Daniel Fischer

11:17 AM

@BalarkaSen Maybe.

Balarka Sen

@DanielFischer Munkres wants me to find $p : A \to B$ a covering map, $q : B \to C$ another covering map such that $q \circ p$ is not a covering map. I think I have already said this before but $p : S^1 \times \Bbb N \to S^1 \times \Bbb N$ as $(x, n) \to (x^k, n)$ and the projection $q : S^1 \times \Bbb N \to S^1$ as $k \to \infty$ works. But is there no such example when $A, B, C$ are path connected and locally path connected?

This should induced inclusion of fundamental groups via the maps $\pi_1(p)$ and $\pi_1(q)$

UserX

@BalarkaSen gimme a problem

Balarka Sen

@UserX Have you studied normal subgroups?

Hippalectryon

@Chris'ssis $$\int_0^{\pi/2}\arctan^3(\sin(x))dx=4\sum_{k,n,m=1}^\infty(\sqrt{2}-1)^{2(k+n+m‌-1)}\dfrac{\frac{-1}{8k+8m+8n-12}+\frac{1}{8k+8m-8n-4}+\frac{1}{8k-8m+8n-4}-\frac‌{1}{8k-8m-8n+4}}{(2k-1)(2n-1)(2m-1)}$$

@Chris'ssis (The latex doesn't render properly in chat, just see the source)

Balarka Sen

@DanielFischer I have no idea what to do then. Maybe $q \circ p$ is a local homeomorphism but I can't see this.

Daniel Fischer

11:25 AM

@BalarkaSen No connected counterexample springs to mind.

UserX

@BalarkaSen not really. Just definitions, haven't even seen worked out problems yet.

Daniel Fischer

@BalarkaSen A composition of local homeomorphisms is a local homeomorphism.

Proof: ...

Balarka Sen

Might be.

I believe it :P

Daniel Fischer

@BalarkaSen Don't believe, prove. Start with the definition.

Hippalectryon

1./ Proof 2./ ???? 3./ Profit

Balarka Sen

11:28 AM

@DanielFischer $f : A \to B$ is a local homeomorphism if for all $x \in A$ there is an nbhd $U$ around $x$ such that $f$ restricted to $U$ is a homeomorphism.

Erm. I am missing something, aren't I?

Daniel Fischer

@BalarkaSen Incorrect.

Balarka Sen

Yeah, right. OPEN MAP.

$f(U)$ is open.

Daniel Fischer

Yes, the point is that (the restriction of) $f$ is a homeomorphism between open neighbourhoods of $x$ and $f(x)$.

Balarka Sen

So well this is obvious @DanielFischer. $f, g$ be local homeos from $A \to B$ and $B \to C$. Composition of homeos are homeos and $f(g(U))$ is open.

Daniel Fischer

@BalarkaSen Not so fast, grasshopper.

You need to restrict the neighbourhoods, generally.

Balarka Sen

11:33 AM

$U_x$ be in $A$. $f(U_x)$ is open and $f|_{U_x}$ is a homeo. Now $f(U_x)$ is an nbhd of $f(x) \in B$

Daniel Fischer

Good, and then?

Balarka Sen

So $g|_{f(U_x)}$ is a homeo and $g(f(U_x)) = g \circ f (U_x)$ is open.

Daniel Fischer

@BalarkaSen No. You can't conclude that $g\lvert_{f(U_x)}$ is a homeomorphism.

Hippalectryon

Could someone verify my above integral with Mathematica ?

Balarka Sen

@DanielFischer Well $g$ is also a local homeomorphism and $g|_{U_y}$ for all $y \in B$ is homeomorphism by definition.

As $f(U_x)$ is an open nbhd of $f(x) \in B$, it follows that $g|_{f(U_x)}$ is a homeomorphism

Daniel Fischer

11:36 AM

@BalarkaSen The point is "for all $y\in B$ there exists a neighbourhood $V_y$ of $y$ such that ..."

Balarka Sen

Ah OK.

Daniel Fischer

That neighbourhood can be very very very small.

Random Variable

@Venus I don't remember. But I used to keep a notebook (which I still have), and it's in there.

Balarka Sen

In that case assume $U_y \subset f(U_x)$.

Daniel Fischer

@BalarkaSen \subset

@BalarkaSen And then?

Balarka Sen

11:42 AM

Ack. We have to restrict the nbhds too.

Will Hunting

@DanielFischer As small as my XXX.

M.N.C.E.

@Venus Hi.

Daniel Fischer

@WillHunting smaller, much smaller.

Venus

@RandomVariable Could you take picture of it & send it to me ^^

Please...

@M.N.C.E. Hi too...

Will Hunting

@venus Is your boyfriend from Mars?

Random Variable

11:46 AM

@Venus I'd rather just post something on the other forum.

Venus

@WillHunting No, he is an earthling

@RandomVariable Really? Can't wait then. Please let me know when it's done :-)

Balarka Sen

@DanielFischer We have to pick some $V_x$ around $x$ in $A$ such that $f(V_x)$ is inside that $U_{f(x)} \subset f(U_x)$, right?

beginner

$\prod \limits _1^{p-1}\gamma_v = \prod \limits _1^{p-1}\alpha\gamma_v=\alpha^{p-1}\prod \limits _1^{p-1}\gamma_v\implies a^{p-1}\equiv1\pmod p$ is so clever!!

Committing to a challenge

12:26 PM

I decided I will try to sleep and get up early.

But first, Coffee vs study done Anova imgur.com/2PVACRO

Strong statistical evidence that coffee is correlated with study done

Daniel Fischer

@BalarkaSen Yes. Typically, one takes the intersection $f(U_x) \cap V_y$, and its preimage in $A$ and image in $C$.

Committing to a challenge

Which really means little, since pages done is the real indication of work done: imgur.com/jEexYP6

Anyway goodnight

Anonymous

@Committingtoachallenge What were you reading?

Committing to a challenge

@AshwinGokhale I was doing Cohn, but I decided I should sleep, since I may be seeing people tomorrow

Anonymous

@Committingtoachallenge Good night!I am beginning with Artin now

Committing to a challenge

12:32 PM

@AshwinGokhale Best of luck :). Talk tomorrow

Random Variable

@Venus I edited my post.

Venus

@RandomVariable Thanks. I'm stuck in the second line. It never crossed to mind to multiply it by $\dfrac{\cos2x}{\cos2x}$

Integrator

Is @Advaitha 's account suspended?

Venus

@RandomVariable It's indeed a cute one. Do you have other cute integrals that can be done in a few lines using elementary methods? Maybe you can provide some links

@Integrator Yeah. I had little argument with him. I'm quite surprised he acted like a child & also rude

@Integrator Is he your teacher or lecture?

It looks like someone post a monster

0

Q: Very Messy Integral.

Find: $$\large \left|\int_{-\frac1{\sqrt3}}^{\frac1{\sqrt3}}\frac{\ln|x|^x+(\arcsin x)^3(\arctan x)-(1+x^2)\arctan x}{(\ln|x|)^2(x+x^3)}dx\right|$$

integration

Anonymous

12:57 PM

@Chris'ssis Have you solved your integral?

Random Variable

@Venus No, not really.

Integrator

@Venus No, Not my teacher!

Daniel Fischer

1:18 PM

@Venus It's just pretending to be hard.

Venus

@DanielFischer Why did you delete your answer sensei?

Daniel Fischer

@Venus Because I noticed the OP spamming no-effort questions in short succession.

Not a behaviour I condone.

Venus

He has edited his post

Daniel Fischer

@Venus Now I have to think whether to undelete.

Venus

Haik sensei! Noted

It should be undeleted since he got it from exam

Besides, he seems a nice user by seeing his profile

Daniel Fischer

1:25 PM

undeleted

Venus

I doubt he didn't give any attempt to solve it

HAHAHAHAHAHAHAHA @DanielFischer

@eaxdpiotnyeantial Have a tough day at school buddy? :-)

eaxdpiotnyeantial

what do you mean?

Chris's sis

Greetings

Venus

Good day @Chris'ssis

@eaxdpiotnyeantial See the history chat

Chris's sis

@Venus Hi

eaxdpiotnyeantial

1:29 PM

oh it's a week old @Venus

Chris's sis

@AshwinGokhale Sure. :-)

Venus

@eaxdpiotnyeantial Ganbatte!!

eaxdpiotnyeantial

@Venus Is it an english word, sorry english too is my second language

Venus

@eaxdpiotnyeantial What's your first language?

eaxdpiotnyeantial

@Venus oh i don't post regularly, i collect them and post them in a go, so that might had made you think that. BTW Hindi.

leo

1:32 PM

@Venus what's yours?

Venus

@Chris'ssis Yup

@Chris'ssis I know. Let me see it first

Chris's sis

@Venus OK. Check that. I consider that piece a masterpiece that I'll add to my book. :-)

Venus

@leo You first

@Chris'ssis You're an author? What book?

leo

@Venus spanish

Venus

@leo Hola, como esta senor? :D

Chris's sis

1:35 PM

@Venus Yeah, I'm going to publish a book about integrals, series and limits, mainly things created by me.

Venus

@Chris'ssis Sounds great! Let me know if it has been published

leo

@Venus ha ha ha, good

Venus

@leo Malay

Balarka Sen

OK, @DanielF, but that doesn't solve my problem.

Venus

@leo Where do you live? Madrid? Barcelona?

leo

1:38 PM

n should go with ~ so it become ñ

eaxdpiotnyeantial

@Venus,@Chris'ssis it came in the exam with ^2014, ^2013 and many dirty paper, very tough . :( I scored 120/320

Venus

@eaxdpiotnyeantial I can relate that

Chris's sis

Still working on it, and at the same time I do research every day for discovering other great results such that all buyers will totally love my book. Sure, out there will be always criticism ... (for me it's important to be a successful book)

eaxdpiotnyeantial

@Venus/@Chris'ssis/etc. what are you people currenltly doing, still in highschool like me?

Venus

@Chris'ssis Don't forget to give a little story for every problem. It'll look good I guess

leo

1:40 PM

@Venus Neither of those. Far from them. Look

@Venus nice

Daniel Fischer

@BalarkaSen By "solve your problem" you mean finding connected coverings such that $q\circ p$ is not a covering?

Balarka Sen

right

Chris's sis

@Venus Yeah, I also like the stories for the problems, but sometimes it's hard to find when a problem appeared first.

Venus

@leo I can't locate in which city do you live by seeing that map

leo

@Venus San José, Costa Rica

Daniel Fischer

1:42 PM

@BalarkaSen Well, you know that if $q$ is finitely-sheeted, then it's a covering. So take an infinitely sheeted covering.

Chris's sis

@Venus In his book, Inside Interesting Integrals, Paul J. Nahin talks about the name of some integrals, some of them seem to be very old.

Venus

@Chris'ssis You're right but at least for the famous problems you should give it

Balarka Sen

@DanielFischer The converse is totally not true.

Chris's sis

@Venus The book will mainly contain original creations, things that came from my mind. I don't need a book like the rest, but something totally different.

Venus

@leo Oh my, I forgot many countries in America also use Spanish

Daniel Fischer

1:44 PM

@BalarkaSen You mean that if $q$ is infinitely-sheeted, then $q\circ p$ can still be a covering? As if I didn't know.

But, @BalarkaSen if you want to construct a counterexample, that's what you need to start with.

Venus

@Chris'ssis At least you can give your personal story when those problems came to mind

Balarka Sen

I know, but I was rather thinking on the lines of proving that no counterexample exist, @Daniel

leo

@Venus no problem

Chris's sis

@Venus They come to mind every day, while sleeping, while doing accounting, while tutoring, while going shopping, jogging ...

Venus

@Chris'ssis Are you below 30? If I may know

Daniel Fischer

1:46 PM

@BalarkaSen I think it's a little early to say that.

Balarka Sen

Oh you're thinking about connected.

Chris's sis

@Venus Around 30.

Balarka Sen

I was assuming path connected and locally path connected stuff, @DanielF

Venus

@Chris'ssis Still young.

Daniel Fischer

@BalarkaSen Yes, and? Still too early to say "doesn't exist". Anyway, trying to construct an example if it fails may give hints as to how one could prove none exists.

Venus

1:49 PM

Anyway, I use your method to answer integral problem here. I hope you don't mind @Chris'ssis

Chris's sis

@Venus Please do not use the method I just showed you. That's why I deleted the post.

Venus

@Chris'ssis I just did. Sorry

I'll delete mine

Chris's sis

@Venus No need, you can let that one now.

Venus

Sorry, I can't delete it. It's accepted

Chris's sis

@Venus No problem. No need to delete it.

Venus

1:51 PM

Are you sure?

@Chris'ssis Heave a sigh of relief finally

Chris's sis

hehe, it's OK.

Venus

Sorry, next time I won't do it again. I promise ^^

Chris's sis

@Venus Did you see this one?

10

Q: An integral $\int^\infty_0\frac{\tanh{x}}{x(1-2\cosh{2x})^2}{\rm d}x$

I would like to enquire about the possible methods of computing the following integral $$\color{blue}{\int^\infty_0\frac{\tanh{x}}{x(1-2\cosh{2x})^2}{\rm d}x=\ ?}$$ A possible way I see of doing this is to apply the substitution $x\mapsto-\ln{x}$, which yields $$-\int^1_0\frac{x^3(1-x^2)}{(1+x^2)...

integration special-functions contour-integration closed-form

Venus

@RandomVariable Does user Yourself have an account on MSE?

@Chris'ssis Yes. I've upvoted too. Why do you ask?

Is there something wrong in David's answer?

Or you have a shorter answer for that problem?

Chris's sis

@Venus Can we do that in a fast way?

Venus

1:58 PM

@Chris'ssis I don't know. I can't think of it at the moment. But it seems your question implying so

@DanielFischer If I may know, where do you live sensei?

Balarka Sen

senpai, @Venus

Daniel Fischer

@Venus Northern central Europe.

Venus

@BalarkaSen I respect to him as a guru

Balarka Sen

We all do.

Venus

@DanielFischer Scandinavia countries?

Chris's sis

2:02 PM

@BalarkaSen On this site I only respect @robjohn as a guru. Sorry, but this is my personal opinion. It's not OK to speak in the name of all.

Venus

Sorry if that too personal. You may not answer it

Daniel Fischer

@Venus A bit less north.

Mike Miller

Cute problem, @Balarka

Balarka Sen

@MikeMiller All of my problems are either very interesting or trivially false.

Venus

@DanielFischer What did you mean by a bit less north? It's Arctic ocean?

Mike Miller

2:04 PM

I wouldn't go that far, @Balarka.

Daniel Fischer

@Venus A bit south of Scandinavia.

Venus

@DanielFischer Either UK or Germany, but I bet Germany

Balarka Sen

Hmm. Map $\Bbb R \times \Bbb R^{+} \to S^1 \times \Bbb R^{+}$. Then map $S^1 \times \Bbb R^{+}$ to $S^1$ by projection.

Whoops. The composition is a covering space.

Nevermind.

Venus

@DanielFischer Your family name sounds familiar. I've heard it somewhere

Daniel Fischer

@BalarkaSen If you mean to take the discrete topology on $\mathbb{R}^+$, but then it's not connected. If you take the standard topology, $S^1\times \mathbb{R}^+ \to S^1$ is not a covering.

Balarka Sen

2:08 PM

Oh right.

Hrm.

Daniel Fischer

@Venus Not exactly a rare name.

Mike Miller

Why would a sane person take the discrete topology, @DanielFischer?

Daniel Fischer

@MikeMiller It can be useful.

Balarka Sen

I am not quite sane, @MikeM

Venus

@DanielFischer I google it, it shows up this one in Wiki: en.wikipedia.org/wiki/Bobby_Fischer

The Great Bobby Fischer

Balarka Sen

2:10 PM

@Venus The one who catches fish with booby traps?

Sawarnik

@BalarkaSen Oh.

Daniel Fischer

@Venus The great Emil Fischer

Venus

@DanielFischer Is he your ancestor?

Sawarnik

Arthur Fischer, the mod.

Mike Miller

@DanielF Fair enough, I suppose.

Daniel Fischer

2:13 PM

@Venus Neither one is a relation (within reasonable limits, of course if you go back far enough, we're all related). Nor is the great Ernst Fischer.

Or the other great Ernst Fischer.

Venus

@Sawarnik I forgot Arthur. The strictest mod in MSE :D

Balarka Sen

LOL. We're bringing up a whole load of Fischers.

Venus

@DanielFischer Lots of the great Fischers. It strongly indicates you're a Germany. ^^

Sawarnik

Hehe, he's not a nation.

Daniel Fischer

@Venus I'm half of Europe thrown in a mixer.

5

Venus

2:17 PM

If I have a son I'll give him name Daniel & if a girl, I will name her Daniela :D

@DanielFischer But you do have a nationality right?

Daniel Fischer

@Venus No, just a citizenship.

Hippalectryon

@Chris'ssis Did you see my post earlier ?

Chris's sis

@Hippalectryon Thanks, yeah. I'll check the details as soon as I finish the proof I', working on ... that is ...

10

Q: An integral $\int^\infty_0\frac{\tanh{x}}{x(1-2\cosh{2x})^2}{\rm d}x$

I would like to enquire about the possible methods of computing the following integral $$\color{blue}{\int^\infty_0\frac{\tanh{x}}{x(1-2\cosh{2x})^2}{\rm d}x=\ ?}$$ A possible way I see of doing this is to apply the substitution $x\mapsto-\ln{x}$, which yields $$-\int^1_0\frac{x^3(1-x^2)}{(1+x^2)...

integration special-functions contour-integration closed-form

@Hippalectryon in one line

Hippalectryon

:O

@Chris'ssis Isn't trigamma $\psi^{(3)}$ ?

Nevermind

Chris's sis

@Hippalectryon No, that answer is correct.

Hippalectryon

2:28 PM

Indeed. I forgot for a moment that $\psi_0$ was digamma and not gamma

Chris's sis

OK

Balarka Sen

@DanielFischer Does that half of Europe contain bits of Western Europe?

If so, then the mixer is bound to blow up at some point.

Venus

@BalarkaSen I suspect @DanielFischer is a Super Aryan

Hippalectryon

@Venus Arya ?

Balarka Sen

Explanation of the joke : the poles are in Western Europe.

Daniel Fischer

2:34 PM

@BalarkaSen Yes, some Dutch and French. Why should that cause the mixer to blow up?

Balarka Sen

=P

Random Variable

@Venus He does, but I don't know if he would want me to share it with anyone. He uses a generic username. Actually, I don't even know what it is off the top of my head.

Daniel Fischer

@BalarkaSen Not all Poles. Some are still in Poland.

Balarka Sen

Well the semi-poles are quite dense around the western Europe.

Venus

@Hippalectryon en.wikipedia.org/wiki/Aryan_race

Hippalectryon

2:36 PM

Oh that

Balarka Sen

2:50 PM

whoa whoa whoa