Mathematics - 2014-12-16 (page 1 of 5)

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12:00 AM

The inly units in $\Bbb Z$ are $\om 1$.

if there isn't a point in the image of $f$, then there is a point on the boundary of $f$ which will violate the open mapping that invariance of domain implies

@cxseven: The degree of composition ...

oh, the way i thought to prove that is with smooth approximants

Still takes work for smooth.

it's not just the chain rule and checking that your approximants are close enough?

12:04 AM

It takes some applications of the inverse fn thm and other stuff ...

if you know an inverse of a continuous $f$ exists, regardless of whether it's smooth, if you take a close enough smooth approximant of $f$ and its inverse, then their composition will be close enough to the identity map to know that the composition has degree 1

you can show the composition is homotopic to the identity

i suppose the same argument would work for triangulations

i remember there exists a german word for the problem of finding consistent sub-triangulations, but forgot what it was

one more tiny odd and end problem in pdes AND I AM DONE!

@Behaviour Do you have a guess on the number of edits, age of posts, &c?

@MikeMiller No idea. I knew of a tag that needed cleaning, so I went and did it. Some posts were from 2013.

12:38 AM

Got three in quick succession. Appropriately, back to #3. Feels good.

Ugh. "Hats".

hats are fun

I got a (-1) in my post after a long time ! :D but this time its a HAT for me :P LOL

@KajHansen you are not wearing any ? ;)

"Hats" are below me. :P

hahaha !! lol

12:44 AM

Look at my hat. You're trying to tell me it's below you?

I don't think so, bub.

It's too powerful.

You look like some native shaman.

LOL, what inane task did you complete for that one? ;)

I got 20 other hats.

@MikeMiller several of them "secret hats"; I wonder what they are

It's a secret.

12:56 AM

@KajHansen When we met, you didn't resemble Matt Damon that much.

I hate Matt Damon now, after Interstellar.

Coward pig.

it's impossible to have 20 hats because of the hat you get for having 20

@PedroTamaroff, I don't think so either. That's Jasper who says that.

@cxseven I think you actually get it for having 20 hats on a single site (so that it's possible to have only 20) but I'm not sure.

Jorge Fernández

@MikeMiller you have an iphone and an android?

that's so fancy

I have an iPhone. I borrowed my roommate's android.

Jorge Fernández

1:00 AM

I dedicate this song to you youtube.com/watch?v=O-zpOMYRi0w

@Hippalectryon LOL :P

1:16 AM

I finally got a cool hat

@user130018 Cool !! :-)

@MikeMiller You are the pride of Math.SE right now, tied for the 3rd place network-wide.

"Tied". This is unacceptable @Mike

I'm 13th network wide, 2nd here. And trying to earn a hat for finally getting final exams graded...

How do we win hats for grading finals?

1:31 AM

It's an imaginary hat.

Ah, does it have a simple pole?

@TedShifrin I am inclined to say 'hat' .. (a gesture of saying .. you must be joking in my mother tongue) =P

Ok @r9m

I can't get that hat for another couple days, @Behaviour. I think I have the references to write an answer to a questionI posted, so I'll hit #2 when I get home.

@TedShifrin's succumbing to hats! :D

1 hour later…

Jorge Fernández

3:07 AM

I have a question, suppose I re-tag a question I solved previously, would I get the medal for the respective tag?

Like for example, suppose I re-tag all of my questions algebraic number theory today

would that give me the algebraic number theory bronze badge?

Probably, and it would probably also get you a scolding.

Jorge Fernández

no, it doesn't, I just rettaged some questions and it didn't change my score on here

It might take some time to recalculate or something.

Jorge Fernández

oh, makes sense

I'm going to answer a combinatorics question and see if it changes.

3:28 AM

@JorgeFernández SEDE only updates every few days, I think.

Jorge Fernández

oh, I see, thanks

ugh, the answer I was hoping would get me a Naruto pulled an upvote.

4:09 AM

@JorgeFernández, for some reason there is a delay.

I have legitimately retagged some of my answered questions in the past, and it took about a day for the score to transfer. Idk why.

I haven't examined things that closely though because, ultimately, I care very little. So don't take my word for it.

4:36 AM

Back to #1.

Hi @Ted

I hate my answer to this question. You would surely be able to write a better answer than me, so I encourage you to do so.

Seems like everyone's thinking about all the wrong things 😛 ah, politics ...

0

Q: $h(z)=\frac {k}{2\pi} \int_CF(\theta)e^{ikz\cos \theta}\,d\theta$

Suppose this integral $$h(z)=\frac {k}{2\pi} \int_CF(\theta)e^{ikz\cos \theta}\,d\theta$$ $$0\le\theta\le\pi$$ $$|z|\le l$$ We are in complex $\theta$ plane.Assume we have knowledge of $F(\theta)$ The questions are, Is it possible to find the value of $h$ at specific points? Is it possible to ...

integration complex-analysis

@TedShifrin Beat this problem for the sake of god ^

@Mike: Thanks for the plug, but that OP has done Hubbard and Axler. He/she wants more advanced.

I didn't write that recommendation for the OP, I wrote it for anyone else who wanders onto the question. The first one was more reasonable. Note that OP posted this a year ago, so one can't know what they actually want now.

Anyway, that's why I called on you.

Oh ... Berger's book on geometry (meant for French high school teacher trainees) is fantastic.

4:45 AM

Post it as an answer!

Tomorrow ... No fun on iPad.

@TedShifrin @MikeMiller Do not bombard poor OP with tons of books :)

Well, as long as you do it. :)

That's what the OP wanted, @FreeMind

@TedShifrin Yeah, but not any book should be told, however, check my question :(

4:50 AM

@Freemind: Complex $\theta$ plane? Um ... What is $C$?

Contour

I don't believe you.

user image

@TedShifrin ^

Then we should not use the letter $\theta$.

Anyhow, it's past my bedtime ...

@TedShifrin :|

4:55 AM

So it looks like we're using Kaplansky's text for point-set @Ted. I'll have to check that out from the library tomorrow.

@KajHansen Kaplansky's cool.

That's what Dr. Clark was saying :)

Yay! @DanielFischer Has 500 up-votes!

@KajHansen The best way to lean point-set is to suddenly wake up knowing it, and never have to read any of the books written about it.

Well that's the easiest way to learn any subject, right @MikeMiller?

Anonymous

5:03 AM

@KajHansen WHere is @Committingtoachallenge these days?

Maybe he committed to the challenge.

@Ashwin, finals perhaps? It's only been 2-3 days since I've seen him.

I know I was out for that long due to my finals, haha

Anonymous

@KajHansen How much score do you expect in the Putnam?

At least 10. Not much higher though. I figure I should be satisfied with that given sleep deprivation and having not studied.

I should've gotten >20 easily, but I misread something on one of the questions.

Anonymous

@KajHansen Have this amazon.com/Putnam-Beyond-Razvan-Gelca/dp/0387257659 ?

5:06 AM

I was planning to buy it to have something to do over the summer.

Anonymous

@KajHansen looks like you take a lot of caffeine,do you?

LOL, how'd you guess?

Anonymous

@KajHansen DO buy this one too amazon.in/Problem-Solving-Strategies-Problem-Books-Mathematics/… !

Oh cool, I was previously unfamiliar with that one. Thanks for the recommendation :)

Anonymous

@KajHansen Sleep deprivation is due to caffeine actually!You mustn't take more than 300mg's

Anonymous

5:08 AM

One wonderful think about these books is that they come with the solution :D

Though I actually do own caffeine in pill form, I usually just stick to soda.

Are there solution manuals?

Or just answers to selected questions in the back.

@r9m Isn't it fashionable?

UnadulteratedImagination

I'm a graduate student (not in maths) and I need to use abstract algebra frequently in my research. I took an introductory graduate level abstract algebra course in maths but I still don't feel confident in solving algebra problems. The book we used was Dummit and Foote. Can somebody suggest other book/resource which I can study from to get a better understanding/confidence in solving problems.

@Ashwin Don't you have some hat Eh?

5:14 AM

@UnadulteratedImagination, perhaps try Michael Artin's "Algebra"

I'm also fond of Shifrin's Abstract Algebra: A Geometric Approach because that's what I learned most of my Algebra from.

Shifrin's book won't have the breadth of material that the other two have, though. It depends what you need.

May I ask what kind of research uses Abstract Algebra @UnadulteratedImagination?

Anonymous

@skullpatrol They come with solutions to all questions I guess!

@skullpatrol, my dad uses group theory for quantum mechanics

There's a decent number of applications in chemistry/physics.

Anonymous

@KajHansen You tried this book amazon.com/Visual-Classroom-Resource-Materials-Problem/dp/… ?

Right, I keep forgetting about physics and computer science :(

5:23 AM

I haven't @Ashwin, but I'd be interested in reading it. A lot of my groups I've grounded in geometric intuition.

@UnadulteratedImagination What's wrong with Dummit and Foote itself?

I.e. $S_3$ is the symmetry group for a triangle, $S_4$ a cube, $A_4$ the tetrahedron, $A_5$ the Icosahedron, etc.

@MikeMiller, I love D&F, but I could definitely see it being a little too ambitious for someone seeing algebra for the first time.

I caught @Integrator red-handed.

I've been trying to earn Naruto all day to no avail.

:O

5:25 AM

Red Baron and Breaking Bad and 30 Minutes or Less also seem out of reach , at present.

UnadulteratedImagination

what I would like is while reading I can make note of all the techniques used to solve questions ... Dummit and Foote does that in the questions. And sometimes it's not very clear.

I want to have a repository of results which can refer to while solving problems

Anonymous

@MikeMiller It looks like you got both android and iOS phones!

@Ashwin I do not own both. I borrowed one.

@MikeMiller Breaking Bad is easy. Look through your list of votes to close. Click through to the questions. Find one where the OP posted some input in comments. Move it to the question. Vote to reopen.

Anonymous

@KajHansen Reading the first four chapters of Visual Group Theory will take you no more that 5 hours.

5:27 AM

@UnadulteratedImagination Unfortunately, I've got no advice.

That's actually pretty cool @Ashwin. You're expanding the book list for when I visit the library tomorrow.

Swapnil Tripathi

Anyone interested in Sturm Liouville? Differential eqn?

@MikeMiller Naruto should not be so hard for me, given I have 100+ 0-score accepted answers. But no luck so far. I considered revisiting my 0-score un-accepted answers to badger the OP, but perhaps it would not count since the answer is old...

Anonymous

@KajHansen What was the first boook you read on Group Theory?

@Behaviour Most things involving answers require them to be new; selfie, for instance, requires a question posted after the bash started.

5:29 AM

@Ashwin, Ted's book that I mentioned above. It had a very geometric flavor that's stuck with me.

Swapnil Tripathi

@Behaviour I got one :D

I felt quite confident that one would give me Naruto for a while, but it ended up grabbing an upvote maybe two hours into its lifespan.

@Behaviour :p

If anybody wants Business in the front, party in the back, just tell me.

hahaha, "please give me downvotes, I want a hat!"

5:33 AM

@Behaviour This question is the best I could find that fits that description, but I'm not quite sure the OP's comment is really worth reopening over.

@HatMan very Cool ;)

@HatMan I got one ! no need for another :P

@MikeMiller That one is still unclear, because we don't know what a proof would mean there... You should have closed more. :p

@Behaviour You say that, but I ran out of votes in UTC 12/15. It's just that I seem to have only close-voted the most egregious garbage, instead of the border cases.

@HatMan Wait I'll give you a reasonable answer to down-vote.....

I just earned 34 chameleon hats. Can somebody tell me the secret behind the hat?

@Integrator Okay!

5:37 AM

@HatMan Think about what you did, and you'll know why you now have 34 of them.

@HatMan math.stackexchange.com/a/1070184/190645

@HatMan Done??

@Integrator the heck is that ? :O

Anonymous

@KajHansen Shifrin's is a great book,but it's an introductory one,right?

@Integrator yes.

@r9m See it's revision history!

5:39 AM

Did you get a hat?

@Ashwin, yeah

@Integrator HAHAHA !! :P u crazy :P

@r9m :)

Anonymous

@KajHansen i actually got Strang's Lienar Algebra and Its Applications

Anonymous

I will add this one too!

5:41 AM

@Integrator what happens when someone gets all the hats ?

@MikeMiller No idea. I swear, I didn't do anything.

Anonymous

@r9m That makes him such an idiot :D

@r9m He'll get suspended! :D

@r9m winterbash2014.stackexchange.com/leaderboard

@HatMan A chameleon can change what it looks like...

I got it now!

Chameleon = Change your avatar!

5:43 AM

@HatMan cool ! :D

slides away to change his avatar and see if anything happens !

@MikeMiller You are one short of begin the leader. Want some help?

@HatMan Two hats short. And there's no hats I don't have you can help on.

5:59 AM

@HatMan works !

@r9m By the way, how did you get Fascinating and Waffles?

@HatMan fascinating I'm not sure ! .. waffles I guess it must have been me retracting my only downvote from a 6 month old question (edited after 6 months .. got me the excavator badge and that hat)

@r9m Waffles might be for getting a bronze badge.

@HatMan ah ! maybe ! :)

I started a bounty on my own question to get Benefactor badge.

6:08 AM

I see !!

But I have to wait 1 day before I can award it.

This hits close to home: http://spikedmath.com/557.html
Except it's more like "Damn, that was trivial. I'm dumb." for me :)

That's what I have! I usually break mine in half and take a half at a time.

Anonymous

@KajHansen Were you inspired by Erdos?

Absolutely @Ashwin. It's hard to not be.

Anonymous

@KajHansen I hope you don't take all the stuff he took.

6:23 AM

LOL, don't worry. I don't take amphetamines.

Anonymous

@KajHansen And one more thing.Your photo suggests sleep deprivation.

Unlike seemingly half the student body here rolls eyes

Anonymous

I must leave.Cya!

I took that photo shortly after moving into my current apartment, which was after a 4 hour drive and moving all my stuff, haha

See you later man. Thanks for the book suggestions; I really appreciate it

2 hours later…

7:55 AM

@MikeMiller, I just unlocked a secret hat :)

Hi @Kaj

Hey there Balarka

Just wake up?

Yeah haha

I need to be going to bed soon. Busy day tomorrow.

I am trying to recall what prof said to me a long long time ago about genus of groups.

I think I remember bits of it.

8:00 AM

What about the genus of orientable surfaces? :P

No, genus of groups/

Surfaces!

You take a group, have a nice enough generating set and draw up the Cayley graph.

I'm just giving you a hard time haha

The genus of the group is the genus of the least genus surface on which the Cayley graph embeds without intersection.

@KajHansen I know that

:P

8:03 AM

I came up with an interesting question earlier. It's not well-formulated by any means, but for general groups with certain stipulations, what is the minimal size of the generating set for groups that fall into the category with a given order?

I don't understand the question.

OK, so a generating set for $S_p$ would be a $p$-cycle together with a $2$-cycle.

Sure.

Given a prime $p$. That is the minimal size of a generating set for such groups.

OK.

So you're asking what the minimal size of the generating set is for a given group?

8:05 AM

So it would be interesting to look at a set of groups and ask for a lower bound on the size of the generating set.

Makes sense.

Sure, but you'd have to put enough hypotheses on here so that the problem isn't trivial.

I.e. "all groups" is boring since then cyclic groups are lumped in and the answer would be $1$.

Yes, sure.

Haven't thought about this at all beyond what I've already said. I'm sure you could spin this into an interesting problem.

Looks interesting enough.

I'm interested in an upper bound, rather.

Lower bound is bound to be 1 or 2 or 3. Not interesting.

8:09 AM

I guess we need to rigorously define "generating set". Because wouldn't the whole group work as an upper bound?

@KajHansen That's not a hard job.

You are looking for the minimum $d$ such that $G$ embeds in $\Bbb Z^d$

Not saying it is. Just wondering what we mean by an upper bound.

Ok, makes sense.

I am sure there have been a lot of work on this.

I have a tendency to come up with previously studied, yet unsolved conjectures.

lol

8:13 AM

For example, one thing I was briefly interested in a few weeks back was to characterize the function $\displaystyle g(n) = \operatorname{min}_{\pi \in S_n}(|\pi|)$

I googled -- it's called rank of a group.

That's pretty cool. Link to Wiki/Wolfram/Article?

en.wikipedia.org/wiki/Rank_of_a_group.

Awesome. I have another thing to read over the break :)

Anonymous

I got a big cap :)

8:16 AM

"Rank problem is undecidable for word hyperbolic groups" interesting.

@Ashwin A dunce cap?

@Ashwin, I unlocked a secret hat without even trying :D

why don't you wear it then

I prefer just seeing my regular avatar, lol

Anonymous

@BalarkaSen He is sleeping now,so he will wear it just after he wakes up

Thinking about changing my avatar to a more recent picture of me.

8:17 AM

I have no face. I am a bot.

Anonymous

@KajHansen A picture of you infront of a blackboard doing Ramsey thoery would be good

Unlike Dr. Sonnhard Graubner, @BalarkaSen can pass a Turing test!

LEL

I duneven know what Ramsey theory is. What's it about?

I have some, like here: https://fbcdn-sphotos-h-a.akamaihd.net/hphotos-ak-xpa1/t31.0-8/1622534_10204325257998722_1109839245956157012_o.jpg

But they don't transfer into good avatars with the restrictions on size.

Anonymous

@KajHansen I saw your G+ profile

8:20 AM

Oh you have a G+, @Kaj?

@BalarkaSen, in short, Ramsey theory takes combinatorial objects and searches for low-level order within them as the complexity of the objects grows.

Anonymous

@BalarkaSen Kaj has a bunch of videos on Ramsey Theory.You mst watch them

LOL, I do but I rarely use it outside of my gmail.

Me too but let's add you on circles.

:P

Anonymous

@BalarkaSen Did you mean dunce cap or dunce hat?

8:21 AM

I have a decent (if I do say so) 1-hour intro to Ramsey theory that I presented to the math club here. I'll get around to uploading it soon.

@KajHansen lol that leaves me completely unsatisfied.

Anonymous

@KajHansen Were you inspired by Alan Turing?

@Ashwin, I wouldn't say so. I just don't know much about him.

Anonymous

Anyone wanted to watch Walter Lewin's videos?

Anonymous

I dont fuckin' understand why MIT removed all his videos :(

8:24 AM

You know the story though, right @Ashwin?

OK, this, right?

haha, yep that's me

Anonymous

@KajHansen He was my friend actually.

Anonymous

@KajHansen MIT is blocking all the mails now.

I actually added Sanath to my circles a while back.

8:25 AM

LOL I see that

what is the story?

slips away from whatever discussions going

I don't even know the story. But it sounded a bit like guilty-until-proven-otherwise from what I saw. Could be wrong?

OK, added, @Kaj

It still depends on what "informal" means.

Are we talking Feynman-esque?

Anonymous

8:29 AM

LOL Feynman's case was worse

AH! It's the 84(g-1) theorem I was looking for!

Aw yeaah.

So this was the thing @Kaj.

Take PSL(2, 7) say

Draw the cayley graph with an appropriate generating set

:19094785 I've removed this because it had a flag. In general, it may be better not to talk about third parties in less-than-polite ways, especially when it can be misinterpreted due to lack of context.

Anonymous

@jmac I understand.

"Draw the cayley graph with an appropriate generating set"

How does this differ from "Draw the Cayley graph" @BalarkaSen?

Enjoy the rest of your day.

8:33 AM

@KajHansen Well two graphs with different generators may vastly differ.

they might not even be isometric

Take Z with <1, -1> and Z with <1, -1, 2, -2>

Draw it.

Forgive my ignorance. What does it mean for two (graphs?) to be isometric?

@KajHansen There is a map from one graph to another which preserves the length of the edges.

You'll see a similar definition in topological spaces later.

Oh man, what do you mean "length of the edges"?

Just assign the edges a length

1, say.

Ok, sure

8:35 AM

Then you have a canonical way to measure "distance" between two nodes.

(When you study topology, you'll see we have made this graph into a metric space)

Oh, so by isometric you really mean to say any two nodes have the same distance after the mapping is applied?

@KajHansen yes.

a distance preserving map

a rigid motion is such a map, that's how we define isomorphism of two triangles/polygons in euclidean geometry.

anyhow, drawing Z with <1, -1> and Z with <1, -1, 2, -2> is an exercise for you.

Draw PSL(2, 7) and draw up a Cayley graph

Sure. I'm rather intimately familiar with isometries in Euclidean space after all of Ted's courses. Just making sure the graph-theoretic notion is the same.

@KajHansen haha, yes.

Ok, and why will $PSL(2, 7)$ be interesting?

8:39 AM

yes, back to what i was saying

draw some cayley graph of PSL(2, 7)

you'll get a very complicated self-intersecting graph

Huh, interesting. And how will that relate to the rank of the group?

now, if you look at the smallest genus surface on which the graph embeds without intersections, you'll see that it's of genus 3.

@KajHansen nah it doesn't :P i was referring to the thing i was looking for, what prof said to me.

Ohhhhh!

this surface is also known as the Klein quartic (google it)

That's some mind-bending shit right there.

8:42 AM

there is a theorem that says there are no more than 84(g-1) automorphism of some "special kind of" surfaces of genus g

This guy has genus 3.

So number of auts 84(3-1) = 168

PSL(2, 7) has order 168!!!!

Oh wow

So I guess if you have a finite group G with genus g, then |G| \leq 84(g-1)!!!

You can think of embedding Cay(G) in the genus g surface, and then G acts transitively on the nodes of the graph as well as open nbhds around the nodes in the surface.

That's pretty awesome.

I want to better understand this, haha

So G precisely injects onto the group of symmetries of the surface!

Are you working out of a book of some sort?

8:45 AM

nope. i came up with this notion of genus and the prof said something.

i guess this was what he was blabbering about

pretty cool.

I think he was mumbling something about topological group theory.

hmm

:P

OK, this

What university do you visit @BalarkaSen?

Thank god you sent that while I was here at UGA so I can bypass the paywall

@KajHansen You won't be familiar with the name. It's quite famous as a forte (:P) of geometers in India but not that well known.

If you really want to know, I have my professor in my circles.

I was just making conversation more than anything. I'm not super-duper interested.

8:51 AM

Tucker starts with "a finite graph is a 1-dimensional CW complex" SHEESH

why not just say "a finite graph is a finite graph"?

this looks interesting. Imma read this. You should get some sleep @KajHansen.

I'll go in a bit

Yikes the groups even act on nonorientable surfaces

I guess that's possible. Try drawing the Cayley graph of the infinite dihedral group, with directions. It looks like skeleton of a moebius strip.

Let's see the Klein Bottle!

@KajHansen do you see the cayley graph of the infinite diherdal group?

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