Mathematics - 2015-05-11 (page 4 of 4)

Math Man

21:00

Nope I followed your link and clicked download and nothing happened.

avid19

You have to enter in a code

Math Man

Yeah I entered the anti robot's code. Then it just refreshed the page and went back to where I started.

avid19

Weird. I just downloaded it from there

maybe try again? Maybe your computer is just weird

Mike Miller

@Balarka: How have you done the third chapter if you don't know cohomology...?

@RobertCardona Derivators? I think that's wildly hard stuff. We have a seminar here on them that's been going pretty much all year now.

Enjoy!

Math Man

@avid19 It started now. I guess I just had to enable cookies for that site.

avid19

21:04

Cool, I wish you well on learning topology

Math Man

Thanks! I'll enjoy it.

avid19

To be honest if you have a decent background in analysis point set topology is super straightforward IMO. It can be hard if you're not used to it.

used to analysis*

Math Man

Not much /: Done a little.

avid19

The main idea of topology is generalizing continuity/limits further. In analysis you work with metric spaces. You talk about continuity/limits in terms of distance. $s_n\to L$ is $s_n$ can be made "close" to $L$. It's easy to go from "close" to "open balls". In topology, you work directly with open sets.

If you get that main idea, the rest of point set topology is filling out all the technical details

Math Man

I don't know what metric spaces are yet so I guess I haven't done hardly any analysis. But I am good with continuity/limits from calc 3. (for some reason the mathJax doesn't auto resolve for me here in the chat room like it does on the main site)

Javascript is enabled

avid19

21:17

You know what metric spaces are, you just haven't called them that. ;)

A metric space is some set, $S$, with some "metric", $d$, on it. A metric is a type of distance satisfying a couple axioms.

So for example, $\mathbb{R}$ with the metric $d(x,y)=|x-y|$ is a metric space

Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined. Those distances, taken together, are called a metric on the set. The most familiar metric space is 3-dimensional Euclidean space. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them. Other metric spaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere...

Math Man

Wow, that's a really powerful Idea!!!

avid19

Before you study topology, study this stuff more seriously. By the time you're ready for topology, it will be in context and not really hard.

Math Man

Sounds good!

Ted Shifrin

Goodnight @MikeM

Mike Miller

Morning @Ted.

Math Man

21:26

Good afternoon @mikeM and @ted.

Ted Shifrin

@MathMan, are you someone we used to know by another moniker?

Math Man

No I'm pretty new to the chat rooms.

Been here only about 3 days.

Ted Shifrin

ah, well, good day to you.

Karim Mansour

@TedShifrin

hi

your here ?

Robert Cardona

@MikeMiller, can you recommend any papers/texts on the subject? I've come across this collection.

Karim Mansour

21:30

do you want to laugh for lil bit

Ted Shifrin

No, @Karim, I'm not.

Seth Kitchen

*youre

Karim Mansour

:D

Ted Shifrin

applauds Seth

Karim Mansour

so here we have a department member that doesn't like one member and one of them asked me to work with him during some semester

Seth Kitchen

21:30

bows

Karim Mansour

so I m closer to other department member he told me tell the other member that you work with him and tell him that I don't want to work with the other department member

lool

@TedShifrin

I didn't expect adults to behave like this

Ted Shifrin

@Karim: Were you the one with the professor who hated the department head, or was that someone else?

BTW, I can't follow what you wrote.

Karim Mansour

no

it was someone else

but I was in this conversation too

Ted Shifrin

There are lots of petty personality conflicts in academia. Academics can be as immature as ... the random politician.

Karim Mansour

yeah

Jeff

21:36

Hi folks. Is there a consistent, algebraic way to find an answer to something like $\cos x = \cos 3x$?

Ted Shifrin

Sure, @Jeff.

First decide when $\cos x = \cos y$.

Jeff

I mean, if it was algebra, something like $x = 3x$ I would cancel the $3$s. Can't do tha there.

@TedShifrin Everywhere

Ted Shifrin

Not everywhere. Specifically, if $x$ is fixed, what $y$ will give you that?

Jeff

@TedShifrin If $x$ is fixed, then when $y$ is $\frac 13$ of $x$, and then every $2\pi$ before and after that.

Ted Shifrin

No. Forget your problem. Answer my question.

Jeff

21:39

(wait, I think it's actually every $\pi$ before and after that.

Ted Shifrin

And it's not just $y=x+2\pi k$.

user147690

So what's the plan in regards to math now @Ted?

Chris's sis

Bad. Too less discussion on chat about integrals, series and limits.

Ted Shifrin

@AlexC: Not worth your worrying about, since I'm clearly not.

Jeff

Oh, I misunderstood question. For $\cos x = \cos y$, with fixed $x$, then the two sides are equal when $y=x \pm \pi$.

user147690

21:40

@Ted Not worried, just curious

Jeff

No, that's wrong!

Ted Shifrin

No, @Jeff.

Think about the graph of $\cos$.

Chris's sis

A nice integral to take (all to be done in the spirit of the art - of course) $$\int_0^{\pi/4} \frac{\cos (2 x) }{1+\sin ^2(2 x)}\log (\cos (x)) \, dx$$ I might add it to my book (although I already have tons of questions all prepared for that).

Jeff

It's when $y=\pm x \pm 2 \pi$.

Ted Shifrin

OK, @Jeff, but, indeed, any integer multiple of $2\pi$.

Jeff

21:42

@TedShifrin Yes, forgot the $k$. When $y=\pm x \pm 2 k \pi$. I think I see where you're going with this, but please continue.

Ted Shifrin

With the $k$ you don't need the $\pm$ there.

OK, now that you've decided that, @Jeff, now go back to your original question.

Jeff

@TedShifrin OK. So, answer we'll work with is $y=\pm x + 2 k \pi$. But if $y$ is $3x$, then both sides are equal when $x = \pm 3x + 2 k \pi$. Yes?

Ted Shifrin

Well, it would be easier to actually put $y=3x$ as you yourself said.

hi @PaulP

Mike Miller

@RobertCardona: No, I can't; I'm not participating in the seminar. I think people use Groth.

Jeff

@TedShifrin Oh, I did the wrong side (this is what happens when I rush math). I should have $3x = \pm x + 2 k \pi$.

Ted Shifrin

21:45

OK, now solve for $x$.

Paul Plummer

Hello @TedShifrin

Robert Cardona

@MikeMiller, He's one of those giving a talk.

Jeff

@TedShifrin Thanks, Ted. Just the way I wanted to learn to solve it. I got that $x=2k\pi, \frac 23 k \pi$. I'm going to check it on my calculator now.

Mike Miller

I see. Well, enjoy!

Jeff

@TedShifrin Have I missed anything else important?

Ted Shifrin

21:47

I don't quite agree with what you wrote.

Redo your algebra, @Jeff.

Stan Shunpike

@TedShifrin Hey Ted! So, I gather I rather bungled part A. But I am confused because although $f$ and $g$ are from $\Bbb{R}^n\mapsto \Bbb{R}$ $\psi$ maps from $\Bbb{R}\mapsto \Bbb{R}^n$. Why is $\psi$ not an inverse for $g$?

Jeff

Sorry, the answer I gave you was for the correct problem (I also typed the original problem in wrong). The answer I gave was for $\cos x = \cos 2x$ sheepish-grin

Ted Shifrin

LOL, @Stan, because smooth maps from $\Bbb R^n$ ($n\ge 2$) to $\Bbb R$ can't be one-to-one.

growls @Jeff

$\psi$ is giving you the constrained optimum point as a function of budget, @Stan.

Jeff

@TedShifrin Uh oh. Did I anger you? Why growling at me?

Ted Shifrin

LOL, @Jeff. So what's the answer for the problem we were discussing?

Stan Shunpike

21:49

@TedShifrin ohhhhh, I got too into the math and threw out my economist thinking cap lol. That didn't occur to me about the budget constraint. facepalm

Ted Shifrin

It's ok to keep the concrete situation to provide intuition, @Stan.

Even abstract math people should do so.

BTW, @Stan, did you ever get my email about the LM problem on your econ test? I don't understand the garbage your prof was telling you.

Mike Miller

@Ted: We decided the problem is just poorly written and we have no idea what the point of the question is. We think it really just meant "A hypersurface of contact type is orientable." This is trivial.

Jeff

@TedShifrin Still working on it. I'm getting $x=k\pi$ or $x=\frac 12 k \pi$. But it's not quite checking out.

Ted Shifrin

@MikeM: "We" = you+Jacob?

That sounds correct to me, @Jeff.

Mike Miller

+ Ko's other student

Ted Shifrin

21:53

Oh, wait, no, @Jeff.

Jeff

@TedShifrin But $\cos k\pi \neq \cos 3k\pi$.

Stan Shunpike

@TedShifrin I did and I am trying to find a polite way to phrase my complaint when I discuss it with him that conceals my irritation as a mathematician. It sounds like the problem is poorly phrased and he wants us to do something cheeky. Confusion has been a reoccurring theme of my class.

Ted Shifrin

No, it's fine, @Jeff. Sorry.

Jeff

@Ted let me do this more carefully. important thing is I got the method down.

Ted Shifrin

OK, @MikeM. That makes me feel better.

You have it correct, @Jeff. Try putting in values of $k$.

Mike Miller

21:54

We all agreed with the calculation on $T^*S^1$.

Not always the best writers...

Ted Shifrin

Well, @MikeM, you have me slightly as a duck out of water, but my intuition didn't buy what you were claiming at all.

Jeff

if $k=1$, we have $\cos \pi = \cos 3 \pi$ Oh, yes, that is right. I had $cos \pi$ confused with $\cos 0 \pi$ in my head. slaps forehead

Ted Shifrin

One of my FB friends from years and years just posted he was cleaning out his office and throwing out stuff from grad school, and he posted my syllabus and problem set from a complex geometry course he took from me (he was at Harvard, but it was an MIT course). I asked if he was jumping on the roast wagon, @MikeM :P

And $\pi/2$ works fine, too, @Jeff.

It is important to know the basic values. Visualize the unit circle, @Jeff.

Stan Shunpike

@TedShifrin one to one means bijective right? Why can't maps from R^n to R be bijective?

Jeff

@Ted the right one checks out, too.

Ted Shifrin

21:56

No, @Stan. One-to-one means injective.

Jeff

@TedShifrin Yes

Ted Shifrin

Smooth maps cannot, @Stan. We don't talk about arbitrary maps here.

Cool, @Jeff. Now you have the method down.

cand

:v

Jeff

@TedShifrin I agree. I like the unit circle method too. But I visualized it wrong. OK.

Ted Shifrin

And good for you for not just punching calculator buttons. @Jeff.

Jeff

21:57

@TedShifrin Yup. Thanks much.

@TedShifrin I left my calculator in my car (but might not have used it anyway).

Ted Shifrin

Someone might break into your car for that, @Jeff.

Jeff

@Ted I mean, I prefer to do things by hand so I learn 'em, but I'm also in a little bit of a rush, so I might have used it.

@TedShifrin At 120 per calculator, that's more valuable than my car :D

Ted Shifrin

We see people in all sorts of college math courses who have been screwed by calculators ... and cannot do anything algebraic or conceptualize. So you stick to it, @Jeff.

Mike Miller

@Ted: Speaking of your stuff, was Kobayashi's book in your bonfire of the vanities?

Ted Shifrin

Which Kobayashi?

Jeff

21:59

@TedShifrin Me, too. I teach at community college, so I see it more than you. But my issue is that I never took math undergraduate courses, so I sometimes don't know how to do stuff that I understand the ideas (or something -- i'm not sure how to explain my situation).

Ted Shifrin

Ah. You teach math?

Or chemistry or something mathy?

Jeff

@TedShifrin @Ted Yes. You probably wouldn't expect that out of someone who just asked that question, huh?

Mike Miller

Nomizu, differential geometry pt1

Ted Shifrin

LOL, I'm a little surprised, @Jeff, but I'll be happy to do anything I can to make you a better teacher :P

Jeff

@TedShifrin I'm teaching Calc 2 this summer. I know how to do integrals and I know how to find the area between two curves.

Ted Shifrin

22:01

@MikeM: I think I decided to hold onto it just in case I engage in math on MSE in the future.

Don't forget concepts and reasoning, too, @Jeff :P

Jeff

@Ted One of the example problems I'm going to do next week is find the area between $y=3 \cos x$ and $y=3 \cos 2x$ from $0 \leq x \leq \pi$. I was preparing my notes now and had to find where they intersect when it occured to me that I never learned how to find where they intersect. :D

Mike Miller

Damn, @Ted.

Ted Shifrin

@MikeM: KN is not my favorite book, but once one knows most of it, one can teach out of it :P

Mike Miller

It's on my buying list, as it's the most general and broad reference I know...

Jeff

@Ted I'm good at explaining the big picture and the intuitive understanding of things. I do have to work out how to the problems in advance (as you may have just noticed), though, so I can show them correctly

Ted Shifrin

22:03

Ah, cool, @Jeff: Well, I applaud you for trying to learn and be a better teacher. I wish more people did that.

cand

its cool :v

Ted Shifrin

I am a bit more arrogant and rarely work things out ahead ... But I have a lot of experience. :)

Jeff

@Ted Me, too. Though I was pretty happy with most of my grad school teachers. Anyway, I have to go now.

Ted Shifrin

Well, had ... :P

Take care, @Jeff.

@MikeM: Yeah, I might give mine up if you're patient.

You really should learn basic moving frames stuff, with computations, before dealing with KN.

cand

what is KN?

Ted Shifrin

22:04

E.g., read my notes that you say you can't read, @MikeM.

Kobayashi-Nomizu @cand

Mike Miller

@Ted: Silly question, because I think McDuff did something like this for a minute. Sometimes one wants to calculate, say, a form pointwise. So the trick was to extend the vector fields so that locally it's just normal coordinates and the Lie bracket vanishes. Is this what you mean?

the silence is worrying.

Ted Shifrin

Anything that's tensorial you can compute, say, in normal coordinates, where the Christoffel symbols vanish.

Mike Miller

Right

Ted Shifrin

You're confusing Lie brackets with geometry here.

Same thing shows up in Kähler geometry.

Mike Miller

No, I know what you meant. I just didn't include the word connection because I was cautious I was totally wrong. :P

Ted Shifrin

22:08

I don't remember our talking about this, though. Am I getting more forgetful?

Mike Miller

Sorry, bad phrasing. I know (was pretty sure) that it included what you said.

Ted Shifrin

What I said when?

Mike Miller

There's too much I'm bad at and not enough time to remedy it. I get distressed about this now and then.

Ted Shifrin

(worrying about dementia here)

Mike Miller

@TedShifrin here

Ted Shifrin

22:10

LOL, oh, but you made it sound like we'd discussed this before ... losing it

Mike Miller

Sorry.

Chris's sis

$$\frac{\zeta(2)}{\Gamma(2)}+ \frac{\zeta(3)}{\Gamma(3)}+ \frac{ \zeta(4)}{ \Gamma(4)}+\cdots=?$$

Ted Shifrin

Yes, in the moving frames language, in Riemannian geometry we can always choose a local moving frame $e_1,\dots,e_n$ near $p$ so that $\omega_{ij}(p) = 0$. Don't even need normal coordinates for that.

Chris's sis

@robjohn did you meet the series above before?

Ted Shifrin

OK, I'm going to cook dinner now. Keep me posted, @MikeM

Mike Miller

22:12

On what?

Ted Shifrin

On what you're now working on, silly.

Mike Miller

General distress is all...

Ted Shifrin

Calm down, kiddo, seriously.

Mike Miller

:P

Ted Shifrin

You don't know how impressed I am with how much math you've absorbed in this year.

Stan Shunpike

22:15

@TedShifrin Whats on the menu for tonight?

Chris's sis

Anyway.

Ted Shifrin

Something like pork piccata, @Stan ... with artichoke and green beans.

Keep me posted on your learning/progress :)

pjs36

If we're foolhardy enough to have rings without identity and consider the ideals $(2)$ and $(3)$ "subrings" of $\Bbb Z$, then they are isomorphic, right?

I ask because this question has me so confused about what the OP is talking about.

Chris's sis

@robjohn I just discovered an incredibly mind-blowing series!!!

$$\huge \text{The best one during this year!!!}$$

pjs36

I guess they aren't isomorphic, and consequently rings without identity are ... unpleasant.

Swapnil Tripathi

22:26

I have a question on neighborhood basis (Topology).

@TedShifrin

Kaj Hansen

Hmm, I could take a stab at it?

Swapnil Tripathi

Hey @KajHansen :D

Kaj Hansen

Hey @SwapnilTripathi. Long time no talk

Mike Miller

@pjs36 They're not, of course. An isomorphism would have to be an isomorphism on the additive groups, and WLOG we may take $f(2) = 3$; now we must have $f(2+2) = 3+3$, and at the same time $f(2^2) = 3^2$. Oops!

Swapnil Tripathi

Yes, hectic semester! I just wanted to ask whether the elements of the neighborhood basis need to be open (belong to the topology).

I guess, no.

Closed balls of radius 1/n where n runs through the set of natural numbers would do the trick, right?

@KajHansen

pjs36

22:30

@MikeMiller Yeah, fair enough. I don't know why I find it so surprising. It just "felt like" two prime ideals should pretty much be "the same".

Kaj Hansen

@SwapnilTripathi, no not necessarily

Mike Miller

Well, rings without identity don't exist, so there's no need to fear.

Kaj Hansen

The only stipulation is that they must be neighborhoods

Mike Miller

There is no point in the distinction. If you have neighborhoods, just pass to the interior of each, and you have an open neighborhood basis. I never understood why people bothered with the notion of non-open neighborhoods.

Kaj Hansen

In other words, if $U$ is an element of a neighborhood basis for a point $x$, then $x \in U^\circ$.

pjs36

22:31

OK, there we go! Vindication at last

Kaj Hansen

@MikeMiller, in some instances should the distinction be important though? E.g. when talking about local compactness? For the most part I see what you're saying and I agree.

Swapnil Tripathi

Ok. The basic elements are (or I should say, used to be) open, so got a little bit curious.

Mike Miller

Sure, fair enough.

Swapnil Tripathi

Thanks.. :)

theage

I'm having an issue with a proof in Atiyah-Macdonald proposition 1.6 - the proposition says that if $m$ a maximal ideal and $1+m$ consists of units then $R$ is local. They let $x\in R-m$, then the ideal generated by $x$ and $m$ is $(1)$ so that there is $y\in R, t\in m$ so that $xy+t=1$, so $xy=1-t\in 1+m$ and $xy$ is a unit. But why does it follow that $x$ is then a unit?

Rammus

22:40

I just flagged this question math.stackexchange.com/questions/1277843/hard-logic-puzzle-help as using foul and abusive language. The question received a downvote and the OP reacted by calling the downvoter(s) ***** ******* multiple times in the comments section (I'm sure you can use your imagination). The comments have since been deleted by OP. Will mods be able to see these deleted comments?

Follow up: question has been deleted.

Mike Miller

Mods can see deleted comments, yes.

robjohn

23:25

Am I missing something here? I think my answer is good, but I am just unsure enough not to know if something is wrong.

@MikeMiller All your secrets are known to us ;-)

Américo Tavares

Is it admissable to add the (calculus) tag to this question?math.stackexchange.com/questions/150242/…

Admissible

33

A: Teenager solves Newton dynamics problem - where is the paper?

In the document Comments on some recentwork by Shouryya Ray by Prof. Dr. Ralph Chil and Prof. Dr. Jürgen Voigt (Technische Universität Dresden), dated June 4, 2012 it is written: Conducting an internship at the Chair of Fluid Mechanics at TU Dresden, Shouryya Ray encountered two ordinary d...

@robjohn What's your opinion?

Thanks in advance!

Mike Miller

I don't see an error, @robjohn.

robjohn

@AméricoTavares very few differential-equations questions are tagged calculus, so I would not do it myself.

@MikeMiller Thanks for looking.

Américo Tavares

@robjohn Thanks for your opinion!

robjohn

@AméricoTavares doesn't mean it is the last word :-) just my opinion

Paul Plummer

23:36

@robjohn I am guessing it was downvoted because it is attached to a question they don't think should be answered

Américo Tavares

@robjohn I understand that. Let's wait for other opinions.

Mike Miller

@PaulPlummer: I don't think so. The other answer received a downvote but that's because it was incorrect, I think.

If this was a philosophical downvote both should have been hit

Paul Plummer

Well the question was also downvoted

And both were downvoted

robjohn

@MikeMiller the other answer only recently received a downvote. Mine was downvoted much sooner.

cap

Is there a matrix $A$ (over the complex numbers) for which $\dim\ker A^2>2\dim \ker A$? I can't find an example, but I can't contradict it (all that comes to mind is the dimension theorem)

robjohn

23:40

@MikeMiller mine was downvoted, but recently received two upvotes (assumedly from the people who looked at it; however, I edited a bit not too long ago, so that could have some effect).

Mike Miller

One was me.

I assumed the other was Paul.

robjohn

@PaulPlummer Never mind, that was about a different question :-)

Mike Miller

@robjohn: I think I told Pedro about this and he told the mod room. Here is another example of a recent problem-statement differential geometry question with a huge number of upvotes in a short period of time.

5 upvotes in 2 minutes. I find it very difficult to believe this isn't sockpuppetry.

robjohn

@MikeMiller Lemme take a look

Paul Plummer

It is strange that the question also has 3 stars

Mike Miller

23:47

That's a good point. The next time Data Explorer reindexes, us unpowered masses can find out exactly who favorited that.

@robjohn: I can find the other questions I was referring to, if that would help.

robjohn

@MikeMiller can't tell much. We'll have to wait to see if the votes are reversed by the script tonight.

Mike Miller

OK. Thanks.

robjohn

@MikeMiller I don't think there has been enough activity for us lowly site mods to see anything yet. The CMs might, but I don't think there is much to be done yet.

@MikeMiller favoriting may only mean that someone wants to see when changes are made.

Mike Miller

Here and here are the other two. Interestingly enough the first one has the same gravatar.

robjohn

@MikeMiller they are both coming from a pretty big university, so it is not too surprising.

Mike Miller

23:53

The first one was favorited by two other generic users, but the questions they've posted are in tags not related to differential geometry.

robjohn

@MikeMiller their department may be using a single IP

Mike Miller

I guess this might just be a witch hunt; the number of upvotes just struck me as odd.

robjohn

@MikeMiller at this point, we can only wait for the script to run at 8 PM PST

Mike Miller

right

Paul Plummer

It could be a couple of class mates wanting their homework done, so they collaboratively upvote. I am not sure if that is against rules...

I am guessing it is not against rules though

Mike Miller

23:56

@PaulPlummer that was sort of my suspicion

robjohn

@PaulPlummer people are supposed to vote on the merit of a post, but in small quantities, there is not much of a way to determine a trend.

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$Mathematics$ Mathematics