Mathematics - 2015-02-16 (page 1 of 3)

MLM

00:56

What does a comma separated pair mean for an angle?

Mike Miller

The angle between those two vectors.

MLM

hmm, It comes from step #4 here: w3.org/TR/SVG2/implnote.html#ArcConversionEndpointToCenter

Antonio Vargas

01:20

@MLM, read that line as "theta_1 equals the angle between the vector (1,0) and this other vector."

MLM

"in between" = in the middle of those two?

Or just a range to know that it is in

Antonio Vargas

maybe saying "the angle formed by" would be more clear?

for example, the angle between the vector (1,0) and the vector (0,1) is 90 degrees

does that make sense?

MLM

Ahh, that is clear now

@AntonioVargas Thank you for the help :)

Antonio Vargas

@MLM Sure thing :)

MLM

Looking at Mike's message, he mentioned that but it didn't click for me

Arkamis

01:55

I have an idiot question

JMoravitz

No such thing as an idiotic/stupid question @arkamis

Victor

02:58

math.stackexchange.com/questions/1149955/… please take a look

Semiclassical

i'm afraid that that question is so open-ended and broadly-posed as to be unanswerable. @victor

Victor

@Semiclassical - I can't find those website, now i delete the question and try to ask it here.

Arkamis

@JMoravitz Surely this question is stupid

But I cannot remember the convention

Does $f\circ g$ refer to $f(g(\cdot))$?

Or vice versa?

Committing to a challenge

Yes that is the way

The left to right order is the same

Victor

Hi, anyone know where can i find a list of open problem in ordinary differential equation

Arkamis

03:05

gotcha. I dont know why I thought it was the other way

Committing to a challenge

Did you ever work with $xf=f(x)$, that is why I had forgotten that one above awhile ago

Arkamis

I've never seen that notation

Committing to a challenge

Really?

$xfg=g(f(x))$

Victor

@Arkamis - Are you familiar with ordinary differntial equation?

Arkamis

the $xf$ notation, yeah. Never come across that.

Committing to a challenge

03:07

Some say it is superior because it reads left to right, just like in English, so it is more natural to beginners and also doesn't require parentheses

Victor

@Semiclassical - Can you answer that on the chat right now?

Committing to a challenge

(it is also used solely in Cohn - Classic/Basic/Further algebra)

Semiclassical

@victor: if you're referring to the question you linked earlier

math.stackexchange.com/questions/1149955/…

Victor

@Semiclassical - Thanks in advance

Semiclassical

as I said before, that question is so broad as to be unanserable

Victor

03:11

@Semiclassical - If i ask are there any breakthrough recently, will that be better?

Semiclassical

it's an improvement, but it really needs to be focused more.

'breakthroughs in ordinary differential equations' seems just too big a topic

Mike Miller

there's a breakthrough in math every day; if you specify that to a given field i have to reduce my statement to 'every week'

Antonio Vargas

@Victor Sometimes mathematicians will publish survey articles for their field in which they describe open problems which are the subject of current research or have had new progress. The Bulletin of the AMS publishes these kinds of papers sometimes.

They're also published in standalone issues of certain journals as monographs - for example, take a look at the CBMS-NSF Regional Conference Series on Applied Mathematics: bookstore.siam.org/…

there are numerous issues there on topics related to differential equations

Committing to a challenge

How does Famous blue raincoat cause deletion within the day on questions he has downvoted?

Arkamis

The comments on this thread are remarkable

io9.com/…

Step 1: Brouwer fixed point theorem.
Step 2: done.

Victor

03:19

@AntonioVargas - Thank you. but are you able to find out the open problem relate to the classes of ordinary differential equation that are unsolvable?

Antonio Vargas

@Victor I'm not certain the question is well posed, but if it is then I think that kind of information could potentially be gathered with some effort.

How many weasel words can I fit into one sentence?

Semiclassical

maybe another question is in order. what motivated you to ask the question? that might help focus you to more specific problem

Committing to a challenge

@Victor Basically they are saying, you are not ready, and you shouldn't be seeking open problems in ODEs

Antonio Vargas

I wasn't saying that

Victor

@robjohn - Are there infinite number of class of ordinary differential equation that are unsolvable

Committing to a challenge

03:24

Please watch this Victor youtube.com/watch?v=FOZO0WWULLA

JMoravitz

I was going through my recent flags, and came across one of the questions I had flagged as lacking enough information. The post had been closed and was edited and reopened. I felt bad that it was never answered, so went through the ugly process of trying to answer it.

0

A: Probability Of 2 consecutive dice throw for 4 players

This can be described using markov chains using 13 states, with states labeled $A$ for if it is player $A$'s first turn or player $A$'s turn immediately after one of player $D$'s turns, state labeled $A^+$ if it is player $A$'s extra turn after having rolled a six, and $A^{win}$ if player $A$ suc...

A thirteen state markov chain. >_< bleh

Mike Miller

@Arkamis One needs to develop some basic notion of the fundamental group or winding number to actually give a proof of that. It was the motivation for the first few weeks of a topology class I took.

Arkamis

Have an upvote for your efforts, and so that question may never need to arise again ;)

Antonio Vargas

@JMoravitz That diagram...!

Arkamis

@MikeMiller Yes, but it is a nice hammer to wield

Mike Miller

03:27

No, @Arkamis, I actually meant the problem they're trying to prove :)

Committing to a challenge

@Victor 2:54 youtube.com/watch?v=FOZO0WWULLA is probably sufficient

JMoravitz

It was posted by a new user, one which hasn't posted anything since, so I don't even know if it will be seen. Still, it was an interesting question once fixed

Mike Miller

of course, their problem is easier than brouwer's

Arkamis

Yes

There are no non-niceties in their problem

I do love that theorem, though

It is one of the sine qua non theorems of numerical methods, game theory, and a few other things.

Committing to a challenge

@Victor Did you watch the video?

Victor

03:34

@Committingtoachallenge - I think i understand what you mean. However, someone told me that ordinary differential equation is a childish-play and have no open problem at all.

Committing to a challenge

@Victor Have you done phase analysis?

@Victor Calculus of variations?

@Victor Then get back to the books for another year and a half, and then you can dream about open problems, but only dream, do another year and you can start looking at journals

Hey @Kaj, how many hours of study are you averaging a day atm?

robjohn

@Victor That depends on how coarsely you make the classes and whether you mean solving in a closed form or solving numerically.

Semiclassical

for me the issue isn't so much 'do you have enough background in ODEs' as 'what particular subfield within ODEs are you interested in'

because there are a lot

Committing to a challenge

I think it is more important to realise that he hasn't been exposed to enough actual math yet, and thus he is basing his 'interest' choice on what he knows

Kaj Hansen

Probably 5 for math stuff @Committingtoachallenge. Takes about 10 hours a week for both my complex and topology sets. Then there's misc studying.

It adds up with my core classes though.

Committing to a challenge

03:41

@KajHansen 5 hours a day is a pretty damn good effort

@KajHansen (assuming we are only counting focused study)

Semiclassical

to be sure, it's hard to really appreciate how big a field ODEs are without the necessary background

Kaj Hansen

In reality it's more like 6-7 on weekdays and not as much on weekends because I sleep a lot on weekends :P

JMoravitz

I hear ya there @Kaj

Victor

@robjohn - solving In infinite series.

Committing to a challenge

@KajHansen That sounds awesome haha

Kaj Hansen

03:46

mhmm. I can't wait for spring break myself. Doing 2 problem sets a week feels like quite a bit with my core classes and whatnot on top. Plus I'm going to be working on my REU applications for the next few weeks.

Committing to a challenge

@KajHansen REU?

Research experience undergraduates okay, got you

Kaj Hansen

"Research Experience for Undergrads". It's one of the most common ways undergrads go about getting involved in research. Potentially the only way if you want to travel to another school.

robjohn

@Victor most can be solved by a series approximation.

Semiclassical

i guess if i were to pick, with substantial bias, one aspect of ODEs I find interesting and open, it'd be understanding non-perturbative aspects

Victor

@robjohn Thank you, does that apply to partial differential equation as well?

Arkamis

03:50

@Semiclassical THat's funny; I enjoy perturbation methods much more ;)

Semiclassical

heh. well, perturbation theory is something that is interesting but also fairly well-understood

on the other hand, looking for what happens when perturbation theory breaks down...

robjohn

@Victor Yes, but I am assuming you have no unknowns other than the variable of differentiation.

Committing to a challenge

You realise he is a freshman right?

Semiclassical

@Arkamis: let me find a few (quite biased) references that I have in mind

Committing to a challenge

As someone who was once where he is, I have to say, feeding the delusion of being capable of anything at that point is not good

Victor

03:54

@robjohn - Thank you again. good night

Semiclassical

@arkamis: mmkay, here's a recent arxiv preprint: arxiv.org/pdf/1501.05671v1.pdf

note that that's a math-physics paper, with a lot of field theory stuff in the background which i dont' actually know

Arkamis

I saw "gauge theory" and immediately my soul wept tears of sorrow.

Semiclassical

lol

i don't touch the field theory stuff

Arkamis

Math-physics to me is a field of skepticism

Semiclassical

but, check out the first few sections wherein they're talking about a certain ODE

Arkamis

04:00

I almost half wonder if they're just begging the question in a very complicated way.

Semiclassical

wouldn't surprise me. but when they get neat calculations i take more notice

Arkamis

I simply don't believe that simultaneously a.) our knowledge can be that complete and b.) the universe is that complex

But that is far from an rigorous breakdown

Semiclassical

heh.

i like such things as mathematics, with examples motivated by physics, rather than the othe way around

Arkamis

My cat is such an asshole

In the last 45 seconds, he has knocked two beer bottles and two books off my desk

For no reason.

Semiclassical

oof

Antonio Vargas

04:06

@Committingtoachallenge I remember being in undergrad and wondering what kinds of open problems were being studied and how to find them. I didn't need to be told to study more - opening a journal and realizing that I couldn't read a thing in it was enough.

Committing to a challenge

@AntonioVargas Haha, too true

Semiclassical

if i were to summarize why i find that particular stuff interesting, it's because it's pretty cool to me how much interesting stuff there is to be found in a differential equation that might seem simple as $-y''(x)+a \cos(x) y(x)==\lambda y(x)$

or, for an even simpler to write case, take $-y''(x)+$(x^4-a^2x^2)y(x)=\lambda y(x)$

it's easy enough to say generic stuff about that, and the perturbative aspects are well-known. but there's still stuff to be said about its exact features

see for example math.univ-angers.fr/~delab/AnnPhys97.pdf (full disclosure, i can't actually read enough of that paper to understand it)

i guess i think that, while one doesn't want to encourage proto-crank behavior, insisting that they don't know enough to even ask the question seems counter-productive (even if the question doesn't actually make sense) @Committingtoachallenge

(done rambling about random math-physics stuff i know about, btw)

JMoravitz

Can you just make life easier please and tell me the answer — Jeremiah Bradley 6 mins ago

sigh

Arkamis

@JMoravitz I say that to the universe every day,

lol

Committing to a challenge

04:29

You can't control how dumb you are, but you can control whether you give up. Why are you choosing to be dumber than you have to be? – MJD

Arkamis

04:41

@JMoravitz math.stackexchange.com/questions/1150069/…

JMoravitz

probably one of the fastest closed questions on the site.

Semiclassical

that's just depressingly bad

JMoravitz

Well, it has garnered moderator attention now it seems, comments are being purged.

Good thing too imo.

Committing to a challenge

'plugging in random numbers for hours but no luck.' I don't think this is true, I hope this isn't true

Semiclassical

it's always a bit frustrating when someone gives a great partial answer, says they'll eventually get around to finishing it all the way, and then never does :(

all the more so now that said question has somehow shown up in my actual research, hah

Arkamis

04:50

@Semiclassical But that's my jam.

Modded Bear

@ThomasAndrews I was snarky? I did everything you asked and I even apologized.

Committing to a challenge

05:16

@ModdedBear If you didn't delete the comments, we would know the context

Modded Bear

I wish I could un-delete the comments

I don't know if I can.

Committing to a challenge

@ModdedBear You can't, but I am 35% sure that moderators can see them

Modded Bear

35%?

Committing to a challenge

I have heard something, but I have heard more contradicting stories

Modded Bear

I am 100% sure they can

Committing to a challenge

05:19

I am 100% that the big admins can

Modded Bear

I am 100% the site admins can

9

A: Are deleted comments destroyed?

There is a dialog that moderators can open to reveal all of the comments on a thread, whether they are deleted or not, along with other information about the comments (who deleted the comment if it is deleted, who a comment is a reply to if it used an @reply, etc.) There is (currently) no way to...

Apparently they can't be brought back

It was actually a pretty good Idea to wait for me to delete my comment and then say it was snarky

Arkamis

I am 400% certain they can, but I am doing probability on algebras and not $\sigma$-algebras.

Committing to a challenge

@Arkamis Good to know on both accounts

Arkamis

I really enjoy solving MATLAB questions, but I question the pedigree of like 95% of them.

infinitesimal

Don't try and fight the powers that be...know matter how sure you are right.

Arkamis

05:32

Skully

My pats

infinitesimal

That was quite a game.

Arkamis

I can't even watch the replays of that game

I still think that Seattle is going to score.

Julian Rachman

@Arkamis what sport?

Thomas Andrews

You really want to do this here, @ModdedBear? You wrote, "I'm sorry you wasted your time." It wasn't just me that I was talking about. Possibly several people wasted their time reading your wrong answer that you knew was wrong. And after you wrote that comment, it was many minutes before you deleted your wrong answer, and only after you had posted a second wrong answer. So, as I said, you weren't really sorry enough, because more people came and read your wrong answer, wasting more people's

infinitesimal

American football

Thomas Andrews

05:36

time

And then, here, you imply I'm being deliberately dishonest. keep it up. You are a class act. @ModdedBear

Julian Rachman

@infinitesimal Ok nice. I am a hockey kinda guy

Modded Bear

I wrote "I'm sorry I wasted your time"

infinitesimal

You can't do more than that^

Arkamis

@JulianRachman Me too. Who's you're team?

Thomas Andrews

You sure? I remember "you." But in any event, your actions showed you weren't sorry, nor were you concerned about not wasting anybody else's time. @ModdedBear

Julian Rachman

05:41

@Arkamis Im a local Kings fan. But also the Flyers I like. You?

Arkamis

Bruins for life.

Julian Rachman

haha. Are you in bruins territory?

Lucic is a goon

Thomas Andrews

And it doesn't forgive you coming here and suggesting I was being deliberately dishonest. @ModdedBear But keep feeling victimized.

Arkamis

I'm from New England

And Lucic is awesome

Bergeron! Bergeron!

But I live in Virginia now

Julian Rachman

Ya ya. One hit wonder :) . The kings are on a role

oh Nice!

See Bergeron has hands that's why

Arkamis

05:45

Bergeron has more than hands.

Bergeron is perfect.

Julian Rachman

he's dirty

Arkamis

If Bergeron wanted to study math, he would prove the Riemann Hypothesis.

For fun.

Bergeron could teach the Stig how to race.

Julian Rachman

Lol Fosho

Arkamis

Chuck Norris thought about playing hockey and being an enforcer, but he decided he never wanted to face Bergeron.

Julian Rachman

I didnt know he was good at poetry?: youtube.com/watch?v=TLwG59D_38E

Modded Bear

05:47

I was trying to fix the solution when you first commented. And I did eventually delete the answer. I also said I was sorry. Then I asked you to forget it happen and deleted my comments. And after that you said I was "snarky" (which is a word I had to look up), and you didn't answer me on here. I don't have any problems with you. But I feel like you are extremely harsh with all of my activity ever since the pythagorean triple incident.

Julian Rachman

@Arkamis have any individual players you like?

Other than bergy? ;)

Arkamis

As in, non-bruins?

Julian Rachman

ya

Arkamis

Hm.

Johnny Boychuk

But he's a former bruin, maybe that doesn't count.

Basically I hate everyone that's not on my team... because I'm a Boston fan and that's how we roll... so no.

There are some guys I respect.

Hard to hate Toews. Or Stamkos

Class acts, they are.

Julian Rachman

haha ya. I am a Kaner fan. Those hands cant be washed

Arkamis

05:51

I was watching the Go Pro video that they released around the all-star break, let me try to find it

Julian Rachman

tries to put hands through car wash. still filty

Arkamis

Oshie does something that blows my mind and I need to learn how to do it.

He fakes a slapshot... but he actually takes it and catches it.

Like

He catches his own damn slapshot.

Julian Rachman

LOL. ya

he won a shoutout for the US in the Olympics

Arkamis

Here, at 3 mins: youtube.com/watch?v=F1jpCBPiWS4

Like, I need to learn how to do that.

Thomas Andrews

@ModdedBear I don't recall the Pythagorean triple thing. I recognized your name, but I have a hard time remembering names.

Julian Rachman

05:55

@Arkamis Haha ya. Do you play some rec hockey?

Arkamis

I've played since I was about 6

I play in a local beer league now; gotta try to stay active while I grow old.

Modded Bear

@ThomasAndrews OK, then, I will delete answers whenever that happens from now on. I hope you have a nice day.

infinitesimal

I played hockey too when I was six :-)

Arkamis

All I have to say is "Pythagorean Triple Incident" would be a great name for a band.

Alright, my wife has officially sent the dog in to get me to go to sleep

so I shall go to sleep.

Julian Rachman

@infinitesimal @Arkamis Nice! I am currently playing travel hockey for the 2nd best team in the nation now

infinitesimal

05:58

Later pal

Julian Rachman

@Arkamis The band will have @Ted as the lead singer

Bye @Arkamis

Thomas Andrews

06:12

@ModdedBear I probably would have forgotten your name after today if you hadn't accused me of being a calculating liar.

That's pretty hard to forget. @ModdedBear

infinitesimal

Didn't a physicist once say "shut up and calculate?"

:D

Thomas Andrews

You mean, I shouldn't interrupt your hockey talk with defending myself from a slur? @Internetsheriffabc123 :)

Committing to a challenge

@ThomasAndrews Why did you send smiles to the sheriff?

Thomas Andrews

Have no idea what you mean @Committingtoachallenge I was just tweaking him, but added the smiley to indicate I didn't take offense.

Committing to a challenge

@ThomasAndrews Tweaking is a different way of saying 'bumping'? Just getting him into the chat?

Or you are implying Infinitesimal = internetsheriff? Or it was a mistake tagging wrong person

Thomas Andrews

06:26

@Committingtoachallenge Ah, yes, I tagged the wrong person. Didn't see that. Sorry, @Internetsheriffabc123

Committing to a challenge

Ahhh Okay, that makes sense haha, hence my confusion

Thomas Andrews

Tweaking is a variant of "teasing." It has a physical origin - a tweak is a pinch with a twist. Usually meant as harmless or friendly.

infinitesimal

The infinitesimal can not be set equal to anything :)

And it was suppose to be Super Bowl talk.

user1667423

@anon That is true, and it squares with intuition, but is the boundary of R3 not empty? I'm missing something here.

anon

07:10

@user1667423 why do you think you're missing something? the hypothesis of the theorem is that either the domain is compact or the differential form compactly supported anyway, no?

Fargle

07:30

Okay, so let's say I'm considering the quotient space $F[x]/f(x)$ where $f(x)$ is of degree $n$ as a vector space over $F$. If I want to show that the collection
$$\{1 + f(x), x + f(x), \dots, x^{n-1} + f(x)\}$$
is a basis, is it alright to simply assume f is monic, show that $x^n$ reduces to a polynomial of degree $n-1$ (because $x^n = x^n - f(x)$), and then

show that $x^{n+k} = x^k(x^n - f(x))$ therefore reduces to a polynomial of degree $n + k - 1$, and thus eventually to one of $n - 1$ inductively?

Karl Kronenfeld

08:12

The notation $x^k+f(x)$ is confusing, since $x$ has two different meanings in that expression, @Fargle. Better to define $I=\langle f(x)\rangle$, and write $x^k+I$. It is also conventional to say $\bar x^k$ instead of that, when the ideal with respect to which you're taking the quotient is understood/obvious.

Now, the inductive argument is fine, but I'd do strong induction without a basis step instead.

Fargle

@KarlKronenfeld Yeah, sorry. I hastily transcribed from my notes, which do have it written as an ideal.

@KarlKronenfeld I just wanted to make sure the argument held water. I was PRETTY sure, but not entirely.

Chris's sis

Greetings

@robjohn Have you seen this one? $$\int_0^{\infty} \displaystyle \sum_{n=1}^{\infty}\left(2\sin\left(\frac{x}{3^n}\right)-\sin\left(\frac{x}{2^n}‌\right)\right)\frac{1}{x^2} \ dx$$ I created it yesterday. Maybe I should add it to my book.

Chris's sis

08:35

OK, yeah, I'm going to add it there.

Fargle

08:47

If $\alpha : G \rightarrow H$ and $\gamma : G \rightarrow K$ are homomorphisms, and $\beta : H \rightarrow K$ is such that $\gamma = \beta \circ \alpha$, is $\beta$ necessarily a homomorphism?

Karl Kronenfeld

Nope!

Fargle

I see the problem.

Only if $\beta$ is a monomorphism, right?

Er, rather, injective.

(If it was a monomorphism it'd already be a homomorphism. -_-)

Actually, no, I misstated my original question entirely. Let me retry.

If $\beta : H \rightarrow K$ and $\gamma : G \rightarrow K$ are homomorphisms, and $\alpha : G \rightarrow H$ is such that $\gamma = \beta \circ \alpha$, is $\alpha$ necessarily a homomorphism?

And it seems to be the case only when $\beta$ is injective.

Committing to a challenge

So $\gamma(x)=\beta(\alpha(x))$?

Fargle

Yes.

Karl Kronenfeld

@Fargle yep

Committing to a challenge

09:01

$\gamma(x\cdot y)= \gamma(x)\cdot\gamma(y)\implies \beta(\alpha(x\cdot y))=\beta(\alpha(x))\cdot\beta(\alpha(y))$ so only $\beta$ seems to necessarily be homomorphic

Karl Kronenfeld

well, in the reformulation $\beta$ was assumed to be a homomorphism. In the previous formulation, you can let $\beta$ be any non-homomorphic function mapping identity to identity and $\alpha$, $\gamma$ trivial maps. @Committingtoachallenge

Your argument is flawed only because $\alpha$ is not necessarily surjective @Committingtoachallenge

Committing to a challenge

I overstepped my boundaries, I am just going to go do some analysis haha

Karl Kronenfeld

There's nothing wrong with being wrong. Or, wait...

Committing to a challenge

Where can I learn beginning measure theory?

Karl Kronenfeld

No idea

Committing to a challenge

09:06

@KarlKronenfeld Haha I didn't realise my argument lay on surjectivity

An introduction to measure theory - Tao T, hmmm. A little too intense for me atm

Mike Miller

stein, shakarchi, real analysis is nice

Fargle

I was trying to prove that $\theta : Hom(V,V) \rightarrow Hom(V^*,V^*)$ defined by $\theta(T)(f) = f \circ T$ is an isomorphism. I decided the easier way was just to show trivial kernel.

In this case, $G = H = V$, $K = F$ (the field which $V$ is over), $\alpha = T$, $\beta = f$, and $\gamma$ is the image of $f$ under $\theta(T)$.

robjohn

@Chris'ssis The method I came up with for your integral yesterday, I was able to apply to this answer, and can be applied to that integral :-)

That is, $$\int_0^\infty\frac{\lambda\sin(x)-\sin(\lambda x)}{x^2}\,\mathrm{d}x=\lambda\log(\lambda)$$

Chris's sis

09:22

@robjohn It seems so.

Committing to a challenge

@MikeMiller Thanks, looks good

A Beautiful Mind

@Committingtoachallenge Hi, I just woke up, trying to get a fresh perspective on my problems.

Committing to a challenge

@ABeautifulMind That is great to hear

@ABeautifulMind Buy all the gym equipment, and turn your problems into bulging muscles?

A Beautiful Mind

@Committingtoachallenge No money, but can go to the gym.

Committing to a challenge

Could be fun

A Beautiful Mind

09:27

I hope I get my miracle in the email I told you about.

Committing to a challenge

@ABeautifulMind I hope so also

A Beautiful Mind

@Committingtoachallenge My mum will be off from work the next 3 days, so I can talk to her more.

robjohn

@Chris'ssis Actually, I can simplify the proof...

Chris's sis

@robjohn The proof is OK I think.

Committing to a challenge

@ABeautifulMind What motivated you to answer questions here in the past?

A Beautiful Mind

09:32

@Committingtoachallenge I was not feeling so bad. Somehow my condition worsened the last couple of years.

Committing to a challenge

@ABeautifulMind Can you try to force yourself to do one really easy one?

A Beautiful Mind

@Committingtoachallenge I can, but it won't help me feel better, so there is no point.

Committing to a challenge

@ABeautifulMind Can you force yourself to do pages of a textbook?

A Beautiful Mind

@Committingtoachallenge Nope, that is out of the question.

robjohn

@Chris'ssis I have updated the answer

Committing to a challenge

09:37

@ABeautifulMind What about just one page?

A Beautiful Mind

@Committingtoachallenge Nope. There are several reasons, all related to my problems.

Chris's sis

@robjohn I'll look at it a bit later. I need to finish something now.

robjohn

09:49

@Chris'ssis You know, I didn't notice that that was your question until just now... doh!

Chris's sis

@robjohn Yeah, it's a cute question that is worth to be posted on main.

robjohn

10:06

@Chris'ssis Yes, it is.

Chris's sis

@robjohn I keep all very hard questions for me. :-)

A Beautiful Mind

@Committingtoachallenge I just showered, back now. =)

Committing to a challenge

@ABeautifulMind I just worked out, but not back yet haha

Chris's sis

At a certain point I'm going to continue my work on $$\sum_{n\ge 1} \psi^{(1)}(n+1) \psi^{(2)}(n+1) \psi^{(3)}(n+1)$$

@robjohn I'm not far from the answer.

Sayan Chattopadhyay

10:24

Hi

Fargle

Huh, that's curious.

$det((1 + \delta_{ij})_n) = n + 1$

Committing to a challenge

@Chris'ssis What is $\psi$ here?

Chris's sis

polygamma function ...

Committing to a challenge

@Chris'ssis No need for the dots, they could very well be costate equations...

Chris's sis

@Committingtoachallenge I don't think so.

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10:30

@Chris'ssis For constructing the hamiltonian...

Chris's sis

@Committingtoachallenge No, they can't be.

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@Chris'ssis and why not?

Chris's sis

@Committingtoachallenge Well, you can use them the way you want to. Note that $\psi$ also has the order specified.

Committing to a challenge

@Chris'ssis Why did you dot me for my question?

@Chris'ssis I have only seen $\psi^{(1)},\psi^{(2)}$ so on in the context of costate or pseudo costate equations

Chris's sis

@Committingtoachallenge Because I can dot.

robjohn

10:32

@Fargle not too hard to compute.

Fargle

Yeah, I found a recursive relation that helps.

Committing to a challenge

@Chris'ssis You don't have to dot me

Chris's sis

@Committingtoachallenge Say "thanks" that I answered back your questions after the stalker background I found you have.

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@Chris'ssis Thanks for answering my question

@Chris'ssis I deny your further claim though

Fargle

$det((1 + c\delta_{ij})_n) = (c+1)(\frac{c}{c+1})^{n-1} det((1 + (c+1)\delta_{ij})_{n-1})$

robjohn

10:35

@Fargle subtract the first row from the rest then each other row from the first and you are left with a lower triangular matrix with $n+1$ in the upper left corner and $1$s on the rest of the diagonal

Chris's sis

@Committingtoachallenge You can do anything you want to. I have the lunch now with my dogs. :-)

Committing to a challenge

@Chris'ssis Okay have fun, I'll watch

Fargle

@robjohn Cool! While I have you, can I ask for a slight nudge in the right direction for proving that Vandermonde's matrix has determinant $\prod_{i<j} (x_j-x_i)$?

I can kind of see it intuitively, but proving it is a whole other beast.

A Beautiful Mind

@Committingtoachallenge @Chris'ssis youtube.com/watch?v=bWtUM1J9eSc

3

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Andrea Bocelli is an amazing singer

A Beautiful Mind

10:51

@Chris'ssis I am also a stalker.

Committing to a challenge

@ABeautifulMind Why are you also a stalker?

A Beautiful Mind

@Committingtoachallenge I don't know.

robjohn

@Fargle It is easy to see that the determinant is a polynomial of degree $n(n-1)/2$

@Fargle if any two variables are equal, the determinant is $0$

@Fargle Thus, $x_i-x_j$ must be a factor for all $i\ne j$

The only thing to compute is the sign of the product...

Alizter

11:08

@KajHansen I have an interesting problem that you might be interested in.

Fargle

I see.

Chris's sis

@ABeautifulMind Nice song.

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11:32

What's an example of a non compact set?

Fargle

@Committingtoachallenge Topologically speaking?

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@Fargle Metric spaces I am on

Fargle

$(0,1] \subset \Bbb R$ is non-compact.

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What is $X$ here? $\Bbb R$?

Fargle

In fact, any non-closed or unbounded set in the reals is non-compact by Heine-Borel.

$X$ in this case is $(0,1]$ seen as a metric subspace of $\Bbb R$.

Another example of a non-compact space is just $\Bbb R$ itself.

(Heine-Borel states that a subset of the reals is compact if and only if it is closed and bounded.)

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11:35

"By an open cover of a set $E$ in a metric space $X$ we mean a collection $\{G_\alpha\}$ of open subsets of $X$ such that $E\subset \cup_\alpha G_\alpha$"

Fargle

I see.

By this formulation, $X$ is $\Bbb R$.

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Why are there no $G_\alpha$ that cover $(0,1]$

Fargle

Non-compact does not mean no open covers exist.

It means there is some open cover which does not have a finite open subcover.

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I don't understand that last bit sorry. What open cover here doesn't have a finite open subcover?

Fargle

In this case, $\{(\frac{1}{2}, 2), (\frac{1}{3},1), (\frac{1}{4},\frac{1}{2}), \dots, (\frac{1}{n+2}, \frac{1}{n}), \dots\}$ does not have a finite subcover.

Excluding $(\frac{1}{2},2)$ would exclude $1$, and excluding any $(\frac{1}{n+2},\frac{1}{n})$ would exclude $\frac{1}{n+1}$, so in fact, NO proper subcollection of this cover is a subcover.

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11:42

Brilliant, that makes perfect sense, thank you so much

Fargle

No problem. Compactness was one of my favorite properties to try to grasp in topology.

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@Fargle Are metric spaces topological in some sense?

Oh I see, most general topological spaces are metric spaces, I wish my book would inform me of this haha

Chris's sis

12:04

@robjohn in my book I'm going to add this version I think

Fargle

@Committingtoachallenge Metric spaces are a specific class of topological space, where the topology is the collection of all possible unions of open balls.

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@Fargle All possible unions of open balls of all possible sizes?

Fargle

@Committingtoachallenge Yes, and with all possible centers.

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@Fargle Very cool, thanks again

Fargle

@Committingtoachallenge No problem. Topology forms a very neat generalization of the concepts of openness, connectedness, and compactness, where none of these rely on distance.

Committing to a challenge

12:16

@Fargle I look forward to seeing it :)

Fargle

@Committingtoachallenge Haha, I'm excited FOR you.

Chris's sis

@robjohn did you continue to work on this one? $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{\sin(x)\sin(y) \sin(x+y)}{x y (x+y)} \ dx \ dy$$

@robjohn I was also thinking of the variant in 3 variables.

The journey to the answer is definitely amazing. I don't even mention the more advanced variants.

Chris's sis

12:53

1 hour of tutoring - away

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@Chris'ssis You are leaving to tutor for an hour?

Chris's sis

@Committingtoachallenge Yeah. Soon I cannot answer back.

Transcript for

$Mathematics$ Mathematics