Mathematics - 2017-05-07 (page 3 of 3)

Kasmir Khaan

16:00

@TedShifrin I know how much i moved it , its in 3 d but you can help me understand the 2 d version, i can calclute how much i moved in x direction and y

Daminark

How's it going @Eric?

Anonymous

"Since the matrices M and N commute with each other, any polynomial expressions involving them are amenable to the same laws as for any two real numbers." Since $M$ and $N$ commuted we didn't even need to prove that $M$ and $N^2$ commute. That's a cool trick I just remembered :) @BalarkaSen

Daminark

@blue Have you seen eigenvalues and diagonalizability yet?

Anonymous

@Daminark The former, yes

Eric

@Daminark alright, I finally have time to take a breather, i've had so much work recently. How bout you?

Ted Shifrin

16:02

@Kasmir: If you know where the machine was and where you moved it, then you adjust coordinates by adding $\Delta x$ and $\Delta y$. Draw pictures.

Anonymous

Yeah, you can prove this property with the characteristic polynomial I think

Kasmir Khaan

@TedShifrin is it linear regression ?

Daminark

OK, well, for the latter, an operator on $V$ is diagonalizable if you can find a basis of $V$ consisting of eigenvectors (eigenbasis for short).

Ted Shifrin

Regression? Huh? You're just making a translation to your coordinate system.

Balarka Sen

@blue Sure, you can prove it by proving $M^m N^n = N^n M^m$ in the same fashion I proved $MN^2 = N^2M$.

Ted Shifrin

16:04

Oh oh, it's the famous procrastinator @Astyx.

Daminark

Which turns out to be equivalent to the statement that the minimal polynomial is a product of distinct linear factors

Balarka Sen

It's nothing fancy.

Kasmir Khaan

@TedShifrin hmm how can I do that with extrem good accuracy ?

Ted Shifrin

(Comment dit-on procrastinate en français? J'ai oublié.)

Astyx

Am I famous ? Cool !

Ted Shifrin

16:05

@Kasmir: You told me you knew how much you'd moved the machine. This doesn't make sense.

Kasmir Khaan

@TedShifrin tergiverser

Astyx

Procrastiner tout simplement

Anonymous

@BalarkaSen Yep, I get it now :)

Astyx

Hi chat btw

Daminark

Now based on this, here's a fun problem: let $V$ be an n-dimensional vector space and let $T$ be an operator on $V$ with $n$ distinct eigenvalues. Show that any transformation that commutes with $T$ is a polynomial in $T$.

@blue

Ted Shifrin

16:05

Ça vient du latin en tous les deux cas, @Astyx.

Alessandro Codenotti

The mean value property of harmonic functions feels like black magic

Daminark

Hey @Astyx!

Astyx

Sûrement

Balarka Sen

@Alessandro It should be a consequence of MVP for complex functions.

Ted Shifrin

@Alessandro: It's equivalent (morally) to the Cauchy integral formula for values of a holomorphic function.

Anonymous

16:06

@Daminark Thanks for the question. I'll try later. Have an exam coming up :P.

Ted Shifrin

@Balarka: Not in $n$ dimensions :P

Balarka Sen

Harmonic functions are real part of holomorphic functions after all

@Ted Ah, he's working high dimensions.

I know nothing about that.

Adam Warlock

@Ted Thank you so much for your answer. It's brilliant.

Ted Shifrin

It's Stokes's Theorem, of course, @Balarka.

@Adam, hardly, but you're welcome. It's just what I told you to do :)

Daminark

And @Eric yeah same, I've finished midterms and my TAPS professor, has, in fact, toned down the load slightly (which was rather heavy), so I've also got breathing room

Astyx

16:07

Fun fact : I'll have the choice to vote for Cedric Villani in the next legislative of my département

Balarka Sen

@Ted That makes sense.

Ted Shifrin

But it's good to get good at that kind of algebra, @Adam, particularly if you're going to take complex variables sometime.

Daminark

@blue No problem! And yeah that's fair

Kasmir Khaan

@TedShifrin yes lets say i moved it a units in x -axis and b units in y -axis , does that just tell me the new coordintes is ( intial +a , intial +b ) ?

Ted Shifrin

No, new coordinates are $(x-a,y-b)$. Draw a picture. But in terms of the new coordinates, the original coordinates are $(x+a,y+b)$.

Kasmir Khaan

16:09

Okay thanks Ted! :D

Alessandro Codenotti

@TedShifrin Can the mean value property for harmonic functions be generalized to sufficently nice open sets rather than just open balls?

Eric

@Daminark whoops looks like a pset jsut got uploaded so breathing room over :P

Daminark

Which class?

Adam Warlock

@Ted I will try to master that art, haha

Eric

Geometry

Daminark

16:12

Fun

Ted Shifrin

No, @Alessandro, it absolutely depends on the uniform distance.

LOL @Eric

Alessandro Codenotti

I see

Ted Shifrin

Tell me if there are any good problems, @Eric.

Daminark

Lol I mean I've got two psets looming but no longer the whole, 2 midterms, a pset, a TAPS assignment, essay thing

Also I think I'm being slowly but steadily drawn in to the geotop side of things. I was wondering a few of the theorems Neves said we'd be doing, like Hopf degree theorem and Poincare-Hopf, and once I looked it up I was actually surprised

Kari

Hey @Ted :-)

Ted Shifrin

16:14

Um, hi, whoever you are.

Kari

Poincaré-Hopf has to do with Euler characteristic and indices of singular points right?

Balarka Sen

Yep

Daminark

Yup

Kari

It's been ages since I've been on here, so I'm not surprised you don't remember @Ted!

No worries :-)

Ted Shifrin

Yeah, I barely remember who I am.

Kari

16:15

Hahahahaha!

Do you at least know how fast you're going?

Or where you are?

Ted Shifrin

Heisenberg uncertainty generalizes vastly.

Kari

:-b

Eric

oop there's a pretty cool one about finding a metric on $S^{3}$ with volume bounded below by $1$ which has negative sectional curvature outside of some $\varepsilon$-ball @Ted.

Daminark

But yeah like, it's been quite some time since I've had that strong of a feeling that something was witchcraft, and that's pretty fun

Kari

@AlessandroCodenotti It's without a doubt the most black magic theorem I've seen

Ted Shifrin

16:17

@Eric: Can we start by doing that on $S^2$?

Daminark

Hey @Mike!

Mike Miller

hi

Kari

Would you mind helping me out with a small technical issue in a complex analysis proof, @TedShifrin?

Ted Shifrin

Demonark: Looks like we're turning this entirely into a geometry room :P

Daminark

How's everything going?

Eric

16:18

is it true on $S^{n}$ generally? @Ted, I have to think about it

Ted Shifrin

G'night, @MikeM. Someone just upvoted an old answer of mine on which I gave you credit for pointing something out to me!

@Eric: I'm not sure.

Mike Miller

woo

Ted Shifrin

That was back in 2014, @MikeM, when we were both younger and smarter :D

Daminark

Lol @Ted, so it seems. Though I'll make sure to keep a healthy of measure theory... for good measure :P

Ted Shifrin

What's the question, @Kari?

Balarka Sen

16:19

I have to do physics homework. Aaah

Kasmir Khaan

@TedShifrin hmm immagine we have a square with 5 points , 1 at center and 1 at each corner , we calculate the distance from 1 point to all others, and then we add point by point and do same thing calculate distance from that point to all others, we sorta create a network, we cant do this for the infinitly many points, but can we get sort of formula ( with least error) to if we stand at any point we can find the distance to any point we wish using the data collected from other points ?

Ted Shifrin

@Kasmir: I do not think about this kind of stuff. And I don't want to now. Too much going on.

Daminark

Though really I won't commit totally to anything yet until I actually see the stuff, still the stuff going on in difftop is rather fun

Kari

2 hours ago, by Kari

Mike Miller

I don't see how to do it for $S^2$

Kari

16:20

It's the choice of $\widetilde{r}$, @Ted (I can't see why it's chosen like that. I feel like it's got to do with fitting discs inside one another for convergence)

Kasmir Khaan

@TedShifrin Sorry =p and thanks for your time :)

Mike Miller

I can do it with say three isolated balls of positive curvature

Ted Shifrin

@MikeM: Probably not possible, if the $\epsilon$ ball is measured in the metric. I wasn't sure about that.

Mike Miller

Right, we had the same thought

so 3 is special

but I still don't know how to do it there :)

Ted Shifrin

Well, odd is special. I dunno about $3$.

It's a good question. :)

Balarka Sen

16:23

Are you getting 3 from Gauss Bonnet or something

Ted Shifrin

@Kari: No, no, it's just done for a triangle inequality estimate to guarantee nonzero.

@Balarka: Maybe he's trying to geometrically close up a saddle surface.

So some sort of Morse theory going on?

Kari

A triangle inequality estimate, @Ted?

(Also, am I right in saying that the supremum should have $|z-z_0| \leqslant r/2$ instead of $|z| \leqslant r/2$?)

Ted Shifrin

$1+w \ne 0$ if $|w|<1/2$. @Kari

Eric

Why is $S^{3}$ frequently kind of special

I feel like Neves uses it for a lot of examples

Ted Shifrin

I don't see what you're talking about, @Kari.

That's way too vague, @Eric.

Do you mean dimension 3? Or a round sphere in particular?

Kari

16:27

Ted Shifrin

Oh, right. Yeah, that should be $|z-z_0|$, of course.

Eric

dimension 3 I guess, it wasn't really intended to be a serious question I was more just wondering an unrefined thought out loud

Kari

Where it says the power series converges on $D(z_0, r/2)$, was there any special reason for the choice of $r/2$, @Ted?

(That's my only remaining problem now after you cleared up the 'h being non-zero' choice for $\widetilde{r}$)

Ted Shifrin

@Eric: Did you guys prove Schur's Theorem in class or in homework? (every point isotropic implies constant curvature)

No, @Kari. Just anything less than $r$.

Kari

Is there anything that'd go wrong in the case that we picked $r$, @Ted?

Ted Shifrin

16:30

Yes. You need a compact disk inside the open disk to get that $M$ bound.

Eric

In homework @Ted

Ted Shifrin

Cool, @Eric. Of course, that's a 2-line proof using moving frames :P

Eric

oh that's sweet

Balarka Sen

I better finish my physics homework so I can get to work

Ted Shifrin

I invite you to do it in two lines :)

Eric

16:32

i'll think about this after some breakfast

Ted Shifrin

Breakfast? It's lunchtime!

Eric

that's true, but I've just gotten out of bed lol

Kari

Oh yea, extreme value theorem. Is the conclusion that $h$ is holomorphic on the open disc correct, @Ted? (The lecturer proceeds to write down a closed disk $\{ |z-z_0| \leqslant \widetilde{r} \}$ which is weird)

Ted Shifrin

NO, he just says it doesn't vanish on the closed disk.

Kari

I'm confused by the conclusion

$h$ doesn't vanish on $\{ |z - z_0 | \leqslant \widetilde{r} \}$ but where is it holomorphic?

Ted Shifrin

16:41

it's holomorphic on the original disk of radius $r$. That's where the original power series converges. He just factored out $(z-z_0)^k$.

Kari

Ah of course, I'm being slow today

Thanks for the help, @Ted! You've cleared up a lot of doubts I had :-)

Ted Shifrin

Nothing very serious.

Mike Miller

@Ted I don't believe you. $\Bbb{CP}^n$ is isotropic.

Unless you meant e.g. constant Ric

Ted Shifrin

No, it's a classic theorem, and $\Bbb CP^n$ is constant holomorphic sectional curvature, not Riemannian isotropic!!

Kasmir Khaan

@TedShifrin Ted can you please just tell me if it can be done or not ? my previous Question

@TedShifrin finding distance is a local thing not global , could not come far with that Q :(

Ted Shifrin

16:55

@Kasmir: I will not think about it.

Kasmir Khaan

:((((((((((

okay:(

Ted Shifrin

hi @EricS

mrnovice

17:11

hi all

W is the vector space of all polynomials with coefficients in R. Consider the linear map D:W->W given by D(p(x))=p'(x)

Eric Stucky

ello Ted

mrnovice

could someone explain why this linear map is surjective and not injective?

Ted Shifrin

Think about examples of derivatives, @mrnovice. Basic calculus.

Eric Stucky

Can you do either statement, nov?

mrnovice

Well I'm considering the kernel and images of D

Faraad Armwood

17:15

@mrnovice $d/dx (p(x)) = d/dx(p(x)+ constant)$

mrnovice

I just dont understand why the kernel of D is the constants

Ted Shifrin

SShhh, @Faraad.

mrnovice

constant polynomials*

Faraad Armwood

@TedShifrin: Oh, i'm sorry

mrnovice

So okay taking examples p(x) = 3x^2+x+1

then Dp=6x +1

@TedShifrin Not really sure how to procee

SteamyRoot

17:17

@TedShifrin Isn't $\mathbb{C}P^n$ also isotropic (I assume we're using the Fibini-Study metric ?)

N3buchadnezzar

Duplicate =) math.stackexchange.com/questions/2270176/…

Ted Shifrin

@mrnovice, what happens with $q(x)=3x^2+x+17$?

@Steamy: Riemannian isotropic? Hell no.

mrnovice

Then Dq = 6x +1

oh

Ah thank you :)

SteamyRoot

What's the difference between isotropic and Riemannian isotropic?

Ted Shifrin

I think you're doing holomorphic sectional curvatures only?

James Bond

17:26

This room is turning into a @ted room. =)

Ted Shifrin

Not my fault, Mr. Bond.

user228700

Hello, everyone :-) I have a quick question about the first derivative test to find the extrema of a function; does it only give the maxima/minima inside the domain of the function excluding the end points?

Daminark

Soon we'll all be complex algebraic geometers

It's just a matter of time

Ted Shifrin

Right, @Kaumudi.H, and only at points where the function is differentiable inside the interval.

user228700

Eek, excuse me for phrasing that quite so poorly.

James Bond

17:27

@SteamyRoot Do you mean Fubini?

SteamyRoot

@JamesBond Woops, typo. You're right

user228700

@TedShifrin Right, right. Ah, that explains the question I'm trying to solve. Alright, thank you! :-)

Ted Shifrin

@Kaumudi.H: Yes, on closed intervals you must always check the endpoints.

Daminark

Well or differential geometers, you get to choose!

By the way Ted, which of the two do you think you worked more in?

user228700

Right. Dang it, I found out too late. Thanks!

Ted Shifrin

17:30

Definitely complex geometry, Demonark. Not that I published all that many papers. But I taught undergraduate diff geo, diff top, and graduate diff geo (not complex geometry) most of my career. I only taught one complex geometry course, really, and that was as a postdoc.

Of course, I also wrote an algebra book and taught algebra five times or more.

Daminark

Ah, nice

James Bond

You wrote two algebra books, in fact. =)

And one calculus book. =)

Ted Shifrin

The "calculus" book is truly not a calculus book, mr librarian.

James Bond

Only one out of those three books are currently on Russian servers. =)

SteamyRoot

@TedShifrin I think your definition and mine (and Mike's ?) of isotropic are different. The definition I recall, is that for any $p \in M$ and any two tangent vectors $v_1,v_2$, there's an isometry fixing $p$ and mapping $v_1$ to $v_2$ ?

Ted Shifrin

17:35

No comment on Putin's servers.

@Steamy: The usual definition I know is that at each point all the sectional curvatures are equal.

SteamyRoot

I see

Yeah, under that definition, the only complex space form being isotropic would be $\mathbb{C}^n$...

Weird... maybe I just misremembered the definition, or confused it with something else

Eric

That is also the definition of isotropic I was given

that is the one Ted gave

Ted Shifrin

Of course, you mean unit tangent vectors, @Steamy.

SteamyRoot

Right

... I'm so used to working with unit vectors only, that I'm now wondering if I ever mentioned in my thesis that I assumed all vectors to be normalised...

Ted Shifrin

But you have to be careful about holomorphic versus real when you work in complex land.

Again, you're talking about holomorphic tangent vectors?

SteamyRoot

17:41

I don't even know what a "holomorphic" tangent vector is :/

Daminark

@Eric Emailed Marianna just now, she said that even if there's no time to do the full year of granola, it's fine to do complex as long as I've seen the undergrad-level stuff

SteamyRoot

I only ever really worked on totally real submanifolds of almost-complex manifolds

Secret

Small question: When we build the sequence $\omega,\omega^{\omega},\omega^{\omega^{\omega}}$ are we acting $\omega$ on the left or on the right to get to the next term?

Ted Shifrin

If you do complex manifolds, you often work with $TM\otimes\Bbb C = T^{1,0}M\oplus T^{0,1}M$.

SteamyRoot

Never worked with that, I'm afraid

Ted Shifrin

17:44

OH well..

SteamyRoot

Geometry department of my uni wasn't a big fan of vector bundles and sections and all that

Eric

@Daminark Schlag showed us more stuff in the bootcamp than I learned in my undergrad complex analysis class

my undergrad class was kind of piss poor though...

Ted Shifrin

No comment, Steamy.

Astyx

Hi again chat

Ted Shifrin

@Eric: I reiterate that people like you guys should take graduate complex, not undergrad ... Although I would hope you still have to compute a few contour and residue integrals :)

Re salut Astyx.

James Bond

17:45

@Eric Ah, we wrote a book on complex analysis and Riemann surfaces which is very good. =)

Astyx

Finally arrived at Poitiers

Ted Shifrin

Félicitations, @Astyx. Hike yet?

It's almost dinnertime!

Daminark

Yeah I was thinking Schlag's thing plus maybe supplementing from Markushevich or Narasimhan, at this point I do not intend to take the undergrad complex class

SteamyRoot

@TedShifrin Fair enough. I don't really have an opinion on the matter, since I never had the chance to learn.

Astyx

It's almost past dinnertime you mean :p

Eric

17:46

@Ted in retrospect my undergrad complex class was a waste of time, but it was my first year of university so whatever, I'll be taking the graduate course next year

Ted Shifrin

But if someone does complex geometry at all, one has to do a lot of that stuff. Weird.

Weird to do that first year, @Eric. Were you an engineering major at the time?

James Bond

@Daminark However, my favourite complex analysis book is Conway's Functions of One Complex Variable 1 and 2.

Ted Shifrin

Blah, Bond. Very blah.

Eric

Uchicago doesn't really do engineering, most of the people in undergrad complex were math/physics majors

Ted Shifrin

You continue to show questionable taste, Bond.

Daminark

17:48

There is basically no engineering major, Eric just took a bunch of math classes first year

Eric

I did it first year because I had loads of space

SteamyRoot

@Astyx Si tu n'as pas visité déjà, faut regarder l'église Nore-Dame la Grande!

James Bond

@Daminark I find Narasimhan too short and Markushevich too long.

Astyx

@SteamyRoot A Poitiers? Je suis pas vraiment la pour faire du tourisme mais pourquoi pas si je trouve le temps !

Daminark

Lol @James from what I've heard about Conway... Eh...

James Bond

17:50

@Daminark What bad things did you hear?

Eric

@Daminark I took even more math classes this year than my first year :P I'm gonna slow down next year cause I have to do gen ed stuff still

Daminark

I've heard that it drags on a lot, to the point of putting you to sleep

James Bond

@Daminark The two volumes of Conway are only over 600 pages, compared to Markushevich which is over 1000 pages and cover less material. =)

It is longer than usual because it proves lots of theorems and proves them in lots of ways, and I was looking for those theorems.

Daminark

So I've got a kind of Markushevich abridged

James Bond

For example, it proves Runge, Riemann Mapping, Little Picard, Big Picard, Bloch, Schottky, and Pompieu formula.

Daminark

17:54

Retitled "Introductory Complex Analysis", by Silverman

James Bond

@Daminark Covers too little material but good for a single undergrad class.

Daminark

I've also got some other things floating around, like Ahlfors, I really just need stuff to supplement what I'd be learning in a bootcamp

I'll reserve the heavier stuff to when I actually take complex analysis

James Bond

Ahlfors is the classic, but strangely neither Ahlfors nor Rudin proves Big Picard. They only prove Little Picard.

Daminark

shrugs

James Bond

I like Conway because it also goes well with his functional analysis text.

Daminark

18:01

Hey @Paul!

And I mean, so I also don't have Conway's functional analysis

Paul Plummer

Hello @Daminark

Daminark

To be totally honest I don't ever really work out of one book

James Bond

I don't like Munkres, but many people here like it, so they throw eggs at me when I say that, lol. For topology I mean.

Daminark

Like, I'll have a main one of sorts, but then afterwards for whatever is necessary I'd just keep others as references (I mostly work out of pdfs anyway). For that reason, the thing I'm most concerned about in my main book is whether it's the sort of thing I could read

Munkres seems pretty good, though I've heard of one called Willard which is supposedly even better

How's it going @Paul?

James Bond

I just use Bredon's Topology and Geometry for both general and algebraic topology in one book.

Paul Plummer

18:05

I guess it is okay. Trying to figure out what I want to do/think about @Daminark

Daminark

Nice, what do you have in mind so far?

@James isn't it a bit light on the point-set though?

James Bond

@Daminark You are right that it is light, but it is enough unless you want to specialise in general topology itself. Enough for all other mathematics and also for an undergrad course.

Paul Plummer

@Daminark There is a reading group in a different chat room reading a survey on "Measured group theory", which I would like to learn about, and there are a couple of other things related to my reading course with a professor, in particular reading and thinking about Thurston metric on Teichmuller space/stretch maps, or reading about hierarchically hyperbolic spaces

James Bond

@Daminark Willard I consider the best book on general topology itself.

Daminark

Nice @Paul

And perhaps @James, I mean with books there's no one size fits all sorta thing.

James Bond

18:15

@Daminark Indeed, just like different people have different sized underwear, lol.

Daminark

Not sure if that's quite the analogy I was going for but sure

James Bond

18:26

@Daminark Willard being a Dover book is also very cheap compared to Munkres...

Eric Stucky

Hey chat, do you have any recommendations for 'multiplayer Paint'-type programs? I mean, where I could work with someone on math online, who's not so comfortable with LaTeX yet?

Daminark

Well I don't usually buy books, I either get them from my school's library or pdf

James Bond

@EricStucky Math should not be done online. =) Just kidding.

@Daminark Aha. The Munkres book can cost 20 times as much as the Willard book.

Daminark

Again, moot point since I have both already

James Bond

I have been thinking very hard about books for Riemann surfaces and several complex variables and complex manifolds but it seems I still am quite undecided.

Partly because the different books focus on quite different aspects of the subjects.

SteamyRoot

18:32

My Riemann Surfaces course used Miranda

James Bond

Yeah, but that book doesn't treat the uniformisation theorem, it seems.

Hello Mike.

I think I will make some coffee for us both.

Paul Plummer

@JamesBond Teichmuller theory by Hubbard has uniformization in the first chapter

Waiting

19:10

@ShaVuklia is baaaaccccckkkkkkk!!! :-) How are you doing?

Cubic

Hello, I'm sorry if this isn't quite on topic. I'm reading a paper on circular arc graphs, and it mentions a "C*" graph. I think that it is supposed to mean an induced cycle + an isolated node, but I can't find references anywhere and the paper doesn't define it

Ted Shifrin

@Astyx: Félicitations! Grâce à Dieu! :)

Sha Vuklia

Hahaha @Waiting, I'm back because I'm stuck on a mathematical little thing that I want to solve from a physics calculus perspective :P

But I'm doing well! I love physics so much! I'm really happy. How are you?

If anyone could help me out:
Say it holds that $\sum_{i=1}^N \vec r_i’\Delta m_i=0$. How can I show that $\sum_{i=1}^N \vec r_i’\times \vec V\Delta m_i=0$? ($\vec V$ is constant)
I would like some calculus justification to understand the physics, so I am really looking for the most intuitive/elementary approach. I wouldn't like to use the definition of the determinant.
So we know that $\vec r_i'\times \Delta m_i\vec V=r_i'\Delta m_i V\cos(\theta_i)$. So we get
$$
\sum_{i=1}^N r_i'\Delta m_i V\cos(\theta_i)=V\sum_{i=1}^Nr_i'\Delta m_i\cos(\theta_i).

how can it be ambiguous when I denote all the vectors with an arrow above their symbol?

SteamyRoot

Also, cross product is sine, not cosine

Waiting

@ShaVuklia Nice for you, you're always in the stuff :P Well, you wouldn't like to know how I'm (right now) :P. Many events here of all kind, but I try to handle with each of them.

SteamyRoot

19:18

Oh, wait, you fixed it compared to the previous post you deleted, sorry

Sha Vuklia

@Steamy oh yea, you're right

@Waiting oh wow, well I have no idea what to think of that then?? I hope you're handling it indeed as you're saying!

Waiting

@ShaVuklia Yeap, I do. :-) What's news today? Other nice activities there at you in a beautiful Sunday? :P

SteamyRoot

@ShaVuklia You can move constants from one factor to another in a cross product

Waiting

@ShaVuklia I went jogging for a while, but not that long as usual, it's a rainy day here.

Sha Vuklia

@Steamy but they're not all constants. The $m_i$ and $r_i$ depend on $i$. It's just that this entire sum equals zero.

SteamyRoot

19:22

doesn't mean they're not constants

Just move all the $\Delta m_i$ to the other side

and use linearity

Sha Vuklia

@Waiting lol it's not a beautiful Sunday! At least not now, the sky is purple and it's cold :P And cool, I didn't know that you jogged. I never have energy for that :P

@Steamy okay I'll try

Waiting

@ShaVuklia And I also slept for half an hour then? Probably. Yeah, it's refreshing going jogging, and actually it happens you may realize after that you have more energy than before. There was a dark sky here, but even so I did my best to take benefit of it. Actually I like when it rains, and that sound of drops ... :P

Sha Vuklia

@Waiting yea that is true that you have more energy afterwards! not sure I like rain, but I'm happy that you took benefit of it:)

@Steamy where should I use linearity, tho? You mean this right: $(a+b)\times c=a\times c +b\times c$

Waiting

@ShaVuklia and I'm happy you're happy of my story of the day :P

SteamyRoot

Yes

Waiting

19:32

What a great story! :-) Most of the day I actually worked on my mathematical stuff.

Sha Vuklia

haha :P @Waiting What kind of things do you work on actually?

@Steamy but I don't have a sum?

Cubic

nevermind, it's defined in one of the referenced papers

Sha Vuklia

ohhh

lol the whole problem was the sum

right, I think I got it

heyy it's @Astyx

Astyx

No it isn't

I'm not here I'm sleeping (supposedly)

What's up with you ? :)

Waiting

@ShaVuklia Some mathematical problems involving (very) difficult infinite series.

Sha Vuklia

19:36

lol, I feel honoured to be in your dream XD @Astyx

I'm doing PHYSICS!!!! :) so I'm very energetic and happy. physics is cool. how are you?

@Waiting ahh, is it for work or for yourself?

Astyx

Wasn't it you who said she hated physics two days or so ago ?

Sha Vuklia

hahahahahah, yes I'm slightly bipolar

Astyx

I'm 10 hours away from the last four days of written exams of this year (of my life?)

Sha Vuklia

I have love-hate relationships with everything and everyone in this world :P

Waiting

@ShaVuklia For my projects. :P

Sha Vuklia

19:38

whoaaaa, good luck then!!:o @Astyx

@Waiting ahh okay, cool:)

Waiting

I have to return now to my work.

@ShaVuklia Take care! ;)

Sha Vuklia

alright, good bye! and good luck:) @Waiting

so you think you will pass everything? @Astyx like, it's competitive, no?

@Daminaaaaark!!!

lol, I already greeted you

but hi:P

Astyx

It is. I'm pretty confident for Centrale (the one I'm taking tomorrow ), but I kinda messed up X ENS (which is the most important, yay). I chose not to worry about that cause in any case it's behind me. As for the Mines - Pont, I'm pretty sure I'll be admissible (ie I'm alowed to take the oral exams)

If I don't get X ENS I'll probably do another year prep and retake the exams next year

Sha Vuklia

ahh, well I'm hoping you pass everything at once!:)

Astyx

Thanks, so am I :p

Anyway, I probably need to sleep so I'll find another mean o procrastination than this chat, good night to you ! (it is night, right ?)

Sha Vuklia

20:00

haha, well it's night if you're about to go to sleep :P night! @Astyx

Anonymous

20:11

$n$ apples and $n-1$ oranges are placed in a row (all distinct). The probability that the
number of apples before every orange (in the row) is at least one more than the number of oranges before it, is? I've solved this manually for when n=3 and n=4 but beyond that it seems to complicated. Is there any general method to solve such problems?

Anonymous

@BalarkaSen Any idea about this ^ ?

Balarka Sen

Not sure if I understand the question, but generally I'm terribad with probability so maybe I pass.

Anonymous

Oh ohkay

Anonymous

I'll ask on the main site then

Astyx

@Ted Félicitations pour quoi ? Les élections ? Quel soulagement!

@blue you can number the oranges according to their order of appearance, which makes the problem easier

Then of course multiply by the number of ways you can organise these oranges (or not depending on how you choose to proceed)

Anonymous

20:24

@Astyx Okay say we've got O1,O2, O3,......,On-1 and A1,A2,A3,.......An

Astyx

For instance, no need to order the apples I believe

Anonymous

Total number of ways of arrangement is definitely $\binom{n+(n-1)}{n}$(n)!(n-1)!

Astyx

Agreed

Anonymous

Now how to apply the constraint ?

Astyx

So start with only the oranges in the row and add the apples one by one respecting the constraint

So the first apple has to go first

Then the second apple can go before or after the first orange

Actually you might end up counting things twice if you're not careful enough

I you let f(i) =j when the ith apple is between the j-1 and j-th orange

You can impose f to be increasing

And such that $f (i+1)-f (i) \in \{0,1\}$

Anonymous

20:31

@Astyx Okay, makes sense

Astyx

So you end up counting up to permutation of the fruits the cardinal of $\{0,1\}^n $ or something similar (I'll let you correctly the details)

Which is of course $2^n $

Anonymous

@Astyx Umm, I see. Okay, am trying

Astyx

I have to go now. Hope I could help

Bye

Anonymous

20:54

I guess I've found an easier way. The favourable arrangement is precisely balanced arrangement of $n$ left parenthesis and $n$ right parenthesis!

Anonymous

(By supposing we add 1 more orange)

Zophikel

21:33

Anyone have the book Applied Asymptotic Analysis

Daminark

I imagine the author has it, if that answers your question. :P

Zophikel

@Daminark I was wondering if anyone had a copy of the book

well y naught

Daminark is being snarky and taking "anyone" quite literally.

Evinda

Hey @SteamyRoot Are you familiar with differential equations?

Simply Beautiful Art

@Evinda Depends on the DE

James Bond

21:48

Yeah, so many DEs that you need to specify the question.

Evinda

My question is why when we have the de $u_{tt}-u_{xx}=0$ in $\mathbb{R}^2$, we need to have two initial conditions and not 4.

user193319

Conjecture: If $S$ is and $T$ are linearly independent sets, and $S \cup {t}$ is linearly independent for every $t \in T$, then $S \cup T$ is linearly independent. Would the following be a counterexample: Let $S = \{(1,0,1,0),(0,1,0,0),(0,0,1,0)\}$ and $T = \{(0,0,0,1),(0,0,1,1)\}$. Clearly they satisfy the hypothesis, but notice that the vectors $(1,0,1,0)$, $(0,1,0,0)$, $(0,0,1,0)$ alone span $\mathbb{R}^4$, so that if $S \cup T$ were linearly independent, its dimension would be $5$,....

which is absurd.

Alessandro Codenotti

how can $3$ vectors span $\Bbb R^4$?

user193319

Sorry. I forgot to include $(0,0,0,1)$ among the three.

Alessandro Codenotti

Looks right now

user193319

21:52

@AlessandroCodenotti Thanks!

James Bond

@Waiting I did not know it was you! Jas here, lol. I haven't talked to you for so long! =)

@Waiting I am waiting for the publication of your book! How is it going now?

@PaulPlummer OK. Anyway, my three favourite books on Riemann surfaces at the moment seem to be Forster's Lectures On Riemann Surfaces, Napier and Ramachandran's Introduction To Riemann Surfaces, and Varolin's Riemann Surfaces By Way Of Complex Analytic Geometry, if you are interested in the subject.

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$Mathematics$ Mathematics