Mathematics - 2017-01-18 (page 3 of 5)

hi all, i need to understand this formula used for combinatorics, (n+m-1)!/n!(m-1)! this was used to solve this situation of placing 9 indistinguishable marbles into 4 boxes. which formula is this referring to? thanks

s.harp

Binomial coefficient @MubB

Ted Shifrin

7:20 PM

@MugB: Are you asking why that is the right counting formula?

MugB

thanks @s.harp yes and why is this being used :) @TedShifrin

Ted Shifrin

Let's say you have a collection of n different objects and you wish to pick out a subset of k of them (without regard for order). You can choose the first one n ways, the next one n-1 ways, and so on, down to n-k+1 for the kth. So you have n(n-1)...(n-k+1) possible ways of choosing them. Right?

MugB

okay @TedShifrin i got that part

Akiva Weinberger

To activate LaTeX rendering, use the link on the top-right. @MugB

Ted Shifrin

BUT we don't care about the order. So each actual k-element subset has been counted k! times. So we divide by k!. But the first product I wrote down was n!/(n-k)!. So you're left with n!/(n-k)!k! ...

Akiva Weinberger

7:23 PM

That way, it'll turn things like $\binom ab$ into cool math notation

Ted Shifrin

I forced myself to skip dollar signs this time, DogAteMy :)

Akiva Weinberger

Yeah, but they should know it exists

Ted Shifrin

How did your Rudin test go, DogAteMy?

MugB

thanks will try that @AkivaWeinberger newbie here

Vrouvrou

@TedShifrin please what i can do to $\sum_{j=1}^n p_j\xi_j^2$ in relation with $|p|$ and $|\xi|$

Ted Shifrin

7:26 PM

@Vrouvrou: All I can see is $\left|\sum p_j\xi_j^2\right|\le |p||\xi|^2$.

MugB

but how did we get to (n+m-1)! that part im not very clear. why are we doing (9+4-1)! @TedShifrin

Akiva Weinberger

@TedShifrin I haven't done that yet

Vrouvrou

@TedShifrin how to see this please

and what about $\sum_{i,j=1}^n p_j\xi_j\xi_i$

please

Ted Shifrin

OK, @MugB. First, to clarify: Can you put 0 marbles in some box?

Akiva Weinberger

Triangle inequality? @Vrouvrou

I assume $p$ means $\sum p_j$

MugB

7:29 PM

@ted

Ted Shifrin

@Vrouvrou: $\left|\sum p_j\xi_j^2\right|\le |p|\sum\xi_j^2$, since $|p_j|\le |p|$.

MugB

@TedShifrin no

Ted Shifrin

So $p$ means the vector of $p_j$'s, DogAteMy.

Akiva Weinberger

Oh.

s.harp

@AkivaWeinberger $p=\sqrt{\sum_i p_i^2}$

Akiva Weinberger

7:30 PM

$\vec p$ (or $\bf p$)

s.harp

$\boldsymbol p$

Ted Shifrin

OK, @MugB. So imagine putting 3 dividing lines between various of the 9 x's. What's to the left of the first goes in the first box, between the first and the second goes in the second, etc. Now can you count?

Akiva Weinberger

Or $\bf\vec p$

s.harp

since $\bf\theta$ doesnt work but $\boldsymbol\theta$ does :D

Vrouvrou

@AkivaWeinberger $|p|=\sqrt{\sum_{j=1}^n p_j^2}$

Akiva Weinberger

7:31 PM

@s.harp Weird.

Ted Shifrin

No, @s.harp, I really think he means the vector $p=(p_1,\dots,p_n)$. This is just Cauchy-Schwarz, but easier.

Akiva Weinberger

@s.harp Aw :(

Vrouvrou

@TedShifrin and what i can do to $\sum_{i,j=1}^n p_j\xi_j\xi_i$

Ted Shifrin

Don't you need \boldsymbol for theta?

s.harp

yes :D

$\boldsymbol\sum$ $\sum$

SoumyoB

7:32 PM

hi does anyone here know a bit of stochastic calculus?

s.harp

apparently bold symbol doesn't bold every symbol :/

SoumyoB

I'm facing a problem

Ted Shifrin

Same answer, @Vrouvrou. Think of this as the inner product of the vector $(p_1\xi_1,\dots, p_n\xi_n)$ and the vector $\xi$.

MugB

@TedShifrin yes that will equal to 9X8X7X6

Ted Shifrin

No, @MugB. You need to have 9+4-1 slots, and the 3 dividing lines will go in some 3of them.

For example, @MugB, with 5 marbles and 2 boxes, you could have xx|xxx or xxx|xx or xxxx|x or ...

So this would be n=6, k=1 in my original notation.

MugB

7:36 PM

@TedShifrin okay i think i get it. although still quite foggy maybe ill need a bit more practice to see how it works

Ted Shifrin

If you google "bars and stripes," you'll see this is a standard counting trick in combinatorics.

MugB

@TedShifrin thanks for the tip. will have a look.

Akiva Weinberger

You mean "stars and bars"?

Ted Shifrin

Oh, maybe I messed up. Listen to Akiva. :P

Akiva Weinberger

Because otherwise I'm pretty sure "bars" and "stripes" mean the same thing

MugB

7:38 PM

haha yeah i think thats what Ted meant. got it

Ted Shifrin

I discovered the counting technique myself in grad school and only learned the nickname recently when I taught probability. So I messed up. shrug

DogAteMy, I was looking forward to finding out some good problems you solved on your Rudin test. The PDE thing you asked me about is all about Mean Value Theorem not applying on disconnected domain to prove f'=0 -> constant.

Oh, I guess I can go back to math mode now :)

MugB

wow this technique is cool! @TedShifrin thanks a lot for pointing me to it. or else i think im gonna take another few mths to understand :D

Akiva Weinberger

What PDE thing? The one whose domain was the plane minus the positive real axis?

I mean, the positive x-axis

Ted Shifrin

Right, DogAteMy. The question from section 3.6.

@AkivaWeinberger DogAteMy: How many matrices do you know that do not live in a plane? :)

Astyx

@Akiva Okay, I posted my answer on the same question Daniel Fischer replied, if you're interrested

Or rather my answers

Ted Shifrin

7:46 PM

@Astyx: I take it you're feeling 100% better.

Astyx

Except for the fact I still dont know wether () or [] come first when sharing a link, much better yeah (although my stomach still hurts from time to time)

How are you by the way ?

Akiva Weinberger

Derp. Plain. @TedShifrin

Ted Shifrin

I'm fine, thanks. Just came back from having a test to make sure my heart still works. Sadly for you all, it does.

LOL, I know, DogAteMy ... but I felt I owed you a carp.

Akiva Weinberger

So you are, at least technically, not heartless

Astyx

What a shame indeed

Akiva Weinberger

7:48 PM

"Carp"?

Astyx

The fish ?

Ted Shifrin

To carp = to complain/whine.

Akiva Weinberger

Huh. Today I learned.

Ted Shifrin

Ah, I should have said "to find fault in a picky way"

Akiva Weinberger

So you just carped your own definition.

Ted Shifrin

7:49 PM

I suppose we could make gefilte fish out of our carps.

I did.

We need to carp diem, too.

Liad

Hi, i got a question:
Suppose $f: U \to R$ , where $U $ is an open and convex set of $R \ ^ n $ s.t $ D_1 f(x) = 0$ for all $x \in U $.

Given $x,y \in U $ , s.t $x_i=y_i $ for $ i=2,\dots ,n$
i need to prove that $ f(x)=f(y)$

I defined $g : [0,1] \to U $ s.t $ g(t) = tx +(1-t)y$.

I want to say that f composition g has derivative zero using chain rule but i dont know that $f$ is differential.

So, how should I continue?

Astyx

Not sure if I should laugh or cry

Lucas Henrique

Ok, so I did that script

Ted Shifrin

@Astyx: So perhaps I am heartless after all.

Liad

@Astyx Hi! :)

Astyx

7:51 PM

@Liad Hi ! How are you ? :)

Lucas Henrique

It's Python 3

Liad

@Astyx fine, and you?

Ted Shifrin

@Liad: It's just a one-variable problem and you know that $D_1 f = 0$. Do you know $f$ is continuous?

Liad

@TedShifrin no

Ted Shifrin

Oh, but you do know it's continuous on any closed line segment varying just $x_1$. Why?

Lucas Henrique

7:52 PM

AFAIK it's idiot/typo proof so it's a nice and simple script

pastebin.com/wXzq3S89

Astyx

@Ted Technically if you don't have a heart, it's working. Your doctor might have been pedantic

Liad

but i see that f composition g is just $f(tx_1 +(1-t)y_1 ,x_2, \dots , x_n) $

Ted Shifrin

@Astyx: Probably so. He was trying to get rid of me. This was my third time going to the office for this appointment (they kept screwing up), and this time he was a half hour late.

So why is that function of $t$ continuous, @Liad?

Balarka Sen

Hi @Ted

Ted Shifrin

Actually, @Liad, to avoid saying the words "chain rule," you should just consider $g(t)=f(t,x_2,\dots,x_n)$, $t\in [x_1,y_1]$.

Hi @Balarka

Adeek

7:55 PM

hi @TedShifrin

Lucas Henrique

@Socrates done

Ted Shifrin

Hi, Karim.

Extrarius

Is there a body of theory around using 'composite' fields aside from prime powers, such as (eg) the cartesian product GF(3) x GF(5)?

Liad

@TedShifrin alright, i thought of this function too.

Ted Shifrin

@Extrarius: A product of fields isn't a field ?

Liad

7:56 PM

@TedShifrin is it correct that the derivative of this function is $ D_1f $ ?

Adeek

I am just proving the following I + J is generated by gcd of n and m. Don't tell me answers though no spoilers. So I proved that I + J is the smallest ideal containing n and m. Then, now I will show that gcd is the smallest ideal containing n and m. Their was characterization of gcd in terms of their linear combination right ?

Ted Shifrin

Of course, @Liad.

Extrarius

What differentiates a field from a ring?

Ted Shifrin

But why does the Mean Value Theorem apply to tell you the function is constant? @Liad

Adeek

that is ax + cy = d for integers x and y right ?

Ted Shifrin

7:57 PM

Karim, unless $I=\langle n\rangle$ and $J=\langle m\rangle$, I don't know what you're talking about.

Adeek

@TedShifrin

Balarka Sen

@Extrarius Existence of inverse.

Adeek

Yeah I am talking about that @TedShifrin

Akiva Weinberger

@Extrarius Inverses and commutativity.

Ted Shifrin

So you mean the gcd generates the ideal $\langle m,n\rangle$.

Language matters, Karim :)

Adeek

7:58 PM

yeah right .. :) Yeah

Ted Shifrin

But this is something you should have proved as an undergraduate.

Tobias Kildetoft

@BalarkaSen Made any progress on the puzzle?

Krijn

You can take composites of field in a different way though @Extrarius

Ted Shifrin

It's OK to reprove it now.

Extrarius

Is there anything analogous to a composite field besides prime powers, such as a combination/mixture of two prime fields?

Adeek

7:58 PM

no spoilers though @TedShifrin

Akiva Weinberger

The integers are not a field because they don't have inverses; the quaternions are not a field because they don't commute. But they're both rings. @Extrarius

Balarka Sen

@TobiasKildetoft I stopped thinking about it.

Astyx

Do you guys often give downvotes ?

Tobias Kildetoft

@BalarkaSen Probably a good idea :)

Ted Shifrin

@Balarka: Did you stop thinking about moving frames too? I'm sure you're still sick :(

Akiva Weinberger

7:59 PM

@Astyx downvotes

Adeek

ohh rightt

it is through bezout identity right

Ted Shifrin

Sure.

Balarka Sen

Three of them are clearly godfather, miserable and education, but the middle one doesn't match up with the number of slots.

Akiva Weinberger

@Astyx That's a good answer, in the link you sent

Astyx

Cause I've ever only given 2 while I gave 140-something upvotes, I don't know how to feel about that

Krijn

8:00 PM

@Extrarius See en.wikipedia.org/wiki/Composite_field_(mathematics)

Adeek

yeah @TedShifrin my undergrad university sucked. My knowledge I guess mainly came from self-study.

Tobias Kildetoft

@BalarkaSen you are on the right track at least

Astyx

@Akiva Thanks, I thought it might be a bit long and ununderstandable at the end

Ted Shifrin

That stuff was the main theme of the algebra course I taught, Karim.

Balarka Sen

@TedShifrin I computed curvature of the sphere and the pseudosphere with it by hand; it's extremely efficient at quickly computing stuff

Adeek

8:01 PM

That is important @TedShifrin. Some profs just don't take teaching seriously.

Ted Shifrin

Isn't it, though, @Balarka? You'll end up agreeing to be a fan, in the end :P

@Balarka: Not to mention proving Gauss-Bonnet :P

Karim: Research universities do not promote and give pay raises according to teaching.

Adeek

@TedShifrin today I helped with help center here the experience was so amazing.

Balarka Sen

Yeah, it's pretty great. I still need to get used to doing the bookkeeping with those forms, but I am working on it.

Astyx

@Akiva I really like Daniel Fischer's answer also

Liad

@TedShifrin this is why i wanted to look at f composition g, so that the function will be from R to R and then i know that if the derivative is zero then the function is constant

Socrates

8:02 PM

@LucasHenrique congratulations

Adeek

I like seeing people joy when I hear that they understand what I expalined @TedShifrin

I see @TedShifrin

Tobias Kildetoft

@BalarkaSen I can give you a hint of the type the hunt had build in (one could pay for these with gold): A yes/no question about the puzzle.

Lucas Henrique

@Socrates $\begin{bmatrix} \, & i & \, & t & \, \\ w & o & r & k & s \end{bmatrix}$, even tho it's weird to work in the first time for a programmer (usually get errors and bugs and meh).

Ted Shifrin

@Liad: Well, you're still doing the same thing, without needing to look at the chain rule. As you pointed out, you can't officially apply the chain rule. But be careful. You need continuity on the closed interval.

Karim: That's good. Explaining things helps you get better at understanding and explaining!

Balarka Sen

I don't actually know how to prove G-B. I suppose one integrates $d\omega_2$... Stokes' says it's sufficient to integrate $\omega_2$ along the boundary of the surface.

Adeek

8:03 PM

yeah

Ted Shifrin

You mean $\omega_{12}$.

Balarka Sen

I apologize, yes.

Ted Shifrin

I have the proof sketch at the end of the section. But you should figure it out :) The key thing is to figure out where $k_g$ comes from.

Balarka Sen

So that should be the geodesic curvature along the boundary.

Ted Shifrin

Oh, and Euler characteristic :)

Depends how you choose the frame field, @Balarka.

Good luck with that. :)

Astyx

8:04 PM

@Akiva However how do you easily show that all the mass have same parity ?

Balarka Sen

Hmm, right.

Ted Shifrin

One hint is to start with a vector field with finitely many nondegenerate zeroes (which you certainly can prove exists).

Liad

@TedShifrin ok, in order to use the MVT , i need to have a differential function. and i dont have it :/

Akiva Weinberger

@Astyx Suppose not. If the total mass is even, choose an odd pebble; the masses of the remaining pebbles sum to an odd number, and so they can't be divided into two groups. Similarly, if the total mass is odd, choose an even pebble.

Ted Shifrin

You do, @Liad. You need a continuous function on a closed interval that is differentiable on the open interval. Go back and be careful. But the function $g(t)$ I wrote down has both those properties. Make sure you can say why.

Astyx

8:07 PM

Oh that was quite easy indeed .. :p

Balarka Sen

@TedShifrin So from an arbitrary frame you want me to get to one which is a Darboux frame along the boundary, right? And that's supposedly where the $\chi$ factor comes in.

Liad

@TedShifrin Sorry, but why $ g$ is continuous ? i dont know that $ f$ is.

Ted Shifrin

The $\chi$ factor comes from what I wrote up there ^^^ ... but, yes, at some point, you'll need to relate different frames.

@Liad: But you do know (or can prove) that $g$ is continuous (i.e., $f$ is continuous along that closed line segment). Why?

@Liad: Hint: It's important to understand something about the set $U$ in the problem.

Liad

@TedShifrin huh, because $ D_1f(x) $ exists for each $x \in U $ ?

Ted Shifrin

So why does that tell you $g$ is continuous?

Liad

8:12 PM

because g is differential

Balarka Sen

Hmm, trying to understand how such a vector field is relevant to the story (of course you want me to do some sort of Poincare-Hopf here).

Socrates

@LucasHenrique was python your first experience?

Lucas Henrique

Can someone help me with a question? If it's not that easy I'll post in the site.

(Mathematical Circles) Eleven students formed five study groups. Prove that we can find two students, say A and B, that every study group that includes student A also includes student B

Ted Shifrin

Right, differentiable implies continuous. (You need to know that $x$ and $y$ are interior points of $U$. Which they are.)

Perturbative

Hey everyone, hope you're all keeping well!

Liad

8:13 PM

yes because U is convex.

Ted Shifrin

@Liad: AND open.

@Balarka: You need to play the game we've played in complex analysis and differential topology all the time.

Lucas Henrique

@Socrates Nope. Java (ugh) was my first experience, then I started to program in C, learn OOP (C++) and then Python

Perturbative

Astyx

Caml 4 life

Perturbative

In the definition of the subbasis for the product topology above, in the collection $\mathcal{S}_{\beta}$ are we taking only one open set $U_{\beta}$ from every $X_{\beta}$?

Ted Shifrin

8:15 PM

No, all possible open sets.

(or basis elements for the topology of $X_\beta$)

Extrarius

@krijn: On that page, it sounds like composite fields must have the same characteristic. Am I incorrect in that understanding? I'd like to "mix" two fields with different characteristic

Krijn

@Extrarius You are correct in that understanding and you'll have a hard time to mix two field with different characteristic into a proper field

Perturbative

@TedShifrin Hmmm, I thought that the use of the same indexing set for $U$ and $X$, meant that we would only be taking one open set from every $X$, because if we change $\beta$, then we are taking an open set $U$ from a different $X$ (I hope what I said makes sense, it's a bit difficult to convey my reasoning here)

Ted Shifrin

quantifiers, @Perturbative: Set-builder notation means you consider a collection formed by taking all possible open sets $U_\beta\subset X_\beta$ ($\beta$ fixed).

Mary Star

Hello!! I want to find the maximum of $f(x_1, x_2)=-(x_1+3)^2-(x_2-2)^2$ with the constraints $x_1, x_2\geq 0$. I thought to calculate the extrema as we would not have constraints and we pink only the positive $x_i$'s. Is this correct?

Extrarius

8:22 PM

Must the additive and multiplicative identity elements in a field be unique?

Krijn

@Extrarius You can easily prove that for a group already

Ted Shifrin

@MaryStar: You must also restrict the function to the boundary and use single-variable calculus to see what happens there.

OK, I'll be back later. Have fun finishing that, @Balarka.

Mary Star

@TedShifrin What do you mean?

Perturbative

@TedShifrin Ah so $\beta$ is fixed for $X_{\beta}$, but not for $U_{\beta}$

@TedShifrin Thanks for all the help!

Balarka Sen

@TedShifrin Yeah, will think about it.

Extrarius

8:24 PM

@Krijn: You seem to be knowledge about group theory. Are there any comprehensive introductions to group theory that you would suggest?

Krijn

@Extrarius Sorry to say that I learned it from a Dutch source, but I'm not that knowledgeable on group theory

Tobias Kildetoft

@Extrarius comprehensive and introduction do not really go together here

Extrarius

@TobiasKildetoft: Well, I'll take one or more of each if you have suggestions =-)

Tobias Kildetoft

For introduction there are tons out there. Most of them will be for algebra in general rather than only group theory, but that is probably a better place to start anyway

I have heard good things about Artin's book, though I have not read it myself

Lucas Henrique

@Astyx (insert puke emoji)

Astyx

8:27 PM

Caml is better than it seems !

Even Facebook is starting to use Caml nowadays

Extrarius

I'm a computer programmer (with a basic computer science education) that is trying to learn areas with heavy mathematical underpinnings. (I recently posted about working to understand the building blocks behind elliptic curves)