Mathematics - 2015-05-11 (page 1 of 4)

Seth Kitchen

00:02

Does anyone know how to present a Concept when it is not in a lattice?

0

If I have a set of objects G={Shark, Penguin, Bat} and a set of attributes M={Breathe Underwater, Can Fly, has Vertebrae} and I make a chart to represent my context (G,M,I) where I is the incidence relation (If my object g has attribute m, I mark an X in the chart) | Breate Underwate...

Mike Miller

00:28

@TedShifrin: Inspired by a question I just saw: do you know any place in topology where one can show that two spaces are homeomorphic without fundamentally constructing a homeomorphism? Certainly there are cases where you can use abstract results to guarantee that there is one, like h-cobordism, but even that fundamentally writes down a homeomorphism.

Seth Kitchen

Absolutely differentiable algebras ?

web.mst.edu/~wjcharat/publications/absolute-differentiation.pdf

pjs36

00:49

That's an interesting question, @MikeMiller. You've got me thinking about even something as basic as group isomorphisms... I can't think of any theorems that don't have an explicit isomorphism lurking in their proof (i.e., a group modulo the kernal of a homomorphism and the image of the homomorphism)

mixedmath

hi @MikeMiller and others

Mike Miller

Hi @mixedmath.

Joe Stavitsky

how can I show summation of x/n=x?

Seth Kitchen

Need more context

anon

@JoeStavitsky the summation of what now?

Joe Stavitsky

01:00

well, there is the rule summation of c=cn

anon

what?

gotta say it in English first before we can put it into math

Joe Stavitsky

Sigma summation, I don't know the mathjax

sum of n terms from i to n, etc

Stan Shunpike

@TedShifrin hey Ted!

anon

$\sum_{i=1}^n a_n$ produces $\sum_{i=1}^n a_n$ (do you have chatjax bookmarked and running?)

Seth Kitchen

if c=2 and n=2 c does not equal cn because 2 does not equal 4

Joe Stavitsky

01:04

ok, the book says
$\sum_{i=1}^n c=cn$

Seth Kitchen

oh well that has the summation haha

anon

yes, $\underbrace{c+c+\cdots+c}_n=nc$

Joe Stavitsky

but I need to show that
$\sum_{i=1}^n c/n=c$

Mathcad says this is true

Seth Kitchen

I'd use induction

base case

Joe Stavitsky

OK, there is no simple algebra way? cause this is calc 1 :)

Seth Kitchen

01:10

seems like you convert it to integrals

i forget how

:P

@anon ideas?

anon

for realz

Joe Stavitsky

wait no I think I am doing it wrong

anon

$\frac{c+c+\cdots+c}{n}=?$

Joe Stavitsky

grrrrr

anon

$\frac{c}{1}$, $\frac{c+c}{2}$, $\frac{c+c+c}{3}$, $\frac{c+c+c+c}{4}$...

Joe Stavitsky

01:12

yeah n/m they make us pull it out of the summation first. sry.

Seth Kitchen

YAYZ

Joe Stavitsky

It seems like an intuitive thing to show tho...

Seth Kitchen

"and then, a miracle occurs"

Joe Stavitsky

bwahahaha I love that cartoon

was that larson or not?

Seth Kitchen

yeah!

Does anyone want to solve the Euler Brick Problem for me?

Thanls

Joe Stavitsky

01:18

And now for something a little less intellectual frogbucket.com/myImages/PGC237.gif

Seth Kitchen

That's not what I remember in Good Will Hunting

Joe Stavitsky

Deleted scene

Seth Kitchen

You right

Joe Stavitsky

nope,I was wrong. It comes out to $1/n \sum_{i=1}^n a_n$

er rather $\sum_{i=1}^n c$

$1/n \sum_{i=1}^n c$

wow I cannot type

there

Paul Plummer

@evinda I don't really know much about algorithms. This is why you just ask, not ask to ask, someone could have actually answered you.

2

Clarinetist

01:25

How often do grad students update CVs?

Seth Kitchen

When was evinda here?

Paul Plummer

3 hours ago, by evinda

Hey @PaulPlummer
Could I ask you something about an algorithm?

@SethKitchen

Seth Kitchen

lol 3 hours ago :)

@PaulPlummer are you a computer scientist?

Paul Plummer

"I don't really know much about algorithms"

Are there computer scientists that don't know much about algorithms?

Seth Kitchen

Yes :)

Paul Plummer

01:28

Welll, I am not one of those then

Mike Miller

@Clarinetist: As often as you need it to apply for stuff.

I applied for conference funding recently, and updated it then, but up until then not since I applied for grad school.

Clarinetist

K thanks

Ted Shifrin

02:00

Mr @Pedro

Pedro Tamaroff

Hello.

How's the crowd behaving?

Ted Shifrin

I just got home

Pedro Tamaroff

Where'd you been?

Ted Shifrin

Friends for dinner

Pedro Tamaroff

Nice.

What food?

Ted Shifrin

02:03

Yummy middle Eastern turkey burgers with zucchini and awesome spices ...

anon

@PedroTamaroff friends

Mike Miller

lol

Ted Shifrin

Silly anon ...

Goodnight Mike

Pedro Tamaroff

@anon Oh, I forgot how wicked @Ted was.

Mike Miller

morning

Stan Shunpike

02:04

Good #TimeofDay

Joe Stavitsky

when showing integral by limit definition, how is choice of Ci made?

Ted Shifrin

Hi @Stan

user147690

@JoeStavitsky You really should give more details when you ask us these questions

Ted Shifrin

@Pedro, you never got back to me, so I assume you don't need me ...

Pedro Tamaroff

@TedShifrin Yeah. I'll let you know.

Ted Shifrin

02:09

Ok

Mike Miller

@Ted: I'm learning that I'm atrocious at tensor calculus and working with connections... :s

Stan Shunpike

@TedShifrin Hey Ted! Okay, firstly, your thing was a great help. I was able to read Rudin's comments on the IFT. And it makes much more sense. But I am still unsure what the bordered hessian has to do with the IFT.

Joe Stavitsky

@AlexClark, we are asked to prove definite integral by limit of sums, but the book's choice of ci is inconsistent

Ted Shifrin

@Mike ... There's a reason some of us teach courses and assign exercises ...

Mike Miller

lol

Ted Shifrin

02:12

@Stan: Because certain equations are defining someone as a function of someone else. The bordered hessian was for defining a max/min test? I've forgotten.

Stan Shunpike

Yeah, along those lines. Apparently the sign on the submatrices are important. But the determinant of the BH has to be 0.

Ted Shifrin

No, the bordered hessian has to have nonzero det. That's supposed to be the hypothesis of the Implicit Fn Thm.

Stan Shunpike

That makes sense. That's what I thought but I talked myself out of it.

Ted Shifrin

Get back to me :)

Stan Shunpike

I will, I'm still working on it.

I don't want to ask for help unless I've tried my best

So I think about it and then come back and talk it over.

SpawnKilleR

02:45

Hey guys I have an integral to calculate for an exercise. so the function is integrant is (e^min{x1,x2,2}) - 1 ) / ( e^{x1+x2} ) and i am integrating from 1 to infinity for both x1 and x2

Anyone knows how this can be calculated

anon

split into cases

SpawnKilleR

So I integrate over x1 first and hold y_2 as a constant and take cases thats what i thought

x_2*

Damn basically i have (x_1 ... x_5) It's gonna take days :(

So the integrals are from 1 to infinity

all of them

So can i take an arbitrary ordering like x_1 < x_2 < .... <x_5

and take the integra,y3 ls boundaries to be [1, y_2] , [y1 , y3] .....

something like that

I mean x not y soz

and take the integral boundaries to be [1, y_2] , [y1 , y3] ..... *

and take the integral boundaries to be [1, x_2] , [x1 , x3] ..... *

Mike Miller

03:14

@Ted: Are you ready for today's silliest question?

mixedmath

@MikeMiller that's a mighty claim

Mike Miller

The silliest one I'm willing to provide.

Looks like he's not ready. I can't blame him.

pjs36

Dammit Ted, answer the man! I have to know about this silly question.

I'm sorry, I lost my cool there...

Ted Shifrin

Apparently you think I'm here when I'm not. @Mike

Mike Miller

The chat says you are, @Ted.

I trust the Chat blindly.

Ted Shifrin

03:28

I don't log out ... iI just close Chrome on my iPad

So?

Mike Miller

@Ted: It's not hard to show that (nonconstant) conformal maps are the same thing as (nonconstant) holomorphic functions by thinking about the Cauchy-Riemann equations geometrically. Instead, I'd like to do it by mindless symbol-pushing :)

Ted Shifrin

With nonvanishing derivative ...

Mike Miller

Sure, you know what I meant. Sorry for the think-o.

Seth Kitchen

vanishing like harry potter? poof

Ted Shifrin

You need a characterization of conformal linear maps. Other than that it is mindless.

Mike Miller

03:32

So suppose I have a diffeomorphism of surfaces $\phi: (\Sigma_1, J_1, g_1) \to (\Sigma_2, J_2, g_2)$, where the $J_i$ are almost complex structures (integrable, automatically, of course), and the $g_i$ are compatible in the sense that $g_i(J_ix, J_iy) = g_i(x,y)$. I want to show that $\phi$ is conformal iff it's $J$-holomorphic (= holomorphic); that is, $\phi^*g_2 = \lambda g_1$ iff $J_2 \phi_* x = \phi_* J_1 x$ for all $x$.

Trouble is the mindless symbol pushing isn't working out so well.

I can write, say, $g_2(\phi_* J_1 x, \phi_* y) = \lambda g_1(J_1 x, y) = \lambda g_1(x, -J_1y) = g_2(\phi_*x, -\phi_x J_1 y) = g_2(J_2 \phi_*x, -J_2\phi_* J_1 y)$; now if I could replace this last term with $\phi_* y$ I'd be set, but it seems I'm no better off than where I started.

Any mindless thoughts on symbol-pushing?

Ted Shifrin

Not at this hour. Maybe tomorrow.

Mike Miller

Sure. Don't worry about it, this was just a vague curiosity.

Ted Shifrin

Night.

Mike Miller

Have a good morning.

SpawnKilleR

04:02

Yes, and this integral is convergent. — cosmoscalibur 5 mins ago

ops soz :(

0

Q: Calculation of integral including exponentials

So I have this integral \begin{equation} \int_1^{\infty}...\int_1^{\infty} \frac{ e^{min\{x_1,x_2,...,x_n,c\}}}{e^{x_1+x_2+...+x_n}} dx_1dx_2...dx_n \end{equation} where $c>1$. I thought that what I can do is break the integrals sequentially from 1 to $c$ and to $c$ to $\infty$. However that woul...

integration analysis

Anthony

@MikeMiller You thar?

Mike Miller

Depends who's asking.

Anthony

Your best friend that has a dumb question about Hilbert spaces <3

Mike Miller

I can think about it if it's not too long, but I'm working right now

Shoot

Anthony

Okay, I lied a bit, it's mostly about Banach spaces, and I think it's short.

Whenever my prof talked about maps from the space into the dual space, the "thing to show" was that the map was an isometry, and that it was onto.

Is the fact that a map like that is into obvious?

Mike Miller

04:09

Yup.

Anthony

Can you enlighten me as to why?

Mike Miller

Ah, there's the key question.

Anthony

In the case of a Banach space, is the functional we're considering integration against the element?

Mike Miller

This is more general. If I've got a distance-preserving map $f: V \to W$, where these are just normed spaces, what can you tell me about an element of the kernel?

Anthony

Is it pretty empty?

Mike Miller

04:13

Yes, but why? There's a pretty good reason.

anon

go the distance, anthony

Anthony

If it had an element, then that element would map to zero, and then it would have norm zero in both...

Mike Miller

There you go.

Anthony

@anon <3

@MikeMiller <3

anon

all I see are ice cream cones for two or else triangle butts

Mike Miller

04:14

triangular butts are in nowadays

Anthony

And the kernel only having zero means that the function is into?

Mike Miller

That's the definition of into.

If $f(v)=f(w)$, then $f(v-w)=0$, since we're working with linear operators.

Anthony

I should learn my definitions better.

pjs36

Well... "into" isn't really a thing, you probably mean "one-to-one" or injective, right? Or is "into" a thing in Banach space?

Anthony

Oh, sure, because our functions are linear, right?

pjs36

04:16

I wouldn't know, I've never been :P

Anthony

noooo that was to Mike.

Mike Miller

@pjs36 Some people say into for those words in other contexts.

@Anthony Yeah.

Anthony

Sorry, my internet and brain are lagging.

Thanks @MikeMiller.

pjs36

That's a new one for me. I'd call any map $f : V \to W$ a map "into" $W$, but to each their own.

anon

when you put things into other things you usually try not to break the thing you're storing

mike eustace

04:21

Hello, can anyone tell me what the empty box means after a set eg. Z+ = {1,2,3...10} []

the two square brackets in my text book look similar to the empty box I'm talking about.

Thank you

anon

screenshot, context, reference?

I assume you've seen this notation multiple times, not just once, and every time was right after a set?

mike eustace

It's an old school paper textbook. I've seen it multiple times. The full context.

"The set of inputs for this function is the set of integers {0,1,2,...60} and the set of outputs is {A,B,C,D,E,F}. []"

But those square brackets are a closed, empty box.

anon

square boxes usually mean conclusions

are you sure you've only seen them after sets?

look through the textbook

mike eustace

Just found one not after a set.
"...p^T = p so w = p as in the previous example. []"

Mike Miller

it means "end of proof", or generally, end of an argument, or similar

mike eustace

04:27

Ah, thank you both.

Balarka Sen

@MikeMiller trust me, i wasn't picking on you. my "should"s are all quantitatively milder.

Mike Miller

@Balarka: I wasn't offended.

Balarka Sen

Good, 'cause that wasn't my intention.

I couldn't go to school today since I slept at almost 4 o'clock :(

Remember me

04:42

Hi@BalarkaSen same here I slept at 5 o clock yesterday....when I woke up it was 8 in the morning.... :p

Balarka Sen

hi.

yeah, I woke up when I was supposed to wake up, but feeling too tired to do anything.

Remember me

I am feeling so drowsy..... Better part I won't have to give a talk on moles and all...so boring

user147690

I slept at 3 o clock, so I guess we need the next person to have went to sleep at 2 o clock

user147690

I feel great though, since I had coffee :)

Mikhail

05:08

Does a matrix whose rows are identical or columns are identical have a fancy name?

Balarka Sen

@MikeMiller Random thought : In the Galois correspondence of fields, the associated $\mathbf{Fix} : \mathbf{Grp}_{\text{Gal}(\overline K/K)} \to \mathbf{Field}_K$ functor and it's derived functor gives me Galois cohomology. In the Galois correspondence of groups, the associated $\mathbf{Fix}$ functor and it's derived functor tells me about group cohomology. What does it tell you about in the Galois theory of covering spaces?

I have a feeling that the right generalization would give me singular cohomology, but I have no idea. It doesn't make sense to talk about exact sequences in $\mathbf{Top}$

Mike Miller

Something is off here, because fields over K don't form an abelian category, so you can't take derived functors. But I don't know anything about Galois cohomology so I'm not going to think about this.

Balarka Sen

@MikeMiller don't they?

Mike Miller

No. What are your kernels?

user147690

@Mikhail What do you mean? Like it is a 4x4 matrix, and every row is a,b,c,d?

Mike Miller

05:16

OK, google says Galois cohomology is literally just group cohomology of Galois groups; $H^k(K'/K,M)$ is by definition $H^k(\text{Gal}(K'/K),M)$ (where $M$ is a $\text{Gal}(K'/K)$-module). But again, I don't know anything about this. (In the paragraph above I think I smell big words without much meaning. If you want to talk derived functors, learn derived functors.)

Balarka Sen

OK, I just know that I can get a long exact sequence trying to extend my left-exact honest-to-god sequence $1 \to \text{Fix}(A) \to \text{Fix}(B) \to \text{Fix}(C)$ to the right, and that's about it.

Mikhail

@AlexClark Like a matrix where the columns are identical, for example : [a,b;a,b;a,b]

ADG

hello math.stackexchange.com/a/1276651/67609

Balarka Sen

I'd love to learn some homological algebra, but I guess I just don't have enough background.

user147690

@Mikhail Why would someone care about such a matrix? where did you find this?

Mike Miller

05:19

Probably you do, but you've got a hundred other things you'd pledged to study. Pick one or two, study them in detail, 'finish' them, move on ...

user147690

@BalarkaSen And hurry up with making your blog while you are at it, so we can read some expositions

Balarka Sen

@MikeMiller yeah, true, but i just can't stop thinking about these. ignore me.

:P

no, @AlexClark

Mikhail

@AlexClark In my research! I have something like (I-NT)u=u where T is toeplitz: in CG-FFT N is the matrix

Mike Miller

Well, instead of thinking, do. If you want to try to get somthing to work, try it!

user147690

@Mikhail I can't find any documentation for a name and don't know one for it sorry

user147690

05:23

@BalarkaSen Where do you keep all of the things you learn? Solely in your head? e.g. are you documenting them on paper?

Balarka Sen

I write them up when I learn, but I can never find them afterwards.

user147690

@BalarkaSen That's exactly how I felt, and then I started throwing out all the paper I use, since I never refer back to any of it ever

Balarka Sen

Well, the good side of it is that I never needed any of those. I usually don't forget the stuff I have understood thoroughly.

user147690

@BalarkaSen Yeah I suppose that is somewhat true for me aswell, usually I forget aspects of it, but once it is learned thoroughly, I find it easy to restore

Balarka Sen

Right.

user147690

05:29

But it would be nice if you could share your insights... on a ... blog :)

Balarka Sen

No, no, no.

If you want to know my insights, ask me, I'll tell you if I have one.

user147690

Because it is too much effort?

Balarka Sen

But not a blog.

@AlexClark No, it's because I don't know anything interesting I could write on.

user147690

05:43

@BalarkaSen Do you have an Erdos number of 4 now?

Balarka Sen

Me? No!

Why in the world would I have any Erdos number.

user147690

@BalarkaSen You mentioned something about it a year ago, but I couldn't remember for sure

Mike Miller

You're confused, he said he was 4 degrees from Kevin Bacon

Balarka Sen

I don't recall, but I might have talked about something I and a guy did a few years ago. But that was elementary number theory, and I was kidding.

Mike Miller

Jesus christ who has a memory like this

Balarka Sen

05:46

Oh, yeah, that was it. I was obnoxious, sheesh.

user147690

@MikeMiller I just ran Erdos - Balarka on the search

Paul Plummer

He isn't a person, @AlexClark is just a computer on the other end

Mike Miller

Yeah but how did you remember he ever said that

user147690

@MikeMiller I was just thinking of how young Balarka is and how well he is doing haha - and I was remembering that he said he was publishing such a paper

user147690

I remember weird things lol

Balarka Sen

05:47

I could have, @AlexClark, but as I said : it was very elementary number theory.

It was on a generalization of the Liouville's identity and some bounds on the maximal element of tuples satisfying sum-of-cubes-equal-to-square-of-sum property, based on this paper.

user147690

@PaulPlummer Do I actually say these things often :P?

Mike Miller

yes

Balarka Sen

does he? I haven't noticed.

user147690

Nor have I, maybe the Chris'ssis things

Paul Plummer

@AlexClark You seem to pull up comments very quickly, and remember strange things

user147690

05:51

If only I remembered the concepts in math so well haha

Balarka Sen

i guess @AlexClark has psychic abilities.

user147690

@BalarkaSen That would be nice

Paul Plummer

So your super power is to remember obscure things that happen in mathematics chat room...

user147690

Unfortunately not a very helpful one lmao

Remember me

Let T be a linear operator of $\Bbb{C^2}$ defined by T(x_1,x_2)=(x_1,0) Let B be a standard ordered basis for $\Bbb{C^2}$ and let $B'={{\alpha}_1,{\alpha}_2} be a basis defined by ${\alpha}_1=(1,i),{\alpha}_2=(-i,2)$

WHat is the matrix of T realtive to the pair B,B'

user147690

05:54

That was when I was a lurker too, before I started chatting

Remember me

this is my first question so i want to check if i am right or not

Balarka Sen

I didn't know you used to lurk, @AlexClark.

But then, your username is changing frequently.

user147690

It was on an account none of you know was me that I started talking here

user147690

Can someone lecture me quickly on continuity of functions via topology?

Paul Plummer

Preimage of open sets are open... done

Balarka Sen

05:57

Functions such that pullbacking open sets give open sets.

user147690

Looks like we have a function from the topological space $(X,\tau_X)$ into the topological space $(Y,\tau_Y)$ and $f$ is continuous if for every $T\in\tau_Y$ the pre-image(?), $f^{-1}(T)\in\tau_X$

user147690

Okay that is what I thought I suppose

user147690

Since sets are open, if they are an element of the topology

Mike Miller

now why should that be what you call continuity?

user147690

Let me think

Mike Miller

05:59

think metric spaces

Balarka Sen

think about a sequence in a metric space with the points cluttering around a particular point.

mike eustace

06:17

Hello, I understand constant functions, linear functions and quadratic functions. Well, I understand constant functions give a constant output eg f(x)=5
Linear functions are of the form f(x) = ax+b
Quadratic functions are of the form f(x) = ax^2 + bx + c

But, why is f(x) = x^2 quadratic?

I mean, there's no + bx and there's no +c.

Is anything quadratic when it has a ^2 in it?

Thank you

Remember me

You can add something like this f(x)=x^2+0x+0@mikeeustace

That is still the same equation and also it is quadratic

mike eustace

Thanks. But it says a,b,c fixed numbers and a not equal to zero

ah! a = 1

I was being a moron thinking 0*5^2 = 25

Thank you @Rememberme

Remember me

@BalarkaSen what do you think about the question I just asked about the matrices one will the matrix be something like this:
(1,0)
(0,0)

That's the matrix ....

Np@mikeeustace

Balarka Sen

I can't read the question.

Remember me

Can you please scroll back a bit.....I just wrote the question

Balarka Sen

06:28

Yes, I know, but it's poorly formatted.

I can't decipher it.

Remember me

Let T be a linear operator of $\Bbb{C^2}$ defined by T(x_1,x_2)=(x_1,0) Let B be a standard ordered basis for $\Bbb{C^2}$ and let $B'={{\alpha}_1,{\alpha}_2} be a basis defined by ${\alpha}_1=(1,i),{\alpha}_2=(-i,2)$

WHat is the matrix of T realtive to the pair B,B

Balarka Sen

It's still written badly, not ok. So have you used the change of basis formula?

Remember me

Thats what i am thinking?

Balarka Sen

I am not sure what the question is, then.

Remember me

Let T be a linear operator of $\Bbb{C^2}$ defined by $T(x_1,x_2)=(x_1,0)$ Let $B$ be a standard ordered basis for $\Bbb{C^2}$ and let $B′={\alpha}_1,{\alpha}_2$be a basis defined by ${\alpha}_1=(1,i),{\alpha}_2=(-i,2)$

Now i think it is fine

Balarka Sen

06:35

@Rememberme $\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$ is the matrix of $T$ with respect to $B$, yes.

Remember me

okay thanks

Now if i have to find the matrix of T relative to B',B i have to use change of basis formula right @BalarkaSen

Balarka Sen

Yes.

Remember me

Fine peace!!!! Let me do the other questions.....

cap

can the dimension of kerA^2 be larger kernel of A?

Remember me

no

Wait....

Balarka Sen

06:38

@Rememberme Well, what is the matrix of $T$ w.r.t $B'$?

You haven't finished that yet.

Remember me

$\begin{bmatrix} 1 & -i \\ 0 & 0 \end{bmatrix}$ @BalarkaSen

no wait

Balarka Sen

@cap Hint : T be a linear operator on a vector space V. if something is in the kernel of T, then it is of course in the kernel of T^2. is the converse true?

Remember me

@BalarkaSen Is that right??

Balarka Sen

I haven't checked.

Remember me

@cap if rank(A^2)=rank(A) then $\text{ker}(A^2) \subset \text{ker}(A)$...

cap

06:50

got it, thanks fellas

Remember me

Well @BalarkaSen I am talking about the matrix of the pair B,B' and thats what i dont understand do i have to write the matrix for both B,B' individually or just for the pair B

Hi@TobiasKildetoft!!!

Tobias Kildetoft

@Rememberme Hi

Remember me

Hi@Chris'ssis

Balarka Sen

oops, sorry, was busy commenting on a question.

@Rememberme What is B? What is B'? Matrix of what?

Hello, @TobiasKildetoft

Tobias Kildetoft

@BalarkaSen Hi

Transcript for

$Mathematics$ Mathematics