Mathematics - 2014-07-27 (page 3 of 3)

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Alizter

23:00

then I claim its a field

not sure about that or anything

anyway

$D:F(\Bbb R)\to F(\Bbb R):x\mapsto x'$ is a surjection

Balarka Sen

@Alizter Isn't $\mathbf{Hom}(\Bbb R, \Bbb R)$ in general a field?

Alizter

Then I consider $F(\Bbb R)/\Bbb R$

@BalarkaSen I have no clue

Balarka Sen

i.e., functions from R to R?

no, of course it's not.

nor is F(R) one.

$f(x) = x^2$ is differentiable. The inverse $\sqrt{x}$ is not a function

Alizter

@BalarkaSen Who said function composition was an operation?

Balarka Sen

@Alizter your operation is composition right?

oh darn

Alizter

23:04

addition and multiplication

my stuff works up to here

Balarka Sen

Ah. then $\mathbf{Hom}(\Bbb R, \Bbb R)$ is a field too.

Alizter

We could probabalyly generalise later

Balarka Sen

i.e., just functions. no differnetiability.

Alizter

Anyway considering the cosets

Balarka Sen

@Alizter OK, continue.

user3222184

23:05

Alizter

Bascially meaning that $f+c=f+a$ even $c\ne a$

with me? @BalarkaSen

user3222184

$f+c=f+a$

Alizter

@BalarkaSen For $a,c\in\Bbb R$

Balarka Sen

this is getting confusing. continue from the beginning, @Alizter

in a formalized way

Alizter

also assuming that $\Bbb R \subset F(\Bbb R)$

@BalarkaSen I will need to decipher the rest

Ill formalise for tomorrow

Balarka Sen

23:07

what were you doing? F(R) \to F(R) right? what's the map?

Alizter

differentiation

user3222184

Balarka Sen

f(x) goes to f'(x)?

Alizter

yes

Balarka Sen

@Alizter So that F(R) of yours is functions from R to R with infinite differentiability, not just differentiability?

Alizter

23:09

No.

those are functions

Balarka Sen

then $f \mapsto f'$ is not defined.

Alizter

however in a way such that $\Bbb R$ is a subset

Balarka Sen

consider a function f which is differentiable but not twice differentiable.

f' isn't in F(R)

so f \to f' is not defined.

Alizter

yeah there are so many holes in this

@BalarkaSen I would stick to polynomial functinos

Balarka Sen

so you need F(R) to be elts in Hom(R, R) and infinitly differentiable.

@Alizter Then you just have R[X]

And no field.

Alizter

23:11

@BalarkaSen Told you its a bunch of crap

Balarka Sen

why not let F(R) to be functions which are infinitely differentiable?

Alizter

Or just consider something completely different

A limit for the nth derivative of a funciton :)

Balarka Sen

ok, so what next? you consider maps from F(R) to F(R). then?

(F(R) is now redefined with infinite differentiability)

Alizter

$\displaystyle f^{(n)}(x)=\lim_{\Delta x\to0}\frac1{\Delta x^n}\sum^n_{k=0}\binom{n}{k}(-1)^{n-k}f(x+k\Delta x)$

that is one of my nicer findings

proven and all

Balarka Sen

yeah, that should be obvious

Alizter

23:14

@BalarkaSen But yeah I think I was trying to set D up as an automorphism

around a field

then extend it with some fnuctions or something

GALOIS

Balarka Sen

D means differentiation?

Alizter

stuff

@BalarkaSen Yes I am lazy

Balarka Sen

ok so F(R) \to F(R) is an automorphism defined as f \to Df.

what next?

it is an automorphism.

Alizter

Pretend F(R) is the least set such that it is a field

Balarka Sen

uh oh. not quite.

Alizter

23:16

then I wanted to study extensions

Balarka Sen

it's the differential operator.

indeed you are kinda getting at differential extensions here.

keep it up and you'll get to differential galois theory quickly.

Alizter

in particular some interesting galois groups that arise from D over the polynomials extended with logarithms or some stuff liek that

Balarka Sen

@Alizter oh?

how do you define field extensions here?

just the usual notion?

but what about the differential?

Alizter

yes

logarithmic polynomials?

Balarka Sen

so you are extending something by logarithms.

Alizter

23:18

Yeah that would be ncie

Balarka Sen

right it makes sense.

Alizter

maybe extending rational functions would make more sense

Balarka Sen

@Alizter you have the field of rational functions over R.

@Alizter exactly what i was trying to tell

R(X) be your field. you are extending by logarithms. maybe call it R(X, log)? can you preserve differentials?

think about it. you'll make much more sense if you can develop this idea.

in fact, i'd ask you to forget about R. think about C.

Alizter

Difficult

Balarka Sen

well, C has much more structure.

especially in the sense of analytic structure.

Alizter

23:25

Well one preservation would be the solution of solving $f'+1/x=f$?

smashes head on table

I MUST SLEEP @BalarkaSen

YOU TOO

Balarka Sen

is it late there?

heya @blue

blue

heya

Balarka Sen

@Alizter you can define the differentials as d(log(x)) = dx/x. Simple enough, no?

1/x is in your F(R).

i think i really need sleep now. bu-byes.

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