Mathematics - 2016-11-13 (page 5 of 5)

Alessandro

23:02

Just curiosity, I was wondering wether there are spaces such that every open cover (not using the whole space as a set of the cover) is countable, or above some cardinal and I stumbled upon those other spaces when thinking about the former

(I quickly realized that there are no spaces like that, if a space admits a covering with open sets that doesn't use the whole space as a set it clearly has a finite cover of open sets too)

Mahmoud

Hello.

DHMO

hi

Mahmoud

Is $\{a\}=a$ ? Just wondering.

DHMO

no

Alessandro

No, it's a set containing $a$

DHMO

23:06

i think it is one of the axioms or something

Mahmoud

I read that $f(\{a\})=f(a)$

Alessandro

$\{\{3,4,5\}\}\neq\{3,4,5\}$, assuming I didn't mess up the brackets on my phone the one on the left has a single element and the one on the right has 3

That's correct, how is f(A) defined when A is a subset of the domain of f?

@DHMO it follows from the axiom of regularity (in ZFC)

DHMO

@aless i see

Alessandro

I have to go now, but if you recall that $f(A)=\{f(a):a\in A\}$ you should see why $f(\{a\})=f(a)$ @Mahmoud

Mahmoud

Thanks @Alessandro $:)$

saturatedexpo

23:18

@DHMO but isnt factoring out "hidden" devision? since $a+b=x(\frac{a+b}{x})$

@Mahmoud $\{$apple$\}$ is like a bag containing an apple. you wouldnt call a bag an apple or?

Mahmoud

No, it's not the same thing,

saturatedexpo

exactly ;)

Mahmoud

Thanks :D

DHMO

@saturatedexpo no it isn't

you just need to use distribution, which is one of the axioms

saturatedexpo

ah ok ;)

the "gathering" haha

Mahmoud

23:28

This function is strange $f : \mathbb R \rightarrow \mathbb R | x^2+y^2-2\sqrt{xy}=0$

saturatedexpo

edit the / to a \

DHMO

strange is subjective

Mahmoud

It shows the symbol of infinity :D

DHMO

can you prove that it is a function? @Mahmoud

saturatedexpo

@DHMO exactly what i thought

Mahmoud

23:31

It takes elements from $\mathbb R$ And maps them in a way expressible with the equation above. Right ?

DHMO

no

Mahmoud

I didn't know if it maps into or onto

saturatedexpo

the normal notation is: $a\mapsto b$ in plaintext: a\mapsto b

Mahmoud

Isn't it a function then ?

saturatedexpo

(for the elements)

Mahmoud

23:32

But we use $\rightarrow$ for the domain and range. Right ?

DHMO

@Mahmoud the first statement is false so the second statement does not follow

saturatedexpo

yes!

DHMO

the notation is wrong then

Mahmoud

What statement it's just the definition of a a function.

saturatedexpo

@DHMO well, some profs are vague on that, $f:R\to R$ doesnt mean necassary for them that its bijective

arctic tern

23:33

presumably mahmoud is considering the set of all ordered pairs for which the relation holds. the first thing to do would be to investigate if every x value has one and only one corresponding y value.

Mahmoud

Yes, @arctictern I didn't know how to express $y$ as an expression of $x$.

arctic tern

based on W|A's plot, y is not a function of x

in any case you can solve for y in terms of x with algebra

move the radical to one side, square, solve the polynomial equation

saturatedexpo

@arctictern by symmetry its also no function for y right?

so no function

DHMO

x^4+2x^2y^2-xy+y^4 = 0 then

arctic tern

well, "y is not a function of x" does not imply "x is not a function of y," but the graph in this case indicates neither x,y is a function of the other

Mahmoud

23:37

Why ?

DHMO

@arctictern by symmetry

since x and y are interchangeable

y is not a function of x iff x is not a function of y

Mahmoud

What do we call this now ?

saturatedexpo

relation

since its a subset of RxR

arctic tern

@DHMO this statement is false for relations. for instance the graph of x=constant has y not a function of x, but x is a function of y.

Mahmoud

Thanks.

DHMO

23:38

@arctictern x and y are interchangeable here. he said "by symmetry"

arctic tern

now, if you or saturated meant symmetry in the original equation...

DHMO

yes

Ted Shifrin

Heya tern!

arctic tern

heya

saturatedexpo

its all cool guys

Ted is here

Mahmoud

23:39

Hello @TedShifrin Long time no see :D

Ted Shifrin

Not cool :)

saturatedexpo

lets throw a party, i order 1 beer

Ted Shifrin

Hi @Mahmoud.

If the beer is for me, I'll take a martini :)

DHMO

how to show that $\Bbb R$ only has 2 ideals?

arctic tern

it's a field

Alessandro

23:40

It's a field

saturatedexpo

its the preciousest of rings!

DHMO

how to show that a field only has two ideals?

arctic tern

suppose I is an ideal that contants a nonzero element. then think about this.

Alessandro

You show that in a ring $R$ if and ideal $I$ contains a unit then $I=R$

Ted Shifrin

I'll chime in: It's a field. ... Precious, but boring as a ring!

Alessandro

23:41

And you get the fact about fields as a consequences

Good evening @Ted!

Ted Shifrin

Hi @Alessandro

Mahmoud

I have a long way to go in order to understand this.

DHMO

@Alessandro how would i show that?

@Mahmoud what?

Ted Shifrin

Definition.

arctic tern

@DHMO review the definition of ideal and unit

DHMO

23:43

ideal is absorber right

Ted Shifrin

Can't study math without truly knowing the definitions!

Alessandro

Start with the easy case where you have the identity in the ideal instead of a generic unit

arctic tern

it absorbs ambient multiplication, yes

DHMO

unit is exists multiplicative inverse

Alessandro

That should show you how to prove the general case too

saturatedexpo

23:43

@TedShifrin or at the very very least having them somewhere ready?

DHMO

@Alessandro oh! thanks

Ted Shifrin

No, KNOW.

saturatedexpo

that is a bold statement, i follow

Ted Shifrin

@Alessandro, ages ago we had an unresolved question, but I don't recollect.

DHMO

is it true that every nonzero element in a field is a generator?

arctic tern

23:46

what do you mean by generator?

Balarka Sen

Hi @Ted

Ted Shifrin

What do you think?

DHMO

i suspect yes

saturatedexpo

is an element in R^2 of the form (a,b)?

arctic tern

if you mean Fx={fx: f in field}=F for all nonzero x in F, then yes

Ted Shifrin

23:46

Good night, Balarka!

DHMO

nice

Alessandro

I'm full of unresolved questions @Ted, do you remember the topic? I remember discussing minimal generators of $\sigma$-algebras with you but maybe it wasn't that?

Balarka Sen

@Ted: Back home, finally?

Ted Shifrin

Oh yeah ... what did you decide, @Alessandro?

@Balarka: No, driving all day tomorrow.

Balarka Sen

Ah

DHMO

23:48

1 in L -> forall x in R x•1 in L -> R=L

Alessandro

That I can't really understand the paper answering my question right now, but it's on my "list of things I should read"

DHMO

u in L -> u/u in L -> 1 in L -> R=L

Ted Shifrin

Ah, and our next query? :)

Better to write $u^{-1}$ for $u\ne 0$.

DHMO

what would a field look like if the additve inverse doesn't exist?

@TedShifrin i like to abuse notations

arctic tern

a division semiring

DHMO

23:50

i see

Ted Shifrin

Your teacher doesn't like it.

Alessandro

I want to prove that $|\bar{F}:F|=\infty$, where $\bar{F}$ is the algebraic closure of $F$, implies that the degree of the irreducible polynomial in $F[x]$ is unbounded (Mike says it's true) but I'm not sure where to begin @Ted

DHMO

i am my teacher

arctic tern

I can't think of any examples that are not subsemirings of [0,infinity) though

DHMO

@arctictern then use ordinals?

arctic tern

23:52

dunno how you intend to use them

DHMO

never mind, just use the complex

C\0 right

arctic tern

in C you get rotation around 0, so you'll end up with additive inverses just using multiplication

C-{0} is not a semiring, they need 0

DHMO

what? no way

Alessandro

However I should really go to sleep now since I have lectures early in the morning, goodnight @Ted and everybody

arctic tern

in any case, C\{0} is not even closed under addition, which is also a requirement

DHMO

23:53

you can't get 0 from non-zero

arctic tern

1+(-1)

Balarka Sen

@DHMO 2 + (-2)

Ted Shifrin

@Alessandro, I don't know what you mean by "the irred poly".

DHMO

in that case

Ted Shifrin

Night!

DHMO

23:54

what would be an example?

Danu

Ed van der Elsken - Een fotograaf filmt Amsterdam - My Amsterdam (1982)

►

Hi @TedShifrin!

Ted Shifrin

Hi @Danu.

arctic tern

of a division semiring that is not a subsemiring of [0,infinity)? I can't think of any. But if you want less interesting examples, just intersect [0,infinity) with any subfield of R.

DHMO

Q>0?

why do we need 0?

arctic tern

definition

DHMO

23:55

??

oh, identity

Danu

@TedShifrin Is there any shortcut to computing Todd classes? Even doing the first one was already a pain.

arctic tern

you have the ability to look up "semiring" on wikipedia.

might as well look up the def of things if you're going to talk about them

DHMO

maybe

Ted Shifrin

Nope.

Danu

Sadface.

Ted Shifrin

23:56

Formal power series computation.

Danu

Yeah, that's what I did.

DHMO

what would a field with 3 operations look like?

Ted Shifrin

But look at Hirzebruch for all such matters.

DHMO

i know exponent is not commutative

Danu

@TedShifrin Do you think it might be a good idea to read that book at some point? It's referred to all the time...

DHMO

23:57

so basically (R,*,^) is a ring right

Danu

In particular, nobody seems to be willing to prove HRR :D

Alessandro

I meant irreducible polynomials by the way, "the" was not the right article there

arctic tern

@DHMO that question is ill-defined. without saying anything about the third operation, it's just a field with a completely random and unrelated third operation on the field's underlying set.

Ted Shifrin

Look at it ... don't read it thoroughly.

DHMO

@arctictern vague questions make for creative ideas

but anyway

3 abelian groups?

Danu

23:58

Alright.

DHMO

like (R,+) is an abelian group

(R,*) is an abelian group

Ted Shifrin

Oh @Alessandro; I'll ponder that on my 8+ hour drive tomorrow.

arctic tern

uselessly vague does not make for creative ideas, they make for frustration in getting someone to add more definition to the question

Danu

In other news, I realized that I don't really know how to induce connections on $\bigwedge^k E$. Can you give me a hint? I was thinking perhaps I can just antisymmetrize the definition for the tensor product somehow?

DHMO

(R,^) is an abelian group

arctic tern

23:59

(R,^) is not a group

DHMO

and then we can explore relations between the three operators

look, ^ is another operator

R is not the reals

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