Mathematics - 2016-10-23 (page 2 of 5)

Mahmoud

10:08 AM

Oh thank you @Fargle

user228700

Hi everyone :-)

Mahmoud

Hi @Kaumudi

user228700

I've a quick geometry doubt...

Mahmoud

My level isn't good enough as far as I know, but do it, test me :D

user228700

I'm trying to find the length of the chord of contact of a circle (ie. The length of the chord connecting two points through which, if tangents are drawn, meet at a common point P $(x_1,y_1)$. I've been given that point P, and the equation of the cir le.

user228700

10:16 AM

Here, I've drawn the diagram:

user228700

user228700

Oh, shucks, I forgot to label things; $L$ is the length of both tangents (equal as they are, in length) and $R$ is the radius of the circle.

user228700

I think it would be best to work with the general equation of a circle given by $x^2+y^2+2gx+2fy+c=0$

user228700

Where the center of the circle is given by $(-g,-f)$ and the radius by $\sqrt{g^2+f^2-c}=R$

user228700

I have to use geometry to find the length of the chord of contact, whose equation is give by $xx_1+yy_1+(x+x_1)+(y+y_1)+c=0$

user228700

10:24 AM

And you see, I kind of suck at geometry, so...anybody?

user228700

@Mahmoud: ..? (Thought u said u'd try..? :-P)

user228700

Sigh. Gots to try till I die, I guess...

Mahmoud

10:55 AM

I wasn't able to do it @Kaumudi

user228700

Oh :/ Thanks for trying :-)

Mahmoud

11:07 AM

Anyone able to help ?

Mahmoud

12:01 PM

Anyone wanting to have a brief discussion about proofs ?

Martin Sleziak

@Kaumudi If you know a bit about Power of a point, it should be relatively easy.

Mahmoud

@MartinSleziak I'm having a math exam tomorrow, it's about Discrete Math class, and I was taught proving techniques such as proof by contradiction, contrapositive proof, proof by construction, direct proof, equivalence proof etc.

Martin Sleziak

Basically, what is says it that if the line PS intersects the circle in the points A, B and if T is the tangent point then $|PT|^2=|PA|\cdot|PB|$.

Mahmoud

I know how the proving system works and I can understand the sample proofs in my text to a sufficient extent. However, whenever I tried proving on my own, I got stuck, with no advancement of ideas in my head. How do you remedy this solution? Should i practise proving as much as possible?

Martin Sleziak

That's difficult to say.

One thing that would be doable would be to try to write some proof and have somebody near you who can point out when you do something wrong and why it is wrong. Or who could give you small hints.

However, I am not sure whether this is doable via chat. And I do not really have that much time today. (I have stuff to do before tomorrow too.) Sorry.

Maybe you will find some posts on the might that might help you with your situation @Mahmoud.

How to be good at proving?

What should I learn to increase my skill to find proof?

How to learn developing the formal proof of statements or theorems?

And you can probably find more similar posts.

Mahmoud

12:28 PM

Thank you @MartinSleziak

user228700

12:40 PM

@MartinSleziak Huh. I do know about this. Will try to do it using that concept, thanks :-)

Hiroto Takahashi

1:33 PM

Hi there

Mahmoud

Hello ?

Hiroto Takahashi

somene can help me to understand this notation..

let $g'$ be some distribution on the sphere $C^{- \infty}(\mathbb{S}^{n-1})$

why $-\infty$¿

?*

is that mean that $g'$ is a fast decreasing function?

s.harp

@DHMO I computed the residue in the message below the one where I pinged you, or what do you mean with "how to do the question"?

DHMO

@s.harp i have read that, thanks

Balarka Sen

1:49 PM

@Kaumudi Don't you mean $xx_1 + yy_1 + g(x+x_1) + f(y+y_1) + c = 0$?

Nevermind that, I misunderstood.

Simple Art

2:11 PM

@DHMO 24/7 chatter?

DHMO

@SimpleArt maybe

Mahmoud

:D

Simple Art

:P

@Mahmoud $\aleph_\omega$?

Mahmoud

I have a hard time raping my mind around that for now.

Simple Art

@Mahmoud Consider the sequence $1,2,3,\dots$

Mahmoud

2:25 PM

$Aleph_0$ Is enough for now :)

Yes please

Simple Art

Then consider the sequence $1,2,3,\dots,\omega$

Similarly, consider the sequence $\aleph_0,\aleph_1,\aleph_2,\dots,\aleph_\omega$

I only know about such things to know the difference between transfinite and absolutely infinite

That is, I do believe any thing like $\aleph_{2\omega}$ is not actually larger then $\aleph_\omega$

Ofc, we have other amazing things

Mahmoud

But what makes $ℵ_0 and ℵ_1$ different @SimpleArt

Simple Art

@Mahmoud That $\aleph_2$ is larger then them

$$\huge\aleph_{\omega_{\omega_{\omega_\ddots}}}$$

Mahmoud

How ? they are both infinite ?

Simple Art

@Mahmoud Uh, please review what $\aleph_1$ is?

Ok, so if $\aleph_0<\aleph_1$, then $\aleph_0$ isn't infinite in the sense that nothing is bigger than it

The same argument follows to $\aleph_1<\aleph_2$

DHMO

2:31 PM

@SimpleArt "infinite" doesn't say that "nothing is bigger than it"

Mahmoud

And this ! $$\huge\aleph_{\aleph_{\aleph_{\aleph_{\aleph_{._{._{.}}}}}}}$$

DHMO

mathworld.wolfram.com/Infinity.html

Simple Art

@DHMO Ofc, it depends on your definition of infinite

DHMO

> Infinity, most often denoted as $\infty$, is an unbounded quantity that is greater than every real number.

@SimpleArt for what I know, no mathematicians use your definition

Simple Art

@DHMO shrugs

Tobias Kildetoft

2:32 PM

@DHMO Indeed, since that would mean we had no infinite things at all, which would be impractical

Simple Art

Absolute Infinite

The Absolute Infinite is mathematician Georg Cantor's concept of an "infinity" that transcends the transfinite numbers. Cantor linked the Absolute Infinite with God. He held that the Absolute Infinite had various mathematical properties, including the reflection principle which says that every property of the Absolute Infinite is also held by some smaller object. == Cantor's view == Cantor said: The actual infinite was distinguished by three relations: first, as it is realized in the supreme perfection, in the completely independent, extrawordly existence, in Deo, where I call it absolute infinite...

Mahmoud

George Canto did @DHMO

Simple Art

Transcends the transfinite lol

DHMO

@Mahmoud then I take back my claim about mathematicians

Tobias Kildetoft

I never find it particularly relevant what mathematicians of that long ago thought.

DHMO

2:33 PM

@TobiasKildetoft exactly

Simple Art

@DHMO Well, that's a different area of math. We're talking set theory

Mahmoud

Mathematicians at that time didn't like what Canto suggested. @DHMO

DHMO

@SimpleArt you just said that it is not true that $\aleph_0<\aleph_1$...

Tobias Kildetoft

math has evolved in the mean time, so their view of it is no longer relevant

Mahmoud

They humiliated him.

Simple Art

2:34 PM

@DHMO No, I said that if $\aleph_0<\aleph_1$, then $\aleph_0$ is not infinite in that sense.

Tobias Kildetoft

@SimpleArt no set is infinite in that sense, which makes it not a very useful sense

Simple Art

You imply that it is indeed infinite, thus coming to the conclusion that $\aleph_0\require{cancel}\cancel<\aleph_1$

s.harp

smbc-comics.com/index.php?db=comics&id=2047#comic

Mahmoud

Yes but what makes it different

Simple Art

@s.harp IMO, this is all just for trivia

Tobias Kildetoft

2:36 PM

@SimpleArt No, we are saying that your notion of what it means to be infinite is nonsense

s.harp

(actually Mahmoud's comment reminded me of that comic)

Mahmoud

Vsause did a great video about this, anyone wanting to see it ?

Simple Art

Actual infinity

Actual infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics, the abstraction of actual infinity involves the acceptance of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given objects. This is contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces an unending "infinite" sequence of results, but each individual result is finite and is achieved in a finite...

:/

@Mahmoud Yeah, I like the animations

Mahmoud

:D

Ok I got it now @SimpleArt

Now to the greatest unanswered problem in Mathematics...

Simple Art

@Mahmoud ?

Mahmoud

2:40 PM

$$\text{Continuum hypothesis}$$

Simple Art

The greatest unanswered problem is "should I really be learning complex analysis before real analysis?"

DHMO

@SimpleArt your point being?

Simple Art

@Mahmoud \text{Con...}

Tobias Kildetoft

How is the continuum hypothesis an unanswered problem?

Simple Art

@TobiasKildetoft ......

Mahmoud

2:41 PM

That was embarrassing

We can prove that :

1. We can never prove that it's true.

2. We can never prove that it's false

.

DHMO

Continuum hypothesis

In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: There is no set whose cardinality is strictly between that of the integers and the real numbers. The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. Τhe answer to this problem is independent of ZFC set theory (that is, Zermelo–Fraenkel set theory with the axiom of choice included), so that either the continuum hypothesis or its negation can be added as an axiom to...

Simple Art

@TobiasKildetoft Second paragraph

mathworld.wolfram.com/ContinuumHypothesis.html

DHMO

> either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent

Mahmoud

Can you explain it to me ?

DHMO

Does that mean that there is no answer?

Simple Art

2:43 PM

@Mahmoud Basically, we proved that if we look at it one way, its true, but looking at it another way, its false

Mahmoud

So mathematics came to a limit ?

Simple Art

by looking at it, I mean using different axioms

DHMO

@Mahmoud look, that goes down to the essentials of what mathematics is

the difference between mathematics and science

Mahmoud

Okay that's better than a contradiction.

Studentmath

@DHMO No, they only stress in that line that it is independent of ZFC

Simple Art

2:44 PM

@Mahmoud The question becomes which axioms are better, which we don't know yet.

Mahmoud

Can you show me the axioms ? @SimpleArt ?

Studentmath

I.e. assuming it's true or false will not affect the consistency of ZFC..

Simple Art

@Mahmoud Not my field

But the link i gave

Mahmoud

Wikipedia ?

Simple Art

wolfram

DHMO

2:46 PM

@Studentmath uh... then what's the answer?

Mahmoud

Ok then

I'll be right back.

Simple Art

@DHMO He basically said the answer doesn't have an impact of ZFC

DHMO

@SimpleArt but wolfram also says that it is undecidable

Balarka Sen

@DHMO What do you mean by "answer"?

Studentmath

42

3

DHMO

2:47 PM

@BalarkaSen touche

Balarka Sen

lol, @Studentmath

Studentmath

@Balarka :)

Simple Art

I should spend more time in the chat for this lol

Studentmath

I have a somewhat less philosophical question

Balarka Sen

The answer to every question.

Studentmath

2:48 PM

@Balarka I never had the courage to write it down in an exam where I didn't know the answer

Simple Art

lol

Studentmath

Usually it was in some literature or some other thing where I knew I could get most of the points by just writing random unrelated things that sound impressive..

Balarka Sen

I once trolled in an exam, but that was a political one, on a geography exam in high school.

Simple Art

hehe

Studentmath

Political geography exam? actually that makes sense, yeah

Simple Art

2:49 PM

I'm gonna try to integrate contour integrals into my improper integrals test

Balarka Sen

No, I mean, the exam was on geography, the troll was political.

Studentmath

@balarka lol

Simple Art

@BalarkaSen The jokes are bad

Mahmoud

:D

BrB

Simple Art

@Mahmoud You keep saying that, but I don't feel like your leaving

Studentmath

2:52 PM

Say I have some field A, and its extension B. also [B:A]=3. So I know that for every $t\in B, t\notin A$ $A(t)=B$ and there is the minimal polynomial $p(x)$ of degree 3 where $p(t)=0$.

Balarka Sen

for the record, I marked out the bits of India which are taken by two neighboring countries (which is obviously a controversial matter).

Simple Art

XD

Studentmath

Did the teacher like it?

Balarka Sen

Fortunately, yes, he laughed a lot. But he's a friend.

Studentmath

So, I take some member of $B$, $b$. Can I have some certain combination, using members of A, so that for every $t\in B, t\notin A$ I can express $b$ with said members and $t$?

If $b$ is a member of $A$ too, it's trivial. Assuming it's not, I am given some formula of expression that supposedly holds and I have no idea how they got to it

Balarka Sen

2:57 PM

@Studentmath Does that allow powers of $t$? Then that's precisely what the primitive element theorem says, i.e. pick $t$ so that $A[t] = B$.

It holds for any separable field extension of finite degree.

Simple Art

math.stackexchange.com/users/-1/community

we have that? ^^

DHMO

@SimpleArt it's kinda a system thing

Simple Art

creepy...

DHMO

it is who theoretically controls the community wikis

and who marks spam answers/questions as spam

user3502615

Hey can anyone help me with a small little thing I have to show for a proof of mine?

Studentmath

3:00 PM

@Balarka thanks, I am looking at that now.. maybe that will get me to the formula they wrote

Balarka Sen

Creepier yet, there is a chat-bot in the sci-fi stackexchange chat, who even posts .gif's when told to

3

user3502615

Given group $G$, identity element $e$ and non-identity element $g$, suppose that $g^n = e$ for some integer $n$. Then the order of $G$ divides $n$

that's the question;

for my proof, i have to prove that $g^{-|g|} =e$ given that $g^{|g|} = e$

Balarka Sen

@user3502615 That's not true... take $\Bbb Z/2 \times \Bbb Z/2$. Any non-identity element has order $2$, but the group has order $4$.

What you probably mean is that $n$ divides $|G|$, not the other way around

DHMO

@user3502615 multiply both sides by $g^{-|g|}$?

user3502615

Balarka Sen what is $\mathbb{Z}/2$?

Balarka Sen

3:03 PM

In which case, it's Lagrange's theorem

@user3502615 cyclic group of order 2

user3502615

yeah

wait one second

@BalarkaSen

Shay Ben Moshe

Hi, are there any cases where two spaces can't be distinguished by their ordinary cohomolgy, but can by some other cohomology theory?

user3502615

I meant the order of $g$ not $G$

Balarka Sen

@ShayBenMoshe What's "ordinary cohomology"?

Shay Ben Moshe

H^*(X,Z)

Integer cohomology ring

atiyah hirzebruch suggests that other cohomologies are determined by ordinary cohomology, but that's not quite right since the differentials can be different for different spaces

Mike Miller

3:14 PM

@ShayBenMoshe Right, things are not determined by ordinary cohomology. Depending on what you mean by 'cohomology theory', it's true but not exciting that if you have a map $f: X \to Y$ that induces an isomorphism on every cohomology theory, $f$ is a homotopy equivalence. It's also uncheckable.

Balarka Sen

@ShayBenMoshe Take $\Bbb{CP}^2$, glue a 3-cell by a degree $2$ map to the 1-skeleton, and take $M(\Bbb Z/2, 2)$ wedged with $S^4$.

These have isomorphic $\Bbb Z$-cohomology, but not $\Bbb Z/2$ cohomology.

Mike Miller

to the 2-skeleton, not 1-skeleton

Balarka Sen

Sorry, yeah.

Mike Miller

nice example tho

Balarka Sen

It's in Hatcher, I just had trouble remembering it. It's a modification of the $\Bbb{CP}^2$ and $S^2 \vee S^4$ example mod 2.

Shay Ben Moshe

3:20 PM

Interesting, do you happen to remember where it is in Hatcher?

Balarka Sen

I can't find it anymore, unfortuntely.

It's certainly in the chapter 3 somewhere.

Shay Ben Moshe

3:31 PM

Well, I've searched for the string "moore space" and didn't find it :/

Balarka Sen

Odd.

Mike Miller

$\Sigma\Bbb{RP}^2$ is a Moore space of the type Balarka wants. So. You could just use that.

What I'd like is an example where the cohomology rings with respect to all coefficients agree but the K-theory or bordism groups or something don't. But I don't really know how to calculate the latter.

Balarka Sen

Are all homology spheres nullcobordant?

Mike Miller

I'd like my example to be simply connected.

Balarka Sen

Interesting question.

Shay Ben Moshe

3:45 PM

Maybe example 6 can provide an example: mathoverflow.net/q/55365/33121

(example 6 in the question itself)

Studentmath

@Balarka thanks, I think I figured it out. They claimed every member can be expressed as the division of two linear expressions of $t\in B/A$. So I think the proof is not that hard, I have the primitive element theorem stating $b=a_1t^2+a_2t+a_3$. So I have to show $(x_1+x_2t)/(x_3+x_4t)=a_1t^2+a_2t+a_3$ where $x_i,a_i\in A$

I open it up and I get a 3-variable 3-equations linear thingy. I do the det, and show it can't be 0.

Mike Miller

@ShayBenMoshe Good call.

I was hoping to get examples that looked like $\Bbb{CP}^n$, but any simply connected closed manifold with the same cohomology ring as $\Bbb{CP}^n$ is homotopy equivalent to it.

(You could presumably work with things with higher-dim cells that aren't homotopy equivalent but that's not my domain.)

Shay Ben Moshe

@MikeMiller I myself don't understand it :)
Do you know anything about the K theory of $B(S^1\times SU(n-1))$ there?

Mike Miller

Let me think.

ncatlab.org/nlab/show/topological%20K-theory lol at motivation

Shay Ben Moshe

(The K theory of BU(n) is on wikipedia, though I don't know how it's derived)

Mike Miller

3:55 PM

It's likely done using the Atiyah-Segal completion theorem.

That or the AHSS.

Shay Ben Moshe

The completion theorem is what I'm trying to understand these days, and the AHSS is what I learned yesterday, so that's nice

What did you see in the motivation at nlab?

Mike Miller

Not look at, but lol at

Ted Shifrin

G'night, @MikeM

Shay Ben Moshe

Actually, it seems reasonable that the completion theorem would differentiate them

But probably hard to calculate, maybe for n=2 or something

Mike Miller

Yeah, I'm trying n=2.

But I'm a little worried that they're actually the same.

Keep in mind that there is an n-fold cover $S^1 \times SU(n) \to U(n)$. Does the representation ring differentiate between covers?

Shay Ben Moshe

4:00 PM

R and S^1 have different rep rings

Mike Miller

Finite covers, I mean

Shay Ben Moshe

though R is not compact so...

Mike Miller

Do SO(3) and SU(2) have different rep rings?

I should know this but I don't.

Shay Ben Moshe

I have no idea

Well, I'll think about it too

Pichi Wuana

In this plot I have two equations of two respective triangle sides. The equation between the blue and the red point is an altitude to side BC. The red point is the orthocenter. Point B is $(-5,4)$. How can I find side $AC$ / point $A$ / point $C$? Thanks if someone could help me!

Mike Miller

4:02 PM

@ShayBenMoshe Have you been around before? I'm surprised we haven't talked.

Hi @Ted.

Shay Ben Moshe

Yesterday was my first time actually

I asked for a reference for computing the cohomology of U(n) using serre ss

Mike Miller

Ah, and I gave you an incorrect approach. :P

Shay Ben Moshe

I've done it using U(n-1)->U(n)->S^2n-1

Hiroto Takahashi

Hi someone can tell what does $g \in C^{- \infty}(\mathbb{S^{n-1}})$ mean? I was thinking that $g$ is a fast decreasing function on a sphere, but not sure..

Shay Ben Moshe

Well, I'm gonna go now, see you around @MikeMiller

Mike Miller

4:06 PM

@ShayBenMoshe I think they do disagree. Look at the original Atiyah-Segal paper for inspiration.

See ya.

Ted Shifrin

I have no idea what that would mean, @Hiroto.

Hiroto Takahashi

@TedShifrin alright, it's fine.. that thing that I don't understand is $- \infty$

Ted Shifrin

Either it's a typo or there's an explanation somewhere in what you're reading.

Balarka Sen

4:30 PM

Hi @TedShifrin

Ted Shifrin

Hi @Balarka

Balarka Sen

@MikeMiller You mentioned a map which is isomorphism on every cohomology theory is a homotopy equivalence. How does the proof go? something something Brown representability.

Mike Miller

@BalarkaSen [X,Z] is a cohomology theory.

If you really want to be able to add things, [X, \Omega Z] is a cohomology theory.

Balarka Sen

For all Z? Hmm.

Ah, yes, it is.

Mahmoud

Hi

:)

Studentmath

4:47 PM

Say, I can't necesserily claim that if $a^3=b^3$ then $a=b$ in every field, right?

Balarka Sen

Take the finite field of order $3$.

Studentmath

@Balarka cheers

Mahmoud

Field ?

Balarka Sen

For positive characteristic take $\Bbb Q(\zeta)$ where $\zeta$ is one of the two complex solutions of $z^3 = 1$.

Studentmath

@Mahmoud do you know rings?

@Balarka thanks again

Balarka Sen

4:50 PM

Sure, no problem

Mahmoud

@Studentmath No :(

They are a mathematical object just like numbers ?

Studentmath

@Mahmoud well, they are algebraic structures - Think of the Natural numbers

I don't get it

@Studentmath!

@TedShifrin !

4:53 PM

Hi @Mahmoud

Mahmoud

Hi :)

Studentmath

We try to define it a bit more 'abstractly' - it's a Set of positive integers (and 0), where we have defined some operation - say addition

Mahmoud

Do we add them all at once ?

Studentmath

We take things like that and we try to generalize them - then you get to algebraic structures. I think.

@Ted !!!

Nah, it's not a number or something of that sort - it's a set of numbers.

I think wikipedia actually deals with it well @mahmoud, the explanation there is nice and you can go down the rabit hole with the definitons

Mahmoud

@TedShifrin I'm trying to change the path of my life and be part of the Mathematics community, the question is where do I start ?

Studentmath

4:55 PM

@Balarka sadly that makes my proof more annoying. I hate not being able to jump into false conclusions

Mahmoud

Ok @Studentmath :)

Ted Shifrin

You are in fact good at false conclusions, @Studentmath :)

Balarka Sen

@Studentmath Haha, I sympathize.

Ted Shifrin

@Mahmoud: Just keep learning.

Studentmath

@Ted haha it's my specialty :)

Ted Shifrin

4:56 PM

Well, it's hard to narrow down your specialty to just that. :P

Dragon

Good, morning, i just want to ask quickly, does anyone here know how to find the formula for the length and width for the A series paper size? or do i have to ask it as a question.

Mahmoud

What field is worth learning in my free-time @TedShifrin ?

Ted Shifrin

Google is your friend, @Dragon.

Dragon

@TedShifrin i can find the formula but none of the pages shows how they've got the formula.

user227867

Hello @TedShifrin. Do you have any book recommendation for the calculus of variations? I am thinking of getting Gelfand and Fomin.

Ted Shifrin

4:58 PM

What do you mean by formula? It's two numbers.

I haven't studied them carefully, Jasper. There's also Bliss. I personally am partial to a little book by Phil Griffiths on Calculus of Variations and geometric problems. Of course it's all using differential forms.

user227867

It's one of those obscure topics not often taught in schools.

Transcript for

$Mathematics$ Mathematics