Mathematics - 2017-02-28 (page 1 of 5)

Topologicalife

00:01

@AkivaWeinberger it has two real roots since $f(0) = -1$ and because we showed that $f'(x) > 0$ for all $x$.

Right?

Hi @Semiclassical

Ted Shifrin

00:15

@Topologicalife: That looks like a Spivak question. Think using Rolle's Theorem to show at most two roots.

Hi @Semiclassic, DogAteMy

Topologicalife

Yeah @TedShifrin

But he already solved it.

I was asking just a detail about his/her proof.

Ted Shifrin

Oh, ok.

Topologicalife

I think because we showed that all the roots are in $[-1,1]$, and $f(0) = -1$ and $f$ is increasing for all $|x| > (0,1)$, then $f$ must have two roots.

It makes sense on my mind :D

Err.

I messed up that msg.

I meant: I think because we showed that all the roots are in $[−1,1]$, and $f(0)=−1$ and $f$ is increasing for all $x\in(0,1)$, then $f$ must have two roots.

Maks

@TedShifrin Could you explain a theorem to me ?

More than the theorem, a consequence of it

Ted Shifrin

Like what?

Maks

00:26

There's a theorem of subspaces

Ted Shifrin

I dunno what you're referring to.

Maks

that say is V and W are subspaces with dimention $n < \infty$
Such that $ V = \langle b_i \rangle$ and $ W = \langle c_i \rangle$
Then there's only one linear transformation T:V -> W such that $T(b_1) = c_1$ for all $i$

Ted Shifrin

Those are bases for the respective subspaces. OK, sure.

Zach Hauk

hi!

Maks

With that theorem

If they gave you any linear transformation

Ted Shifrin

00:29

hi @Zach

Zach Hauk

just had some really good meatballs

Maks

And the vectors they give you on the domain are a basis of all the space on the domain, then,by theorem, you know its a LT

For example

Ted Shifrin

Well, no, you define it on other vectors to make it be a LT

Maks

If I'm asked if there is a LT such that $T(1,0) = (1,2,3)$ and $T(0,1) = (0,4,2)$ by theorem I can say yes, because $(1,0)$ and $(0,1)$ are a base of $\mathbb{R}^2$

Zach Hauk

@Ted Akiva told me how he took the AP calc BC test right before high school

Ted Shifrin

00:31

Yeah, actually, you need the $v_i$ to be a basis for $V$, but the $w_j$ can be anything.

Maks

@TedShifrin Yay, exactly

Now, why ? I dont see the relation

Zach Hauk

do you think I should ask the administration about doing something like that?

Ted Shifrin

@Zach: Only take it if you're going to ace it. But what does it gain you at your high school?

Zach Hauk

I don't know yet

to ace it i would only need to do some review

Ted Shifrin

I would recommend learning Spivak's book and knowing way more than the BC ... well. But that's just me.

I taught a high school kid the Spivak course and then he took the BC for the formalities ...

Zach Hauk

00:33

i mean

Spivak before anything else?

like, before i continue with algebra and stuff

Mike Miller

Do whatever you want man

Ted Shifrin

Depends on your life plan. I still don't see the point of being a graduate student before you go to college. But it's not my life.

Maks

@TedShifrin BC ?

I have the hoffman

Oh, I thought you were talking to me, sorry

Zach Hauk

@Ted I don't know where to go even

I don't have anyone to really talk to about math besides the people in this chat, so aside from what I have right now, I don't know what to study or what to do

Ted Shifrin

@Maks: A linear transformation is uniquely determined by where it sends basis vectors in the domain.

@Zach: We had discussions about this months ago. I am more in favor of building solid foundations, but my opinion is not universal.

Zach Hauk

00:38

I mean, specifically?

sorry, as in, should I be studying Spivak's after this?

Ted Shifrin

I told you to learn Spivak and a rigorous multivariable calculus course before all this algebra.

Zach Hauk

You did? sorry, i don't recall.

Ted Shifrin

Yeah, you weren't much interested in my thoughts.

Zach Hauk

Are your lectures rigorous enough?

Ted Shifrin

As you know, math is all about the exercises you do.

5

Zach Hauk

00:44

@Ted am I allowed to use a computer for the "diagonalizing quadratic forms" question?

Ted Shifrin

I told you to skip that question once you know what's going on.

Zach Hauk

how do you mean?

Semiclassical

rehi

Ted Shifrin

It's just straightforward computation. So do one or two if you want.

Semiclassical

I think it's not a bad exercise, once you've done a few by hand, to figure out how you'd get Mathematica to do the computation.

Zach Hauk

00:47

oh, I misread

thought you said "Skip that question until you know what's going on"

For #6 i just looked at cross sections and so far I have 1 sheeted hyperboloid, 2 sheeted hyperboloid, paraboloid, and ellipsoid

are there any others I'm missing?

Ted Shifrin

Are you getting that from the math that you've learned here?

Yes, there are degenerate cases for sure.

Topologicalife

Is there a proof that norms on R^n are equivalent, which is "allowed" to use some topological notions like compatness, but "not allowed" to use other topological notions like continuous functions?

Ted Shifrin

Nope, continuity is essential.

Topologicalife

Why?

Zach Hauk

I looked at all the cases for $ax_1^2 + bx_2^2 + cx_3^2 = 0$

Ted Shifrin

00:52

You needn't have $0$, @Zach.

Zach Hauk

(which you can obtain by diagonalizing your quadratic)

Mike Miller

@Topologicalife How do you talk about equivalence without continuous maps?

Zach Hauk

oh, nevermind :P

so when $a,b,c > 0$, we have an ellipsoid obviously

Ted Shifrin

Not with a $0$ on the right.

Semiclassical

Not with anything nonpositive, for that matter.

Zach Hauk

00:54

oh

Topologicalife

@MikeMiller well, for example, saying they generate the same topology.

Zach Hauk

with a 0 that form isn't an ellipsoid

Mike Miller

@Topologicalife Which is nothing more than saying that the identity map in both directions is continuous.

Semiclassical

With zero there's hardly any solutions in the first place. (So long as we're over R^3, I mean.)

Zach Hauk

ok, tell you what, multiply both sides so that the constant is positive, that will make things much easier.

Topologicalife

00:55

Or if there are constants $a$, $b$ such that $a \|x\|_1 \leq \|x\|_2 \leq b\|x\|_1$

Semiclassical

You can probably make things even simpler: multiply both sides so that the constant is 1.

Ted Shifrin

unless it's 0.

Zach Hauk

^

Semiclassical

...hrm.

Ted Shifrin

Which is a case to look at.

Semiclassical

00:55

Yeah, fair point.

Zach Hauk

so, we know that an ellipsoid is a possible one, though

Ted Shifrin

Everything you listed was ok. I'm not convinced you've listed everything.

Topologicalife

@MikeMiller I'm talking about continuity if I say that tere are constants $a$, $b$ such that $a \|x\|_1 \leq \|x\|_2 \leq b\|x\|_1$?

Mike Miller

Yah.

Zach Hauk

now, for one negative $a,b,c$, we have something like $ax_1^2 + bx_2^2 - cx_3^2 = 1$ (i'll check the zero case in a bit)

so notice that when $x_3 = 0$, we have an ellipse

Ted Shifrin

00:58

BTW, you're not going to get paraboloid from this question.

NO. That is totally wrong.

Zach Hauk

b-b-but

Topologicalife

@MikeMiller why? I don't see notions of continuity there.

Zach Hauk

wait, how is that wrong? I'm just taking the xy-plane slice?

Semiclassical

Probably better to say: The $x_3=0$ cross section is an ellipse.

Ted Shifrin

It's still a surface, @Zach.

Zach Hauk

01:00

?

but, it's in the plane

Mike Miller

@Topologicalife Those are equivalent to saying that the identity maps $(X, \| \cdot \|_1) \to (X, \| \cdot \|_2)$ and vice versa are continuous.

Zach Hauk

the $x_1$-$x_2$ plane

Ted Shifrin

Zach, no, it is not.

This is a fundamental error.

Topologicalife

@MikeMiller right.

Thank you.

Zach Hauk

then how do i take a cross section?

Ted Shifrin

01:01

I'm asking you what the surface is ... not what its various cross-sections are.

Zach Hauk

but if i look at the cross sections i can analyze the surface... right?

Semiclassical

I don't think he was claiming that...

Zach Hauk

he = who? @Semiclassical

Ted Shifrin

This is very false.

You said "when I have $c=0$ I have an ellipse."

Semiclassical

You.

Zach Hauk

01:03

i edited it

to mean $x_3 = 0$

Ted Shifrin

It just displayed my comments in reverse order.

So, again, when $c=0$, what is the surface?

And also you did see my comment that you can't get paraboloids.

Zach Hauk

oh, it's some weird degenerate infinite cylinder thing

@TedShifrin yeah i saw

and I realize why :P

Ted Shifrin

It's a right elliptic cylinder, yes.

OK, I'm outta here for now.

Zach Hauk

alright

im gonna analyze these surfaces

sorry for being a hassle lol

@Akiva want to analyze some surfaces with me

Akiva Weinberger

Don't really have the time at the moment

Zach Hauk

01:14

heh, i was kidding

Adeek

hi everyone

I was wondering let us say I have the following definition for projective modules

P is said to be projective if whenever we have a surjective morphism $g : B \rightarrow C$ and $f : P \rightarrow C$ then f factors through B.

why are free moudles projective using this definition ?

hi @TedShifrin

Adeek

01:33

nvm I figured it out

Zach Hauk

hi Adeek

Adeek

hi @ZachHauk

Zach Hauk

01:46

im just classifying some surfaces

Secret

Political use of mathematics: Using metric geometry to give a notion of district compactness in helping to combat gerrymandering

Pretty hard to become a juror nowadays. They need to know the law, statistics, and now geometric group theory

3

Daminark

02:03

Hey everyone!

Zach Hauk

hi @Daminark

Simple

use generating function to show that for a symmetric random walk $P(S_1S_2,\dots,S_{2n}\neq 0)=P(S_{2n}=0)$ for all $n\geq 1$

@Semiclassical do you mine give me a hint to do this problem

Daminark

How's it going?

Semiclassical

Not sure what that's supposed to mean. S_n is the position after the nth step?

Zach Hauk

i jus thave a headache

Daminark

02:14

:(

Hope you feel better

Simple

@Semiclassical yes

Zach Hauk

i'm classifying surfaces which are the solutions of setting a quadratic form to a constant

ZION

Hi i'm new here also can someone give me tips to improve my mathematical writing ability:math.stackexchange.com/questions/2129016/…

Semiclassical

Mmkay.

And it's step size of +/-1 I presume

Simple

It is symmetric random walk, I think so

Semiclassical

02:16

It doesn't terribly matter what the step size is, come to think of it, since you only care about zero vs. nonzero.

Simple

Since this is symmetric, the probability of returning to 0 sometime is equal to 1?

Semiclassical

I think so.

Well, suppose we denote $P(S_1,S_2,\cdots,S_{2n}\neq 0)=p_n.$

Then $p_{n+1}=p_n\cdot P(S_{2n}\neq 0)=p_n(1-P(S_{2n}=0))$.

Simple

yes

Semiclassical

Which, if that's supposed to be $p_n$ as well, means we'd need $p_{n+1}=p_n(1-p_n)$.

I'm not convinced that helps much, though.

I mean, $p_n^2$ is kinda a pain as far as GFs go.

Simple

I am still not sure how to apply generating function

Semiclassical

02:23

Nor I, if I'm honest. I was hoping this would lead to one deriving the GF for the symmetric random walk, and then work backwards to deduce the above from the GF.

Simple

$P(S_1,S_2,S_3,\cdots,S_{2n}\neq 0)=\sum_{k>n}f(2k)$?

Semiclassical

Yeah.

Simple

?

Semiclassical

Convinced myself what I wrote was not quite right.

ALannister

@Secret I like that. F$&@ Gerrymandering. That's why we've got Orange Julius Caesar in office.

Semiclassical

02:31

(I think the Catalan numbers could be connected to this, via $P(S_1,S_2,\cdots S_{2n-2}\neq 0,S_{2n}=0)$. But ehhh)

Simple

Do you think we can use $P_0(s)=1+P_0(s)F_0(s)$?

Semiclassical

How are those defined?

Hmm, this answer might be useful? math.stackexchange.com/a/1196154/137524

Simple

$P_0(s)=\sum_{n\geq 0}p_0(n)s^n,F_0(s)=\sum_{n\geq 0}f_0(s)s^n$

Semiclassical

Okay, and what are $p_0,f_0$?

You've probably had those defined somewhere, but I wouldn't know.

Simple

$p_0=P(S_n=0)$,$f_0=P(S_1,\cdots,S_{n-1}\neq 0,S_n=0)$

Semiclassical

02:40

Ok.

Let's write $q(n)=P(S_1,\cdots,S_{n}\neq 0)$.

(I see no good reason to take the 0 subscript in this case.)

Obviously $q(n)=1$ whenever $n$ odd, so I'm meh on that being the best definition to use. But w/e.

I guess one thing to note is that we can focus on $f_0(2n)=P(S_{2n}=0)$ by considering the even part of $F_0(s)$.

Simple

03:05

I still confuse

If I use the definition of the generating function directly, I can have$\sum_{n\geq 0}\sum_{k\geq 2n+2}s^{2n}f_0(k)$

Not sure how to compute that sum

Simple

03:26

$\sum_{n\geq 0}\sum_{k\geq 2n+2}f_0(k)=\sum_{n\geq 0}\sum_{k\geq 0}s^{2n}f_0(2k)$

Balarka Sen

05:42

@ZachHauk If you learn multivariable calculus I'll talk to you about stuff. If you learn algebra we probably won't talk much. :)

That doesn't mean you should do calculus and not algebra, of course.

Fargle

I got lent a copy of Steen and Seebach's Counterexamples in Topology today by a professor. This is the oddest giddiness I've ever felt.

Balarka Sen

Hah, I always wanted to get that book

lol, I just notice that google says the genre of the book is "non-fiction".

Fargle

Not inaccurate.

The first quarter of the book is pretty much just an onslaught of properties.

Balarka Sen

It brings strange images to the head nonetheless

Fargle

Do you happen to know what filters are used for?

(The topological kind, not the coffee or faucet kind)

Balarka Sen

05:53

I think it's the right notion of sequences in abstract topological spaces.

But I dunno

Fargle

Ah, okay, that makes some sense.

Daminark

@Fargle Not the donut kind?

Well you didn't say coffee mug so I guess that joke doesn't apply

Also wait aren't the generalized sequences called nets?

Fargle

Filters are a generalization of nets.

Daminark

Oh god

Paradox 101

06:12

Hi, does anyone here know fixed point iterations?

Fargle

Perhaps the greatest misnomer in math is the naming of category theory "abstract nonsense". There's plenty of abstract nonsense without needing to involve category theory, haha.

Secret

06:26

How should we view about mathematics: By baking it...?

and is it true we focused too much on proving theorems and not doing enough on making connections. Personally I don't see that kind of problem, but maybe I have not learn enough maths

Fargle

@Secret Depends on the course, the lecturer, and the purpose of the course.

But I have had courses like that.

Secret

> Unfortunately, the kind of maths we teach in school is often not in any way useful for most people’s lives – people say “When am I ever going to need to solve a quadratic equation in my life?” The kind of maths I teach is about logical thinking, thinking your way through situations, understanding what is causing something to happen and working out how things fit together.

I kinda agree with this quote. Perhaps we should pull the topic "logic" down to the high school curriculum and emphasize it more cause it will benefit decision making and programming in the long term, both will be essential daily skills

> Yes, definitely! Mathematicians really like making up their own rules that make sense for particular situations, and we hate having rules imposed on us.

Really??

Learning user

Hi

@Paradox101

@BalarkaSen

I have one question

Secret

06:42

> Mathematics is actually all analogies. What I am trying to do is provide the ideas and the way into something. Unfortunately, a lot of people derive their feeling of self-worth from the fact that they can understand things other people can’t. I don’t believe in that.

My self worth is the ability to understand things other people cannot, AND help those people to develop a understanding of the topic by themselves

Things are more interesting when you understand them, rather than keeping it to youself

Learning user

Hi @CompulsiveMathurbator

Compulsive Mathurbator

Hi

Learning user

yes i will send question

How are you?

Compulsive Mathurbator

I'm going to die in three months

Just kidding I'm fine, you?

Learning user

I am fine, i got really shocked

Paradox 101

06:47

Hi @Learninguser

Learning user

A person spends 15 percent on rent , 12 percent on grocery , 13 percent on maintenance,5/6 on shoppping,3/24 on others and saves his mutual fund on 1330,How much he got total income

15,8,9,15,32 ___ what is the next number

Secret

@Learninguser You need more context. Any number can be the next number as there always exists a polynomial to generate an arbitrary next number and fit the first finite number of given numbers in a sequence

Compulsive Mathurbator

Assume his income to be x. Then 0.15x + 0.12x +0.13x + 5x/6 + 3x/24 +1330 = x. Solve the linear equation

Learning user

ok

Compulsive Mathurbator

As to the second, like secret said, there are infinite possibilities

Learning user

06:54

can you guide me one possibility

Compulsive Mathurbator

Try recurring series

Learning user

Recurring series

?

Secret

Actually, are their general guidelines to the most natural choice of some given sequence. For example, when presented with 1,2,4,8,_, we tend to think the next number is 16, even though there are other choices. Another example is 1,2,3,4_, and we will usually say the next one is 5

I wonder if there's a mathematical notion of choosing and defining the most natural number will be for some given sequence S. Perhaps, are these seemed the most natural choice simple because we get used to them the most?

Compulsive Mathurbator

"(Math.) an algebraic series in which the coefficients of the several terms can be expressed by means of certain preceding coefficients and constants in one uniform manner."

Alessandro Codenotti

@Fargle they're equivalent, but that's where my knowledge of nets ends

Learning user

06:59

A Jar contains 7:3 milk and water ,Another jar conatins 1:4 milk and water.These two mixtures are added to the new jar finally it contains 100 litres of water and 150 litres of Milk.How Much the Jar a contains milk initially

15,8,9,15,32 ___ for this , what is the pattern behind this

arctic tern

@Learninguser How much does which jar contain initially? There are two jars. It'd make more sense to ask how much both jars contain initially.

Suppose the first jar has x liters and the second jar has y liters. Then you can set up a system of two equations in two variables: the one equation expresses how much water is in the third jar in terms of x and y, and the other equation expresses how much milk is in the third jar in terms of x and y.

@CompulsiveMathurbator you mean recursive sequence?

Balarka Sen

07:18

@Alessandro Did you figure out what's up with the cone on the earring?

Alessandro Codenotti

I can't do math while sleeping (yet) :P

Learning user

How Much Jar A contains initially milk

Balarka Sen

@Alessandro You should work on it

Learning user

can you provide the equation

@arctictern

@CompulsiveMathurbator guide me

Alessandro Codenotti

On the earring or on the ability to sleepmathing? I'm workinh on the former

Balarka Sen

07:24

The latter

I think I solved a cute number theory problem someone gave me

Fargle

@BalarkaSen What was it?

Balarka Sen

For each integer n construct a circle in R^2 containing exactly n lattice points

Fargle

@BalarkaSen Oh Jesus.

Alessandro Codenotti

It's not obvious to me that this is even possible for every n

Balarka Sen

I played around with a few things in desmos before understanding how it worked. Try stuff with irrational centers.

Learning user

07:28

@arctictern I have tried

7/10 *x + 1/5*x = 150

Is it correct?

arctic tern

you mean (7/10)x+(1/5)y=150, but yes

Learning user

both are milk , how you are representing another new variable y

Paradox 101

Hi. Do you know how to find order of convergence of secant method? @BalarkaSen

arctic tern

@Learninguser what do you mean both are milk?

Learning user

ok'

arctic tern

07:31

x is how many liters total are in the first jar. y is how many liters total in the second jar.

7/10 of the first jar is milk, 1/5 of the second jar is milk

Learning user

ok

we have to solve two equations

for finding X and Y

Learning user

07:56

@CompulsiveMathurbator guide me solution mixture sum

Compulsive Mathurbator

I'm not sure what you just said

Learning user

A Jar contains 7:3 milk and water ,Another jar conatins 1:4 milk and water.These two mixtures are added to the new jar finally it contains 100 litres of water and 150 litres of Milk.How Much the Jar a contains milk initially

for this sum

15,8,9,15,32 ___ what is the pATTERN

you told recurring series

what is that?

Compulsive Mathurbator

@arctictern I was going with that but it would appear this sequence isn’t one

@Learninguser which jar are we talking about?

Also arctic tern's answer is fine, I don't see the problem

Learning user

7:3 Jar conatins how much milk

Compulsive Mathurbator

140 dm^3

That's what you get when you solve the equations, like arctic said.

I don't see what you need me to do

Balarka Sen

08:08

@Paradox101 I don't

Compulsive Mathurbator

08:20

@Learninguser Found this after a quick google search
(15 x 0.5) + 0.5 = 8
(8 x 1) + 1 = 9
(9 x 1.5) + 1.5 = 15
(15 x 2) + 2 = 32
Similarly,
(32 x 2.5) + 2.5 = 82.50.
Incidentally where is this problem from? You are, from your use of the currency rupees, statistically likely to be Indian, and from your previous questions most likely in middle school. I don't mean to be creepy, but when I was (presumably) in your position I don't recall having to deal with these types of arbitrary sequences.

And while I don't have much praise for the Indian education system, I don't think they would give such a question without some hint as what the pattern would resemble.

LeGrandDODOM

@DanielFischer I found a different way of computing the Frechet derivative we discussed the other day imgur.com/Z94bF7I and imgur.com/Lb1ZrH9

Paradox 101

ok

Topologicalife

08:48

@TedShifrin @MikeMiller actually I proved it without using continuity

Learning user

09:06

ok Thankyou

YOUSEFY

11:22

hello, I want to know how Walsh Transform works, for example in this picture I want to follow the arithmetic but it takes time for me see this en.wikipedia.org/wiki/Hadamard_transform#/media/…

for example, suppose W_f (000) then what is f(x) inside the Walsh function? what is its value?

let's take for example w = 000, then using the Walsh Transform, we have the summation \sum -1^{f(x_1,x_2,x_3} xor <x,(000)> then what is f(x_1,x_2,x_3}?

Learning user

12:03

Hi

Learning user

12:15

A Deer and Rabbit can complete a full round on a circular track in 9 minutes and 5 minutes respectively.P,Q,R,S are the four consecutive points on the circular track which are equidisant from each other .P is opposite to R and Q is opposite to S.After how many minutes they will meet together for the first time at the starting point,when both have started simultaneously from the same point in the same direction

Zach Hauk

@BalarkaSen I know multivariable, but not very rigorously

like, I know surface and line integrals and change of coordinates and the likes

I'd imagine a more rigorous course would go more in depth with concepts like the Jacobian, however.

Balarka Sen

@Zach There's a lot more to multivariable calculus than you think.

It's actually a gateway to differential geometry/topology

Zach Hauk

really?

Balarka Sen

Yup. I thought it was basic until I learnt it like, a year ago.

Zach Hauk

anyways sorry i have to leave for school

Balarka Sen

12:24

Bye.

(Note that I said gateway, not a prerequisite. Learning the rigorous multivariable calculus - i.e., mathematician's multicalc, not engineer's - you'd already secretly know some ideas of differential geometry/topology. Talking to someone about them would make the picture clear)

Alessandro Codenotti

We're doing a lot of stuff in analysis that looks like differential topology, it's all regular surfaces, parametrizations and getting a basis of the tangent space from the differentials of those

Balarka Sen

Right, but sometimes you actually have nontrivial ideas involved. Eg, implicit/inverse function theorem, second derivative test and Green's theorem. The three of them are secretly about manifolds. The last two point towards Morse theory and differential forms respectively.

Alessandro Codenotti

We had Gauss, Green and Stokes (special cases of) last week

Balarka Sen

Neat

Alessandro Codenotti

12:40

Anyway I think I don't understand simple connectdness, I'll ask you a couple of questions after lunch

Balarka Sen

Sure. Bon appetit

and apologies for all the missed ticks above the e's

Mahmoud

13:06

Hello !

In Math terminology, we say right-hand derivative, right ?

And for a slightly technical question, when referring to the set of the real number, which is convenient to write ? $R$ or $\Bbb R$, apart from aesthetics, of course.

Alessandro Codenotti

@BalarkaSen I don't know French either so I wouldn't have noticed :P

So for the Hawaiian cone if I imagine everything in $\Bbb R^3$ with the earring on the $xy$ plane and the vertex of the cone along the $z$ axis I think the loop obtained as the intersection of the cone with a plane perpendicular to the $xy$ plane through the vertex won't be homotopic to the constant loop

Balarka Sen

@Alessandro Ah. What's the analogue in Italian?

Alessandro Codenotti

Buon appetito, quite similar

Balarka Sen

Got it. Also I don't believe what you claim to be a loop is actually a loop.

Alessandro Codenotti

I'm not sure it is either

Balarka Sen

13:18

I mean, even for the standard cone it is not a loop.

It's half a loop.

Alessandro Codenotti

So, simpler case, if I take a finite wedge of circles and do the same cone construction the result will be simply connected, right?

Balarka Sen

Yes.

Alessandro Codenotti

Hm, can't I just return to the starting point walking backward on this half loop? It doesn't need to be injective right?

Akiva Weinberger

What's the easiest way to prove this? Suppose we have an infinite chessboard. Some squares are painted blue, some are painted orange, and many aren't painted all. No blue and orange squares are adjacent, even diagonally. If A and B are two unpainted squares, and there's a path from A to B that avoids all blue squares and another path from A to B that avoids all orange squares, there's a path between them that avoids painted squares of both colors.

Balarka Sen

That's nullhomotopic, @Alessandro.

Akiva Weinberger

13:21

Seems obvious enough. (A path is a sequence of squares, each adjacent to the last, not allowing diagonally.)

Alessandro Codenotti

Oh, right

Balarka Sen

You "backtrack" the loop.

Akiva Weinberger

The reason is, I want to prove that if I have two disjoint closed sets in the plane whose complements are connected, the complement of their union is connected as well. The proof I have in mind would start by overlaying a sufficiently fine grid onto the plane.

Balarka Sen

@AkivaWeinberger You mean blue is not adjacent to orange? Can blue be adjacent to blue?

Akiva Weinberger

The more general reason is I'm trying to think up a new exposition of proving Brower's fixed point theorem, which would use the above as a lemma.

@BalarkaSen Blue can be adjacent to blue, yes

Balarka Sen

13:27

@AkivaWeinberger I see. That's a little clearer.

I am not sure if your approach would work if the sets are horrible.

Eg, Cantor set x Cantor set.

Akiva Weinberger

@BalarkaSen Say the two disjoint closed sets are $A,B\subseteq\Bbb R^2$. Say also we want to show that any two points $x$ and $y$ can be connected by a path in the union's complements.

By assumption (the complements of each individual set are connected), there are paths $p_a$ and $p_b$ connecting $x$ and $y$ avoiding $A$ and $B$ respectively.

Balarka Sen

(I feel like a homotopy between p_a and p_b should contain a path which avoids both A and B - or so it would after a small perturbation, but I am not sure how I'd prove it)

Akiva Weinberger

Choose a compact disk containing $p_a,p_b$ in its interior. The intersections of $A$ and $B$ with that disk are compact, so there's a minimum distance $\epsilon$ between them. (TL;DR We can assume they're compact and thus have a nonzero distance, so choose $\epsilon$ less than that)

Furthermore, choose epsilon smaller than the distance between $p_A$ and $A$, between $p_B$ and $B$, and between either path and the disk we put them in.

BAYMAX

Hi all !

Balarka Sen

OK.

Akiva Weinberger

13:41

Then overlay a grid with mesh size $<\epsilon/2$ I think (or $<\epsilon/1000$ to be really sure, I guess), color squares with $A$ points orange, and squares with $B$ points blue

The squares we want to connect are the ones containing $x$ and $y$.

Balarka Sen

Interesting. That does sound convincing enough.

Akiva Weinberger

I suppose the exposition would be cleaner if I assumed they were compact, and then at the end wrote "Oh, by the way, they just need to be closed, 'cause blah blah blah"

Alessandro Codenotti

@BalarkaSen I don't think so, look at my bad drawing

Akiva Weinberger

Putting the weird stuff with the disk at the end.

Balarka Sen

@Alessandro Not disjoint.

Alessandro Codenotti

13:45

Hm, looks like I can't read

I agree it should intuitively work for disjoint ones though

Akiva Weinberger

Yeah

But it kind of feels Jordan curve theorem-y, with it being intuitively obvious until you remember just how strange curves and closed sets can be

Balarka Sen

The problem is the speed of the homotopy. There need not be a path free of every point of $A$ and $B$ inside the homotopy, because you can slowly fit stuff from $A^c$ to $B^c$.

I guess one should start by modifying the straightline homotopy somehow.

Alessandro Codenotti

Yeah, even for very simple A and B you can write dumb homotopies that are always intersecting one of them

Balarka Sen

Right

Akiva Weinberger

The theorem is true; there's a not-too-hard proof with homology.

Balarka Sen

13:55

I know.

It should be a Mayer-Vietoris argument by homology.

Akiva Weinberger

Ya

Balarka Sen

Let $f$ and $g$ be the paths from $x$ and $y$ which avoids $A$ and $B$ respectively. Look at the line $\ell_t$ joining $f(t)$ and $g(t)$ and look at $\ell_t - (A \cup B)$. Union these for all $t$. It boils down to whether this subset of $\Bbb R^2$ is path-connected or not I suppose?

Alessandro Codenotti

I'm definitely missing something which is probably obvious, it seems to me that this Hawaiian cone is a star domain in $\Bbb R^3$ and as such it should be even contractible but that's not true

Balarka Sen

@Alessandro Why's that not true? :)

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