Mathematics - 2015-04-28 (page 2 of 3)

user96977

15:19

is the probability of tossing 3 heads in a row = 0.5^3 ?

Mew

yes

0.125

assuming the coin is unbiased

user96977

what if it is biased?

Mew

a rigged coin, a biased coin is one that is especially made so that the chance of tossing heads is greater than 50%

user96977

or less, presumably

Mew

If the coin is rigged so that the probability of heads is 80% instead of 50%, then the odds is 0.8^3

correct

user96977

15:24

thanks. how does this relate to the binomial distribution?

Mew

well the binomial distribution is where you have a probability of success, p and a probability of failure, q = 1-p

you also have a certain number of trials, n

So the binomail distribution can be specified by probability of success, and number of trials

user96977

Mew

in this case, we have p = 0.5 and n = number of trials = 3

yeah

so N = number of trials = 3

user96977

thanks, i'm starting to see

Mew

P(r|f,N) means what is the probability of r successes

in thise case, n = 3 trials and we want 3 successes, so r = 3 as well

user96977

15:28

so f^r is the probability of r heads, and (1-f)^(N-r) is the probability of the rest being tails

Mew

(3 3) 0.5^3 (1-0.5)^0 = 1 * 0.5^3

correct

e.g what is the probability of 2 tails occuring and 1 heads occuring:

N = 3, but this time r = 2

so P(r|f,n) = (3 2) * 0.5^2 *(1-0.5)^1

(3 2) = 3

so the answer is 3*0.5^3

in this case we still end up with 0.5^3 because (1-0.5) = 0.5

user96977

yes, i see

Mew

oops i meant 2 heads occuring and 1 tails

user96977

i do not see where the binomial coefficient comes into it though

Mew

but in this case it didn't matter

the binomial coefficient is (N r)

user96977

15:31

i see where the two factors come from, and then presumably we multiply them because P(H $\cap$ T) = P(H) * P(T)

Mew

yeah

consider probability of 2 heads and 1 tails

we can get this by: HHT, HTH or THH

that is why we multiply by (3 2) = 3, because there are three ways of getting this combo

user96977

wow thanks

Mew

np

robjohn

@Chris'ssis: chat.stackexchange.com/transcript/36?m=21337791#21337791

I am going back today to move stuff back into her place. There is not too much chance that the fire will flare up now.

user96977

is there another way to use this chat, or is it only available through the browser? also, how do i enable latex for formulas in the chat

robjohn

15:41

@TruthSerum I believe that a browser is the usual way to access chat. See the sidebar where it says "$\LaTeX$ in chat"

Chris's sis

@robjohn Bad news. Sorry.

user96977

@robjohn thanks, it works now

robjohn

@TruthSerum great!

user96977

16:00

what is the difference between a probability density and a probability density function?

user96977

perhaps you evaluate a probability density function to obtain a probability density?

Mew

yep

what are you studying for?

user96977

just trying to broaden my knowledge. probability has always been a mystery to me. currently reading this punkuser.net/vsm/vsm_paper.pdf

user96977

"To approximate such a distribution using a small amount of data, we store the first and second moments: the mean depth and mean squared depth." - so i'm reading about moments at the moment :)

Mew

i see

E[x] is "the expectation of random variable x" which means what is the mean or average of X

simply sum all x and divide by no.

this is the 1st "moment"

user96977

16:05

i see

Mew

Var[x] is the second moment

variance measures how far the values of x are from the mean

user96977

is there some generalization for the n'th moment?

Mew

yes

but it is unneccarily complicated

user96977

ha, ok

Mew

Var[x] = E[(X-u)^2]

where u is the mean , or in otherwords u = E[x]

the 3rd moment, known as skew is given by

Skew[x] = E[(X-u)^3]

user96977

16:07

so Var[x] = E[(X-E[x])^2] ?

Mew

yep

so it is the expected difference between x and the mean value of x

squred

the reason it is squared is because otherwise the +ve differences would canceil with the -ve differences

so we square it to make all differences positive

the n-th moment is thus

E[(X-u)^n]

but usually only the 1st, 2nd and 3rd moments are ever really useful

mean, variance and skew

Mean = average value of X.

Skew = measure of spread of the values of X from the mean

oops i mean variance = measure of spread of the values of X from the mean

Skew: If skew is 0, the distribution is symmetric (like a bell curve). If skew is positive the distribution has a long tail to the right. If the skew is negative the distribution has a long tail to the left.

user96977

i see

user96977

in the equation for variance, what does upper-case X signify? is it the set of samples?

Mew

upper case X means "the random variable X"

So for example X could be the result of a coin flip, and thus X = 1 with 0.5 chance, and X = 0 with 0.5 chance ( if 1 = heads and 0= tails)

E[X] = expected value, or mean of the random variable X

So again if X is a coin flip this is a binomail distribution, for which E[X] = np = n*0.5.

user96977

ah, i must try to develop my idea of what this X represents

Mew

16:16

X represents the outcome of a random experiment

X will follow a certain distribution. E.g. for a coin flip, X follows a binomial distribution.

user96977

or rather, what the notation E[x] means. because E is an operation defined on a discrete set of samples. as far as i understand, it is analogous to the arithmetic mean/average

Mew

yes but X is a theoretical distribution

E[X] is not the average of a sample of results

user96977

hmm, X is a random variable, and X is also a theoretical distribution?

Mew

X represents the random variable before an experiment has been concluded

have you heard of schrodinger's cat in physics?

user96977

yes

Mew

16:18

well X is the state of the cat before measruement. It is both alive and dead

X is 1 with 0.5 probability and 0 with 0.5 probability

Now suppose X is the number of heads that will be achieved after 100 throws

E[X] = 100*0.5 = 50

The expected number of heads is 50, even though we haven't done an experiment

So X is the theoretical distribution of the outcome

user96977

ok, so it can be used to determine if there is a bias?

user96977

i suppose that would be one trivial use for the expected value

Mew

well, when we actually perform the experiment, we might actually count 60 heads instead of 50

So the Sample mean was 60, but the E[X] = 40

E[X] = 50 i mean

We can use E[X] and Var[X] and compare with our actual result of 40

user96977

i see. what conclusions would you make though? that might be within some kind of reasonable error for an unbiased coin

Mew

ahh

well for a coin X ~ Binomial (n, 0.5)

this means X is a random binomail distribution with n trials with success probability of 0.5

For large values of n, binomial distribution approximates the normal distribution to give:

X ~ Normal(np, npq)

This means a normal distribution with a mean of np and a variance of npq

user96977

16:23

why would you need an approximation? is it cumbersome to compute with a hand calculator or something?

Mew

(X - np)/sqrt(npq) ~ N(0,1)

Now,

(60 - 50)/sqrt(100*0.5*0.5) =

2

So looking at a normal distribution table for N(0,1), the probability of getting a value of 2 or more is 0.02275

So thus there is a 2.28% chance of getting 60 heads or more with an unbiased coin

this is quite low, and thus we can conclude that the coin is probably biased

user96977

thanks, nice conclusion

Mew

yeah I wouldn't expect you to be able to do this yet, but it shows the use of E[X]

and how E[X] isn't the mean of the sample but it is the expected value given your assumptions about the distribution (in this case that it was p = 0.5)

evinda

16:42

Hello :)

user96977

16:53

hi

user96977

do i say skew(X) = ... or how is the third moment written?

user96977

on wikipedia it is written as $\gamma_1 = ...$

user96977

i'll just write skew(X) for now :)

Remember me

@evinda!!!!!hi

Remember me

17:13

Hi!!!!!@iwriteonbananas

Mew

skew(X) is fine

I believe γ1 refers to something different, the skewness coefficient

LeGrandDODOM

@Emrakul Yo, I got a riddle for you

user61230

Hmmmm?

user61230

I'm not very good at riddles, but sure!

Ted Shifrin

LeDodo !!

LeGrandDODOM

17:25

@Ted Salut ! 3 ème semaine consécutive de concours ;)

Ted Shifrin

Salut ... Tu ne t'endors pas? :)

Mew

What's the riddle

LeGrandDODOM

C'est fatigant, mais on s'y fait. Hyppa fait la même chose @Ted

Ted Shifrin

On ne voit pas Hippa depuis bien longtemps :)

Remember me

Hi @TedShifrin

LeGrandDODOM

17:30

0

Q: Find the liar in the library

I've just taken a CS exam, which had the following question 6 persons entered a library the day a book got stolen. Each of them entered the library once and only once, and stayed there for some time, then left. If two persons were in the library at the same time, at least one of the two saw t...

logic-puzzle

@Emrakul @Rememberme the link is above (not a riddle, but a puzzle)

ᴇʏᴇs

@LeGrandDODOM I think it would be more fun if this happened in real life and you were trying to figure out who's lying in your school library

GBeau

If $f: A\rightarrow\mathbb{R}$ is a continuous function on the closed interval A, is it true that $f$ has a maximum?

I was thinking in class when a max could be guaranteed and I noted it wasn't true for noncontinuous or open intervals

I do not know rigor of continuous functions, I'm in an intro analysis class; I'm just curious if my suspicion is a true statement

user61230

@GBeau It has a maximum value, but that maximum may not be unique.

evinda

@TruthSerum What is your field of study?

robjohn

I am off to help move my friend back into her place. My answers are not doing too well today anyway; it might be a good time for a short break.

Parth Kohli

17:40

A familiar string of numbers: math.stackexchange.com/users/228265/user17762

user61230

@LeGrandDODOM I like it! A class is starting, so I'll think about it in a bit.

user61230

Interesting problem, though.

GBeau

@evinda any basic reason? Since it's not able for instance that it does not necessarily have a maximum for open intervals

Notable*

Ted Shifrin

hi, @Remember

Hi, @robjohn. Guten Abend, @evinda.

evinda

Guten Abend @TedShifrin :)

user61230

17:43

Also, hi, @Ted!

Ted Shifrin

hi, @Emrakul

GBeau

@evinda also how could a max not be unique?

evinda

@GBeau I think that it is true because in this case the function is bounded.

GBeau

But it is not true for bounded open intervals

evinda

@GBeau I think that the maximum will always be unique, but it can happen that there are more than one values of x such that f(x)=maximum value.

GBeau

17:47

For example on $(0,1)$, the function $1-x$ has no max despite being bounded and continuous

Mike Miller

@GBeau: Consider $\sin(x)$ on $[0,4\pi]$. That $\pi$ and $3\pi$ both map to one is what one might mean by "the max is not unique"

user61230

Oh, yeah, sorry, that's what I meant, @evinda

GBeau

@MikeMiller oh I thought he meant the max

user61230

Sorry, yeah, I should have been clearer.

GBeau

Oh sorry I think I highlighted evinda instead of you on my phone

Andrew Thompson

18:15

Anyone with a copy of Big Rudin closeby? Have a small notational question.

Ted Shifrin

What's the question, @Andrew?

Alonso Delfín

why can´t moderators choose the best answer ? Lately I have found a lot of OP who ask a question gets the answer they want and don´t even up vote. A lot of question have plenty of good answer and none of them is the best one.

Andrew Thompson

Theorem 8.6, equation (7). As far as I can see $X_n, X_m$ has yet to be introduced.

Ted Shifrin

By Big, you mean Real & Complex or FA?

Andrew Thompson

Real and Complex.

Ted Shifrin

18:19

(I am in my office, so I have both ... but am soon getting rid of ... everything.)

Andrew Thompson

Oh, you're moving?

Ted Shifrin

Yup, taught my last class yesterday.

Andrew Thompson

I see. Don't know the circumstances, so pretend my reply was "Ah, that sucks.." or "Brilliant, good luck!" depending on them.

Ted Shifrin

Hmm, this is not fitting with what I have. I'm on p. 156, but equation (7) has $A$ and $B$ defined.

Retiring and moving away, @Andrew ... nothing sucky, I hope :P

Andrew Thompson

Maybe different versions. I am on page 164. It is the theorem before Fubini, right before the product measure is defined on sigma-finite spaces.

Ted Shifrin

18:22

In my edition of the book, this is essentially the Lebesgue Decomposition Theorem.

Andrew Thompson

The equation at hand is $Q_{mn} = Q \cap (X_n \times Y_m)$ if that helps you see it.

Ted Shifrin

oh, ok, I found it, on p. 140.

Ah, it's in the notes after equation (2).

Andrew Thompson

Oh, jeez. Silly.

Thanks for bothering!

Ted Shifrin

It took me a while.

Mike Miller

@Ted: I'm next. Maybe after I leave I can think again.

Ted Shifrin

18:29

Oh, have fun, @Mike. I'm off to a retirement party for one of my colleagues.

Have fun, @Andrew.

Andrew Thompson

Thank you kindly, good luck with the moving!

ᴇʏᴇs

@TedShifrin Did you already have your retirement party

Ted Shifrin

nope, mr eyeglasses. Mine is in a bit over two weeks.

Remember me

Hi @ᴇʏᴇs

@Chris'ssis you are just like fermat posing questions and making people go crazy on them

Chris's sis

@robjohn I think of adding it to my book.

@Rememberme lol, I'm just a poor amateur. ;)

Remember me

18:43

I would really love to know what will be your last theorem lol@Chris'ssis

Chris's sis

@Rememberme :-))))))))) I liked that one.

Remember me

Which will be unsolved for decades....................

good at least someone did i havent been starred since days

Where has Jasper been its been days since i have seen him @WillHunting i hope he is just as fine and working hard.....

columbus8myhw

Hey, I know that $x^4-4x+4$ is always positive…

Since it's equal to $(x^2-1)^2+2(x-1)^2+1$.

But, I wonder… is there a way to write it as the sum of only two squares?

ᴇʏᴇs

Hi @Sayan

Balarka Sen

@MikeMiller It's pretty systematic. I just saw the first lecture (and read up until 1.2. in the corresponding notes) so far, not sure if I should see the next one too. Extended TQFTs seem quite involved.

Remember me

18:56

How are you?@ᴇʏᴇs

hi @Balarka!!!!!

Balarka Sen

"don't know a single useful fully extended TQFT" does that mean you don't know if it's a practical invariant?

I'm actually not even sure how good it is : I have only computed TQFTs of one-dimensional smooth manifolds and orientable surface of genus $g = 0, 1$s so far

$Z(\Sigma_g)$ in general doesn't seem to be too hard to compute.

The cobordism hypothesis is quite nice. I'd love if my TQFTs are all determined by $Z(pt)$.

@Rememberme hello.

Remember me

@Balarka are you free?

Balarka Sen

not completely. what's the questions?

Remember me

@BalarkaSen well i have prove few stuffs are linearly independent that $f_1(x)=1,f_2(x)=e^{ix},f_3(x)=e^{-ix}$ i have to prove these are linearly independent...so i was thinking do i have to use Euler identity in this?

Balarka Sen

19:12

@Rememberme yes, using Euler's identity is definitely a good start.

Mike Miller

Well, one doesn't want them for surfaces. One wants something that tells you about 3- or 4-manifolds.

Balarka Sen

yeah, but i will want to see if it works for surfaces first, not?

yikes, i gotta run.

Mike Miller

Sure.

Remember me

19:38

well i have prove few stuffs are linearly independent that $f_1(x)=1,f_2(x)=e^{ix},f_3(x)=e^{−ix}$ i have to prove these are linearly independent... can anyonw give me some ideas on it

ᴇʏᴇs

19:50

@Sayan you know that $e^{ix} = \cos x + i\sin x$?

Remember me

yes @ᴇʏᴇs

user96977

20:33

and that cos(-x)=cos(x), sin(-x)=-sin(x)

Remember me

i get that $e^{-ix}=cosx-isinx$

user96977

f_2(x) + f_3(x) = 2*cos(x)

Remember me

okay

now how do i prove they are linearly independent

user96977

in the complex plane?

Remember me

it contains complex values so i suppose in complex plane

user96977

20:38

they are not linearly independent, since f_2(x) = f_3(-x) = -f_3(x), i would say

Remember me

they questions says prove that they are linearly independent

JC574

@BalarkaSen Is it true that any simplicial complex is locally path connected?

user96977

as far as i understand, it means to show that there is no k such that f_2=k*f_3. but there is a k, namely k=-1.

Remember me

@JC574 can you help me with a bit of linear algebra

user96977

or well, if i am wrong then you can solve for k and show that it leads to **something that is not a constant

Remember me

20:44

I think you are wrong @TruthSerum

$cosx+isinx \neq -cosx+isinx$ @TruthSerum

user96977

yes, i see, i am wrong

Remember me

So how are you supposed to prove this

user96977

show that k is not a constant

Kelly Blunie

Hello, i have asked a question some time ago, can any can try answer?

-1

Q: Rational,and irrational number between any 2 real numbers.How to prove?

If $x,y$ are 2 real numbers such that $x < y$ how to to prove there is an $r$, belongs to the set of rational numbers, and a $i$, belongs to the set of irrational numbers and hence many more rational numbers and irrational numbers between $x$ and $y$. we want to prove $x < r < y$ and $x < i < y$...

real-analysis irrational-numbers rational-numbers

i try to undersand a very similiar quesrion/answer

15

Q: Is there a rational number between any two irrationals?

Suppose $i_1$ and $i_2$ are distinct irrational numbers with $i_1 < i_2$. Is it necessarily the case that there is a rational number $r$ in the interval $[i_1, i_2]$? How would you construct such a rational number? [I posted this only so that the useful answers at Rationals and irrationals on ...

real-analysis irrational-numbers rational-numbers

but i dont understand the acepet anwser in one step

Remember me

@TedShifrin!!!!!!!

Ted Shifrin

20:52

So what specifically do you not understand, @KellyBlunie?

hi @Remember

Remember me

can you help me with a problem@Ted

Ted Shifrin

maybe ... depends how much you pay :)

Kelly Blunie

ny-nx>1 hence nx<m<ny for an integer m

user96977

if they were linearly dependent, you could write: $f_0=af_1 + bf_2$ for constants $a,b$

Ted Shifrin

@Kelly: So how would you show that if two numbers x and y are more than 1 unit apart, there must be an integer between? (Hint: How far apart are the integers?)

Remember me

20:54

my chat screen suddenly became smaller??

Ted Shifrin

Maybe you hit command keys?

Remember me

how to reset it

Ted Shifrin

depends what sort of computer

hi @Karim

Karim Mansour

Hi @TedShifrin

how is your day ?

Ted Shifrin

getting closer to cocktail time :) how's yours?

Karim Mansour

20:57

good was just running and gym now will eat something then read DF algebra

:D

that sounds fun

@TedShifrin

Ted Shifrin

@Remember: Were you going to ask a question?

Remember me

i cant see what you are writing my screen only for chat has become smaller!!!!!!!!!!!!

Karim Mansour

ask your heart out @Rememberme

:D

haha

Ted Shifrin

Leave MSE and restart your browser, @Remember

OH, he can't see.

This seems like a reductio ad absurdum argument.

user96977

i think there is some corrupted mathjax. i refreshed the page and it's ok

Remember me

21:00

no change

Ted Shifrin

Try quitting your browser, @Remember.

Or look in the "View" menu and make the page larger.

user96977

if i have a cubic polynomial and i know one of it's roots, can i find the other two using the quadratic formula and avoid having to use the cubic formula?

Kelly Blunie

@TedShifrin i understand that ,intuitively but how to prove that there is a integer between nx and ny?

Remember me

got it....yes!!!!!!!!!!

Kelly Blunie

hi @KarimMansour can you try answer my question, math.stackexchange.com/questions/1256400/… ?

Ted Shifrin

21:05

@KellyBlunie: Try reasoning by contradiction. If there's no integer between them, what happens?

Karim Mansour

yeah

Remember me

so @TedShifrin my doubt was $f_1(x)=1,f_2(x)=e^{ix},f_3(x)=e^{−ix}$ i have to prove these are linearly independent...

Ted Shifrin

There are two basic methods, @Remember. What are you trying?

Remember me

please ping me the hints its 4 am over here and i need to sleep or my mom will kill me ................

Karim Mansour

hey

Remember me

21:07

sorry for that

Ted Shifrin

Go to sleep, @Remember.

Karim Mansour

@Rememberme take a scalar a ,b,c and see if that equations has the trivial solution only

Ted Shifrin

I suspect he knows that, @Karim. @REmember: Either plug in three different values for $x$ in your linear combination OR take that equation and differentiate it ...

Karim Mansour

a + b$e^{ix}$ + c$e^{-ix}$ = 0

yeah exactly

plug in values will make it easier

Ted Shifrin

I prefer the second method, @Karim :P

user96977

21:10

the proof is that there are no three values, right?

Karim Mansour

yeah me 2 :D

Ted Shifrin

What does that mean, @TruthSerum?

Karim Mansour

should I take introduction to art as elective or economics?

user96977

the equation is not solvable if the functions are linearly indipendent

Ted Shifrin

NOt so, @TruthSerum. Linearly independent $\iff$ there is only the trivial solution $a=b=c=0$.

Karim Mansour

21:12

it is always solvable

user96977

but it is not in order to satisfy the definition of linear indipendence

Ted Shifrin

No, you need to relearn the definition.

Or we're having a language problem.

user96977

maybe it is some formalism to consider zero as a solution, but it is never accepted in practice...

Ted Shifrin

um, @TruthSerum: Don't say things like that when you're wrong.

user96977

i'm wrong that it is a formalism?

Ted Shifrin

21:22

Unless you have a Ph.D. and have been a mathematician for 30 years.

It is not a formalism. Any homogeneous linear equation always has the zero solution. The question is whether it has any others.

Kelly Blunie

@TedShifrin i understand that there is a rational number between x <y. now how can i modify something in the answer and prove for irrational number be between x<y ?

Ted Shifrin

@KellyBlunie: OK. Find a specific irrational number between $0$ and $1$.

Karim Mansour

1/sqrt(2) for example

Ted Shifrin

hush, @Karim

Karim Mansour

heh

user96977

21:26

@TedShifrin does mathematica ever print out the zero vector as a solution to a system?

Ted Shifrin

I don't know, @TruthSerum. But the way Mathematica is coded is not what runs mathematics.

Kelly Blunie

@TedShifrin 1/(2)^(1/2)

Ted Shifrin

And, yes, @TruthSerum, it just gave me $\{x\to 0, y\to 0\}$ when I told it to solve.

user96977

what about when there is another, non-zero, solution

Ted Shifrin

OK, @KellyBlunie. Now give me an irrational number between any two rational numbers.

@TruthSerum: If the system is homogeneous, it gives you the general solution with parameters, doesn't it?

If you have a homogeneous linear system, it either has just the zero solution or it has infinitely many solutions, with a certain number of parameters.

user96977

21:29

$a(1-1/\pi) + b(1/\pi)$

Ted Shifrin

OK, @Truth. Now you're done. :)

Use what you showed first (between any two numbers there's a rational) and what you just told me to finish the problem.

user96977

i think that was someone elses problem

user61230

Okay, quick question. If $A, B, C$ are matrices, $A$ is invertible, $\det(AB)=0$, and $AB=BC$, can we say anything about the invertibility of $C$? (I've gotten to the point where I know $B$ isn't invertible.)

Ted Shifrin

Oh, hell. I wish people would stop interrupting.

user61230

oh, sorry

user61230

21:31

I thought it was quiet

Mike Miller

@Emrakul: He wasn't talking to you.

user96977

sorry, did i spoil the fun

Kelly Blunie

@TedShifrin 1/sqr(9)<1/sqr(5)< 1/sqr(4)

Mike Miller

@Emrakul: No. Set $B = 0$. Then $C$ can be literally anything.

user61230

...fair enough. Huh.

Ted Shifrin

21:32

@KellyBlunie: That remark was meant for you. Go back and answer chat.stackexchange.com/transcript/36?m=21356941#21356941.

Kelly Blunie

@TedShifrin i mean root square

Ted Shifrin

@TruthSerum: YOu're too busy being wrong about linear equations ... so stop doing other people's work.

user96977

i suppose you are right, zero is solution, so all "homogenous linear systems" are solvable

Ted Shifrin

Yippee. The boy sees the light.

Mike Miller

no need to fight, eh

Kelly Blunie

21:36

1/(9)^(1/2)<1/5^(1/2)< 1/(4)^(1/2) same as 1/3<1/5^(1/2)< 1/2 @TedShifrin

Ted Shifrin

@KellyBlunie: You're supposed to be doing the general case?

Kelly Blunie

yes

Ted Shifrin

So what's the answer to what I said up there ^^^ ? :)

Kelly Blunie

can you point it out where, please

Ted Shifrin

13 mins ago, by Ted Shifrin

OK, @KellyBlunie. Now give me an irrational number between any two rational numbers.

Kelly Blunie

21:44

general case? or particular solution?

Ted Shifrin

General case.

If x and y are rational, give me an irrational between them.

Karim Mansour

okay @TedShifrin in this summer I will only take mechanics as elective so now I have alot of time to do self studying during the summer for math.

I was gonna take one extra elective but screw that I will take that next winter semester now I will just get ready for my fall semester.

Samuel Yusim

21:56

consider the topological space {1, 2, 3} given the topology {{}, {1, 2}, {1, 2, 3}}. from my understanding the sequence (2, 2, 2, ...) converges to 1. is this true?

Mike Miller

@SamuelYusim Yes, and also to 2.

Samuel Yusim

okay, glad I'm not insane

but wow that's weird

Mike Miller

Convergence need not be unique in non-Hausdorff spaces.

non-Hausdorff spaces are weird.

ᴇʏᴇs

@TedShifrin Are your students invited to your party

Ted Shifrin

mr eyeglasses, we did invite some former students and a few current ones, plus all the grad students

Mike Miller

21:59

@Ted: Does a connected 3-manifold with positive curvature have to be covered by $S^3$? Certainly its universal cover would also have positive curvature (and one can take it to be constant 1) but to invoke Killing-Hopf I would need its universal cover to be complete, and I have no idea whether or not I should expect that to be true

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$Mathematics$ Mathematics