Mathematics - 2021-03-02 (page 1 of 2)

The above should be write. The fundamental group of the above space is $\langle a,b,c|aba^{-1}b^{-1}cbc^{-1}\rangle$ if I let $c$ be a line segment connecting the inner circle and bottom left vertex

BigSocks

3:01 AM

'it isn't the klein bottle" i think is the resolution

Akiva Weinberger

What if you took your first picture and just reversed the orientation of the innermost circle

BigSocks

matching 2 and three would be tough

Akiva Weinberger

I mean, it doesn't say which way to identify the boundary circle with the longitudinal circle

BigSocks

true, could "slide inwards"

love_sodam

It seems the same problem happen when I identify 1 and 3 but twisting would give the space

I don't know this is the right one

Akiva Weinberger

3:04 AM

Reverse 3, identify it with 1 by leaving 1 static and "pulling 3 inside" and over, and then bring 2 around

No wait

Do what I just said but don't reverse 3

It's unnecessary

3 mins ago, by Akiva Weinberger

I mean, it doesn't say which way to identify the boundary circle with the longitudinal circle

^This choice doesn't matter

love_sodam

I can't understand the part 3

Jeff

OK, can someone help with this (it's different than what I posted before).
If partial sum of $\sum_{k=2}^{\infty} a_k$ (starting at 2) is given by $S_n=(n-2)/(n+6)$, find the sum (from 1) of $\sum_{k=1}^{\infty} a_k$.

BigSocks

@love_sodam me neither

Akiva Weinberger

BigSocks

still 3 is weird

Akiva Weinberger

3:14 AM

How so

BigSocks

I don't understand where the hole is. like it is "pushed in"?

Akiva Weinberger

In Sodam's third picture, 3 is the dotted lines. Erase them, and it looks like a torus

Yes

love_sodam

Ah

I see what you're doing

BigSocks

yeah I saw it now

love_sodam

Thanks a lot @AkivaWeinberger !

Akiva Weinberger

3:17 AM

No problem

@Jeff Interesting. What's the formula for $S_n-S_{n-1}$? (That should be $a_n$)

If we assume $a_1$ follows the same formula, we can add it on to the rest of the series

Jeff

Not given (that's a different part of the problem). But it comes out to $a_n = 8/((n+5)(n+6))$. However, it does not hold for $a_2$. @AkivaWeinberger

Akiva Weinberger

I don't think we have enough information

$a_1$ could be anything without affecting the partial sum of the sum starting at $a_2$

Jeff

@AkivaWeinberger Not what the boss says.

@AkivaWeinberger Good point

Ted Shifrin

3:39 AM

@Jeff Are you still going on with this? Nuts.

Jeff

@TedShifrin It's not quite the same. I misunderstood the problem (there were no typos).

Ted Shifrin

You’re still typing garbage that I corrected before.

Jeff

Given the sum starting from 2, and the formula for its partial sum, find term a_1, and find the sum from 1.

Yes, but that's how I was given it.

Ted Shifrin

It's crap.

Jeff

OK. It's still how it was given.

Ted Shifrin

3:43 AM

You are not typing partial sum when you go to infinity, and there's no way to know what $a_1$ could be, Be a critical thinker and explain to your teacher why it's not rightf. Stop telling us crap was given ..

There are bad teachers, but you have to hold them accountable. We have no power to mindread.

I say this having taught more than 40 years.

Jeff

Yeah, I tried telling her. She didn't reply until I told her that I gave students credit for the problem because it had a type. Then (and only then) did she reply "There were no typos". But she said nothing else. She is not what you would call helpful.

@TedShifrin :D

In fact, she gave me a problem the week before that which was (that conditionally convergent series) that couldn't be done with the tools given and she insists it's right.

Ted Shifrin

Well, that one might have been right. I need to see screen shots of what she gives before I can say definitively.

Jeff

She and I don't get along very well. She is very micromanaging, so she looks at all of the papers I grade and critiques all my grading (which, I might add, is fine for all the other professors in the department).

Ted Shifrin

Oh, you're a grader, not in the class?

Jeff

Not in the class. Recitation instructor.

Ted Shifrin

3:51 AM

So you are a grad student? You know better than this.

Jeff

Huh?

Ted Shifrin

Complain to the graduate coordinator or head of undergraduate studies. Don't complain to us..

Jeff

I wasn't complaining here. I was asking for help on a problem I didn't understand. She is who she is.

Ted Shifrin

She's in charge of the course and you have to make her responsible. As I said, if she is not responsible, complain to the department .

I spent 8 years being associate department head and dealing with such fun.

Mark

4:39 AM

Where is the issue in the reasoning that $$0 = \Im(-1) = \Im(i^2) = \Im(i)\Im(i) = 1^2$$, where $$\Im(z) = \frac{z - \overline{z}}{2i}$$ is the standard imaginary part.

Thorgott

imaginary part isn't multiplicative

Mark

ok cool. I think I just remembered that both z-> z and z-> \overline{z} are, and decided to assume or something

Akiva Weinberger

comments from a fan

@TedShifrin

Ted Shifrin

Thanks, DogAteMy.

Wolgwang

4:58 AM

Can I edit this question? The edit will turn the given answer useless...but it doesn't answer the question.

Ted Shifrin

What if $\theta$ doesn’t evenly divide 360?

Wolgwang

5:20 AM

> If $n$ is in decimal then only the integral part is taken.

Ted Shifrin

I don't believe that.

Wolgwang

Uhm Why? If $\theta$ doesn’t evenly divide 360 then an incomplete image would be created...

Leonhard Euler

7:10 AM

never thought I would dig so deep in stats

for the sake of my new project...

I am making a scientific python project

I am first making statistical functions

I don't know much about stats so I may ask silly questions...

is there any nice sort of algorithm to calculate variance...?

porridgemathematics

8:12 AM

@LeakyNun thanks for that, made it so easy :)

in a separable metric space, is it always possible to find a sequence of uniformly continuous functions that converge pointwise to the indicator on an open set?

robjohn

@LeonhardEuler You mean like the mean of the squares minus the square of the mean?

Leonhard Euler

yes

nvm I already coded it

robjohn

ok

Leonhard Euler

now I am working on some methods to visualize data

robjohn

@LeonhardEuler Ah, you mean you're learning to lie ;-)

Leonhard Euler

8:21 AM

why

robjohn

How to Lie with Statistics

How to Lie with Statistics is a book written by Darrell Huff in 1954 presenting an introduction to statistics for the general reader. Not a statistician, Huff was a journalist who wrote many "how to" articles as a freelancer. The book is a brief, breezy illustrated volume outlining the misuse of statistics and errors in the interpretation of statistics, and how these errors may create incorrect conclusions. In the 1960s and 1970s, it became a standard textbook introduction to the subject of statistics for many college students. It has become one of the best-selling statistics books in history...

Leonhard Euler

I am going to burst out laughing

robjohn

So many people have lied with statistics that this book was written to show people how to avoid it.

Leonhard Euler

Is it possible to code analytical functions like zeta function?

I mean approximating the value is easy but I am talking about the exact value

Well there is a formula for zeta function at even integers

robjohn

The exact values are only known for a countable number of real points.

Leonhard Euler

8:27 AM

yes

and infinite sums?

robjohn

@LeonhardEuler and negative integers.

Leonhard Euler

I wonder which algorithms mathematica uses

robjohn

@LeonhardEuler for approximating?

@LeonhardEuler this answer does that

Leonhard Euler

@robjohn for getting the exact value

I think mathematica uses a database where many standard infinite sums' values, integral, etc. are stored

it simplifies the given integral or sum and then uses the values from the database

I think this is it

Okay I am rewriting my code in c++

bye

robjohn

have fun

ery cerious matherfunker

8:45 AM

Dear @robjohn I am truly thankful to you for not banning me for 30 min.-Your Sincere Audience William

porridgemathematics

under what circumstances do two measures on metric spaces coincide if they agree on all open or closed sets of the metric space? (here the metric space has the borel sigma algebra)

if the measures are sigma finite, then certainly I can see that this is true

but does it necessarily hold in general?

love_sodam

If $X = S^1\times S^1$ then how can I presentation of $\pi_1(T_f)$ in terms of the induced map $f_*:\pi_1(X)\to \pi_1(X)$ where $f:(X,x_0)\to(X,x_0)$ is a continuous map. $T_f := X\times I/(x,0)\sim (f(x),1)$ for $x\in X$ ?

robjohn

9:07 AM

@porridgemathematics ack! don't make me think about non-sigma finite measures.

porridgemathematics

sorry :D

robjohn

Feb 21 at 21:48, by robjohn

@Jakobian I guess I am saying that non-sigma finite measures are very weird creatures.

@eryceriousmatherfunker what did you do that might require that?

love_sodam

9:33 AM

wow the problem is too hard to me

ery cerious matherfunker

@robjohn Hehe some things should be kept secret. Apologies for wasting your time my lord.

Alessandro Codenotti

9:58 AM

@porridgemathematics No, there are easy counterexamples

There are arguments with the monotone class theorem or Dynkin's $\pi-\lambda$ stuff that give sufficient conditions but I always forget how these work

Look at the counting measure on $\Bbb R$ and the measure that is $0$ on $\varnothing$ and $\infty$ everywhere else for example

If you try to prove that they agree on all Borel then you see that the issue is with complements, you can't say anything when a set and its complement both have measure infinity

robjohn

10:42 AM

@eryceriousmatherfunker just don't get too happy.

user2103480

10:58 AM

@robjohn thanks for saying that

robjohn

I don't always use measures, but when I do, I use sigma-finite.

Naman Agarwal

11:31 AM

Hi I am stuck at this integral.

Here, m is non negative. I tried to take E as a complex variable, complete a contour and integrate, but that didn't work out as the integral doesn't seem to converge at infinity for any non negative t.

Any other suggestions?

I am interested in the t-> infinity limit. So if I solve the integral, or justify limit, both are okay for me.

porridgemathematics

11:49 AM

hi, could someone help me figure out why 'it is clear' that $\mu(K) > 1 - \epsilon$ in this proof? imgur.com/a/75EJMDK

user2103480

@porridgemathematics Let $A$ be open, so that $A^c$ is closed, take $f_n(x) = \min\{1,n\cdot d(x,A^c)\}$. If $d(x,y) < \varepsilon$ then $d(x,A^c) \leq d(x,y) + d(y,A^c)$ and vice versa so $|d(x,A^c) - d(y,A^c)| < \varepsilon$, and capping at 1 does not hurt this

porridgemathematics

oh you're answering an old question of mine, yeah I found one, I went with $(min(1,d(x,A^c)))^{\frac{1}{n}}$

of course, you're works just as well

yours

user2103480

@porridgemathematics I guess geometric series

porridgemathematics

so you're saying if we put $A_n = \cup_{k=0}^{N_n} B(r_k,\frac{1}{n})$, then $\mu(A_1 \cap ... \cap A_k) > 1 - (\frac{\epsilon}{2^1} + ... + \frac{\epsilon}{2^k})$, but why?

user2103480

Is that really what I'm saying

porridgemathematics

12:00 PM

idk, what are you saying

user2103480

I think I'm saying $\mu(X\setminus K) = \mu(\text{union of sets of measure less than eps/2^n}) < \epsilon$

or <= whatever

porridgemathematics

ah okay, yeah that makes sense

thanks :)

user2103480

one can probably also phrase this in a direct way, but like this I find it the easiest

you're welcome!

it's stuff that I have to revisit anyways

user2103480

12:19 PM

@NamanAgarwal are you sure this converges

Naman Agarwal

@user2103480 No. Unfortunately, I am not sure about that.

user2103480

exchange i with $(i + \varepsilon)$ for any $\varepsilon > 0$ and then this should at least converge

Nicolás Castellanos

2:06 PM

How change decimal in binary?

Leaky Nun

How To Convert Decimal to Binary

►

two methods are explained here ^

Monty

2:36 PM

Does anyone know of any interesting diffusion equations where the diffusion matrix is singular?

Akiva Weinberger

2:59 PM

Conjecture: for any sequence of $1$s and $-1$s, there exists a $c$ such that $(-1)^{\lfloor c3^n/2^n\rfloor}$ realizes that sequence

Mike

Hello, can anyone prood these two well-known facts in functional analysis by constructing inverse maps?

0

Q: Prove that for $W$ is closed vector subspace of $V$, $V/W\cong W^\perp$ as well as $V^*/W^\perp\cong W^*$ by a different method

This is a well-known result in functional analysis, in terms of dual of quotient spaces and annihilators of subspaces. Let me formulate the problem first and a new attempt to prove it: Let $W$ closed vector subspace of $V$, $V/W$ the quotient space, and $W^\perp$ the annihilator of $W$. (1) Show ...

Thanks.

user2103480

@AlessandroCodenotti define a counting measure to be a measure on $(\Bbb R, \mathcal{B}(\Bbb R))$ that has values in $\Bbb N$ and is finite on every finite interval. This is equivalent to a an increasing sequence in $\Bbb R^{\Bbb Z}$ that goes from $-\infty$ to $\infty$, where we assign to every interval the number of points in the sequence lying in said interval.

We can define a kind of "weak" measurabability on the space of counting measures $\mathcal M$ by choosing the sigma algebra generated by evaluation functions $\mathrm{ev}_C \colon \mathcal M \rightarrow \Bbb N, m \mapsto m(C)$, a…

anakhro

3:28 PM

@RyanUnger what have you been up to? Are you still doing Riemannian stuff?

anakhro

3:47 PM

@TedShifrin hi!

Alessandro Codenotti

4:08 PM

@user2103480 do you know that there must be such a topology?

Ryan Unger

@anakhro no

user2103480

4:26 PM

@AlessandroCodenotti i mean, one could possibly take the topology generated by exactly the same functions

(Not sure if that isn't bigger than the sigma algebra we aim for)

But expand

Alessandro Codenotti

Not every sigma algebra is the Borel sigma algebra of a topology is my concern

user2103480

Oh, I thought that was a "do you know that this is actually a fact"

No, I dont know whether such a topology exist

Alessandro Codenotti

I see

I'm in a seminar right now but I'll take a look later

user2103480

Ok! Its not important anyways

Monty

Consider a non-linear diffusion equation $ \partial_t u_t = \div( A ( \nabla p(u) + u f ) ) $ where $p$ is thought of as the pressure, $f$ is some forcing term, and $A$ is some matrix. The atypical example is the Porus medium equation. What examples are there of this equation when $A$ is singular ? I cant find ANY interesting examples online

sorry I should of wrote $ \partial_t u = \div( A ( \nabla p(u) + u f ) ) $, $u$ is the object of interest function of time (and say space and/or velocity)

leslie townes

4:38 PM

it's funny to me that LaTeX assigns the least-used symbol of all time to \div

Monty

aha agreed In my macros its set to \divv.

leslie townes

i'm also annoyed that LaTeX doesn't have something simpler than langle and rangle for the inner product stuff. my macros have it as \< and \> but i shouldn't need macros.

robjohn

@leslietownes Is it the symbols you don't like, or the amount of typing?

leslie townes

i love the symbols. it's just weird that you need such strange text to call what many would regard as a very standard thing. more common than a lot of other things, anyway. yet without macros, it's langle and rangle.

robjohn

@leslietownes I guess it depends on your particular focus. I use them once in a while, but I don't find their macro length annoying. I wish \mathrm was a bit shorter, though.

and \operatorname

leslie townes

4:51 PM

i worked in functional analysis. everything was langled and rangled to high heaven

the weird thing is you'd get tex from your collaborators where they didn't introduce a shorter macro for it, and it was literally langle and rangle written out everywhere. weird way to find out that you're working with a psychopath

robjohn

@leslietownes then, yes, a set of macros would be useful. There are some people who preface even the shortest posts with a HUGE set of macros.

leslie townes

i worked with a guy who would verbalize langle and rangle (rhymes with angle) while writing on a chalkboard

robjohn

@leslietownes They probably had a keyboard macro for those.

leslie townes

complete nutcase

Ted Shifrin

I believe I wrote a macro \ip#1#2 for that.

Ryan Unger

4:56 PM

@TedShifrin yep

it's surprising how many (smart) people write $<x,y>$

Ted Shifrin

Of course, in most of my math writing I just used dot for dot product, anyhow, but still ....

That just looks horrid, spaces wrong, etc., etc., etc.

Smartness and LaTeX-awareness and laziness are probably all uncorrelated.

Ryan Unger

yeah, i don't know how you can look at that and think this is the way tex is supposed to look

Ted Shifrin

When I wrote it on the blackboard, I certainly didn't write < and >.

leslie townes

how do people manage spacing with dx in an integral? I usually toss in a \, before the dx. It feels kludgy and not the right way to do it

Ryan Unger

that's what I do

Ted Shifrin

5:00 PM

I usually do that, unless there are parentheses or other things to spread it out. And I always do dx\,dy in iterated integrals. But I do not do \mathrm d for dx.

Ryan Unger

mathrm d is for nerds

Ted Shifrin

I actually don't like the way it looks.

Ryan Unger

that too

Ted Shifrin

Have we turned into the Nitpicky LaTeX morning?

Ryan Unger

I am procrastinating a calculation

robjohn

5:01 PM

@TedShifrin Ah, yes. That is why I don't often use $\langle\dots\rangle$

Ryan Unger

so why not

robjohn

@RyanUnger <kick>

Ted Shifrin

In your prior mathematical days, @robjohn, I would suspect you had lots of inner products in your work.

leslie townes

mathrm d is the devil

robjohn

@leslietownes $d\mathrm{evil}$

leslie townes

5:03 PM

i feel like when people do that they're trying to tell me that they're better than me. i'm working through it. therapist says it will take some time.

Ted Shifrin

Oy vey. Now we need LaTeX therapists.

robjohn

@TedShifrin There were, but mostly, they were expressed as integrals, and we just wrote out the integral for clarity (with $\mathrm{d}x$).

Ted Shifrin

What if you were in a Sobolev space, @robjohn?

robjohn

@TedShifrin I would try to be as smooth as I can.

Ted Shifrin

Uh huh.

Ryan Unger

5:05 PM

I think lots of people use (,) for Sobolev inner products

<> is usually for the Riemannian metric

robjohn

I do remember Hörmander using a lot of $\langle\dots\rangle$s

Ryan Unger

you're right but the people who model their writing after hormander's are probably the same people who insist on mathrm d

robjohn

But back then, there was only $\TeX$, and that was hard no matter how you worked with it.

Ted Shifrin

I started learning (AMS)TeX in 1989, a full ten years after my Ph.D.

I did my thesis on a Hermes portable typewriter and then paid a secretary to type it officially on an IBM Selectric. What a pain (especially with all the integral geometry and indices).

leslie townes

i worked with a guy who would not depart from troff. he had twenty-year-old macros. troff or get off. he should have learned at least one dialect of TeX.

robjohn

5:09 PM

$\LaTeX$ is so much nicer to work with.

Ted Shifrin

About the third revision of my first textbook I realized I needed to learn LaTeX. Manually chasing down changes in Theorem and Exercise numbers got old really fast.

robjohn

@TedShifrin I did a lot of work on an Olivetti typewriter which had a changeable ball.

leslie townes

i had a conversation with dick kadison once about how he had an array of stamps he would use to annotate his typewritten manuscripts with sigmas and integrals and such. stamps, with a pad and ink. it felt so prehistoric.

Ted Shifrin

OMG. And I just used my Parker 51.

I feel like I'm in a math retirement home.

robjohn

Luckily, when I started writing things up, I had a Mac and Word would allow a bit of mathematical typesetting.

Ted Shifrin

5:11 PM

You're a young'un, @robjohn.

Ryan Unger

@TedShifrin lots of people still use fountain pens

robjohn

Before, it was all handwriting (mechanical pencils)

Ted Shifrin

I didn't mean the fountain pen, @Ryan, just this discussion as a whole.

I still have—and use daily—the Parker 51 I bought with hard-saved $10 when I was about 10.

BigSocks

trying to flag a question and it only lets me choose from 3 alternate stack exchange websites, none of which are the one I would put down?

robjohn

@BigSocks flag for migration?

Ted Shifrin

5:13 PM

Just flag for execution instead.

Daniel Adams

Someone at google should invent a latex search engine.

BigSocks

@robjohn I don't know how to do a migration (probably can't because low point count)- I was just doing a flag and hoping something like "stackoverflow" would show up, or the one for theoretical cs

robjohn

@DanielAdams Try Approach0

Daniel Adams

would have to be smart enough to not distinguished between different symbols used to denote the same thing.

Ted Shifrin

Well, TeXShop's LaTeX spell checker works pretty well.

BigSocks

5:14 PM

@TedShifrin possibly a good option, thanks

Daniel Adams

@robjohn ahh its in the works :)

robjohn