Mathematics - 2017-01-16 (page 4 of 5)

Krijn

20:00

@SimpleArt $\sum_{i = 0}^n F_i^2 = F_nF_{n+1}$ where $F_i$ is the $i$-th Fibonacci number

Simple Art

Approximate this sum with +/-0.1 error for all n

$$\sum_{k=1}^n\frac1{\sqrt k}$$

@Krijn Thanks, I might do that :3

Krijn

Or $\sum_{i = 1}^n 2i = n(n+1)$ has a nice visual proof

Simple Art

@Krijn I think more advanced

SteamyRoot

Building on that, prove that $\left( \sum_{i=1}^n i \right)^2 = \sum_{i=1}^n i^3$ ?

Krijn

No

sarcopsy

20:02

You can motivate some simple group theory stuff with counting problems I guess?

Krijn

Draw $n \times (n+1)$ dots

Then look at the triangle with 1, 2, 3, $\ldots$, n dots

Astyx

Numbering the maximal amount of regions you can form with $n$ straight lines ? How many are bounded ? How many triangles do they form ?

Krijn

The other triangle is also the same triangle and they hold $\sum_i i$ dots both, so $\sum_i 2i$ dots in total

Simple Art

I ask this because I've recently made a small chat room full of random people, and I've decided I'm going to have some questions and try to get them to touch on a lot of random topics.

@Krijn Gauss :-)

Krijn

@SimpleArt Close, but simpler than his method

Simple Art

20:04

But I seriously think I can introduce harder concepts than that @Krijn like the gamma function.

Mhm... maybe I should ask them how to define fractional values of the factorial :D

Astyx

It requires quite a lot of pre-reqs to some extent

Simple Art

@sarcopsy I'll have to learn that XD

@Astyx The gamma function?

Astyx

Yes

These are not so random people I figure

Simple Art

I think they all know calculus/integrals, and we're not going to do anything crazy

@Astyx Oh no, they are random :D

Astyx

The amount of people who know calculus/integrals is not that big

sarcopsy

20:07

Isn't it a compulsory thing?

Astyx

That does not mean much

Simple Art

What's compulsory? Learning? Probably

@TheGreatDuck Hi totally random person :-)

Ah... anyways...

TheGreatDuck

well yeah I am very random right now. Incredibly eratic.

Simple Art

lol

TheGreatDuck

dont get in my way right now. I'm in the process of purging my inappropriate questions.

:p

sarcopsy

20:09

@Astyx and @SimpleArt I guess.. haha

Mike Miller

@PVAL-inactive A bit aggressive.

Cameron is very nice.

Simple Art

The camera?

PVAL-inactive

Whats aggressive?

Simple Art

Cameras

Krijn

So, at university I work in a room with 4 other Masters students and sometimes I need to look up a silly definition, like "automorphism". Out of shame I will google it on my phone instead of my laptop because the others will think I'm hella dumb. Today we found out that all five of us are doing this.

PVAL-inactive

20:10

@Mike What's aggressive?

Simple Art

@Krijn XD

SteamyRoot

@Krijn everyone does it

Astyx

I've no idea what an automorphism is

Balarka Sen

@Krijn I do it so very often

Simple Art

@PVAL-inactive Me saying cameras

Astyx

20:11

I always look it up too

Simple Art

@Astyx :P Have you taken real analysis?

Astyx

Sure

Simple Art

lol

I haven't

PVAL-inactive

@Mike Cameron is unbelievably nice.

TheGreatDuck

@SimpleArt gonna lose about 100 rep but I think if the questions have sat dead for this long unanswered they should be deleted. (referring to questions from last year).

Astyx

20:12

@Mike I've read you were going to Cambridge ?

SteamyRoot

@Astyx Do you know about groups and group isomorphisms?

TheGreatDuck

i figure more than 50 questions is a bit... excessive

PVAL-inactive

He's usually in his office all day available all the time and always willing to discuss things.

Astyx

I sure do @SteamyRoot

Simple Art

@TheGreatDuck Nah, don't delete

You are probably referring to products and integrals?

TheGreatDuck

20:12

no

just random silly questions

Simple Art

Oh

No, don't

SteamyRoot

Oh, okay.

TheGreatDuck

like "why does the multiplication operator change shape"

Simple Art

that like ruins the system

Just leave it mate...

TheGreatDuck

if they're unanswered it doesn't matter...?

Simple Art

20:13

@TheGreatDuck :|

I come across plenty of unanswered questions

and they can still be of value to me

And I broke 800 rep today finally

TheGreatDuck

what? I cannot clean up old questions that I find no longer useful to me personally and that in a lot of cases attracted downvotes?

Simple Art

Hm, what do you guys think?

Balarka Sen

Tim Burton on Kevin Smith (after Smith jokingly accused Burton of stealing the ending of Planet of the Apes from a Smith comic book): “Anyone who knows me knows I would never read a comic book. And I would especially never read anything created by Kevin Smith.”
Kevin Smith on Tim Burton (in response to “I would never read a comic book”): “Which, to me, explains fucking Batman.”

Astyx

Lol

Mike Miller

@PVAL The proof by Lagrangian.

@Astyx I'm here for this week.

Simple Art

20:17

You know what

TheGreatDuck

manbatbatmanbatmanmanbat

Astyx

Oh is it only one week ?

Simple Art

I think I'm either going to show my chat buddies some fractional calculus or divergent series

Balarka Sen

David Cronenberg on M. Night Shymalan: “I HATE that guy! Next question.” (@MikeMiller would like this)

Simple Art

lol

man-bat the best

TheGreatDuck

20:19

do you know what I was referring to?

PVAL-inactive

@MikeMiller Here's a more ridiculous one. $H^1(T^n) \ne 0$ as $T^n$ is a Lagrangian inside $\Bbb R^{2n}$ and by Gromov there are no such exact Lagrangians.

Simple Art

@TheGreatDuck Maybe... man-bat is a thing though

Forever Mozart

I have a 2 page paper with a nice little result. Should I send it to Arxiv? will they accept?

PVAL-inactive

Do you have an adviser?

SteamyRoot

@ForeverMozart ArXiv isn't about "accepting".

PVAL-inactive

20:21

Or anyone you are reading research with?

TheGreatDuck

care to guess?

SteamyRoot

You need people (as in, academics), to vouch for you

Forever Mozart

yes I have advisor, but this is just something I came up with over the weekend

TheGreatDuck

...

PVAL-inactive

Ask her/him.

TheGreatDuck

20:22

then it's probably garbage atm, no offense.

nothing that special happens over a weekend. :-)

Forever Mozart

@SteamyRoot to post on archiv I need someone to vouch?

Balarka Sen

Werner Herzog on Jean-Luc Godard: “Someone like Jean-Luc Godard is for me intellectual counterfeit money when compared to a good kung-fu film.”

Err

PVAL-inactive

If they think it might be novel and interesting enough to prove, do a thorough search through the literature to make sure it is actually novel.

SteamyRoot

@ForeverMozart arxiv.org/help/endorsement

Forever Mozart

@TheGreatDuck actually its a result I've had for several months... I just found a way to fully generalize over the weekend

SteamyRoot

20:23

And definitely talk about this with an advisor before submitting.

Simple Art

@TheGreatDuck What? All the special things happen on the weekend

Like the realization that you still haven't finished your homework

PVAL-inactive

If your adviser isn't sure about the novelty, you might want to email an expert or two in the area (ideally one you have had contact with) to ask them.

Forever Mozart

@SteamyRoot oh wow I did not know that thanks

TheGreatDuck

@SimpleArt i finish my homework the day it's handed out.

even if I have to stay up all night to do so

Simple Art

@TheGreatDuck I do that unless its history. Then I put it off till the day its due

:D

TheGreatDuck

20:25

well if I get it done early then I can just pretend it isn't done and then I have an excuse to work on "other things".

i.e. weird axiomatic garbage

Astyx

@TheGreatDuck What about staying up all night and all day long the day after ?

Forever Mozart

Godard is my favorite filmmaker

Simple Art

@Astyx No, weekends are about sleep. Sleep, sleep, and math.

TheGreatDuck

@Astyx I don't take that long to finish my homework.

Forever Mozart

(note my profile picture)

(and my name)

@BalarkaSen

Astyx

20:26

I thrive to do the same as you, but sometimes too much simply is too much

TheGreatDuck

@SimpleArt no the weekends are about never needing to sleep and pushing one as far as possible.

just like the regular week

Simple Art

XD

TheGreatDuck

i only sleep when I have absolutely no homework left

SteamyRoot

Weekends are for realising you slacked off too much during the week and need to catch up on work.

3

TheGreatDuck

period

otherwise I am being irresponsible

Forever Mozart

20:27

Herzog never made a good film that wasn't a documentary, so...

@BalarkaSen Have you seen Hail Mary?

TheGreatDuck

@SimpleArt the point of homework is not to finish by the due date. It is to finish it immediately. It doesn't matter if you have two weeks to do it. You finish it that day. Period. Otherwise, you don't sleep till it is finished.

Simple Art

Lmao

SteamyRoot

Uhhhh

TheGreatDuck

im not joking, man.

SteamyRoot

That sounds terrible.

TheGreatDuck

20:29

why?

Simple Art

Cuz, who wants to finish their homework?

Forever Mozart

Also, Shymalan sucks compared to Cronenberg!

SteamyRoot

Because, you'll eventually get projects that you can't possibly finish in one sitting.

Balarka Sen

@ForeverMozart Nope. I plan to watch something by Herzog at some point

TheGreatDuck

@SteamyRoot I don't need sleep. I rarely do regardless of homework.

Balarka Sen

20:30

I only like one or two films of Shyamalan. The rest is a pile of crap

Forever Mozart

oh, Hail Mary is by Godard

SteamyRoot

Also, taking your time to properly think about something, discussing it with others, etc, can be very valuable.

Balarka Sen

Ya, I was responding to two messages at once

SteamyRoot

It's not just about sleep. It's about concentration and inspiration too.

TheGreatDuck

@SteamyRoot most of the time, I'm the one having to help explain it to others cause they either have no clue what they are doing or need help.

and I'm doing quite well so I must be doing something right. :-)

Forever Mozart

20:31

Pierrot le Fou is possibly my favorite movie

so amazing

Balarka Sen

There also seems to be harsh criticisms of Godard by Bergman.

I thought he was a fine guy, huh

Astyx

I found Pierrot le Fou very weird

Forever Mozart

I mean, Godard and Bergman are like polar opposites

Balarka Sen

In what sense? I am interested, but I haven't seen anything of either.

Forever Mozart

@Astyx the first time I saw it, I hated it.

Astyx

20:34

That must explain

Forever Mozart

@BalarkaSen Bergman is very serious. Godard is very playful.

TheGreatDuck

@SteamyRoot The way I figure, one should never stop working on homework. When you run out, you find something else to work on.

Balarka Sen

hm, I see

SteamyRoot

I'm not sure what you count as "homework".

Astyx

@TheGreatDuck Time will come when you have too much

Forever Mozart

20:35

all of Bergman's films are kind of morbid

excellent, but morbid

TheGreatDuck

@Astyx if I have so much that I cannot finish it by the deadline then I am doing something wrong.

Astyx

That's not the point : you have to sleep by the deadline

TheGreatDuck

alright fine

but i shouldn't do anything in between that except eat

Astyx

Being hardworking is good. Working too hard isn't

TheGreatDuck

and go to classes, obviously

what else would I do?

Astyx

20:37

Rest

TheGreatDuck

and just twiddle my thumbs? No thank you.

Astyx

Resting is very important for learning

SteamyRoot

^

TheGreatDuck

and i do sleep

SteamyRoot

I only learned the joy of napping when I was like, 20.

TheGreatDuck

20:38

I'm 21

napping makes me feel sick afterwards

SteamyRoot

o.O

TheGreatDuck

like literally

i get a headache

i mean, cause then I get too much sleep

SteamyRoot

Seriously, though, with your (compulsory?) work drive - how do you handle writing a large project or thesis?

TheGreatDuck

i was exaggerating about no sleep. I meant normal homework like a sheet of problems

i couldn't possibly stay up /that/ long.

id pass out

XD

Socrates

if you don't sleep after learning, the learning effect gets demenished

if you play videogames or watch tv afterwards it's really bad

TheGreatDuck

20:40

@Socrates I do sleep every night. 8 hours. Though everyone tells me I need to sleep for 10 or 12

idk why.

Socrates

taking a nap after a learning session is the best

your brain needs this to order the stuff

TheGreatDuck

well I don't watch tv

and i don't play video games.

i just move onto the next subject.

SteamyRoot

wut, 8 hours?

Astyx

How do you cope with holidays also ?

Forever Mozart

i still play Mario Kart to relieve stress every now and then

TheGreatDuck

20:42

@Astyx I relax all day like normal people.

SteamyRoot

I probably sleep an average of 6 hours during the week. 7-9 on weekends.

Forever Mozart

running over people is very fun

and relaxing

sarcopsy

I slept 16 hours yesterday to fix my sleep schedule, feels good

Socrates

there is memorizing and learning tho

i'll take learning any day

Forever Mozart

16 hours???? are you sick?

TheGreatDuck

20:43

@SteamyRoot I don't tend to have a lot of difficulty in my homework. Usually it only takes me about an hour or two. However, if I feel like having alone time then it usually ends up "taking" 5 or 6 hours.

but that's only when I have nothing left

sarcopsy

@ForeverMozart No, I was too lazy to get up haha

SteamyRoot

@TheGreatDuck Are you mass-unaccepting answers and flagging them?

TheGreatDuck

@sarcopsy that can actually damage your brain.

sarcopsy

I was sleeping too late so I decided to fix it

Its not like I do this on a regular basis haha

TheGreatDuck

@SteamyRoot I've unaccepted a few of my answers. Not that many though.

and there were 3 or 4 which I flagged once again as they were still there even though they were definitely not answers.

(like, blatantly not answers)

@SteamyRoot to be honest, I work on homework like any other normal person. It's just that early in the semesters a lot of the homework only takes maybe 10-20 minutes.

so in reality me staying up late at night or whatever is just me making up BS reasons to stay up late cause I'm not tired.

SteamyRoot

20:47

Partial answers are allowed (and encouraged) according to meta, though.

Either way, your call. It's just that a load of answers on your posts popped up in review

TheGreatDuck

@SteamyRoot no I mean like, "assuming x is true, prove y is true"

SteamyRoot

And that you have 71 questions with only one of them having an accepted answer

TheGreatDuck

their answer is "x is false, therefore y is false. your question is garbage"

@SteamyRoot they never really did have accepted answers. I only had 3 or 4 questions with accepted answers.

why are you looking at that, anyways?

:/

SteamyRoot

Because one of my answers was unaccepted; and I was going through the review queue

TheGreatDuck

it was probably one of the two or three that were answers to inappropriate questions that now need to be deleted. I cannot delete questions with accepted answers.

SteamyRoot

20:50

Doing certain actions en masse (flagging, unaccepting) may trigger some website stuff.

You can't delete a question with upvoted answers.

robjohn

@amWhy you called?

TheGreatDuck

@SteamyRoot well 99% of my questions are absolute garbage anyways. I need to get of them.

SteamyRoot

That's not how MSE works.

TheGreatDuck

no i mean I made them intentionally vague

sarcopsy

that seems like a strange way to ask questions

SteamyRoot

20:54

It seems like something that would attract many downvotes or close votes.

TheGreatDuck

it does

amWhy

@robjohn Hi!!!! Haven't seen you around too much (you'd help at more life to the math mod chat!) Yes, I'd like to know how to adapt an avatar/gravatar etc with another image. You've done some creative things, in this regard. Perhaps you've even written code to do so (e.g., pumpkin). For now, I'm wanting to "wear" the cheese head hat in honor of the Green Bay Packers.

SteamyRoot

The preferred action would then be to edit and improve your question(s).

Semiclassical

^

SteamyRoot

Either way, I can't be bothered with this anymore, but if you keep flagging answers unjustly you're likely to get suspended

And, well... it's incredibly ungrateful to anyone who put time and effort into answering your questions.

Astyx

21:00

That's a lot of stars on this very subject

SteamyRoot

Right, back to actual work now. I get too worked up about this.

amWhy

@robjohn Or maybe the above that my avatar can wear?

TheGreatDuck

@amWhy any particular reason why you chose to keep this question closed? math.stackexchange.com/questions/1539433/… I'm curious. :D

(I was looking through stuff and noticed it. Thought I'd ask since you popped in.)

amWhy

@robjohn ... or maybe I should simply replace my "standard" avatar with an image?

Socrates

@TheGreatDuck why did you get suspended?

TheGreatDuck

21:05

@Socrates hmm?

Socrates

"this account is temporarily suspended"

TheGreatDuck

not sure

i hadn't seen

i got a message from the mods

none of your business. :p

hadn't noticed that though

been busy yakking in here i guess

Perturbative

Hey everyone, is this proof below a rigorous enough way to show that $0$ is a limit point of $(0, 1) \subset \mathbb{R}$ with the standard topology?

Take $A =(0,1) \subset \mathbb{R}$ with the usual topology on $\mathbb{R}$. Then take $U$ to be a neighbourhood of $0$. Fix $\gamma > 0$ and $\epsilon > 0$, then every $U$ is of the form $(0 - \gamma, 0 + \epsilon)$, and thus for every $U$ we have, $U \cap A \neq \emptyset$, and $U \cap A \neq \{0\}$.y

Astyx

A neighbourhood does not have to be an open interval

TheGreatDuck

@Socrates Probably cause I asked so many close-worthy questions and horrible answers.

Astyx

21:09

It only has to include an open interval in which the point lies

Else it seems fine

Perturbative

@Astyx Thanks for the verification, but in Munkres book he uses the convention (which I've also used above) that a neighborhood of $x$ is an open set containing $x$, not the other convention that some use where a neighborhood of $x$ merely contains an open set containing $x$? Which is the standard convention?

amWhy

Well, I've gotten this far, @Robjohn (see my current avatar)!

Astyx

To my knowledge a neighbourhood is the later, but I've heard the first too. As long as you make clear which one you use, there's no problem here

sarcopsy

well technically an open set in the standard topology is not an open interval either, but a union of open intervals? I think

Astyx

If it is a union of open intervals, it contains an open interval

sarcopsy

21:17

yeah, but the typed out proof says that 'every neighbourhood is of the form (open interval)'

just being pedantic >_>

Astyx

Oh alright :p

sarcopsy

but its a distinction i guess

Astyx

You're right

Vrouvrou

Can someone help me to prove this

Why $(C_{E}A\times B)\cup (A\times C_{F}B)\cup(C_{E}A\times C_{F}B)=(C_{E}A\times F)\cup (E\times C_{F}B)$

sarcopsy

What are the variables supposed to be

Astyx

21:20

Is there an other way of checking that $(GL_2(\Bbb Z/2\Bbb Z), \times)$ is isomorphic to $(\mathfrak S_3, \circ)$ than actually giving an isomorphism ?

Oh yeah there is

PVAL-inactive

If you want to prove there exists something, the best way is to construct that something.

Vrouvrou

can someone help me

Astyx

Since there are only two groups of order 6 (up to isomorphism) my question shrinks down to "Is $GL_2(\Bbb Z/2\Bbb Z)$ abelian ?"

@Vrouvrou Please explain your notations

sarcopsy

@Vrouvrou What are $C_E$ and $C_F$ and the other variables

Astyx

Hi @Hippalectryon

Hippalectryon

21:23

\o

Vrouvrou

$C_{E} A$ is the complement of $A$ from $E$ @sarcopsy

where $E$ and $F$ are two spaces

Astyx

@Vrouvrou the first and last term of the LHS merge into the first term of the RHS

Vrouvrou

??

Astyx

And the second and last into the last

Vrouvrou

i don't understand what you say

Astyx

21:25

$(A\times B) \cup (A\times C) = A\times(B\cup C)$ by writing it down

Does this help ?

Vrouvrou

then $(C_{E}A\times B)\cup (A\times C_{F}B)\cup(C_{E}A\times C_{F}B)= (C_{E}A\times F)\cup (A\times C_{F}B$

Astyx

Yes

Vrouvrou

this don't help me

Astyx

Why not ?

Vrouvrou

i hace to find $(C_{E}A\times F)\cup (E\times C_{F}B$

Astyx

21:31

Hint : $A = A\cup A$

Vrouvrou

we write $(C_{E}A\times B)\cup (A\times C_{F}B)\cup(C_{E}A\times C_{F}B)=(C_{E}A\times B)\cup (A\times C_{F}B)\cup(C_{E}A\times C_{F}B)\cup(A\times C_{F}B) $

Astyx

Exactly

Mike Miller

@Astyx Yes, only this week. There's a workshop.

Astyx

Cool. You don't live in england otherwise, do you ?

Socrates

$A=A\cup A=A\cap A$

Vrouvrou

21:39

@Astyx thank you

Astyx

@Vrouvrou my pleasure

Socrates

@Astyx hi, how are you?

Astyx

@Socrates Hi (we've been in this chatroom for some time today, haven't we ?), I'm fine and you ?

Socrates

@Astyx that is good to hear. I feel ok. We are some time here, but I can't recall that we greeted us :D

Astyx

We didn't, you're right

SteamyRoot

21:46

Oh, @Astyx, another Linear Algebra question: Prove for any square matrix $M$ that $\det(I + M) + \det(I - M)$ is an even number.

Socrates

why is the 0-vector a thing to check for subspaces? isn't closure under scalar multiplication enough?

Astyx

@Steamyroot $M\in M_n(\Bbb Z)$ ?

Socrates

since that implies that the 0-vector is in the potential subspace

SteamyRoot

Uh, yeah, indeed.

Should've definitely mentioned that.

Astyx

No, it ensures the space isn't empty @Socrates

Socrates

21:48

@Astyx ah, then that was the thing, so it could have been any other vector too

Fargle

@Socrates Absolutely not.

Socrates

to check if it's not empty?

Astyx

@Fargle Why not ?

Oh I see what you mean

Socrates

I don't see it

Astyx

Well $0$ is in any subspace

Any other vector isn't

Fargle

21:50

Well, you can find any other element to check if it's not empty, but the zero vector is the only one that automatically determines whether a subset is a subspace or not.

Astyx

It's true that $0\in V \iff \exists x\in V$

PVAL-inactive

@Ted Is there a totally real 2-sphere in $\Bbb C^2$?

Socrates

but 0v=0, so scalarmultiplication rules that?

Astyx

No

Fargle

@Socrates If the set is non-empty, yes, but $0 \in V$ is a much easier condition to check.

$\begin{bmatrix}2 \\ 1\end{bmatrix} \notin V$ doesn't tell you much of anything. $0 \notin V$ tells you everything.

Socrates

21:53

so {(a,1)} with a in a field, will never be a vectorspace

Astyx

No

Fargle

@Socrates Indeed. There is no zero vector. (Also, scalar multiplication by anything but 1 takes you outside the set.)

Mike Miller

@Astyx No, I live in California.

Astyx

@MikeMiller Right. I would have warned you about brexit, but if you're only there for a week, I guess you don't care that much

Socrates

ok, and what tells us $0\not\in V$?

Astyx

21:56

That it isn't a subspace

Mike Miller

I'm certainly aware of Brexit.

Socrates

mmh, I see

so, apart from the nonemptyness {} fullfills the axioms?

or not, because there exists no identity?

Simple

I need help with this question. Let $Y$ be a Poisson $\lambda$ random variable, and define $X=I_{[Y>0]}$. How to fine the mass function of $X$

sarcopsy

@Simple $I_{[Y\gt 0]}$ is an indicator variable?

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$Mathematics$ Mathematics