Mathematics - 2016-05-17 (page 1 of 3)

Mary Star

00:01

Does the generating set have to contain the element $1$ ? @EricStucky @Sebgr

Jessy Cat

@MikeMiller, are you a topology guy?

Eric Stucky

Does it, Mary?

Mary Star

Otherwise, how can every element of $G$ be a finite product of elements in $S$ ? @EricStucky

Eric Stucky

How could they, Mary?

Mario Lamas Espinoza

00:38

Hi

Eric Stucky

hello

Mario Lamas Espinoza

I don't understand this statement: "Consider a weighted digraph $D$. Let $D’$ be a spanning subdigraph of $D$ (i.e. $D’$
is obtained from D by removing some of the arcs). Let $C$ and $C’$ be maximum
closures of D and D’, respectively. Then
$ w(C) \leq w(C') $."

Eric Stucky

So, I imagine 'closure' means that you add edges $uv$ if there are edges $ux$ and $xw$, and do that recursively.

The issue is that you have to weight the new edges.

But probably the weights are just the sum of the weights on $ux$ and $xw$.

But I suppose if $D'$ is not acyclic there's no reason for this to be well-defined.

Mario Lamas Espinoza

I am reading this sciencedirect.com/science/article/pii/0166218X88900303

The statement is in the page 5

I have this definition
La clouse of a weigthed digraph $D$ is the vertex set $C$ of $D$ such that if $x \in C$ and $(x,y) \in$ arcs of D then $y \in C$

Chinatsu-creepy-chan

01:17

Hello there guys!

how are you doing? :)

Rudy the Reindeer

01:31

This = Possible duplicate of Determining the number of subgroups of $\mathbb Z_{14} \oplus \mathbb Z_6$? I think the answer by Henning Makholm is really good. Much better than all the others.

Chinatsu-creepy-chan

D:

J. M.

Hi @Rudy! Haven't seen you in a while.

Puzzled417

01:48

what's the difference between doing something like $x \equiv 1 \bmod 5$ and $x \equiv 1 \pmod 5$?

is there any reason to use parenthesis?

Chinatsu-creepy-chan

afaik the first expression is the same as saying 10 mod 5 = 5 mod 5

Rudy the Reindeer

02:19

@J.M. Haha, that is the pot calling the kettle black! How are you? : )

J. M.

@RudytheReindeer Blacker than a black hole, I guess. (I had to use modly magic to remember who you were.) I'm doing quite fine, thank you.

Rudy the Reindeer

Not much going on here, is there?

J. M.

@RudytheReindeer I certainly miss what we had three or four years ago...

Eric Stucky

Yeah, it's not really the active time of day :/

Rudy the Reindeer

@J.M. Me too. I also miss better quality of questions and better quality of answers.

Mike Miller

02:24

I tend to be satisfied with the quality of Q&A once I restrict to favorite tags.

J. M.

@RudytheReindeer You might remember that I used to be able to catch up to what I missed in my sleep in less than an hour, back then. Now, the front page gets filled just in the span of a lunch.

Rudy the Reindeer

Sometimes I just browse the front page for stuff I might want to answer.

J. M.

@MikeMiller Of course, if the OP is not good at the tagging...

Mike Miller

Then I will either remove their question from my sight in an unnecessarily violent manner, or I don't see it in the first place, and oh well. :)

Rudy the Reindeer

@J.M. I wish it did for me. Sometimes I want to answer something and I go there and nothing happens. : D As if someone froze the front page.

J. M.

02:27

@MikeMiller I was thinking more the second than the first; I've answered questions that were ostensibly not part of my purview, since I thought it'd be fun to learn. That does not look practical anymore.

Mike Miller

I tend to only try to answer things I expect I can figure out in a reasonable amount of time. Sometimes eg yesterday this leads to me spending half a day on it.

Rudy the Reindeer

That is reasonable : D

Especially if, once you hit the post button, a previously posted full answer appears that is better than yours : D

Mike Miller

Usually I don't answer 'competitive' questions. The way I browse MSE when I'm looking to answer is by searching intags:mine answers:all; it's rare I get a question somebody else is looking to answer.

J. M.

Niche stuff can be fun.

Mike Miller

I have learned a lot by answering questions.

Rudy the Reindeer

02:43

I'm going afk for a while. See you later! Good to see you J. M.!

Balarka Sen

04:50

@MikeMiller I hope that's not a reference to me :(

Mike Miller

05:14

@BalarkaSen Hehehe, no, but that would have been funny.

Balarka Sen

Funny is very relative.

Balarka Sen

05:35

Ooh, Sullivan proved that every topological manifold of dimension $\neq 4$ admits a Lipschitz structure. That's pretty darn close to having a smooth structure!

I wonder if anything can be proved using this fact. But even completely theoretically it seems like an exciting result.

Mike Miller

Yes, I think it has actually turned out to be of some use. But I don't know much about that stuff. I've tried to convince my officemate to teach me.

Balarka Sen

Ah, alright.

Is it possible that given a manifold with a lipschitz structure, a bunch of points (of measure 0 - in what sense? dunno) can be chucked out to get a manifold which actually admits a differentiable structure? I know that a lipschitz map is differentiable away from a measure 0 set, thus the question. Probably far-fetched though.

Mike Miller

Lipschitz homeomorphisms (or just maps) preserve measure zero sets, so that makes perfect sense. In any case I have no idea what that would mean.

Balarka Sen

05:50

I was wondering if something like that is true, then the above fact (which actually says every top manifold of dim $\neq$ 4 admits a unique lipschitz structure) would imply something like "chucking out a measure 0 set (in sense of the unique lipschitz structure) from a top manifold of dim \neq 4 gives a top manifold which admits C^1 structure", which would be cool. But this is all pipe dream.

Mike Miller

Is the set of points on which it's differentiable open? I see no reason to believe your differentiable map is actually a diffeomorphism, though.

I also think it's probably possible to delete a measure zero set and get something homeomorphic to an open subset of $\Bbb R^n$.

Balarka Sen

Good point on the first message about diffeomorphism (also; complement of measure 0 set need not be open?)

I am curious about the second message, because it's not immediately obvious to be how to do that.

Mike Miller

the rationals...?

Balarka Sen

ah, oops! I always think of volume when I say measure, sorry about that.

Mike Miller

i don't know any notion of volume such that volume 0 sets are closed

Balarka Sen

06:02

OK, yes, $\{1/n: n \in \Bbb N\}$ is an example of a volume $0$ set in $\Bbb R$ which is not closed I think.

Guess I was being stupid.

Balarka Sen

06:28

@Huy Still cooking I see.

Huy

@BalarkaSen: of course. cooking is awesome.

Lucas

07:04

Hi guys, can anyone give me a hint as to how to find all integers x,y satisfying 1/x+1/y=1/5

Balarka Sen

@Lucas Can you simplify the both sides so that they are integers?

The expression would be less elegant, but doing so would give you more elbow-room.

Lucas

@BalarkaSen, do you mean expressing it as 5(x+y) = xy? or expressing x=5y/y-5 perhaps?

I've tried the first, and found some things about divisibility, but I didn't find it to help with finding solutions x,y

Balarka Sen

07:20

5(x + y) = xy is what I wanted, yes.

5 divides the left hand side. So what does that tell you about the right hand side?

Lucas

5 must also divide the right hand side

so xy is a multiple of 5

Balarka Sen

Yes. Does that tell you anything about x and/or y?

Lucas

hmm

i'm not sure if this is true but they can't both be multiples of 5?

I mean it's not necessarily that they are both multiples of 5

one of them must be though

Balarka Sen

The last thing is what I wanted. Either x or y is divisible by 5.

without loss of generality, you can assume x is divisible by 5. agree?

Lucas

yup

Balarka Sen

07:25

Write x = 5k. Plug that in 5(x + y) = xy; what does that give you?

Lucas

so 5(5k+y) = 5ky

Balarka Sen

simplify.

Lucas

5k+y = ky

Balarka Sen

you can simplify further...

Lucas

hmm

I can express y in terms of k

Balarka Sen

07:29

Well, do it.

Lucas

y = 5k/k-1

Balarka Sen

So now all you have to do is to figure out when 5k/(k-1) is an integer, right?

Lucas

yup

Balarka Sen

Sorry, I had to leave for a minute. Can you figure out when it's an integer, @Lucas?

Lucas

i've played around with it

but I'm trying to think as to how I can show how to find more (or if not show these are the only ones)

Balarka Sen

07:37

You should be able to tell immediately for which k it's an integer.

Lucas

some obvious ones I noted are when k=2

since the denominator will just be 1

Balarka Sen

Yes, I am not looking for iteration.

Give me a proof.

Lucas

hmm

hmm for some reason I am stumped as to how to show when this is an integer. I know that it is when k-1 = 1 or k-1 = 5

Balarka Sen

Think about it.

@Lucas Any ideas?

Lucas

07:53

not sure if this helps but y/5 = k/k-1, so when y=k and k=6

wait this doesn't

hmm

Balarka Sen

Hint: k and k-1 are relatively prime...

except when k = 2, that is.

Lucas

so if k,k-1 are relatively prime except when k = 2

don't we just have y=5 for all k except 2?

wait

Balarka Sen

why?

Lucas

my bad

Balarka Sen

I mean you're not wrong... you just haven't given a proof.

No, you mean y = 6 there.

Lucas

08:03

hmm

still having difficulty as to how the relatively primeness of k and k-1 helps with showing whenever y=5k/k-1 is an integer

difficulty understanding **

Balarka Sen

google "Euclid's lemma"

Lucas

thanks for your patience, I am new to number theory

Balarka Sen

If n divides ab, n is relatively prime to a, n divides b.

Lucas

08:19

why can't y = 10 be too

hmm, i'm thinking that since k,k-1 are relatively prime for all k, no common terms would cancel in the 5k/k-1, except when k=2 and k=6

k=0 and k=-4 would make y an integer too

Balarka Sen

I was away.

OK, I was thinking of positive integers but sure. You still haven't given me any proof.

@Lucas Right, but that's oddly parsed.

Lucas

hmm

Balarka Sen

Anyway, this tell you that the positive integer solutions are 1/10 + 1/10 = 1/5 and 1/30 + 1/6 = 1/5, so you're done.

I have to go now; I hope you have answered your questions.

Lucas

thank you very much for your help

Akiva Weinberger

11:01

@BalarkaSen Isn't 1 considered to be relatively prime to everything?

Balarka Sen

Eh, sure, I was just distinguishing the case k = 2 specifically.

Akiva Weinberger

You don't really need to, though

Balarka Sen

ok. it's a triviality anyway.

Akiva Weinberger

You just need k-1 to be a factor of 5

Balarka Sen

bleh misread.

sure.

Jessy Cat

12:21

Really quick, if anyone is listening.

If $d(x,y)$ is a metric

Is it true that $0 \leq \frac{d(x,y)}{1+d(x,y)}\leq 1$?

lattice

yes

Just multiply by 1+d(x,y)

Huy

ö____________________ö

morning @Mike

Balarka Sen

G'morning, @MikeMiller.

I survived admission.

lattice

I am glad I could help for once xD

Balarka Sen

@Huy I like those faces.

Huy

12:32

@BalarkaSen: thanks, a lot of people like my face.

Balarka Sen

it wasn't your face i was complimenting

Huy

:(

Mike Miller

Congrats. Can I ask where you're going?

Balarka Sen

Going as in?

I'm not admitting to a uni or anything. I am stuck with school for 2 more years.

This was just a promotion from "middle school" to "high school", if that's what those are called there.

Jessy Cat

@lattice, thanks

lattice

12:55

Hey, recently I've been thinking to continue with a PhD in some Algebra- and/or Geometry-related topic after my master/graduate studies. However, I feel like I am still missing a lot of basics, for example:
- Logic (Set Theory, Proof Theory, Model Theory, Category Theory (?))
- Number Theory
Then also there's some more "advanced" topics that I heard about often:
Lie Algebras, Modular forms, Manifolds, Elliptic Curves, Autmorphic Forms, Spectrum Theory and Operator Algebras, von Neumann Algebras

s.harp

short answer: everything except set theory, model theory and number theory :D

lattice

These 3 are not at all connected to Algebra/Geometry?

Then another question: Shouldn't you have at least a basic knowledge of those areas if you do your PhD? Not that it would help for your dissertation, but simply to be a "good scholar" and not to be embarrassed when some undergraduate knows more about logic than you do? :D

this is a rather philosophical question maybe :D

Eric Stucky

@lattice, where are you hoping to do your phd?

s.harp

real set theory is in general connected only to foundations I think, model theory I would say the same but I'm not very knowledgable about it. Lie algebras/Lie groups/manifolds are to geometry as fields and vector spaces are to linear algebra: there are other things with which you do geometry but these are the most basic

Eric Stucky

(Also, I am a grad student and even good high schoolers know more about, say, geometry, than I do; if you let such things embarrass you, you won't get far.)

lattice

13:04

@EricStucky That's another good/important question for me. I heard that it is quite common to do it at the same university where you did your basic studies, so that would be TU Darmstadt (Germany) in my case. But on the other hand, I have often wondered if it wouldn't make sense to do it abroad, for example in the U.S.

and then again, I love Korea, where I am right now as an exchange student for a year^^

Eric Stucky

'basic studies' meaning your undergrad degree or your masters degree?

In Germany you would be expected to do a masters first, I would think.

Not sure about Korea :)

In the US though it is typical for a student to go straight from undergrad to a PhD program, and also more or less expected for a student to transfer universities between the two programs.

Anywho I ask because if you don't have a masters yet, you shouldn't worry too much about not knowing things; this is what the next two years are for (regardless of where you study)

lattice

@s.harp Thank you! And those topics like "operator algebras", spectrum, and functional analysis? Does it make sense for me to study them?

s.harp

dunno, depends what you mean with algebra. I think C* algebras etc are awesome so why not

lattice

by "basic studies" I mean both of them, undergraduate and masters degree (I'm confused sometimes with the English terms, we call it "Bachelor" and "Master" in Germany which together take around 5 years usually). So by the end of your master you usually write the thesis being supervised by some professor who would be your first option for doing a PhD (from what I know).

Eric Stucky

Okay I see.

With a masters degree it is probably trickier, since you would probably be expected to stop taking courses soon.

Or at least, to stop taking 'entertainment' courses :P

Transferring universities may alleviate this problem a little, country-swapping may help more.

lattice

13:14

In Korea they have a similar system as in the U.S. or Germany, but at least where I study now the level is lower than in Germany, I think. So for my PhD I would rather stay in Germany or go to the U.S. where I heard are quite good universities^^

yes, but I want to be good enough to do the PhD and it can never hurt to self-study next to the usual classes, right? :D Also I know almost exactly which classes I will take during the following 2 years (we have a lot of restrictions for our choices), and Logic for instance won't be one of them sadly

Eric Stucky

So, to me it sounds like, the US system would offer you a little more time to sort yourself out, develop an advising relationship without thesis pressure, etc.

But this doesn't sound like a particularly good reason to choose a grad school, honestly.

With as much experience as you have, if you want to learn something new, you can probably just read some basics in books

and then go to seminars for more interesting bits, if you want to keep learning on the topic.

lattice

Yes, when you do a PhD in Germany, you participate at a lot of seminars I think, but you don't study any basics anymore, I think

Eric Stucky

Nobody can prevent you from reading books :/

(at least, not in Germany XD)

lattice

ah... of course I mean to self-study with books^^ so of course I will continue my master studies at my home university in Germany for 2 years, and my goal is to self-study some topics during these 2 years, next to the usual classes

Eric Stucky

As someone who has been going to seminars from the day I entered university, I highly recommend making a habit of attending talks that you don't understand.

It's lower commitment than a class and it can give you the motivation you need to make it through self-study, since of course it can be very hard to keep yourself on track.

And, you will also see what honest researchers are working on

Which will help with the whole issue of not knowing what to study.

Balarka Sen

13:28

Hi @EricStucky

Eric Stucky

Hello bala

This is the only part of your name that we can agree on, it seems :/

lattice

Okay :D I agree, during the (only) two seminars that I attended so far, I somehow got highly motivated. I'm planning to attend more from now on! :P

Eric Stucky

:)

Balarka Sen

@EricStucky Agree on? In what sense?

Eric Stucky

Balarka: Akiva, MatsGranvik

Juan Fran

13:34

sup

Eric Stucky

Jo

Balarka Sen

I certainly would advise using my full name :)

And not twisted Frankensteinish names devised by Akiva and Mats

Eric Stucky

:P

Balarka Sen

Wow, people must really like my answer here.

Juan Fran

nice one

even tho i think hatcher is a dirty book

Balarka Sen

13:39

It's an old answer but got a couple upvotes as of late.

@JuanFran I have heard many people say that. I disagree with them.

Juan Fran

why?

surely it is not functorial enough

Balarka Sen

because i think hatcher is a clear, clean, nice book :)

Juan Fran

clean?

Balarka Sen

yes, I don't like the functorial approach.

Juan Fran

intuitive certainly

and the examples are good

but i dont think it is very elegant

not as concise as it could be

Balarka Sen

13:42

OK, I haven't tried to make things more concise. The overall picture is good, provides lots of intuition, good exercises, and less functors. That's good enough for me.

It's my favorite algebraic topology book. But different people have different tastes.

Juan Fran

why don't you like functors? they are good

even your answer uses in an essentially way the functoriality of the LES

Semiclassical

morning chat

Balarka Sen

I don't dislike functors.

Juan Fran

i see. you think it can be too functorial?

Balarka Sen

They are good as a packaging language. They can also be useful for proving things sometimes, but most of the time I think introducing the concrete topological intuition before describing them as "functors hurr durr" is better. But that's just me.

@JuanFran Naturality of long exact sequences could be interpreted functorially. But in my answer, I don't need that formality because it doesn't add any new intuition or insight.

Semiclassical

13:49

construct twice, functor once? :p

Balarka Sen

That's why I won't talk about it. Most of the time my issues with being too functorial is that. The approach doesn't provide me with any new intuition or insight.

Juan Fran

i see

well in algebraic topology

the functoriality is not as important as in algebraic geometry, where schemes can be viewed as functors

and this is very fundamental there

Balarka Sen

Yeah, I have heard of it. I also think homotopy theorists use them a lot.

Semiclassical

functions v. functionals v. functors. funky

Balarka Sen

ashes to ashes, funk to functors.

@JuanFran Hi, by the way. Are you new to MSE?

Semiclassical

13:54

oh, mathematica. i just love when i realize i don't know how to make you do what i want :/

Juan Fran

hi. well I sometimes find useful information here

I am not strictly speaking new

I have used it before, but rather passively

Balarka Sen

I see. Well, welcome to the community! Are you a grad student?

Semiclassical

'lurker' isn't quite the right word in this context, but close enough

Juan Fran

thanks. No i am still an undergrad, main focus on algebraic geometry.

Balarka Sen

ah, I see. cool.

Juan Fran

13:56

I will start graduate studies in next year

Balarka Sen

I studied some basic (as in classical) algebraic geometry a couple weeks ago. I plan to study more at some point of time.

Although I feel like my main interest lies in complex algebraic geometry rather than schemes.

But I don't know much about either, so :)

Juan Fran

I see

you mean from a complex geometry point of view?

These are very closely related, via GAGA

quite interesting connections

Balarka Sen

Yeah, like, studying topology of algebraic varieties over C.

Juan Fran

schemes are just beautiful

Balarka Sen

I have heard of GAGA a lot. Dunno what it is yet.

iwriteonbananas

13:59

Lady gaga?

Balarka Sen

... @bananas

iwriteonbananas

Hi btw.

Balarka Sen

No hi to you for that bad joke.

Transcript for

$Mathematics$ Mathematics