Mathematics - 2012-10-21 (page 1 of 5)

Gnintendo

00:01

gah, I know I need such a small push to get the answer D;

Gnintendo

00:17

facedesk

Link

01:01

Hi, i was getting into newtonian mechanics, and the normal force. My question is that the normal force is only present when objects are touching right?

Gnintendo

01:28

@Link yes, although this is probably more appropriate for the Physics.SE

Link

01:50

@Gnintendo, thanks, I just didn't want to make what would probably be a very trivial question

user19161

04:44

It just rained heavily here.

2

Dan Brumleve

06:26

i wrote up a riddle that i later figured wouldn't be appropriate on the main site so i blogged it, what do yall think? brumleve.blogspot.com/2012/10/three-wise-men.html

Jayesh Badwaik

@DanBrumleve its more of an english language play than probability.

Dan Brumleve

well it admits a definite answer under very reasonable and universal assumptions within our community (it isn't supposed to have a culuturally relative answer). but it's not appropriate for main simply because it is a riddle that i know the answer to.

i tried to use language whose meaning should be precise in context and solving it shouldn't be about distorting any words. anyway please try to enjoy i'll clarify any aspect that doesn't ruin it.

it's a generalization and sort of an inversion of the three hats problem

Jayesh Badwaik

you can put a self-answering question there. :-) Also, is the answer zero? The questions is "What is the probability that any person who remains in the room
at the time you submit your worksheet has chosen you?". Now, since you are submitting your worksheet, you are exiting from the room, and hence, any guy who had chosen you will now be asked to choose another guy and hence, it is not possible to choose you if you are submitting the answersheet.

Dan Brumleve

the answer is at least 2 because the last two people to exit the room got an A.

Jayesh Badwaik

ahh wait. sorry. I meant, the answer to the probability was 0. Now the answer to the question I will have to work out. that has almost not much to do with the probability I suppose.

Dan Brumleve

06:39

oops i got some of the proportions wrong let me edit. also i will hint that the first large group of people to leave the room did NOT get an A.

the probability depends on the situation in which you leave the room.

Jayesh Badwaik

but what about my explanation? Is there something wrong with that?

Dan Brumleve

your explanation would make a nice comment on my blog and it might take the question beyond what i intended. but my intention is that a person changes their selection immediately after the selection leaves the room, and also that there is no possibility of contention (the universe is simulated on a serial computer).

Jayesh Badwaik

ohh, okay. I get it now.

Dan Brumleve

also when i say "one by one" i mean that the actions of exiting the room and submitting the worksheet to the TA occur atomically :)

another hint is that the 80 who left first answered 1-1/e :)

Jayesh Badwaik

Yup. :-)

Dan Brumleve

06:55

i added a preamble to goad my preferred interpretation: "The rumor was that the mathematics test today would be easy,
or perhaps an hour of amusing and unaccountable activities,
and based on the nuance of the rumor it was even considered
reasonable to some to expect to leave class early and some of
those students who held that expectation had even planned as such."

user19161

@JayeshBadwaik Yo yo! I woke up at 11 plus just now, will try not to take a nap.

Jayesh Badwaik

@JasperLoy yo! yo! yo! :-)

Matt N.

Hi! Does anyone in here know about manifolds?

I want to know the definition of curve.

And also, parameterisation.

anon

look up atlases and coordinate patches. the basic idea of manifolds is being locally homeomorphic to R^n.

roughly a curve is the image of or an isoclass (under homeomorphism) of continuous maps from a real interval into a topological space, and the paremetrization is the map (representative of the isoclass) itself.

Matt N.

07:11

Thank you!

anon

homotopy was the word I was looking for, not homeomorphism

the isoclass idea though is more of a "path" a la fundamental groups in algebraic topology (where particular parametrizations and metric information don't matter)

Matt N.

I suppose. I don't know any differential geometry and I thought I didn't need it to know what a parameterisation is -- I'm doing stuff with topological manifolds.

And was wondering whether an orientable manifold always has a parameterisation.

But then I couldn't think about my question because I don't even have a definition of parameterisation : /

anon

Well, I'm talking about parametrization of curves. You're probably thinking more along the lines of coordinates. Manifolds aren't generally describable with only a single coordinate system, which is why you need to patch some together and make an atlas.

Matt N.

I thought there is something like a parameterisation of a manifold.

@anon Not even if the manifold is orientable?

For example, the torus.

Or a sphere.

anon

what do you mean by parametrisation?

Matt N.

07:17

But this conversation is already undefined for me, since I don't know what a parameterisation is.

Haha, beat me to it!

@anon That's what I'm trying to find out. But the books I have don't have a definition.

anon

why are you asking me if orientability suffices for existence of parametrizations if you don't know what you're using the word to mean...

Matt N.

And I couldn't find anything on the internet.

anon

where did you see the word used, first of all?

Matt N.

@anon You might have a definition.

anon

I haven't studied differential geometry.

user19161

07:19

Just read Lee's Topological Manifolds and then Smooth Manifolds and then Riemannian Manifolds. QED. @matt

anon

I think you can immerse arbitrary Riemannian manifolds in a Euclidean space, but not necessarily embed one in a Euclidean space. I am not sure how/if orientability changes this picture.

Matt N.

@anon Well. The noun doesn't appear. But if you open Lee's Intro to topological manifolds on page 2...

... he uses parametrically without giving a definition. It's also not in the index.

I would think "described parametrically" = "exists a parameterisation"

anon

(a) that's just the same idea as coordinates (b) don't introductions always reach ahead of the definitions and so forth?

BenjaLim

@MattN. Hey

Matt N.

@anon Coordinates = parameterisation?

Hi Ben

BenjaLim

07:24

how's goin

Matt N.

good

anon

in general. for your particular case, parametrization is using a map from a real interval into a space in order to describe the image (which is a curve).

Matt N.

@anon Is there any difference?

: )

user19161

I just realised it can be spelled as parameterisation or parametrisation.

Matt N.

Or parameterization.

anon

07:25

@MattN. one is a very special case of the other

Matt N.

@anon Ok. Thank you!

I'll see you all later! : )

anon

I pronounce meter as "meater" so meterisation sounds funny

user19161

@MattN. Bye!

anon

actually, I think en.wikipedia.org/wiki/Whitney_embedding_theorem is what should be referenced. don't trust me for your diffeogeofacts folks!

user19161

I was thinking about anon just now...

user19161

07:29

@anon Or anything else for that matter. =)

Dan Brumleve

@Jayesh and anyone who likes a riddle, i think i am done with my edits. i posted to my fb too so maybe someone there will try to answer. http://brumleve.blogspot.com/2012/10/three-wise-men.html
if there is some easy way to adapt this to main i would do so and post it, am i right in thinking it is fundamentally inappropriate there due to restrictions on the puzzle tag i.e. no riddling?

BenjaLim

@anon I have a question about formatting

Consider my answer here: math.stackexchange.com/a/215503/38268

I put the word "minus" in a math expression because I don't know how to get the big minus sign. Do you know how to get it there?

anon

you can put a font size in a single array entry and it stays localized (like \large, \Large, \LARGE, \huge, \Huge, or whatever they are)

since each expressions spans it's own line it might be best to just keep the text "minus" though

BenjaLim

07:45

ok.

@anon How do you type in the minus sign in latex apart from just doing

-

?

anon

why does it matter what the input is if the output is the same and the "-" key is likely shorter than anything else? :confus: Or do you want a dash or different type of horizontal center line thingie character?

or maybe your keyboard is broken and the "-" key needs to be manually done with a clickey-clickey on-screen window

you might be able to combine \mbox{} with the -, --, and --- things

I cannot into brain no more, anon is closing up for the night

Old John

Good morning

Dan Brumleve

hola old john

Old John

@DanBrumleve hola

Dan Brumleve

i just blogged a riddle and i'm curious if anyone has feedback about it or knows how or if it could be adapted to main

Old John

07:58

@DanBrumleve where is it?

Dan Brumleve

brumleve.blogspot.com/2012/10/three-wise-men.html

Old John

thanks

Dan Brumleve

it is my "math blog" err really just a gesture at starting one

Old John

@DanBrumleve Fascinating question - but I am generally hopeless at solving them :(

Dan Brumleve

glad u think it is fascinating. :)

my intended answer is "6"

Old John

08:05

@DanBrumleve I thought the answer had to be a probability?!

Dan Brumleve

that's the question for the students. the question is how many students answered correctly?

note that the probability each student is supposed to estimate depends on the time that student submits the worksheet and also on the time each other student leaves the room. :)

Old John

@DanBrumleve Ah - sorry!

@DanBrumleve Yep

Dan Brumleve

it is the same as the problem where 3 students are all exposed to a blast of fossil fuel exhaust and have smudges on their foreheads, and the professor asks the first one who knows whether or not there is a smudge on his forehead to please raise his hand.

Old John

@DanBrumleve I thought that old one was OK - but can't yet see how the same sort of argument works here ...

Dan Brumleve

well the first 80 students apparently answered 1-1/e since they were sure about it all at once and spread a justified argument around the classroom. but that won't earn an A since the last student answered 0 and was exactly correct.

so there it differs from the old one in that there is a more continuous relationship between knowledge and time. although there are other discontinuities arising from the nature of the problem.

Old John

08:16

@DanBrumleve Hmm - I will think about that when I have more time - but it is a nice problem

Dan Brumleve

thx

Jayesh Badwaik

@DanBrumleve I agree with oldjohn on both of the counts. Its a really nice problem. I will think about it, when I have more time. :-)

Dan Brumleve

thx^2

Jayesh Badwaik

@OldJohn hi!! Wassup?

Old John

@JayeshBadwaik Hi - just deciding what to do today - and it looks like clearing out the garden shed :(

Jayesh Badwaik

08:26

@OldJohn :-(

Old John

Been putting it off for far too long

Jayesh Badwaik

Ohh.

Jonas Teuwen

@OldJohn But then you have a nicer garden 8-)).

Old John

@JonasTeuwen we like it - especially in summer

Jayesh Badwaik

@JonasTeuwen Hehaahehaahehaa

Jonas Teuwen

08:30

@OldJohn So you are preparing the garden for summer liking!

Old John

@JonasTeuwen Indeed :)

@DanBrumleve Did you see the part-solution I posted to your irrationality question?

Dan Brumleve

yall are making me wonder if an almost certain conclusion can be cast in doubt by more florid writing

@oldjohn not yet blog comment?

oh irrationality question

yeah i think i did but didn't have time to look

Old John

here

Dan Brumleve

17

Q: What is the simplest way to prove that the logarithm of any prime is irrational?

What is the simplest way to prove that the logarithm of any prime is irrational? I can get very close with a simple argument: if $p \ne q$ and $\frac{\log{p}}{\log{q}} = \frac{a}{b}$, then because $q^\frac{\log{p}}{\log{q}} = p$, $q^a = p^b$, but this is impossible by the fundamental theorem of ...

Old John

I bought an old copy of Chrystal's Algebra a while ago, and stumbled across it while reading his account of continued fractions

Dan Brumleve

08:34

i had read it and just absorbed "cfs"

ahh tanh eh

Old John

yep

Dan Brumleve

i gave up at e^3 which seemed an unfavorable subject to pursue elementarily

Old John

@DanBrumleve agreed

Jonas Teuwen

There are like 14 hours left in this day, bloody holy monkey.

Old John

@JonasTeuwen but there are 24 more tomorrow :)

Jonas Teuwen

08:35

HOLY COW.

Dan Brumleve

thanks i'll follow up the ch connection whether or not i think it is "elementary", also curious to discover if there is some way to simplify it down to tanh specifically via which i definitely think so.

Old John

I think it is a solution which would have been seen as elementary back in the 1890's - but not so much so today :)

Dan Brumleve

although i did finally wrap my head sufficiently around the proof of pi's irrationality using that polynomial form (1-x)^n * x^n although i still find it a bit hard and unmotivated

at least i can see how it's possible to express in FO number theory by expanding the definitions of integral and limit into that language

Old John

@DanBrumleve me too - I like Chrystal's argument for $\pi$ too - but then I quite like continued fractions (for stuff like Pell's equation etc.)

Dan Brumleve

i have a dover book on cfs by khinchin at work maybe i will pick that up next time i'm back there

Old John

08:46

@DanBrumleve Yep - sometimes there are some little gems in the "old stuff" :)

but I think that Khinchin only considers CFs with all numerators 1

Jonas Teuwen

Khinchine inequalities? Those are hardcore.

Old John

@JonasTeuwen Yep - but his (if it is the same "he") books are nor so hard-core (i.e. I can understand them)

Jonas Teuwen

Well, the inequalities are not so hard you cannot understand them :-).

Old John

yep- it is the same guy

@JonasTeuwen probably - I have never actually looked at the proof, though

Jonas Teuwen

It is very cute, with some probabilistic thingies :-))).

Have you seen the Gowers - Maurey paper?

That is the type of mathematics I like 8-).

I usually do not read books, just check wikipedia and start computin'. More fun.

Old John

08:57

@JonasTeuwen nope - pretty sure I haven't come across that

Jonas Teuwen

Oh noes! A Fields medalist for your country with one of his most intricate papers!

Old John

Indeed :(
I see that his work would interest you - functional analysis and combinatorics :)

Jonas Teuwen

I does interest me :-). Also Villani's work.

Old John

Right! - time to tackle the shed ...

Jonas Teuwen

09:14

@OldJohn Enjoy!

Nimza

09:41

Nice morning

user19161

10:47

@jonas How is your paper coming along?

Gustavo Bandeira

11:00

@JasperLoy Can you help me?

@JasperLoy I'll need "Analytic Geometry 1" - Can you recommend me a good book on this? The topics I need to cover are these - It's in portuguese but I guess you won't have much trouble, there are a lot of cognates.

Noah

hi

Hi, Jasper

And hi Gustavo'

Actually hi everyone

Gustavo Bandeira

@Noah Hello. =)

Noah

Whats new

Gustavo Bandeira

You?

Not much.

Noah

Looks like I am coming down with the flu.

This is gonna take the strength out of my body.

Gustavo Bandeira

11:16

=(

Amith KK

Hey

Noah

What's up Mr. double K?

Amith KK

LOL

Hello @Noah

:P

Noah

How are you @AmithKK

Amith KK

I'm fine

I'm doing a maths project on the "Survey of various types of bank accounts, rates of Interest offered"

And I wanted to know what would be the relevant graph

Noah

11:19

@GustavoBandeira should probably answer that.

But what kind of data do you have from these surveys?

Gustavo Bandeira

@Noah ?

Noah

@GustavoBandeira
I'm doing a maths project on the "Survey of various types of bank accounts, rates of Interest offered"

Gustavo Bandeira

@Noah Nice, I'm planing something small for the near future.

"How much money can I earn by picking stocks with a dice roll?"

Noah

@GustavoBandeira That's not me. @AmithKK asked that question. Read the chat

Amith KK

@Noah Like Intrest rates, savings and stuff

Gustavo Bandeira

11:23

@Noah Oh, you want me to answer his question?

Noah

@GustavoBandeira Well yeah. Becuase I am not as good as you are at Math.

Gustavo Bandeira

@Noah ???

I am good?

XD XD

I'm on calculus 1, reading about limits.

Imma a noob, trust me

You can even read my questions

Noah

@AmithKK your best bet would be to ask a question on Math.SE

Gustavo Bandeira

@AmithKK What you mean with graph? A plot or something about graph theory?

Amith KK

@GustavoBandeira A plot

Our teacher is not too verbose

she wants graphs

user19161

11:27

@GustavoBandeira They should mostly be in Paul's notes already somewhere.

Amith KK

I can use any data from a bank account I guess

Gustavo Bandeira

@AmithKK I'm not so sure if this is a math question, maybe it has more to do with data visualization.

user19161

@AmithKK Geezis, they have this kind of project these days. What's there to do?

Amith KK

@JasperLoy I know

Gustavo Bandeira

@JasperLoy I thought he would do that on a separate volume entitled Geometry

user19161

11:30

@GustavoBandeira Titles often don't tell much, you need to look at the contents. Weil's Basic Number Theory is not basic at all, and Serre's Course in Arithmetic is not about 1+1=2.

Gustavo Bandeira

@JasperLoy Yes... I tried to read Serre's.

user19161

@GustavoBandeira Why?

Gustavo Bandeira

I opened it thinking: "Wow, should be easy", then I went to the middle of the book: "WTF?!"

@JasperLoy Curiosity.

I usually do that, I open some math book on some random topic and try to see what's there.

sperners lemma

me too @GustavoBandeira

Gustavo Bandeira

Of course, I don't waste all my time doing that,

@spernerslemma It's amazing! You open a math book on.. Cohomology! And you look it as if it were Mandarin!

Pantelis Sopasakis

11:33

Hi Everyone! I was wondering, if $f:\mathbb{R}^n\to\mathbb{R}^m$ is continuous but is not linear, will it have a closed graph?

Just found the answer... it's negative.

take $f:\mathbb{R}\to\mathbb{R}$ be $f(x)=\frac{1}{x}$ which is continuous but without a closed graph. Isn't it?

Mmmmh, maybe I was wrong. The sequence $(x_n,f(x_n))$ with $f(x)=1/x$ and $x_n=n$ we cannot say it converges to $(+\infty,0)$.

Martin Sleziak

Your function is undefined for $x=0$ and the graph is closed.

See e.g. here

Pantelis Sopasakis

Hi Martin. Yes, indeed! I noticed that.

Martin Sleziak

For real functions you have:

$f$ is continuous $\Rightarrow$ $\Gamma_f$ is closed

Pantelis Sopasakis

Thanks!!!

Martin Sleziak

IIRC $\Leftarrow$ is true if you work on closed interval. (And it can be probably generalized to compact spaces.)

Pantelis Sopasakis

11:45

The converse is always true - meaning: closed graph $\implies$ continuous function. But under what assumptions does it hold that continuity $\implies$ closedness of its graph.

Martin Sleziak

See this link at PlanetMath.

The function $f(x)=1/x$ and $f(0)=0$ has closed graph.

It is certainly not continuous.

In that link I've just posted they say that continuity implies closed graphs whenever the codomain is Hausdorff space.

The other direction is true if you add compactness.

Pantelis Sopasakis

OK, it is clear from proposition 1 that the graph is closed whenever the function is continuous.

and that the converse is not always true as you pointed out in the counterexample.

Pantelis Sopasakis

12:09

A set $A$ has the property that $\bar{A}$ is compact. How does this characterise $A$?

$\bar{A}$ stands for the closure of $A$.

Jonas Teuwen

12:21

Depends on what space you are living in.

Martin Sleziak

@PantelisSopasakis In what space are you working? If in $\mathbb R$ then they're precisely the bounded sets.

Jonas Teuwen

In a metric space making the set closed this either means it was already closed or it was open and totally bounded.

Martin Sleziak

sets are not doors

You probably wanted to write it was not closed instead of it was open, Jonas...?

Jonas Teuwen

Yes, yes, let me rephrase.

user19161

@MartinSleziak Or more generally, $R^n$.

Jonas Teuwen

12:23

Either $\overline{A} = A$ and in that case $A$ was closed already. Otherwise $A$ was not closed and totally bounded.

@JasperLoy That is not more general.

user19161

A metric space is compact if and only if it is complete and totally bounded.

Martin Sleziak

I just guessed $\mathbb R$ since his last question was about real functions.

user19161

@JonasTeuwen It is bro! 1 is a positive integer! =)

Jonas Teuwen

Closed it a softer property than "complete" to check.

No, it is just the same, it is a tensor (or whatever) product.

user19161

I remember I drew this huge chunk of arrows showing this implies that for topology.

Jonas Teuwen

12:24

You can reduce it to the case $d = 1$.

Often I see papers where they go through lots of trouble dragging some constants around.

user19161

This is beyond my paygrade bro!

Jonas Teuwen

While you can often quickly see that setting them to $1$ or $0$ will not change the result.

user19161

OK, I think I have now 100 per cent confirmed which LaTeX packages to use. They are lshort, pstricks, pst-node, pst-math, pst-plot, pst-3dplot, pst-eucl and pst-solides3d. QED.

user19161

Also, I will be trying to learn French and German from the "Teach Yourself" series.

Jayesh Badwaik

@JasperLoy what does lshort do?

user19161

12:28

@JayeshBadwaik It is just a LaTeX introduction, hahaha.

Jayesh Badwaik

@JasperLoy No, I mean, the package?

user19161

@JayeshBadwaik Exactly what I answered, hahaha.

Jayesh Badwaik

@JasperLoy hmm. okay.

user19161

@JayeshBadwaik Did you install the full TeX Live?

Jayesh Badwaik

@JonasTeuwen This mendeley is so stuuuupid. I am using koma-script and have a subject, a title and a subtitle, and after that there is a todo list and mendeley thinks that "To-Do List" is the title!! Stoooopid.

@JasperLoy Expect for the langpacks I guess, yes.

Jonas Teuwen

12:31

@JayeshBadwaik Yes, but how would it know...?

user19161

@JayeshBadwaik I burned the DVD and installed every damn thing.

Jonas Teuwen

@JasperLoy Other than plan to study or learn certain things, do you also do it?

user19161

@JonasTeuwen Not yet, not yet. You know why bro, you know why.

Jayesh Badwaik

@JonasTeuwen Well, then why the hell does it try to guess? Get out of my way!!!

Jonas Teuwen

I don't see that as a good reason.

@JayeshBadwaik You can modify it 8-)).

@JayeshBadwaik You could have fixed it already by now!

Jayesh Badwaik

12:33

@JonasTeuwen Yeah. 8-) but bro, I wish it was awesome.

user19161

@JonasTeuwen Well, every case is different bro. You can do it but maybe my case is different.

Jayesh Badwaik

@JonasTeuwen You mean the code? or the title? (what could I fixed already by now?)

Jonas Teuwen

@JasperLoy Nah, I have seen so many people postpone their whole life waiting for something to happen. Even if it happened, something new will pop up.

Life is not so serial as you might think.

@JayeshBadwaik The title.

Jayesh Badwaik

@JonasTeuwen yeah. 8-)

@JasperLoy What Jonas is saying is very true.

user19161

@JonasTeuwen Well, then I would say what you have in mind is not the same as what I am experiencing.

Jonas Teuwen

12:36

Dude.

user19161

Let me give you an analogy.

Jonas Teuwen

Sure, do that.

user19161

If a person takes 2 days to do something when others do it in 1 day, nobody will complain.

user19161

But now say I am taking many years. Perhaps it is because it is 10^100 more serious?

user19161

OK, bad analogy.

Jonas Teuwen

12:37

Really bad one.

user19161

And as you know @jonas, different people have different themes.

Jonas Teuwen

I do know that.

But I also know that postponing and waiting does not help at all.

user19161

OK, another analogy.

user19161

A fractured arm takes a month to heal, it ain't possible in a second right?

user19161

Anyway, what I wanna say is, I ain't procrastinating. I am using the time to resolve certain things in certain ways.

user19161

12:40

I can tell you, I am the last person on earth to procrastinate.

Jonas Teuwen

I believe you.

But I am more talking about the TeX :-).

Finally, after a couple of months you decide which packages to use.

You could have started right away and figure out why some are better than others by running into issues.

And you would know it by now very well.

user19161

I have a secret to share here!

user19161

Well, not really a secret.

user19161

But you know, I got someone to type my undergrad paper for me, haha. I mean just the LaTeX!

Jonas Teuwen

Yes, I was already wondering how you did that.

user19161

12:45

But that was so long ago, I have learnt some LaTeX since then.

Jonas Teuwen

Start typing bro.

user19161

Exactly, don't you see me typing answers here? =)

user19161

@JonasTeuwen Actually it is tedious but quite simple really, no complicated symbols!

user19161

@jonas The poem I wrote about you got the most number of stars compared to the others I write. I conclude that they like Jonas!

Jonas Teuwen

Or they like your poems!

skullpatrol

12:59

Or both.

Transcript for

$Mathematics$ Mathematics