Mathematics - 2012-08-26 (page 2 of 5)

user19161

3:12 AM

@KannappanSampath Oh OK. I still think you are going too fast, you and Ben and Pedro.

Kannappan Sampath

Several variable real anal. not. c'x... @Jasper

Hi @anon

anon

hello

user19161

@KannappanSampath Aha! I also find it weird that there are no fixed courses at your place IIRC.

user19161

The cool anon has appeared.

Kannappan Sampath

Your answer to that question about characters of irreps was indeed nice. @anon

anon

3:16 AM

thanks

user19161

In other news, though this is not interesting to anyone, I am deleting my tex account as well. I will just keep math and english.

Kannappan Sampath

You might want to include Isaacs book. It's very nice and I am struggling with its exercises.

@JasperLoy You! Why!

anon

I haven't looked in Isaacs book. I've strictly used notes from online, but I haven't looked for awhile.

user19161

@KannappanSampath Well, I just don't like too many things lying around. I think I will just focus on these two.

user19161

In case you guys dunno, Isaacs has a grad algebra text as well.

user19161

3:18 AM

The latest edition is published by the AMS.

user19161

The AMS seems to be publishing many great textbooks in recent years.

user19161

I like their GSM and Chelsea series.

Jayesh Badwaik

@KannappanSampath Apart from Isaac's Algebra :A Graduate course, which other book are you taking about?

user19161

@JayeshBadwaik Character Theory.

Jayesh Badwaik

@JasperLoy Okay

user19161

3:22 AM

But I find the AMS books a little more expensive than the Springer ones.

Jayesh Badwaik

Have to look up what character theory is.
@KannappanSampath you have no fixed courses? That is definitely not what they say on the website. Looking at website you would think it is good old english boarding school.

Kannappan Sampath

@JayeshBadwaik The point is the courses are not written rigid... There are standard courses (with standard title) you'll have to take but the content varies widely from instructor to instructor and you end up covering much more than what you thought you'd learn.

user19161

@KannappanSampath At least there are standard courses. There must be some organization!

Jayesh Badwaik

@KannappanSampath Ohh. That is nice. I like that type of stuff.

user19161

Malik has written both algebra and analysis texts I think.

Kannappan Sampath

3:25 AM

@JasperLoy who is malik?

Food time here.

user19161

@KannappanSampath Some Indian mathematician I think.

Kannappan Sampath

Masala Dosa.

user19161

@KannappanSampath Or Tosai.

Kannappan Sampath

@JasperLoy Not anyone widely known then.

@JasperLoy Whatever! :)

user19161

@KannappanSampath Not as famous as Kannappan Sampath.

Kannappan Sampath

3:26 AM

Later folks.

Jayesh Badwaik

Later

Kannappan Sampath

@JasperLoy I am an useless idiot. I am not famous.

user19161

@KannappanSampath a

user19161

Why, when did anon have the GND?

user19161

By the way the Isaacs grad algebra text is really good, way better than Lang IMHO.

Kannappan Sampath

3:40 AM

Back from food.

@JasperLoy I cannot agree more.

user19161

@KannappanSampath You know someone I read said that none of Lang's books are good?

Kannappan Sampath

@JasperLoy False.

user19161

And after a while I sort of realize what he meant.

Kannappan Sampath

Look at his text on linear algebra.

BenjaLim

@anon hey

anon

3:41 AM

hey

Kannappan Sampath

Look at his several variable calc text.

BenjaLim

@anon Can I ask you something?

Both are great!

sure

Hello @Ben . :)

3:42 AM

I have been trying to prove that the kernel of $exp : \mathfrak{g} \to G$ for $exp$ surjective is discrete

@KannappanSampath hey

Jayesh Badwaik

@JasperLoy I personally like Lang. I have not read Isaac and I will plan to do so.
But I like Lang.

BenjaLim

@anon Now I know it is a local homeomorphism

and I can see by applying an isometry that it is a local homeomorphism about any point

so if I have a point $x$ in the kernel

Jayesh Badwaik

@JasperLoy You realized what he meant? As in?

user19161

@JayeshBadwaik I think his best chapter there is the one on Galois theory.

BenjaLim

there is a neighbourhood $U$ about it such that $U$ is homeo to some $V$ about the identity in $G$ yes?

anon

3:43 AM

read that wrong nvm

BenjaLim

No the exponential map is surjective

user19161

@JayeshBadwaik I mean I just feel that they are like copy and paste from his other books. Lumping many topics into one book. Leaving important things to the exercises.

BenjaLim

you see because G is connected and abelian I deduce that $\mathfrak{g}$ is abelian and so the kernel is a subgroup

anon

I couldn't say, I haven't studied that much Lie theory.

(read: any, really)

BenjaLim

oh ok.

Kannappan Sampath

3:45 AM

@JasperLoy why is <<Leaving important things to exercises bad>> if done at the right level? Unlike the Tao's Analysis kind.

BenjaLim

@JasperLoy I always see you discussing books, not maths

user19161

@KannappanSampath Because I might not want to do all of them?

user19161

@BenjaLim Well, what to do, I am hopeless.

BenjaLim

@KannappanSampath I like lie algebras

very interesting

and the ideas in there are very geometric

Kannappan Sampath

@BenjaLim I know nothing about them.

BenjaLim

3:46 AM

Like to prove this map is surjective here

Kannappan Sampath

@BenjaLim I see.

BenjaLim

the idea is very geometric

user19161

Read Lee's book for Lie groups and Lie algebras. :-) (Smooth Manifolds)

Jayesh Badwaik

@BenjaLim Lie Algebra are also used heavily in physics, exactly how I am not too sure.

BenjaLim

@JasperLoy Actually you know what I like about my course?

Jayesh Badwaik

3:47 AM

Probably due to the same reason.

BenjaLim

We learn lie algebras

Jayesh Badwaik

@BenjaLim Lie Algebra are also used heavily in physics, exactly how I am not too sure.

user19161

@BenjaLim What? The guys are cute?

BenjaLim

but manifolds are not the main focus

@JayeshBadwaik yes.

For example the isomorphism I proved above

I heard it's a very important isomorphism

@JasperLoy And I find the ideas are very powerful

for example in the link above

Kannappan Sampath

I need to buy some chalk holders.

BenjaLim

3:48 AM

to prove the associated lie algebra homomorphism is injective do you know how you prove it?

user19161

@benja I will start discussing math and not books when I start studying again. When that happens, it will be too deep for you bro!

BenjaLim

You use the fact that

Jayesh Badwaik

@BenjaLim I agree.

@KannappanSampath How did your talk go?

BenjaLim

If the differential is locally injective, the map is.

@JasperLoy A lot of the ideas in our course generalise to differential geometry

actually we are using a lot of ideas from differential geometry

Kannappan Sampath

@JayeshBadwaik Well, it went well. I could not prove Schur's Lemma this time. So that is where I'll begin after two weeks of hiatus.

user19161

3:50 AM

@BenjaLim Yeah, I think you will like DG. Lee's 3 books are all excellent.

BenjaLim

just implicitly

Jayesh Badwaik

@KannappanSampath two weeks between talks?

BenjaLim

Ok bye guys

Jayesh Badwaik

Bye

BenjaLim

I'm off

user19161

3:50 AM

@BenjaLim Bye!

Kannappan Sampath

@JayeshBadwaik There are mid semesters. So, I won't be able to prepare for the talks well.

user19161

Why do they make undergrads give talks these days?

user19161

Is there a lot to talk about?

Jayesh Badwaik

@KannappanSampath ohh, damn them. They always interrupted my talks as well.
@JasperLoy It is a voluntary thing. At least in my case it was.

anon

I'd love to give a talk about something I knew.

Kannappan Sampath

3:52 AM

@JayeshBadwaik Hmm, true.

user19161

@JayeshBadwaik I only gave one real talk for my final year project.

user19161

@anon I just prefer to give lectures to students.

Kannappan Sampath

@anon Yeah, it is a very useful process.

People: Does anyone use chalk holders here?

4 mins ago, by Kannappan Sampath

I need to buy some chalk holders.

user19161

My teacher did.

Jayesh Badwaik

@JasperLoy It is a very helpful thing.
@KannappanSampath Naah, I always used whiteboards.

user19161

3:53 AM

But in uni we used whiteboards. :-(

user19161

I prefer blackboards, sad panda.

Kannappan Sampath

@JasperLoy Me too.

user19161

Blackboards are for real men, whiteboards no.

anon

I prefer whiteboards. Whiteboard supremacy.

Jayesh Badwaik

@JasperLoy why? I prefer whiteboards. Easy to write, easy to erase, no dust, high contrast.

user19161

3:55 AM

Blackboards don't reflect too much light.

user19161

Chalk allows you to shade.

user19161

Markers stink.

user19161

QED.

Kannappan Sampath

@anon Which university are you at, now?

anon

Uni of Nebraska at Omaha

user19161

3:56 AM

Will you tell us your identity one day @anon?

anon

Maybe.

user19161

OK, you just look so cool to me...

Kannappan Sampath

@anon How is the department? Is it big?

anon

I don't know what to compare it against.

It's got one building for math and science combined, if that indicates anything.

Kannappan Sampath

@anon Perhaps mine.... I am at SMU-ISI-BC

@anon Well, a building could be VERY big.

(But, how does that matter.)

anon

4:01 AM

My professor was talking about me doing REU or FUSE, etc. in the next year or two ish.

(Research Experience for Undergraduates / Fund for Undergraduate Scholarly Experiences)

Kannappan Sampath

That sounds cool.

mixedmath

@JasperLoy Yes, blackboards are great. I would be a fraction of a man without a blackboard in my office.

@anon If I might ask, what year are you?

anon

Freshman. :)

Oooh, you have an office.

Kannappan Sampath

@mixedmath Mixed Math -- A Number Theorist crunching zetas and Galois representations for fun. :)

leo

interesting

same user posted 3 questions tagged as pony

mixedmath

4:08 AM

@anon Well, I share it with a couple other grad students ;p but I'm lucky, as they're never there, so it's almost like I have my own office

leo

it seems that he caused the creation of pony tag

anon

@leo the pony tag was created solely because of that users fanciful requests

mixedmath

I was asking because I did an REU, and it was when I became certain about math. Interestingly, it was when my partner (2 person project) learned that he didn't want to do math.

BenjaLim

@mixedmath Hey

Does this sound right to you?

mixedmath

hey

BenjaLim

4:11 AM

I have an abelian connected matrix lie group $G$

I know its lie algebra $\mathfrak{g}$ is abelian

and so the kernel of the exponential map is a subgroup

leo

@anon when I was about to post mine in meta I was looking for a "posting" tag, I typed "p" then pony pops up and then I tagged mine as pony

BenjaLim

call the kernel $K$. I claim that $K$ is discrete

choose $x \in K$ and $U$ about $x$ that is open in $\mathfrak{g}$ such that $U$ is homeomorphic to $\exp(U)$

then $\exp(U) \cap \{x\} = x$

anon

@Khromonkey @MattN I revised this answer, so you guys can undownvote it :-)

BenjaLim

@mixedmath In our department

there is no place for the maths students to hang out

Kannappan Sampath

@anon Matt downvoted an answer? :-O

BenjaLim

4:15 AM

there is a big meeting/afternoon tea room with big bords

anon

@KannappanSampath I asked them to do so, as well as upvote a question, in order to get my rep at 31415

Kannappan Sampath

@anon !!!!!!!!!!!!!!!! CRAZY !!!!!!!!!!!!!!!!!!

anon

it's on the star panel -->

also Jonas = uberwrench in plans

Jayesh Badwaik

@anon uberwrench?

anon

a very big wrench (I just made that up)

a lovable wrench :)

Jayesh Badwaik

4:17 AM

Ohh. Yeah Jonas is a big big wrench.

:-)

anon

My plan B was to downvote four of my least favorite users answers, but that was too mean :)

user19161

I am a very small wrench.

mixedmath

@BenjaLim that's too bad (no hangout place). Perhaps you should just occupy the afternoon tea room.

user19161

@anon What is special about 31415?

anon

pi

Jayesh Badwaik

4:22 AM

@JasperLoy WTF?

anon

or maybe I should have rounded up...

I like truncation better though

user19161

The movie Pi was terrible. I watched 5 min and gave up.

Jayesh Badwaik

Curiosity would have recognized it from over 9000 thousand miles away.

user19161

My two favourites are still A Beautiful Mind and Good Will Hunting.

user19161

Pi was just meaningless and pretentious.

user19161

4:24 AM

Proof is not bad, the one with Gwyneth Paltrow.

user19161

:5922321 Yes.

user19161

Why delete?

anon

Have you seen Primer?

user19161

@jayesh Your deletions are scaring me...

user19161

@anon No.

Jayesh Badwaik

4:26 AM

@JasperLoy delete?

user19161

@JayeshBadwaik Yeah. Why do you delete your messages?

user19161

I think jay just wants to be mysterious.

Kannappan Sampath

Bye folks. Later.

Jayesh Badwaik

Ahh. This message got deleted? I did not mean to delete it this time. I generally delete when I haven't said something fully and my poor internet gives up on me. I wish the guy would install the wire soon.

And I have the bad methods of IRC.

@KannappanSampath bye.

user19161

@JayeshBadwaik OIC, OK.

Jayesh Badwaik

4:44 AM

@anon I personally think Donnie Darko is a much better time travel film than Primer.

anon

heh

IMDB doesn't even indicate anything about time travel. never heard of the film.

BenjaLim

@anon Hey

I have an idea but I don't know how to formalise it

anon

hey girl

BenjaLim

I will work with just the one dimensional case

@anon Consider for example $\Bbb{R}$ and $H$ a subgroup of it

under addition

suppose that $H$ is closed and discrete

I want to claim that every element in $H$ is an integer multiple of some real number.

Jayesh Badwaik

@anon en.wikipedia.org/wiki/Donnie_Darko

It has a wormhole.

anon

4:49 AM

Can we show $H$ has a least positive element?

BenjaLim

well such an element must exist

anon

well that pretty much settles the dim-1 case

BenjaLim

otherwise you would have every neighbourhood about 0 containing a point other than zero, contradicting discreteness

anon

you might find math.umn.edu/~garrett/m/mfms/notes_c/discrete_sbgps.pdf helpful

BenjaLim

yeah

Jayesh Badwaik

5:05 AM

@anon what are closed subgroups?

anon

subgroups that are closed sets within the overall topological group

Jayesh Badwaik

Okay.

anon

fun exercise: every finite-index closed subgroup is open, and every open subgroup is closed :)

BenjaLim

@anon I think I have an idea

If we have $H \subseteq \Bbb{R}^n$ closed and discrete

Jayesh Badwaik

open subgroup is an open set by definition right?
aah never mind, got it, I guess.

BenjaLim

5:10 AM

by a similar argument as before we can choose elements $x_1,\ldots x_n$ such that $x_1$ has the least $x^1$ coordinate of all the points, $x_2$ has the least $x^2$ coordinate of all the points, etc @anon

anon

a subgroup that is an open set, or an open set that is a subgroup, or whatever you want to say

BenjaLim

@anon I think that would work

But now we have to modify our argument a bit

anon

pick *an element of least distance from the origin, mod out by the cyclic group generated by it, induct. or something like that.

BenjaLim

yes

ah crap I'm going to have to deal with coordinates

anon

why?

BenjaLim

5:13 AM

when you induct

surely coordinates will be coming in

@anon I can't just go for a maximal $\Bbb{R}$ linearly independent subset in my subgroup $H$

and then look at the $\Bbb{Z}$ span of that

anon

okay, how bout this

BenjaLim

@anon wait let me tell you my idea

I think we can choose points $x_1,\ldots x_n$ such that the $x^1$ coordinate of $x_1$ is least, and so on

Suppose the $x^1$ coordinate of $x_1$ is $a_1$, $x^2$ coordinate of $x_2$ is $a_2$, and so on

I now claim that my subgroup will be generated by the points:

$(a_1,\ldots,0), (0,a_2,\ldots,0), (0,0,a_3,\ldots,0)$ and so on@anon

anon

Say it's true for $n-1$, then consider $n$. Pick $x\in H\le \Bbb R^n$ an element of least distance from the origin. Then $H/\langle x\rangle \le \Bbb R^n/\langle x\rangle\cong \Bbb R^{n-1}$ decomposes as $\langle y_1,\cdots,y_k\rangle$ (induction hypothesis) and you can write $H=\langle x,y_1^*,\cdots,y_k^*\rangle$ with $y_i^*$ representatives from the inverse images of $y_i$'s under the quotient maps

BenjaLim

my idea above is not correct.

leo

5:36 AM

g nite!

Matt

7:32 AM

What does CW in CW complex stand for? I think I remember that the C stood for closed. But I might be wrong.

C = Closure-finite and W = weak topology.

Wikipedia actually says it.

BenjaLim

yes that is correct.

Sebastian

@BenjaLim Hi Ben

Matt

@BenjaLim I finally posted an answer to that other question of yours. Meant to do that two weeks ago. That was fun. I've been meaning to look up Seifert-van Kampen for a bout a year. It was mentioned in our course but we never got a chance to apply it.

I thank you for asking the question : )

BenjaLim

7:48 AM

@Sebastian hey

@Matt Hahahahaha van Kampen

Now whenever I get the chance I apply it straight away

@Sebastian The thread is on the main page

Matt

: )

Jonas Teuwen

8:55 AM

Damn.

@Matt Community Wiki? 8-).

Jayesh Badwaik

9:11 AM

inching closer to 500.

@JonasTeuwen haskell with some modifications would be more awesome than it is now.

Jonas Teuwen

Sure.

user19161

9:26 AM

@JayeshBadwaik Waiting for you to do it!

user19161

Haskell sounds like a nice name for a person.

Jonas Teuwen

More awesome... that is like xmonad.

Henning Makholm

@JasperLoy It was Curry's.

Matt

9:43 AM

Fool me once, shame on you...

: D

Jayesh Badwaik

10:34 AM

@JonasTeuwen xmonad are so so clumsy... arrgh.

user19161

Well, I think I will keep my TeX account after all...

user19161

@HenningMakholm Ah, Curry tastes nice.

Gustavo Bandeira

11:05 AM

Jasper?

Gustavo Bandeira

11:23 AM

@JasperLoy The song I made is similar to this

Gustavo Bandeira

11:35 AM

I felt This user talked to me with an air of obviousness which wasn't present to me.

Jayesh Badwaik

@GustavoBandeira He thought the thing was obvious when you did not?

Gustavo Bandeira

@JayeshBadwaik Yep.

Jayesh Badwaik

@GustavoBandeira Hmm. Usually I am on the side of obviousness (if I know what I am talking about that is.)

Gustavo Bandeira

What yo mean?

Jayesh Badwaik

I want to say that everyone can be like that sometimes. But if you asked for a specific explanation and then the guy refuses because he thinks it is obvious, then you should be careful that he may be is pulling a fast one on you.

Gustavo Bandeira

11:43 AM

@JasperLoy Oh, my comment on Nihilism was from [this book](http://www.amazon.co.uk/Nihilism-Key-Ideas-Bulent-Diken/dp/0415452171): "This is how the devil speaks toward the end of Brothers Karamazov,
announcing the ludicrousness of sublimation, of ‘all that is great and
beautiful’, in modern times, and demanding moderation. A banal, normalized
devil that no longer speaks the language of evil, a devil without
evil. This paradoxical, mediocre devil was the nightmare through which
the nineteenth century dreamed of the times to come, a future that promotes

@JasperLoy You said something that reminded me of this somehow.

@JayeshBadwaik Fast one?

Jayesh Badwaik

idioms.thefreedictionary.com/pull+a+fast+one

Gustavo Bandeira

@JayeshBadwaik Yes, I thought so.

skullpatrol

12:09 PM

@GustavoBandeira That is the first time someone has quoted that much Nietzche in this room, congratulations :-)

Gustavo Bandeira

@skullpatrol Achievement unlocked: Nietzche recordist.

skullpatrol

@GustavoBandeira Thanks for the reference.

Gustavo Bandeira

@skullpatrol You're welcome. =)

Jayesh Badwaik

@GustavoBandeira xkcd.com/167 ?

Gustavo Bandeira

lol

Jayesh Badwaik

12:17 PM

I just read again the wikipedia page on nihilism and I cannot understand a thing of it.

@GustavoBandeira Is nihilism considered positive of negative in general or none?

Gustavo Bandeira

@JayeshBadwaik I'm back.

Jayesh Badwaik

Good. I am reading more about nihilism and I am thinking they are mixing up a few things there.
What I feel is, their analysis is correct to a society as a whole but not single individual elements of the society.

And that is by the virtue of the fact that the individual is "inside" the society and hence cannot clearly see the contrivance of the morals.

Gustavo Bandeira

@JayeshBadwaik I still have no idea: Until now, Bulent just said Nihilism is one of the most misunderstood body of ideas.

Jayesh Badwaik

Something similar to earth is reasonably flat .

Gustavo Bandeira

@JayeshBadwaik Yes.

skullpatrol

12:56 PM

@JasperLoy yo sup?

user19161

@skullpatrol Boo!

skullpatrol

@JasperLoy Boo who?

user19161

@skullpatrol Boo is my standard greeting, in case you don't know.

skullpatrol

@JasperLoy Why?

user19161

@skullpatrol It's to scare people when I enter the room.

skullpatrol

12:59 PM

@JasperLoy Do you like to scare people?

user19161

@skullpatrol No, I only use boo in chat.

Nick Kidman

When we do math, we usually put established statements together to form another (here we use logic on the meta-level). Then in the study of logic, there are many versions of it, some use law of the excluded middle (inside the theory, so theory level), some not, etc. I wonder now if there are some meta rule, which are always necessarily used when doing math. That is, even if we study some very abstract variant of logic, do we use logic from a more mild logic to infer?

user19161

@NickKidman Great question. Hope a logic expert answers you. You can even post on main.

skullpatrol

"Pure mathematics is, in its way, the poetry of logical ideas."

user19161

@NickKidman You don't happen to be the actress Nicole Kidman do you? :-)

Gustavo Bandeira

1:14 PM

@JasperLoy lol

skullpatrol

Gustavo Bandeira

@skullpatrol Hahahaha

@skullpatrol The art of finding the perfect image for the situation.

2

Nick Kidman

I wonder if one can gather together the logic rules which really have to be used to do math. Like I expect if 'A' is true and if 'B' is true, then we have to be able conider 'A and B' to do math.

then one could argue a logic calculus without the and symbol can't be about logic

Zhen Lin

Hmmm. Perhaps. But logic isn't confined to the logic of thinking. Linear logic is an especially strange variant...

Nick Kidman

of course you can do symbol calculus without and, but that would then really be just a formalist game

also

user19161

1:24 PM

@NickKidman Is that Nicole as well?

Gustavo Bandeira

@NickKidman Isn't this the Russell's Principia Mathematica?

Nick Kidman

@JasperLoy: Someone's got prosopagnosia

yes, it's her as 16 year old or so

@GustavoBandeira: I don't know, is it? Is this idea in there?

user19161

@NickKidman You use difficult words that I don't know!

Gustavo Bandeira

@JasperLoy Inability to recognize faces.

experimentX

Hi ...

skullpatrol

1:27 PM

hi

Nick Kidman

@JasperLoy: That's what my friends say when I talk about quantum field theory during picnics

experimentX

i would like to ask a question out of curiosity

skullpatrol

askaway

user19161

@NickKidman I only eat sandwiches during picnics.

experimentX

how to evaluate series of form $$ {1 \over n^{2k+1}}$$

Gustavo Bandeira

1:28 PM

@NickKidman I guess Russell and Whitehead tried to do what you proposed but I have no deep understanding of it.

J. M.

@experimentX Are you summing over $n$, or over $k$?

experimentX

more particularly for 1/n^5 math2.org/math/expansion/power2.htm

J. M.

@experimentX Well, there really aren't elementary expressions for odd values of Riemann's function.

hhh

Is Empty set a polyhedron?

experimentX

thanks!! I really need to study this after all ...

hhh

1:32 PM

I think it is because choose any $A$ that is $m\times n$ -matrix and $\bar{b}\in \mathbb R^{m}$ so that $A$ is an empty set so $\{\bar{x}\in \mathbb R^n | A \bar{x} \geq \bar{b}\}$.

experimentX

how do we copy this message link??

Gustavo Bandeira

@experimentX Right button?

J. M.

@experimentX See this as well.

hhh

(I am studying page 42 in the book Introduction to Ilnear Optimization, Bertsimas)

experimentX

thanks @J.M.

hhh

1:35 PM

"The empty set corresponds to the null polytope, or nullitope, which has a dimensionality of −1. These posets belong to the larger family of abstract polytopes in any number of dimensions."

<-- I think this means, Empty-set is a polyhedron because polyhedron is a special type of polytope.

(I cannot understand the meaning of dimensionality -1, though -- how is it determined? What is a polytope of dim -2?)

Gustavo Bandeira

Why the line crosses the $\Pi^3$?

Zhen Lin

Bad typesetting.

Gustavo Bandeira

@hhh It's usually better to make a question.

@hhh When you do it, you expose your question to a more broad audience.

hhh

-1. Is an empty set a polygon?
0. How is null polytope of dim -1?
1. How are polytopes of minus dimensions detemined?
2. What does polytope of -2 dim look like?

<-- trying to formulate different ideas

Gustavo Bandeira

@hhh Sorry, but I have no knowledge on this.

Zhen Lin

1:41 PM

It is defined to have dimension $-1$ because it doesn't even have a point. You could equally well define it to have dimension $-\infty$.

Jayesh Badwaik

@NickKidman About your "and" logic, what if taking "and" of statements is not possible (or rather not defined)?

Gustavo Bandeira

@JayeshBadwaik Still suffering with dial up conection?

Jayesh Badwaik

@GustavoBandeira Dude, it is going to last till monday when I get a new broadband connection.

Gustavo Bandeira

Damn.

@JayeshBadwaik I remember when I used dial up, you need to press F5 several times for the pages to open.

Jayesh Badwaik

@GustavoBandeira Well, you can imagine my frustration, till two weeks ago I had a reasonably fast unlimited broadband.

Henning Makholm

1:48 PM

@NickKidman Actually, one of my favorite formalizations of classical logic doesn't have "and" as a primitive concept. Only "implies" and "false". Then "P and Q" is viewed as an abbreviation of "(P implies false) implies ((Q implies false) implies false)".

Zhen Lin

You may as well be doing combinatory logic in that case. :p

hhh

@GustavoBandeira I think the purpose of the q is to recall just the definition of polyhedron. I would answer "Yes" i.e. empty set IS a polyhedron because it satisfies the definition unless I miss something.

skullpatrol

@robjohn In the English Language and Usage chat room they have posted under the room name the phrase: "aka The Incomprehensible Room." Do you think you could post: "aka The Askaway Room." here? (just to encourage people to not ask "Can I ask a question?")

Henning Makholm

@ZhenLin It's less cumbersome in Girard's system F where you can quantify over propositions, and let "P and Q" mean $\forall\alpha: (P\to\alpha)\to(Q\to\alpha)\to\alpha$.

hhh

@ZhenLin Is the empty set also a definition question? Can you exclude empty set out of polyhedrons somehow?

Zhen Lin

1:51 PM

Sure. I hereby define polyhedra to be non-empty.

hhh

...ofc you can, just define vector $\bar{x}$ as non-empty but my book does not have word "non-empty".

Zhen Lin

There are occasions where one excludes the empty set because it's too troublesome otherwise.

skullpatrol

@ZhenLin What would make the set with no members toublesome?

Zhen Lin

@HenningMakholm I recently worked out how to define interpretation of algebraic terms without structural recursion. I wonder if I should suppress this knowledge.

Henning Makholm

@ZhenLin Can you even define what an algebraic term is without structural induction (or something equivalent thereto)? Tell me more.

Zhen Lin

1:57 PM

It turns out you can do it by abstract nonsense. A term in $n$ variables is an element of the freely-generated algebra on $n$ generators.

Henning Makholm

Oh, right. And then the universal morphism describes an interpretation.

Zhen Lin

Mmm... not quite. It's a bit more complicated than that, because you need to be able to interpret terms with respect to assignments in arbitrary models.

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$Mathematics$ Mathematics