Mathematics - 2012-12-04 (page 2 of 4)

Peter Tamaroff

4:00 AM

It is a little about integration, sequence of functions and constructivism/existencialism in proofs

Patrick Da Silva

You're asking me a question?

amWhy

@WillHunting You're right...I realized that after getting some nourishment!

Peter Tamaroff

@PatrickDaSilva I actually wanted to show you a proof

Would that be OK?

Patrick Da Silva

I'm a little confused, I just want to understand why you wanted to talk to me before we start talking >.>

user19161

@amWhy Haha, I am not called WH for nothing you know... =)

Patrick Da Silva

4:00 AM

Oh. Sure

Are you going to type it here?...

Peter Tamaroff

@PatrickDaSilva OK. So we say a function is "ruled" over some closed interval $[a,b]$ if it is the uniform limit of a sequence of step functions $\{s_n\}$

amWhy

@WillHunting hehehe...New user name tomorrow? (or is it the 4th for you today?)

Peter Tamaroff

@PatrickDaSilva Yes. I might be not the fastest typwer in the world.

user19161

@amWhy Well, I will take a nap later, wake up, change my username, and drop you a comment somewhere to say something...

Patrick Da Silva

I was just wondering. Alright, so "ruled" means uniform limit of step functions... measurable step functions?

Peter Tamaroff

4:02 AM

I aim to show that every continuous function $f:[a,b]\to \Bbb R$ is ruled.

@PatrickDaSilva In Spanish I use "reglada":

Patrick Da Silva

Wait

Peter Tamaroff

@PatrickDaSilva Well, just functions that are constant over intervals

Patrick Da Silva

you know that every bounded measurable function is a uniform limit of measurable step functions, right?

Peter Tamaroff

Linear combinations of indicator functions over intervals

Patrick Da Silva

yes, sure

Peter Tamaroff

4:03 AM

@PatrickDaSilva I know zero measure theory

amWhy

@WillHunting I'll look for it when I'm wake up tomorrow! (Unless I get restless in the middle of the night and call fall back asleep!)

Patrick Da Silva

I have two measure theoretic courses this semester so I kind of assumed, but it's okay

Peter Tamaroff

@PatrickDaSilva But SPivak is giving me a taste of it, it seems.

Patrick Da Silva

go on

Peter Tamaroff

@PatrickDaSilva Well, I have two proofs

KaliMa

4:03 AM

Anyone familiar with pythagorean triplet generation

Patrick Da Silva

@KaliMa : yes, there are many way to span those

user19161

@KaliMa Well, I know the formula for primitive triples.

KaliMa

I am trying to generate triplets of the form a^2+b^2=5c^2 and I have a parameterization for it already but I don't yet have an intelligent way to loop through what I need

it seems like the generator yields both primitive and nonprimitive

Patrick Da Silva

Hm. These are not pythagorean triples. I just know there's a way to generate them using matrix products for instance. I have no idea about those triples you speak of.

@PeterTamaroff I'm still listening

user19161

Oh you call that pythagorean too?

Peter Tamaroff

4:06 AM

The first, which robjohn helped me devise, consinsts in basically splitting the closed inteerval $[m_f,M_f]$ into $n$ parts with a partition $\{t_0,\dots,t_n\}$ and defining for each $n$ and $k$ the $k$ sets $\lambda(n,k)=\{x:t_{k-1}<f(x)\leq t_k\}$ where $t_k=m_f+k\Delta_n$ Here $m_f$ us the minima of $f$

KaliMa

1

Q: Generating Pythagorean triples for $a^2+b^2=5c^2$?

Just trying to figure out a way to generate triples for $a^2+b^2=5c^2$. The wiki article shows how it is done for $a^2+b^2=c^2$ but I am not sure how to extrapolate.

Peter Tamaroff

over the interval, $M_f$ is the maxima

KaliMa

What I am doing is taking each triplet, dividing it by GCD to get its primitive

Peter Tamaroff

and $\Delta_n=(M_f-m_f)/n$

@KaliMa Please use [name](link) so you don't clutter

Patrick Da Silva

@PeterTamaroff : Can we go to a room where it's just us? The other conversation is kind of "annoying" (i.e. I don't want to read two conversations at once)

Peter Tamaroff

4:07 AM

@PatrickDaSilva Yes, sure

Patrick Da Silva

How do you do that

KaliMa

Thanks for calling me annoying

user19161

Aww, relax guys...

KaliMa

jackass

Patrick Da Silva

I wasn't saying you were

I was saying trying to read two conversations at the same time is annoying

sorry if I didn't make myself clear enough... I thought the ( ... text ) was explaining this

Jayesh Badwaik

4:11 AM

Its okay. There is a reason why everyone is allowed to create his own chatroom, no one should feel offended.

Yesh!! I am in the room without being detected on the bar!! @WillHunting Was I there on the right before you saw this?

user19161

@JayeshBadwaik Well, I was thinking of other things so I did not notice you. After all, you ain't Skylar!

Jayesh Badwaik

@WillHunting Duuude. :-( Bros before skylars.

user19161

@JayeshBadwaik No, Skylars are more important than bros.

Jayesh Badwaik

@WillHunting I am being dramatic. :-)

user19161

@JayeshBadwaik Have you woken up already or going to sleep?

Jayesh Badwaik

4:16 AM

@WillHunting I woke up around three hours ago. Went on a run, had breakfast, now studying.

jshin47

5:05 AM

Before I go crazy, it is a true fact that a maximum of two ideals will span a lattice $\mathbb{Z}[\sqrt(-d)]$ ?

Thats not exactly what I mean actually, now that I think about it.

Nevermind.

Gustavo Bandeira

5:28 AM

@PeterTamaroff ?

Peter Tamaroff

5:50 AM

@GustavoBandeira Sire?

Gustavo Bandeira

@Peter it's a really naive comment/question.

user19161

@GustavoBandeira Did you spend your birthday with Maria?

Gustavo Bandeira

Look his comment:

The question should be more clearly stated. $n^\frac13$ is not a polynomial in $n$ because it is not of the right form (a linear combination of monomials in $n$), period. By the same token $\exp(2\ln x)$ is not a polynomial in $x$, even though for $x>0$ one has $\exp(2\ln x)=x^2$. The question should be to show that $n\mapsto n^\frac13$ is not a polynomial function of $n$, where the domain of the function should be specified (for instance the name $n$ suggests integers only, in which case differentiation arguments cannot be used). — Marc van Leeuwen Nov 30 at 8:29

Peter Tamaroff

OK

user19161

Guys I have attained 8k now. I will aim for 9k and then 10k.

Gustavo Bandeira

5:52 AM

@WillHunting Nope, she live in a different city, lookk

@WillHunting And then $\inf$k!

Peter Tamaroff

@GustavoBandeira Do you live in the coast?

Gustavo Bandeira

@PeterTamaroff Yes.

Peter Tamaroff

@GustavoBandeira Silly, he would have to get massively downvoted to get to $\inf$k!

user19161

@GustavoBandeira You mean $\infty$, not $\inf$ dude.

Peter Tamaroff

And that would be like $1$ rep here.

@GustavoBandeira What is disturbing your brain, man?

user19161

5:54 AM

I am going to sleep now, I hope some miracles happen soon...

Gustavo Bandeira

@PeterTamaroff Consider his comment: Well, in this question I can't see the linear combination of monomials, but you did some kind of sorcery here to make them appear.

@WillHunting Like 99999999999999 upvotes?

user19161

@GustavoBandeira Nah, like my wishes X, Y and Z.

Gustavo Bandeira

@Will Are you giving continuity to your math studies?

user19161

@GustavoBandeira Well, everything is just as I told you before.

Gustavo Bandeira

@Will do it, dude! Math is awesome!

Peter Tamaroff

5:58 AM

@GustavoBandeira A function $f$ is a polynomial over some domain $D\subseteq \Bbb R$ if for each $x\in D$, $f(x)=\sum a_k x^k $ for some fixed constants $a_k$

user19161

@GustavoBandeira Therefore, X, Y and Z. QED.

user19161

@PeterTamaroff Wrong again dude.

Peter Tamaroff

@WillHunting Really?

Correct me, then.

user19161

@PeterTamaroff Remember what I told you about the order of quantifiers?

Peter Tamaroff

@WillHunting What is the fuzz now? Don't confuse Gustavo.

user19161

6:00 AM

@PeterTamaroff If one wants to define polynomials, one might as well do it properly and not confuse him by defining it wrongly this way.

Peter Tamaroff

@WillHunting OK, correct me, for the sake of Gustavo.

user19161

@PeterTamaroff Well, I just did. Order of quantifiers. QED.

Peter Tamaroff

@WillHunting You're being stupid now.

Gustavo Bandeira

Order of the quantifiers? But arent "For all", "there exists" quantifiers?

Didn't know "is an element of" is a quantifier.

Peter Tamaroff

@GustavoBandeira Nah, he means that I should have written "for some fixed constants..." first.

Gustavo Bandeira

6:03 AM

Oh, got it.

He went away.

Well, my question is really naive and it's not very well formulated...

BTW, look - It seems there are some was of seeing this linear combination of monomials on some math structures, but I'm unaware of these mathods.

Jayesh Badwaik

@GustavoBandeira May I ask what are you doing currently?

Gustavo Bandeira

I'm not saying that it might be possible to see polynomials in everything - well, perhaps you can see polynomials in everything but then you would need some cocaine for that.

@Jayesh? Specificaly?

Jayesh Badwaik

@GustavoBandeira Yes, as in, working? Enrolled in uni?

Gustavo Bandeira

@JayeshBadwaik Yes, my classes already started but I wont go this week - it's the first week and there's only some presentations.

And I also have to study some piano music.

Jayesh Badwaik

@GustavoBandeira Hmm. Nice. Bachelor's? Master?

Gustavo Bandeira

6:08 AM

@Jay Bachelor.

Jayesh Badwaik

@GustavoBandeira Cool.

Gustavo Bandeira

I'm planning on doing bachelor, then master then Ph.D.

Jayesh Badwaik

Cool. :-)

Gustavo Bandeira

I mean, when I finish the university, I'm going straight to master then straight to Ph.D.

Jayesh Badwaik

Yup, got it.

Gustavo Bandeira

6:09 AM

I'll get so hardcore at math that MIT/Harvard/Oxford are going to send me a letter: Please! We beg you! Come to teach at our shitty university!

A little pretentious...

LITTLE.

Jayesh Badwaik

@GustavoBandeira I hope.

Gustavo Bandeira

And you?

Jayesh Badwaik

I finished my bachelor's in electrical engineering, preparing for entrance exams to Masters in mathematics, after that PhD, after that wherever life takes me.

Gustavo Bandeira

Why didn't you decide to follow the master on elec eng?

Jayesh Badwaik

It was never my intention or desire to bachelors in elec engg. I for sure can't stand a masters in the same field.

Its not I hate elec engg, I find it interesting, but I want to do mathematics, and want a shot at doing it now.

Gustavo Bandeira

6:13 AM

May I ask why did you do electrical enginering then?

Jayesh Badwaik

got it?

Gustavo Bandeira

Yes.

It's strange - I wanted this to happen with me.

Jayesh Badwaik

why?

Gustavo Bandeira

I wish I was something more than I am today.

Jayesh Badwaik

hmm.

Gustavo Bandeira

6:18 AM

I feel like I've spent too much time lost - wandering and procrastinating.

Jayesh Badwaik

yup, it works both ways.

for some people, it works positively, for some it works negatively

Gustavo Bandeira

Yes.

Jayesh Badwaik

depends on how motivated towards a particular thing you are from the beginning. I think, I was sure I wanted to be a mathematician since I was 13-14.

Gustavo Bandeira

Yep.

But why math?

Some specific reason?

Jayesh Badwaik

I gravitated towards it. In school, I would read texts, then go online and read more about stuff and all. I gave a school talk on nanotechnology when I was 14, during my preparation, I read some college physics, and I liked doing it. Similarly, during another school talk on cryptography, and I grew interested in prime numbers, number theory and stuff. So, I used to go to this retired professor with a bunch of friends, and he introduced us to other areas of mathematics, combinatorics. (contd.)

graph theory, graphical geometry (almost everything that could be done without calculus). And I liked that stuff even more.

So, by the end of school, I was set on two things, math and physics. Mathematics more than physics.

And even for physics, I would see stuff, and realize how good math skills were really important for physics, so I wanted to do math for sure.

@GustavoBandeira why math? for you?

Gustavo Bandeira

6:35 AM

@Jayesh I hated math when I was on high school, then I completely ignored it. The problem wasn't math, the problem were teachers unable to handle/teach it properly.

While I was 14-15, I started to make music - then I grew up and started studying piano at 19 - but I alwyas felt that math could have some kind of hidden power which I was curious to know about.

Jayesh Badwaik

okay.

Gustavo Bandeira

I've always searched for alternatives to make more experimental music, while searching the net I discovered some new softwares, one day I discovered the rubato music composer.

This software was programmed by him:

Jayesh Badwaik

@GustavoBandeira Hmm. To tell the truth, I am very very confused by what that problem is. The same teachers under whom I flourished were not rated highly by other students of my class, and this has happened quite a lot with me. Same goes for in

Gustavo Bandeira

Guerino Mazzola

Guerino Mazzola (born 1947) is a Swiss mathematician, musicologist, jazz pianist as well as book writer. He graduated at the University of Zürich in Mathematics, Theoretical Physics and Crystallography and completed his PhD in Mathematics in 1971. In 1980, he habilitated in Algebraic Geometry and Representation theory. In 2000, he was awarded the medal of the Mexican Mathematical Society. In 2003, he habilitated in Computational Science at the University of Zürich. Mazzola has recorded several Free Jazz CDs with musicians like Mat Maneri, Heinz Geisser, Sirone, Jeff Kaiser, Scott Fie...

Jayesh Badwaik

okay.

Gustavo Bandeira

6:38 AM

His software involves knowledge in math and then I decided to learn it for using on his software.

Jayesh Badwaik

okay.

Gustavo Bandeira

Some months ago, I've read a book - a pop science book: Complexity.

I was amazed but how they could use math to describe nature, to think, etc.

Now I'm GREATLY curious about math.

I know I want math, but I'm also wandering on it again.

Jayesh Badwaik

okay.

Gustavo Bandeira

I'm reading math and studying piano, lets see what can I do with it in the future.

@JayeshBadwaik Well, I was a little revolted. =/

Jayesh Badwaik

@GustavoBandeira Hmm. Revolted with teachers?

Gustavo Bandeira

6:41 AM

Whenever conditions were adverse, I just used to say "fuck you" - this cost of this attitude was high.

Yes.

Jayesh Badwaik

Ohh, you mean rebellious.

Gustavo Bandeira

Yes.

Jayesh Badwaik

okay.

Gustavo Bandeira

In Brazil, we have the vestibular:

Vestibular

The Vestibular (from , "entrance hall") is a competitive examination and is the primary and widespread entrance system used by Brazilian universities to select their students. The Vestibular usually takes place from November to January, right before the start of school year in February or March, although certain universities hold it every semester. The exams often span several days, usually two, with different disciplines being tested each day. Structure Several Brazilian universities follow the FUVEST (University of São Paulo's entry exam) pattern, which is divided into two stages or "p...

Jayesh Badwaik

@GustavoBandeira We too have it. In forms of IIT-JEE and AIEEE.

Gustavo Bandeira

6:43 AM

Yep, but it may not hae the same name.

They're the same thing of course.

Jayesh Badwaik

Yeah.

Gustavo Bandeira

One day I asked: "What's the utility of the sine function" - I asked it because I wanted to have a ground to step, I wanted to explore that - he said me that "the sine function is useful for passing the vestibular".

Jayesh Badwaik

@GustavoBandeira Hmm, that's a little short-sighted.

Gustavo Bandeira

I'm kinda revolted with tests, I'm kinda apathic and I've never wanted to study to enter the university and get a job - Of course, I'll need a job and I know that, but for me, studying is a little transcendental.

I always wanted to apreciate the beauty of ideas.

And also colaborate with people who's make this beauty exist.

Jayesh Badwaik

See, thing is, a lot of such explanation requires so much effort. I used to teach my brother, and explaining motivation is one of the toughest thing. I was similar. I used to ask my teachers, "What's the use of grammar?", "What's the use of this?", "Why do I need to do this?". I must say, my teachers were very very patient with me, never getting irritated, always answer honestly. When I taught my brother, I was like "Fuck, they underwent this much of effort?" and I had huge respect for them.

Gustavo Bandeira

6:47 AM

Yep.

Today I see the utility as something contained in itself, which is pretty much like pure mathematics.

Jayesh Badwaik

Hmm.

Gustavo Bandeira

Perhaps I'm talking bullshit, but doesn't matter. =D

Jayesh Badwaik

Its not bullshit. I felt the same way, when I learnt the fundamental theorem of calculus. (Its so beautiful, oh dear god.)

Gustavo Bandeira

Yes.

I've read about the fermat little theorem.

Seeing it working is amazing!

Jayesh Badwaik

yup.

lately there are now many more theorems with the same beauty that I have encountered.

Gustavo Bandeira

6:54 AM

Lately? You mean contemporary math?

Jayesh Badwaik

nopes, I meant, that now, there are a thousand more theorems other than the FTC which get me hooked on now.

Gustavo Bandeira

Oh, got it.

user19161

7:22 AM

@PeterTamaroff Well, I hope you know what difference there could be if we interchange them.

Gustavo Bandeira

@Will wasn't you sleeping?

user19161

@GustavoBandeira Slept for a while, can't really sleep now.

Gustavo Bandeira

Yep.

user19161

@jay Have you tried CentOS?

Jayesh Badwaik

@WillHunting Yes, it was my primary development machine for the coding I used to do, which then used to be run on RHEL clusters.

user19161

7:31 AM

@JayeshBadwaik Do you know of an easy way to get Microsoft core fonts on CentOS other than building the rpm oneself?

Jayesh Badwaik

@WillHunting I believe, due to the licensing restrictions, that is the only way to get them I suppose.

user19161

@JayeshBadwaik On Debian, one just installs ttf-mscorefonts-installer.

Jayesh Badwaik

@WillHunting Also, its not that difficult.

user19161

@JayeshBadwaik I know, and I can tell you that half the instructions on the internet don't work, because I tried them all 9000 years ago, and I did successfully build the rpm before.

Jayesh Badwaik

@WillHunting Yes, I understand, in debian, the installer does that job. Someone would have to write such an installer for CentOS, there should be demand for that.

user19161

7:35 AM

There are various subtleties that need to be taken care of.

user19161

@JayeshBadwaik Maybe a Jayesh?

Jayesh Badwaik

@WillHunting No, I don't personally use MS fonts. I have a script though, an OS-independent part and an OS dependent part. Basically, all the subtleties are in downloading the font files, not much else.

user19161

@JayeshBadwaik There is a package that might be needed called chkfontpath.

Jayesh Badwaik

that's the os dep part

user19161

I think I have finally gotten used to Chrome now. Just need to refresh a bit more.

user19161

7:40 AM

It's the only way to get the latest flash on linux.

Jayesh Badwaik

what's the latest flash version?

user19161

11.5

user19161

If you install from adobe it is 11.2 for linux.

user19161

And I don't think flash is going to die anytime soon.

Jayesh Badwaik

so, chrome has 11.5 you say?

coz I have 11.2 on firefox.

user19161

7:42 AM

Sure html 5 can be used to replace it, but many people prefer using flash to do the job.

user19161

@JayeshBadwaik Yes, that's what I am saying, probably some partnership between google and adobe.

Jayesh Badwaik

@WillHunting Or probably just google's dev team?

Gustavo Bandeira

If I presume that I am the set Gustavo such that Gustavo $\neq$ Gustavo - Will I get invisible?

user19161

I am in weird states of mind these days...

user19161

Ladies and gentlemen, I have capped for the day, thank you for your support.

Gustavo Bandeira

7:51 AM

@will pro!

user19161

Some of my fans seem to forget not to upvote me after capping!

Gustavo Bandeira

If $S \subseteq T$ and $T \subseteq T$, then $S=T$. But how?

I <3 the editing feature. =D

I mean: I could have S={0,1,2,3,4} and T={0,1,2,3,4,5}

Then $S \subseteq T$, right?

But the oposite is not true.

Oh,got it.

Ignore me.

Jayesh Badwaik

@GustavoBandeira $S \subseteq T$ and $T \subseteq S$, a typo.

Gustavo Bandeira

If $A \subseteq B$ and $G \subseteq J$, then $ \text{Im} \subseteq \text{Malasya}$.

\text is not working.

I may be noobing somewhere.

user19161

Learn LaTeX first.

Jayesh Badwaik

7:58 AM

$\text{Im} \subseteq \text{Malasya}$

Gustavo Bandeira

Oh, yes!

user19161

@GustavoBandeira I am only a banana.

Gustavo Bandeira

@will With honey??

Gustavo Bandeira

8:16 AM

Does the empty set contain an empty set?

The book I'm reading says that the power set of a empty set consists of one element: {$\emptyset$}

user19161

@GustavoBandeira The empty set is a subset of itself.

user19161

@GustavoBandeira The power set would contain all its subsets, and in this case, there is exactly one subset, namely the empty set.

user19161

So the empty set is empty but the power set of the empty set is nonempty, having the empty set as its one and only element.

user19161

Over and out, I will be back tomorrow for more reputation points...

user19161

9:31 AM

Hi @old I capped today and also attained 8k!!!

Old John

@JasperLoy wow - you are racing upwards these days - well done!

user19161

@OldJohn So are you!

Old John

@JasperLoy not at you rate - I have not capped for abut a month - but I am too busy for answering questions at the moment

user19161

@OldJohn Yes, you must be secretly working on the millennium problems!

Old John

@JasperLoy I have more sense than that! I am secretly trying to improve my understanding of CA in order to understand ANT better :)

user19161

9:41 AM

@OldJohn So secretive!

Old John

@JasperLoy I have a sort of "no secrets" policy in my life ... keeps things simpler

user19161

@OldJohn Hahaha, actually my secrets aren't really secrets either. =)

Old John

@JasperLoy :)

Jayesh Badwaik

@OldJohn Good Morning. :-)

3-3 in thirty minutes, what a goal fest!!

Old John

@JayeshBadwaik good afternoon!

Jayesh Badwaik

9:44 AM

no goals in the second half

:D

Old John

@JayeshBadwaik the last Man U match?

Jayesh Badwaik

@OldJohn yup

reading 4-3

Old John

@JayeshBadwaik first half was certainly exciting ... second half not so much

they certainly need to defend better in their next few matches!

Jayesh Badwaik

Yes. They have come back to win after being goal down 7/10 times.

More prepared teams won't offer that oppurtunity

Old John

@JayeshBadwaik indeed!

robjohn

10:18 AM

@JasperLoy again?! You're making this a habit.

user19161

@robjohn Well, I just happen to have made some friends here. =)

user19161

11:01 AM

Did I just kill chat?

a Dangerous Idea

No. I did.

user19161

But you were quiet.

Charlie

11:48 AM

Helloooo

@JayeshBadwaik No, they won't

user19161

12:04 PM

@Charlie So many o's.

Charlie

@JasperLoy to see if someone hears me

@JasperLoy hey, Jas, I was going to see some document stuff and the pearson who answered me got astonished when i said i'm 20.... the pearson thought i was 12 :P

user19161

@Charlie You mean the voice?

Charlie

@JasperLoy no, my face

user19161

@Charlie OMG.

Charlie

hahaha

user19161

12:14 PM

@Charlie Only XXX has seen your face I know.

Charlie

yup

Jayesh Badwaik

12:37 PM

@Charlie 12? He was joking I bet.

@Charlie Hello!

N3buchadnezzar

I am 12 and what is this?

N3buchadnezzar

1:07 PM

@JayeshBadwaik When does $T(x) = T(t)$ imply that $x = t$ ?

Jayesh Badwaik

@N3buchadnezzar Injective? Dimension of the operator is n?

T is invertible

N3buchadnezzar

I would say that T is invertible and injective

Jayesh Badwaik

@N3buchadnezzar $T$ is invertible would imply it is injective I guess.

All the above three conditions I listed are equivalent.

N3buchadnezzar

1:26 PM

Haha

Jayesh Badwaik

What?

N3buchadnezzar

I am doing math, and borrowed away my calculator.

He was like, you are doing math. Do you not need a calculator?

Jayesh Badwaik

Hmmm. You need calc to do math?

Ahh,. see, even I said the same thing. :P

N3buchadnezzar

@JayeshBadwaik One does not simply need a calculator when doing serious math

2

Jayesh Badwaik

@N3buchadnezzar Yes! Wolfram Alpha is cool though. :p

N3buchadnezzar

1:28 PM

:p

@JayeshBadwaik Use wolfram alpha to prove the rank nullity theorem, I dare you

Charlie

@JayeshBadwaik many people already said that to me...

Jayesh Badwaik

@Charlie Ehh, hmm.

Charlie

@JayeshBadwaik really weird.... how was youe walk?

N3buchadnezzar

hmmmm

Jayesh Badwaik

@Charlie Good. Run, Charlie, Run. (Not Walk)

@N3buchadnezzar Hmm, might be difficult, but I can prove it partially using some cases, like the four color theorem.

Charlie

1:32 PM

@JayeshBadwaik details

Jayesh Badwaik

@Charlie Not now, I am going out for dinner in a few minutes.

MathJAX gone crazy on me.

@Charlie See you later. Bye.

Charlie

Fine

@JayeshBadwaik i didn't say i wanted details, i said that "walk" or "run" is a detail...

@gustavo are you a little better?

@amWhy hello

Old John

2:03 PM

@JayeshBadwaik maybe at that time your PC had a good connection to MSE, but no connection to the MathJax server?

although I would not expect the browser to fetch a fresh copy of MathJax every time - surely it keeps a cached copy

Charlie

2:21 PM

Hi @oldjohn

Jayesh Badwaik

@OldJohn No, not possible, I have my own local copy of MathJAX which I host using the /etc/hosts file and it is displaying everything else properly. But even the original MathJAX server copy is not displaying properly.

@Charlie Ohh. Walk is not a tiring activity. I can walk for as long as I want probably. On the other hand running is a tiring activity.

Charlie

@JayeshBadwaik hmm interesting

Jayesh Badwaik

@Charlie Yup, previously could run 15-20 km easily. Now-a-days cannot run more than 5-6. Have to get it back up.

Charlie

Hmm

Nimza

2:47 PM

Good day!

Hi @Charlie, @JayeshBadwaik!

Charlie

@Nimza Alexey! Good to see you!

@aDangerous hi hi skull

Nimza

@Charlie I'm too happy to see you too! Do you know, is it possible to say "you might not get it" in sense you might not understand it?

Charlie

I didn't get it...

a Dangerous Idea

@Charlie hi

Nimza

:D then there is a nice joke about UDP but you might not get it!

Charlie

2:51 PM

:(

Nimza

hey :) I found it in runet

@Charlie oh, how difficult is this composition on gender linguistics :(((

@aDangerousIdea hi!

@Charlie wazzup?

Not much

@Nimza Hi!! Sorry, going out for some time. See you later.

Nimza

2:55 PM

@JayeshBadwaik bye :)

Charlie

@Nimza i didn't know that! Interesting.

@aDangerousIdea you?

a Dangerous Idea

@Charlie chillin'

Charlie

@aDangerousIdea hmm

@Nimza what have you been doing, Alexey?

Charlie

3:12 PM

...

Peter Tamaroff

@Charlie Say fo shizzle to my nerdizzles

Charlie

:D

N3buchadnezzar

Is there any reason why we often specify that that a vectorspace has a field $\mathbb{K}$ ?

Where $\mathbb{K} \in \{ \mathbb{R},\mathbb{C}\}$

Why not simply say that the vectorspace has a field $\mathbb{C}$ since $\mathbb{R} \subset \mathbb{C}$ ?

Charlie

I wanna know. that too

anon

3:27 PM

@N3buchadnezzar because a vector space over R is not the same as a vector space over C

In addition to the allowable scalars being restricted in the real case, the dimension is halved and the linear operators do not all have eigenvalues. Though see en.wikipedia.org/wiki/Complexification

It is also a matter of logical refinement to say precisely when results hold over C but not over R. (Same idea applies with other pairs of fields and subfields.)

N3buchadnezzar

@anon I see, I get your point =)

@anon Have anything to say about the difference between $\perp$ and $\oplus$ too? =)

anon

The first is a way of saying two subspaces or vectors are orthogonal (in the presence of an inner product or bilinear form). The second is an operation that puts two (sub)spaces together, but even as an internal operation of subspaces of some given vector space (with the associated metadata that the two arguments are trivially intersecting) it does not necessarily imply orthogonality with respect to any given inner product.

Peter Tamaroff

@anon However...

anon

For example, in $\Bbb R^2$, the expression "$\langle(1,0)\rangle\perp\langle(1,1)\rangle$" says that the $x$-axis and the line $y=x$ are perpendicular (with respect to the standard inner product, say) - of course this statement is false. OTOH, $\langle(1,0)\rangle\oplus\langle(1,1)\rangle$ refers to the subspace of $\Bbb R^2$ spanned by $(1,0)$ and $(1,1)$ (which is just the whole space) and at the same time says that the $x$-axis and line $y=x$ are trivially intersecting.

Old John

3:45 PM

@JayeshBadwaik OK - gotcha

N3buchadnezzar

@anon I thought the $\oplus$ symbol stood for some thing some general than orthogonality. Like orthogonality between sets or spaces, seems i was wrong

Old John

I noticed that Harry Potter seems to have lost his erect nipples ... I wonder what happened to him...

Jayesh Badwaik

@OldJohn The chills of dementors has gone away! :P

anon

"Trivially intersecting" is more general than orthogonality. Orthogonality between subsets would just be $\perp$ again, and I don't see how "orthogonality between [...] spaces" could be "some thing more general than orthogonality." Of course I refer to $\oplus$ as a binary operation; it is more generally defined as an operation on any number of spaces (external direct sum) or subspaces (internal direct sum).

Old John

@JayeshBadwaik :)

anon

3:50 PM

In general, $\oplus_{i\in I}V_i$ (an external direct sum) can be viewed as the vector space of functions sending each $i\in I$ to a corresponding element of $V_i$, for which all but finitely many values are simply the zero vector. As an internal direct sum, it is just the usual sum but with the "metadata" stating that every element in the sum can be uniquely expressed as a finite combination of vectors from each component space.

Also see: en.wikipedia.org/wiki/Direct_sum

One easily checks the view of $\oplus$ as a binary operation is associative and commutative (up to natural isomorphism in the external case), and that its iterations agree with the universal definition in terms of unique combinations, and furthermore that the infinite-ary definition is consistent with a direct limit of finite-ary sums.

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$Mathematics$ Mathematics