2:00 PM
Yes, all standard matrices are affine

Then my problem lies elsewhere.

I was just considering that there are perspective matrices, but they are not the ones that simply use multiply and add.
They go into a higher dimension and project out the extra dimensions.
Thus, applying perspective.
But that is probably not what you're dealing with.

Well, I'm using 4x4 matrices and homogenous coordinates.
(x,y,z,T); discarding T between operations and always treating the vector T as 1.0 when transforming vectors

Whee!

But the matrices I'm dealing with right now are only translation/rotation matrices.

2:03 PM
That is what I see... It will probably disappear when I refresh my browser.

So I don't think that should matter.
Also, congratulations.
Still working on getting that 10k myself.

There, now it is back to 11.5k as it should be.
I wonder how that happened?

Oh, wait; I have 12.9k
that's a lot
but I guess that's my SE total.

@WillihamTotland I just recently got 10K, but I saw the 23k below my avatar. Must have been a server burp.

Indeed.
Neither of those look right at all.
The bottom one is correct, the top one, not so much.
The upper being scaled 0.9 on all axes and translated 1.1 up on the y.
and the lower resting on the origin.

2:09 PM
Just looking at the images, I couldn't tell you what is going wrong.

I think I'm messing up the normals, somehow.
To wit; the object space normals aren't correctly transcribed into world space.

That might explain the funky shading.

The thing is; my hit detection obviously works.
The spheres are right there.
In the correct location.
Normal map.
That don't look right.
That don't look right at all.
Owel, bbl

2:40 PM
hello

Hi.

someone has a buddhabrot avatar

Oh, that's the name of that thing?

yes
instead of plotting the points that don't escape, you make a histogram of the orbits of every point
for some reason you get a cool picture

Intriguing...

2:44 PM

cool people are talking about me :)
wait
let me blow tha tup

I don't really know anything about fractals

and more
dang, you crushed my avatar-chat-blow-up
some more
I wanted to get as big as the picture :)

I have a soft spot for black and white pictures. It's harder to make beautiful pictures in just black and white.
(Okay, there are some grays there, but you get my point.)

And good evening, by the way.

2:48 PM
Good evening. Did jury duty on that. Other potential jurors seem to be idle though.

3:05 PM
@rubenvb so you do graphics I guess
oh no I mixed you up with the other guy
why do you have buddhabrot as your avatar

@QED: nope, plucked mine off the internet.
I think it looks cool
and I once had to write a java program to draw the 2D version :)

:D
so if you apply z^2+c to every point of the buddhabrot it stays the same I guess
same with sqrt(z-c)

3:38 PM
@HenningMakholm I thought that everyone can delete his own question. Is some number of reputation posts needed for that? (Cyril is asking in comments to that question that moderator should do that for him.)

Unregistered users don't have the ability to delete their posts.

I see a certain mark next to his username...

In such a big letters... as if I were blind.
I was looking for it bellow in the profile.
Ok, so it explains everything.

4:34 PM
> in principle, there is no way to actually sum two points on a variety.

Why don't we vote to close this Grandfather's paradox?

haha
I did find it a bit weird that he says "Upper and lower bounds" then does something completely different

@Matt I was teaching today. I need not be entertained.
Ah, very well.

4:49 PM
Good afternoon, all

Hi Srivatsan

5:06 PM
Is the inverse of the product of two matrices identical to the product of the inverse of two matrices?

Yes. (AB)^{-1} is equal to B^{-1} A^{-1} provided A and B are invertible.

Good.
Thanks. :)

(Just in case you want to see the proof, too.)

@WillihamTotland Oh, I forgot to caution you: you must be careful with the order of multiplication. It is not A^{-1} B^{-1}.

Naturally.
AB != BA, after all.

5:12 PM
@Srivatsan I made a new batch of algebra retagging today. I'll leave my hands of it for next two days - so you can go on with the number tag.

just calculate B' A' A B = I to prove B' A' = (AB)'
uniqueness of inverses in a group

@MartinSleziak cool, thanks! I was going to ask you about it myself.

Slightly trickier is showing that order doesn't matter when inverting and transposing a square matrix...

infact this is for any group
not just matrices

Yeah, what's with that recently?

quite odd
people just don't understand freely generated grammars

Who's counting?

As in, since Martin is doing [algebra], is it ok with simultaneously editing posts for this tag as well? Or should we wait?

Well, if some question did have both tags, then kill two tags with one edit, no?

5:27 PM
There were only 2 questions like that. Just killed 'em both. =)

hi all

hi

5:59 PM
It looks like the red notification thingy in the user profile page is back.
2

@Srivatsan Finally.

IA == A?

what

@Srivatsan Yes! Also the yellow background for new reputation entries.

6:07 PM
@WillihamTotland If I is the n-by-n identity matrix and A is an n-by-p matrix, then IA is equal to A. If A is p-by-n, then AI is equal to A. In general, as long as IA or AI is defined, it will be equal to A.
@HenningMakholm We should celebrate. Open the champagne! =)

and if p = n; IA == A == AI, right?

@WillihamTotland Yes. If A is also an n-by-n matrix, then AI = IA = A.

@Williham, what's "=="?

Equals.

6:10 PM
is it different to =?

@QED Programmer's tic. = tends to be used for assignment; == for comparisons.

Indeed.

@JM To add to that, some programming languages (and nowadays some mathematicians) use := for assignment or definition.

confusing

(I know you know that tidbit, JM =))

6:12 PM
what's p and n?

@Srivatsan 79536 is correctly tagged now, I think

@QED You get used to it if you've been writing programs long enough...
@Srivatsan :D

@HenningMakholm In particular, the [natural-numbers] tag is ok?

@Srivatsan are we trying to get rid of natural-numbers for 58085? My best immediate replacement would be elementary-number-theory.

What Henning said.

6:14 PM
I guess we can keep the [natural-numbers] tag for set theory or logic questions on natural numbers.

I have proposed [natural-numbers] as a synonym for [elementary-number-theory] some time ago, but no takers yet.

@HenningMakholm Well, if it is a synonym for elementary NT, then it cannot cover questions from logic or set theory, no? In particular, questions pertaining to the definition of natural numbers?

For 34105 I think it would be just as well tagged without [natural-numbers].
@Srivatsan I would expect questions pertaining to the definition of natural numbers to count as elementary-number-theory, actually.

@HenningMakholm Oh, if that's the case, I am down with the suggestion.

But I don't actually know any elementary number theorists, so what do I know?
@Srivatsan I thought Algol actually got it from an existing mathematical-physical convention.

6:20 PM
"Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc." -- At least the tag wiki summary doesn't cover this definition of natural numbers part...
@HenningMakholm Oh, I thought Algol (but not sure), but didn't know about mathematical physics used it. Thanks.

No, but I'd say that the definition is even more elementary than those things.
@Srivatsan It's not that I have actually seen any pre-Algol math/physics text use it. But I have seen physics texts that follow a long expression with " =: M" and thereby give a name to it.

@HenningMakholm Aw, I don't think the qualifier "elementary" is meant to cover definitions. It signifies difficulty -- or that's my understanding.

iirc, one of my Intro to Modern Number Theory pdfs says that there is "formal number theory" (which relates NT, and diophantine equs in particular, to computation and logic). That might be relevant but I haven't been following the discussion there.

Btw, irrespective of what happens to [natural-numbers], I feel that the [number] tag should go. What's your opinion on that, @Henning?

Agree completely.
Isn't there already an eradication campaign underway? I think I've seen people here discuss the pace.

6:25 PM
That's it for me today. Later, y'all.

Yes, we are doing [number] (& [natural-numbers]) and [algebra] in parallel -- on different days of the week.
@JM Later, JM. Sleep well.

Oh, no....
no chat?
Does anyone have any idea how long the upgrade is going to take?

@robjohn - what are you talking about? =)

wait, what?

Did you not get a notice that chat would be down for a while due to upgrades?

6:37 PM
well, they could have given more notice, yeash.

@robjohn

It appeared over my chat window.

Well, I was going to say that SE was rejecting you like a bad organ.

There... we're back! That was a quick upgrade.

But then we all went offline. :P

6:38 PM
I was going to say we should invade some server's #math channel temporarily but it appears that is unnecessary now.

Did anyone else get the warning?

Yeah, got it.
But not until I reloaded after a message couldn't be sent. :P

I did, but only seconds before did I notice going offline (or it might have only been around for a few seconds in the first place...)

On a related note.

Well, I feel special... I got notified a minute or two ahead :-)
2

6:40 PM
Why will this not work!?

robjohn, veritable prophet of our humble math locale

hah, just Cmd-Tab:ed in; read "math cabal"
Perhaps more appropriate?
Oy.
I'm just moving further and further away from my objective. :(

Ello. Hm... is Asaf around?

He was last seen denying that he needs any entertainment.

Oh.
I thought he and I had agreed that I quit functional analysis and only do set theory from now on.

7:18 PM
[P]=[Q] iff P=Q? So what's the point of the brackets

the point is you want to go in between the group perspective and the freer algebra perspective but keep clear which you're in and when. I suppose it gets more important for derivation and conceptual understanding depending on the specific application.

7:35 PM
Hmm, look at how the Intro to Mathematical Crypto book writes quotients in CRT:

I dislike (a mod m1, a mod m2, ...)
and the use of ---> rather than \cong

It has one group literally over another instead of side-by-side, as I've always seen it. Also, \to and \cong mean two distinct things: the first is referring to some (canonical) morphism map and the lateral says they're isomorphic.

I don't like it. A quotient structure is sufficiently different from a fraction that they should not share notation interchangeably.

7:50 PM
isn't Z/mZ just ASCII for Z "over" mZ
why does it matter if they're vertical

@Matt here?

I can tell you right now it's not "just ASCII" (did you make that up on the spot?), but really my only qualm is that it's not as aesthetic when describing groups to put them in anything but inline except in the case of diagramming.

the quotient notation Z/mZ is just common because you can't type Z "over" mZ
as 3/4 is used to mean 3 over 4

What makes you say that though? Were mathematicians writing group quotienting one over the other before computers that you know of?

The / notation for quotients is chosen because of it's use for fraction

7:53 PM
And I wouldn't say common, I would say near-universal, because I've literally never seen anything else till now.
@QED: So are you aware of any instances of writing group quotients the other way? Do you have historical knowledge that mathematicians wrote it the other way pre-ASCII or at all commonly in history?

the algebra books
seems like maclane avoids this intentionally but it happens in algebra 0 paolo aluffi

What notation does MacLane use? You're still talking about quotient notation, right?

hmm. I guess there is are real instances then. I'm still not sure about the conjecture that ASCII or print-type is the reason we have the "/" notation so overwhelmingly.

8:09 PM
Print type could be the reason. I doubt that ASCII is, though.

Zoom in, enhance (Hat tip: Enthusiasms.org):

Nice. But where did they get that deconvolution kernel from?

8:24 PM
@AsafKaragila Hi Asaf! I asked you a question earlier today and now I've posted it as a comment below my post... Are you quite busy at the moment?

It's a first. Usually Didier comes up with the simple answer using The Law of Total Variance, but his answer looks more like mine usually do and I answered using the Law of Total Variance
@anon That is pretty amazing...
Is that native in Mma, or is that something that someone is selling?

@robjohn Is this a reply to a previous comment? =)

I don't know. I just copied it over from the blog Enthusiasms (which likewise has no explanation). You could email Simen (the author) or Tineye it or something.

No, I'm just amazed that my answer looks far simpler than Didier's

I think Didier is going for general than simpler.

8:31 PM
@Matt I have a minute.

@robjohn Actually, I am not very sure what the difference is between all the answers =)

@AsafKaragila Nice! Look at this, it's easier to read with latex...

There is no real sense to p[{p}].

Oh. Well good then because I couldn't make any sense of it. This means I messed up somewhere while doing the sums. I'll just post the whole thing as a question, that'll be easier.

The canonical name for G, while not being a member of the ground model, is simply the collection of names of conditions in P. So when you interpret it by G you only take those p's which are in G, and you have G.

8:39 PM
Clarification. Isn't every discrete metric an example of this? math.stackexchange.com/questions/87135

@Srivatsan The other way around: in the discrete metric every set is open. He wants no open sets apart from empty set and X...

Oops, you're right. I am confusing between open sets and balls.

The indiscrete topology is not metrisable if there is more than one point. (It's not Hausdorff.)

@ZhenLin Yes yes. I thought the OP wants a metric space where the only open *balls* are \emptyset and X.
Thanks Zhen and Matt.

Hmm, I've overtaken Willie Wong. That looks like a good place to call it a day.

8:43 PM
Same answer. The open balls always form a basis for the topology.

Oh shoo, Mike is active again. He's gonna catch up with me soon!!

@ZhenLin No, the answer is different, no? In a discrete metric, it is the case that the only open balls are \emptyset and X. -- or am I confusing something?

@Srivatsan If you have the discrete metric, i.e. distance either 0 or 1 then every singleton is open because you can take an open ball of radius 1/2 and it will contain just the point (because every other point has distance 1)

No, every singleton is an open ball, of radius 1/2.
The empty set is never an open ball.

Ah, um, ok. =)

8:46 PM
Only in the empty metric space.

Sorry about that. Not my day ;)

When is your day, Srivatsan, Tuesday? Monday? Friday?

Asaf, I'm just about to post some entertainment for you! : )

@Asaf: I thought metric spaces were required to be non-empty?

Yes.

8:47 PM
Bye all.

Bye.

But part of the "Oh, he's a logician, let's bash him." approach some folks here have, I was ridiculed for saying "There are no empty metric spaces because it makes FOL stiffer and less comfortable."
And if folks agreed to accept FOL as a basis for mathematics, perhaps they would be kind enough to accept its rules.

Bye @HenningMakholm!

@AsafKaragila Can you explain? Why does FOL conflict with empty metric spaces? Is it something to do with empty metric spaces or is it do with any empty structure?

If you allow empty structure then every universal sentence is true vacuously.

8:50 PM
And is that wrong?

So you want to require structures to be non-empty. In particular, metric spaces are FOL structures, and so they are required to be non-empty.

Btw sorry if I sound stupid. Mostly, I am. =)

How do you define the metric on an empty domain? I don't think that's possible.

Of course you can change the way you handle things, but then you always have to require that the structure is nonempty and so and so. It's just easier to do it this way.

@Matt: The empty function, of course.

8:52 PM
@AsafKaragila No, I don't get it. Every universal sentence is true for the empty structure -- seems ok to me. What's the issue?

@Srivatsan \forall x . x \ne x holds in the empty structure.

@ZhenLin Is that non-zero and satisfies the triangle inequality?

@Matt: Vacuously.

@Srivatsan You can run into contradictions if you want to say that all models of this and that theory are such and such.

8:55 PM
I think the only problem with allowing the empty model is that you no longer have soundness for the usual rules for \forall elimination and \exists introduction

I am not really sure I follow it actually. May be I should read a book or something.

I wholeheartedly recommend 1984 by G. Orwell..

Me, The Things They Carried or Lord of the Flies

@ZhenLin But for all elimination would say, you can substitute x in that sentence where x is from the structure. But I can still do it, no? Just that I cannot find any x to put in the sentence...

No. That's not how it works.
Deduction is purely syntactic, so you don't know what's in your universe or not. You only have the expressions which can be built in your language.

8:58 PM
Right. Vaguely following.
Btw, any serious suggestions for a book? [I might have read 1984 before. =)]

Girard's Proofs and Types is available online. That's pure logic and computer science though, with a strong bias towards constructivism.

@ZhenLin My professor recommended it last year.

@Srivatsan Heart of Darkness. Not everyone likes it, it's dark and heavy. Much like Apocalypse Now if you've seen it.

@AsafKaragila There's a reason for that similarity ;-)

@AsafKaragila You're still talking about 1984? =)

9:03 PM
No... I am talking about a whole other book.
@Srivatsan: You may also want to consider this question on this very site

@AsafKaragila Oh, sorry, I wasn't reading that comment properly.

What will be a first textbook in this area?

I liked Forster's Logic, induction and sets.

@Asaf, Thanks for the question. I don't understand all the answers, esp. JDH's, but I get some sense of them.

9:11 PM
@Srivatsan Try again tomorrow :-P

@AsafKaragila I should've expected that... =)

But alas... it is not your day and you did not expect that :-)

In any case, I only skimmed through the top 3 answers. I should put in more time if I want to understand...

@AsafKaragila has someone else bought the day? eme diem!

9:21 PM
I am going to drink me a beer.

Can we close this as a dupe? math.stackexchange.com/questions/87143/…

@Asaf, I was wondering if you knew anything about the automorphisms of the complex numbers without choice

I think so.

@AsafKaragila - who were you replying to? =)

@Srivatsan: You.
@JacobSchlather I have to go now, if you'll be here in an hour or so I might come back and we can talk about it.
Ciao.

9:27 PM
Okay

Anyone can help me out please?

What do you need help with?

I'm stuck on this:
$\int_{0}^{\infty}ke^{-3x}dx = k \lim_{x\to\infty}\frac{e^{-3x}}{-3} = k/3$
Why "k/3"

Also know as:
The gifification.

9:34 PM
@Gigili k/3 is the answer. What do you mean «why»?

I don't get how

Make the substitution 3x = y. What do you get?

which part?

Is that equation correct? His limit goes to zero, not one

3 lim(something) = k? you mean

9:36 PM
right, that should be \left[\exp(-3x)/-3\right]_0^\infty, not \lim_{x\to\infty}

@anon \exp(3x)/ 3x? That's not right, is it?

he meant exp(-3x)/-3
I presume

my lord am I glad I can edit

@JacobSchlather Yes

The point is once you evaluate you're going to get that limit

9:38 PM
So, what anon has is correct.

but you'll also get a constant term
which will be that function evaluated at x=0
and then the limit will be zero and you're just going to get the constant term

It's actually, lim_{x to infty} \exp(-3x)/(-3) - \exp(-3 (0))/(-3)

right
times k

But that's written in the book that way I wrote it
it has both limit and what @anon wrote

Then your book has a mistake

9:43 PM

wait

I still don't get how it will be k/3

does it have what srivatsan just wrote?

Gigili: do you know how to evaluate the limit(s) that Sri just posted?

This is one more way of writing it, but I am not sure which of the two your book means.

9:45 PM
None of these

Gigili, what your book has is certainly a typo. The correct way to evaluate the integral is what I posted.

Right, could you please explain the one you posted?

Ok, first take the k outside the integral: it's a harmless constant. You are left with integral of exp(-3x) between the limits ... and ... (let's come to the limits later).
What's the anti-derivative (or primitive or indefinite integral - I don't know which term you're most comfortable with) of exp(-3x)?

-1/3 exp(-3x)?

Yes. And that's, of course, the same as exp(-3x) / (-3).
You should evaluate this anti-derivative at the two limits: 0 and infinity.

9:51 PM
Right, and we should substitute 0 and infinity in it now?
Aha

You plug in 0, that's right. You can't plug in infinity - you take the limit of the function as x approaches infinity.

That's exactly where my problem comes from

(but heuristically you can just plug it in once the underlying theory is second-nature)

Right ..
x =0 : -1/3 ?

9:53 PM
x -> infinity : infinity ?
Um, I don't know

@Gigili You certainly realise that's not the correct answer, right? =)
Well, what's the limit of exp(-x) as x goes to infinity?

what are the sizes of 1/2, 1/4, 1/8, 1/16, ...? now what about 1/e^3, 1/e^6, 1/e^9, ...

0?
Aha, and then the answer is -k/3
Got it
Thank you so much

Almost. (0) - (-k/3) = ?

9:58 PM
@Gigili Not so fast. =) You got a sign flipped...

Aha, right , yes
I was thinking of e^infinity which is infinity all the time
Phew
Thank you again all, @Srivatsan, @anon, @Matt ..