Mathematics

@anon from vector spaces, all the basics: subspaces, generators, bases, linear eqns systems, homogeneous snd NH, triangulation, sum of subspaces, direct sum

from matrices, the basic defintions, the operations, the ring structure, $GL(n,K)$, elemtnal matrices, coordinates

anon

okay, so left-multiply BA=I by the E matrices' inverses (themselves elementary) until you get A = [product of elementary matrices], right?

Peter Tamaroff

1:49 AM

@anon Yeah, I did that =) But it seemed a little "this is obvious, BAM".

Like it was a little informal.

anon

It is obvious. The hard part is the FT you describe, methinks.

Peter Tamaroff

@anon Right.

anon

And that BA=I=> B=E1...

Peter Tamaroff

The text states that given any matrix in $n\times n$ there exist elemental matrices such that $\prod_{i=1}^r E_i A$ is upper triangular

and if there are no zeros in the diagonal, there exists another set such that $\prod_{i=r+1}^s E_i^\prime \prod_{i=1}^r E_i A=I_n$

But it is a "we observe that" thing, not a proof.

anon

@PeterTamaroff is that over any base field?

I can't remember

Peter Tamaroff

1:52 AM

@anon Yeah

Clearly if there are no zeros, $\det(\prod E A)=\prod a_{ii}\neq 0$

so $\prod E A$ is invertible

But no dts are mentioned yet.

Dunno, maybe I have to wait and I'll prove all this rigorously later on.

TRiG

2:17 AM

Coway's Game of Life, played with floating point numbers instead of integers.

►

kahen

2:29 AM

en.wikipedia.org/wiki/Riesz_space - shouldn't "principle projection property" be spelled "principal projection property" in this article?

anon

pretty much every instance of principle in that article should be principal

user19161

3:03 AM

@PeterTamaroff I have been thinking about it for 9000 times, and I think that anon is probably a girl. QED.

anon

user19161

@anon You must be as pretty as your avatar shows.

J. M.

3:22 AM

@JasperLoy Girl or not, that does not at all affect anon's mathematical ability, eh? :)

Jayesh Badwaik

3:58 AM

@PeterTamaroff That is the part of the process of the LU Factorization. You can prove it "algorithmically" by using the fact that left-multiplication by $E_{i}$ is a row operation. And then you can use algorithms like Gauss Elimination to reduce it to $I$. (Or am I being really dense here?)

Rajesh D

5:57 AM

Hi @anon

I have a quick question

anon

okay

Rajesh D

The ridges I was talking about yesterday

Ridge (differential geometry)

For a smooth surface in three dimensions a ridge point occurs when a line of curvature has a local maximum or minimum of principal curvature. The set of ridge points form curves on the surface called ridges. The ridges of a given surface fall into two families, typically designated red and blue, depending on which of the two principal curvatures has an extremum. At umbilical points the colour of a ridge will change from red to blue. There are two main cases: one has three ridge lines passing through the umbilic, and the other has one line passing through it. Ridge lines correspond to cu...

robjohn

@RajeshD I am still not sure I understand what is being asked.

Rajesh D

@anon

anon

whatever the question is that you're preparing to ask, I doubt it's going to be quick :-)

Rajesh D

6:14 AM

@robjohn puzzled

@anon sorry about that. My browser was struck and it showed no messages

anon

I plan to visualize this, watch a stupid youtube video, and then go to bed tonight.

Rajesh D

I have proved that there is arbitrarily close to line x = a for sufficiently large y (y-axis). But this means that tthe two cases shown in the figures are well possible. I want to rule out the case given in left picture. The picture on right is what i want to prove

@robjohn @anon

user19161

@anon Have you solved your spacing problem for the toc?

anon

nope

I'm on the last section of my first draft though. Who know it would take so much work to write six measly pages.

user19161

@anon It might take more work rewriting it, it's true!

Rajesh D

6:25 AM

@anon : Atleast do you understand the problem I am talking about

I am not able to convey this problem to @robjohn

anon

I remember your problem from yesterday, which is what I now assume is what you're talking about at this moment, yes.

Rajesh D

I guess you ll be able to explain him

Hey says that the ridge is already present from what i described and thinks that i am trying to prove P => P.

anon

there's a vertical strip in the plane and a smooth scalar function of R^2, such that every vertical line in the strip has at least one local maximum. the idea is to show there is some kind of "ridge" of maxima approaching the line x=a in the strip (I don't remember why the value a is privileged in the strip however)

Rajesh D

@anon I haven't mentioned it before

but thats a point of interest

anon

"it"?

Rajesh D

6:29 AM

why a is previleged.

anon

wow you misspelled it even worse than I did

Rajesh D

haha

check'

Privileged

@anon, g(y) = f(x,y) goes to infinity with y in the neighbourhood of x= a with same rate of growth everywhere except at x = a, and f(a,y) goes to infinity with y at with much faster growth (by an order higher) than at any point in the neighbourhood of x = a .

BenjaLim

@anon Hey

@anon Do you think my computation here is correct? math.stackexchange.com/questions/108553/…

anon

I don't know what Ext is.

and I don't know homological algebra

BenjaLim

@anon Ok.

Sorry.

Rajesh D

6:40 AM

@anon : "such that every vertical line in the strip has at least one local maximum." well I do not understand what you are saying fully. What is this vertical strip in the plane ?

BenjaLim

@anon A lot of diagrams

anon

@RajeshD $\{(x,y):|x-a|<\epsilon,y\in\Bbb R\}\subseteq \Bbb R^2$ for some $\epsilon$, right?

@BenjaLim I gathered that much :)

BenjaLim

@anon The definition of the Ext functor is ok, and then after that it's just BoOOoOoOm

Rajesh D

@anon ofcourse. I do not understand why you are saying this. $f$ is defined all over $\mathbb{R}^2$. what is the need to mention this. Its implied from context right?

Ah now I get it

A round about way of saying the presence of maxima arbitrarily near to the line x = a for sufficiently large y

@anon thats a messy way to say

@anon : the idea is to show that the ridge curve is asymptotic to the line x = a, as y goes to infinity. Thats how 'a' is privileged

N3buchadnezzar

hi? =)

Rajesh D

6:57 AM

@anon : moreover you are wrong in saying that every vertical lline in the strip need to have a maxima. Just check the left plot of figure I posted. This is all the problem isn't it, this is what robjohn is also presuming

BenjaLim

@anon

@anon Do you remember by two kitties

I have a dilemma

do I name them Schur and Brauer or Ext and Tor?

Rajesh D

@BenjaLim Why complicate life?

BenjaLim

@RajeshD I have two brown kitties

Rajesh D

here we are discussing some math with anon and you disturb by asking what to name your pets

BenjaLim

@RajeshD sorry :(

Rajesh D

7:02 AM

@BenjaLim to add further, I hate pet animals. I am scared of even cats and dogs

user19161

I think anon is being overworked. Guys, please don't ping anon so often for every damn thing.

robjohn

@BenjaLim Schur and Brauer sound more like cat noises.

@BenjaLim However, kittens have no fixed points.

user19161

@robjohn I wonder why kittens seem more popular than puppies.

robjohn

@JasperLoy I don't think they are.

user19161

@robjohn At least in chat, I think kittens appear more often than puppies!

user19161

7:09 AM

Wait, there is no general reference closing option anymore?

robjohn

@JasperLoy kittens 83 - puppies 73

@JasperLoy pardon?

user19161

@robjohn I see the GR option for closing on Eng but not on Math, why the difference?

robjohn

@JasperLoy GR option?

user19161

@robjohn Close voting a question as "General Reference".

robjohn

@JasperLoy I don't think I've ever seen that here.

user19161

7:13 AM

8

Q: why don't we have a "general-reference" close reason?

I realized some SE sites like English Language and Usage have a close reason called "general reference". Why don't we have such a reason to close questions while it's obviously not the same as not constructive and other close reasons. This question, for example.

user19161

Ah I found the meta post!

Rajesh D

@robjohn : From the figure could you spot the difference between the left one and the right one?

robjohn

@RajeshD the image you posted?

user19161

@RajeshD The two certainly look very different.

Rajesh D

yes

if you scroll up its there here too,

user19161

7:15 AM

@RajeshD Not sure what you mean by it.

user19161

They look as different as a cat and a dog.

Rajesh D

@JasperLoy : we are continuing the discussion from yesterday. things may not be clear to you now

@robjohn ??

robjohn

@RajeshD For each $y$, is there at least one $x$ or exactly one $x$?

Rajesh D

@robjohn atleast one x

'there is a maximum' meant that i did not rule out many

robjohn

@RajeshD Ah, then I can construct a $C^\infty$ function which looks like the left side image

Rajesh D

7:25 AM

yes agree

So how can show the case is like on the right side.....what more information about f(x,y) do i have to use..how to proceed with this

@robjohn

robjohn

@RajeshD I would look at the partial wrt x of the function.The ridge is going to be where the surface passes through the plane z=0

Rajesh D

yes

I will plode and probe on this for sometime.

thanks @robjohn

robjohn

If you have a strip where $\frac{\partial^2}{\partial x^2}\lt0$, then your ridge should not stop.

@RajeshD Now ridges could coalesce and split..

@RajeshD ah, but if $\frac{\partial^2}{\partial x^2}\lt0$ then they can't. OK

between two ridges, $\frac{\partial^2}{\partial x^2}\gt0$

Jonas Teuwen

7:41 AM

Good morning fellow mathematicians.

user19161

@JonasTeuwen Morning bro!

Jonas Teuwen

Hi.

BenjaLim

@JonasTeuwen I have computed my first ext group :D

user19161

@BenjaLim Have you done the int as well?

BenjaLim

@JasperLoy int?

user19161

7:54 AM

@BenjaLim Joke, Ben, joke.

BenjaLim

@JasperLoy I am talking about en.wikipedia.org/wiki/Ext_functor

user19161

@BenjaLim I know I know.

BenjaLim

@JasperLoy Ah crap

I thought I had an exact sequence of $\Bbb{Z}/4$ modules:

$0 \rightarrow \Bbb{Z}/2 \rightarrow \Bbb{Z}/4 \rightarrow \Bbb{Z}/2 \rightarrow 0$

where the first non-trivial map was "inclusion"

problem is it isn't a $\Bbb{Z}/4$ - module homomorphism...

@JasperLoy ahhhhh

maybe

@ZhenLin hey

@ZhenLin Can I ask you some homological algebra

@ZhenLin Do we have the map $f : \Bbb{Z}/2 \to \Bbb{Z}/4$ that sends 1 to 2 being a homomorphism of $\Bbb{Z}/4$ modules?

Seems good to me

Zhen Lin

yes

BenjaLim

@ZhenLin thanks

Ok then I can form that ses above

and then I have proven that $\textrm{Ext}_{\Bbb{Z}/4}^n (\Bbb{Z}/2,\Bbb{Z}/2)$ is not zero.

man zhen there are so many maps and diagrams in homological algebra

@ZhenLin I am relying on the fact that $\Bbb{Z}/4$ is a free (and hence projective) $\Bbb{Z}/4$ - module.

Jayesh Badwaik

8:46 AM

@JasperLoy Bill Dubuque's answer is quiet good about the general reference IMHO. I do not like the idea of general reference myself .

user19161

8:59 AM

@JayeshBadwaik I see. I was just surprised that Eng has it but Math does not.

Martin Sleziak

If we have some users with sufficient rep here in chat, this question should be reopened. This question was also discussed at meta in this thread: Another questionable closure.

user19161

@MartinSleziak Done.

Martin Sleziak

Thanks!

@MartinSleziak Thanks for people who voted, it's already reopened.

Rajesh D

@robjohn : What if $f(x,y)$ is monotonously increasing in $y$.

@robjohn : Do you think we need any manifold theory. I have no idea what it is, but just thought if its related to this

Martin Sleziak

9:20 AM

@MartinSleziak The meta thread I should have linked to is this one: Voting to close as duplicate without taking the time to check if they are. Sorry for the mistake.