Mathematics - 2018-03-26 (page 1 of 2)

Akiva Weinberger

12:00 AM

@Semiclassical This seems like something we talking about a long time ago, except we had lines instead of circular arcs

DarkRunner

@Semiclassical I just realized

GFauxPas

For all the talk about how Lebesgue integrals are better than Riemann integrals, we haven't learned how to actually find the integrals , only prove that they're finite

Secret

GFauxPas

i wonder if that will change

Secret

These are incredibly hard to draw by hand

Akiva Weinberger

12:06 AM

Dec 5 '16 at 21:46, by Semiclassical

Dec 6 '16 at 4:24, by Semiclassical

Dec 6 '16 at 4:31, by Semiclassical

Secret

handdrawn versions

ALannister

0

Q: Second derivative of $\phi$ in nondimensionalization problem.

I am reading a section in my differential equations textbook and am trying to fill in the left out steps in their explananation of how to nondimensionalize the second order system $$mr \ddot{\phi}=-b\dot{\phi}-mg\sin \phi + m r \omega^{2}+mr\omega^{2}\sin \omega \cos \phi$$ Where $\dot{\phi}=\fr...

Secret

One reason I exposed myself to fine art is to be able to cultivate an intuition so I can drew these labyrithine patterns easily

because it is almost impossible to drew them with purely analytical mindset

Akiva Weinberger

@GFauxPas If the Riemann integral exists, the Lebesgue integral exists and equals the Riemann integral

so the Lebesgue integral will never disagree with the Riemann integral

$\displaystyle\int_0^1\left(\begin{cases}a,&x\in\Bbb Q\\b,&x\notin\Bbb Q\end{cases}\right)\operatorname d\!x=b$

GFauxPas

we haven't actually proved that and he said we won't, which annoys me, but the Lebesgue integral is only better than the Riemann integral in cases where the latter isn't defined

Secret

12:12 AM

48

Q: Physical meaning of the Lebesgue measure

Question (informal) Is there an empirically verifiable scientific experiment that can empirically confirm that the Lebesgue measure has physical meaning beyond what can be obtained using just the Jordan measure? Specifically, is there a Jordan non-measurable but Lebesgue-measurable subset of ...

mp.mathematical-physics lebesgue-measure

Ted Shifrin

DogAteMy: Your statement is false for improper integrals.

GFauxPas

okay that's an interesting example

Akiva Weinberger

Really? There are integrals where the Riemann integral exists but the Lebesgue integral doesn't?

Ted Shifrin

You can heuristically think about the Lebesgue integral as coming from partitioning the $y$-axis, whereas the Riemann integral comes from partitioning the $x$-axis.

Yes, DogAteMy. $\displaystyle{\int_0^\infty \frac{\sin x}x\,dx}$ exists Riemann, not Lebesgue.

GFauxPas

right Ted I've been shown that and it's interesting, but where are all these integrals that can't be Riemann integrated

well that's "improperly riemann integrable" you could argue, Ted

Secret

12:14 AM

having said that, it is unclear to me if Lebesgue integrals can arise naturally in analysis without introducing the full Borel algebra

Ted Shifrin

I said improper integral, @GFaux.

Akiva Weinberger

@TedShifrin That's so weird. Is it because the Lebesgue integral is defined as the positive area minus the negative area, giving $\infty-\infty$?

GFauxPas

"exists Riemann, not Lebesgue."

Ted Shifrin

DogAteMy: For Lebesgue, $f$ is integrable iff $|f|$ is integrable.

ALannister

I just need to know, if $t=\tau T$, how do I show by using the chain rule that $\frac{d^{2}phi}{dt ^{2}}=\frac{1}{T^{2}}\frac{d^{2}\phi}{d\tau^{2}}$

Ted Shifrin

12:15 AM

Start with first derivative, @ALannister.

Akiva Weinberger

@GFauxPas $\displaystyle\int_0^1\left(\begin{cases}a,&x\in\Bbb Q\\b,&x\notin\Bbb Q\end{cases}\right)\operatorname d\!x$ can't be Riemann integrated if $a\ne b$

ALannister

@Ted I already have that.

that was easy.

Ted Shifrin

OK, do it again (applying to the first derivative).

ALannister

@TedShifrin I don't know how!

GFauxPas

thank you Akiva that's an interesting example, I hope the class does things like that

Ted Shifrin

12:16 AM

Call $g(t) = d\phi/dt$.

Write it out for $g(t)$.

GFauxPas

we did do something interesting last class

we proved Fubini's theorem and then used it to evaluate a Riemann integral

Akiva Weinberger

@GFauxPas To prove it, look at the upper and lower Riemann sums

GFauxPas

I should go google the proof that the integrals agree if both defined, he said he's not going to do that proof "even though he probably should" (???)

Akiva Weinberger

To be honest, I don't know that proof

All I know is that it's true

GFauxPas

or so you're told ;)

Secret

12:18 AM

What's an easiest example of a function that is not riemannian nor lebesgue integrable and is bounded?

ALannister

the first derivative is $g(t)=\frac{d\phi}{dt}=\frac{d\phi}{d\tau}\frac{d\tau}{dt}=\frac{1}{T}\frac{d\ph‌i}{d \tau}$.

Akiva Weinberger

Indicator function of a nonmeasurable set @Secret

Ted Shifrin

You want to allow non-measurable sets, @Secret?

Oops, DogAteMy beat me.

user193319

Dumb question: If a function $f$ is lebesgue integrable, then its absolute value is lebesgue integrable. Proof: $||f|| = |f|$

Ted Shifrin

Now put in $g$ instead of $\phi$, @ALannister.

Secret

12:19 AM

5 mins ago, by Ted Shifrin

Yes, DogAteMy. $\displaystyle{\int_0^\infty \frac{\sin x}x\,dx}$ exists Riemann, not Lebesgue.

But that's lebesgue non measurable?

Ted Shifrin

No, the function is measurable.

Akiva Weinberger

5 mins ago, by Ted Shifrin

DogAteMy: For Lebesgue, $f$ is integrable iff $|f|$ is integrable.

@user193319

Ted Shifrin

You need to go back to the definition of Lebesgue integrals to see that.

Akiva Weinberger

("DogAteMy" is me, to be clear)

ALannister

But what is $\frac{dg}{d\tau}$?

Secret

12:20 AM

ok

Zee

@TedShifrin in terms of abstract algebra, do I need anything beside commtive ring theory to study AG ?

Ted Shifrin

Modules, localization, tensor product, for starters.

Akiva Weinberger

More Truchet tiles: http://xahlee.info/math/i/truchet_tiles_animation_f7c7d.gif

@Semiclassical

user193319

@AkivaWeinberger Haha okay. And that theorem is true because $||f|| = |f|$; i.e., the absolute value of the absolute value is the absolute value?

Ted Shifrin

@ALannister: It's $d^2\phi/d\tau^2$.

Zee

12:22 AM

@TedShifrin is there a book that covers that stuff and only that stuff ?

ALannister

Where does the extra $\frac{1}{T}$ come from though?

GFauxPas

what is AG?

Ted Shifrin

You get a $1/T$ when you compare $dg/dt$ and $dg/d\tau$.

Akiva Weinberger

@user193319 I guess? Or use the fact that he said "iff" and not just "if"

Ted Shifrin

I'm not a good source for questions like that, Zee.

Ugh, I need to be packing stuff for my move tomorrow ...

Zee

12:24 AM

@TedShifrin alright , thx anyway , I shouldn’t have avoided algebra for this long

ALannister

I'm afraid I don't understand what you mean. I'd expect it to pop out from the chain rule or something.

Ted Shifrin

Well, it does.

Did you read what I just said?

ALannister

Yeah. I dont understand what you mean by compare.

DarkRunner

@Semiclassical Is there any way to prove it repeats

Ted Shifrin

Write down the chain rule again, with $g$.

ALannister

12:27 AM

$\frac{dg}{dt}=\frac{dg}{d\tau}\cdot \frac{d\tau}{dt}$

Ted Shifrin

Which gives?

Zee

@TedShifrin are the modules taken over commutive rings in AG ?

Ted Shifrin

Certainly for the AG I know, but I'm not an algebraist.

ALannister

$\frac{d \tau}{dt}=\frac{1}{T}$, but I don't know what $\frac{dg}{d\tau}$ is. It's supposed to give me $\frac{1}{T}\frac{d^{2}\phi}{d\tau^{2}}$ but I don't understand how it does that.

Ted Shifrin

No, you're wrong.

I already told you $dg/d\tau = d^2\phi/d\tau^2$.

And you know that.

This crap all confuses you because we use the same letter for two different functions when we write the chain rule this sloppy way.

ALannister

12:31 AM

Book says $\ddot{\phi}=\frac{1}{T^{2}}\frac{d^{2}\phi}{d\tau^{2}}$

Ted Shifrin

So you have $\frac{dg}{dt} = \frac 1T \frac{dg}{d\tau}$.

Now plug in your first equation.

ALannister

What first equation?

Ted Shifrin

Where you started with $d\phi/dt$.

Daminark

Hey Ted!

Ted Shifrin

Hi Demonark

ALannister

12:34 AM

Plug it in for what?

Ted Shifrin

You really should be able to do this on your own.

ALannister

I'm sorry.

Ted Shifrin

We set $g(t) = d\phi/dt$.

ALannister

I really should, and I'm ashamed because I can't.

$\frac{dg}{d\tau}$

Akiva Weinberger

Index 1 plus index -1, I think is what that is

ALannister

12:38 AM

Is $\frac{d^{2}\phi}{dt d\tau}$?

Akiva Weinberger

Presumably has applications in physics

Ted Shifrin

Yeah, that's the problem with this notation.

We're going to use different letters to get it right.

ALannister

I'm also trying to type with one finger on an iPad and autocorrect keeps wanting to write "dog" every time I write "dg"

Ok...then what?

Ted Shifrin

Let $h(\tau) = \phi(\tau T) = \phi(t)$.

Then $h'(\tau) = \frac{dh}{d\tau} = T \phi'(\tau T)$.

Now $h''(\tau) = T\cdot T\phi''(\tau T)$.

This is $h''(\tau) = T^2 \phi''(t)$ in their notation. Sometimes Leibniz notation is too confusing.

ALannister

And your $h^{\prime\prime}(\tau)$ is doing the same job as my $\frac{d^{2}\phi}{d\tau^{2}}$, then?

Ted Shifrin

12:45 AM

DogAteMy: With its moving around all the time, it's hard for me to tell, but yes, it looks that way.

Yeah, but using the same letter for a function and its composition with another function is too confusing.

ALannister

THANK YOU!

Ted Shifrin

Sure.

ALannister

Yes, it is too confusing. My book uses a lot of bad notation.

Ted Shifrin

This is standard notation in the real world.

You've taught calc 1 ... you didn't make your students be pedantic about different letters.

ALannister

I did. But I don't remember it being this complicated.

Perhaps that's why!

Ted Shifrin

12:48 AM

You didn't do second derivatives.

But the Leibniz notation is convenient but often leads to misunderstanding.

ALannister

Hm.

Secret

mn.uio.no/fysikk/english/research/groups/amks/superconductivity/…

Isa

If $x=c(\epsilon +n)$ and $t=\epsilon -n,$ then $u_{xx}=\frac{1}{2c}(\frac{1}{c}u_{\epsilon\epsilon}\frac{1}{2c}+\frac{1}{c}u_{\‌epsilon n}\frac{1}{2c})$ is correct?

Secret

Physics example of vortex antivortex annhilation

Ted Shifrin

You lost a sign for sure, @Isa.

0celo7

12:51 AM

@TedShifrin yo

Ted Shifrin

Or maybe not, @Isa. I don't know.

Yo.

0celo7

@TedShifrin Thesis will likely be done tonight. At least a good first draft will.

Ted Shifrin

Cool.

ALannister

Good night, everyone!

Isa

@TedShifrin the answer should be this $u_{xx}=\frac{1}{2c}(\frac{1}{c}u_{\epsilon\epsilon}\frac{1}{2c}+\frac{1}{c}u_{\‌‌epsilon n}\frac{1}{2c})$ or this $u_{xx}=\frac{1}{2c}(u_{\epsilon\epsilon}\frac{1}{2c}+u_{\‌epsilon n}\frac{1}{2c})$ but I don't which is the correct answer

Ted Shifrin

12:55 AM

There should also be a $u_{nn}$ term somewhere.

Oh wait.

This is a bizarre set-up. But, yeah, you should have all three 2nd order partials.

Isa

yes I forgot to calculate the other partial, I think the answer is $(1/2c^2)[(1/2)u_{\epsilon\epsilon}+(1/2)u_{nn}+u_{n\epsilon}]$

is it correct? @TedShifrin

Akiva Weinberger

@TedShifrin You'd like this

Apparently it's from MoMath

Abel's theorem book: maths.ed.ac.uk/~v1ranick/papers/abel.pdf

I'm in the middle of another book right now so I'm not gonna check it out but it looks interesting

(No quintic equation)

Xander Henderson

Any here speak and read French really well, and feel like translating Laurent Schwartz's book on distributions?

anyone?

Bueller?

0celo7

@XanderHenderson what do you need from it

Xander Henderson

nothing that I can't piece together from a dozen other sources

but I really would like to be able to understand the original

but the combination of French plus a really Bourbakist style makes is quite hard for me to get anything useful out of it

0celo7

1:12 AM

@XanderHenderson there doesn't seem to be a good book on that stuff nowadays

you can piece together a somewhat complete picture from Grubb, Hormander, Yosida, Reed and Simon

But I don't think any of those have the structure theorem

Xander Henderson

Again, I can piece it together from a bunch of other places

Reed and Simon is good

and Joshi and Friedlander is a nice summary

but it would be nice to have an English translation

Isa

there is a high rep user who speaks and read french very well but he's not online now since it's 2:oo am there now..

Xander Henderson

if only Schwartz wrote in a language I could understand, such as English

or possibly Russian

stupid French mathematicians, writing in stupid French :(

(I'm looking at you, too, Bouligand!) $\ddot\frown$

Zee

1:41 AM

Distributions we invented by the Russians before Schwartz , perhaps you can read the truly original piece on distributions there

Secret

Domain coloring

In mathematics, domain coloring or a color wheel graph is a technique for visualizing complex functions, which assigns a color to each point of the complex plane. == Motivation: four dimensions == A graph of a real function can be drawn in two dimensions, such as x and y. By contrast, a graph of a complex function (more precisely, a complex-valued function of one complex variable g : ℂ → ℂ) requires the visualization of four dimensions. One way to achieve that is with a Riemann surface, another by domain coloring. == Method == For easy visibility, complex values are represented with colors. This...

Adeek

@AkivaWeinberger thank you :)

0celo7

2:09 AM

If people are happy with errors and typos, the thesis is done...

Akiva Weinberger

@0celo7 Congrats!

Now all you have to do is fix the typos and errors, defend it, and accept that there are no guarantees in life

0celo7

@AkivaWeinberger I don't think there's a defense for honors theses

Akiva Weinberger

Oh

0celo7

I already gave 7 seminar talks on it...that should be more than enough

Akiva Weinberger

What's the thesis equivalent of playtesting for puzzle games?

0celo7

2:20 AM

time to do some traditional Riemannian geometry

computing curvature tensors

Akiva Weinberger

Test audiences? I guess that's the seminars

0celo7

fuck

help

Akiva Weinberger

What

0celo7

computing curvature tensors

Akiva Weinberger

I see

0celo7

2:21 AM

$g_1=f^{-(2p/n-1)}(f^2g_2\oplus g_3)$

ugh!

Akiva Weinberger

Just do it (dot gif)

0celo7

Hmm. It's a conformally changed warped product.

There's formulas for these things

Akiva Weinberger

What even is the manifold

(Inb4 "plane")

0celo7

$S^p\times B^q$

I think this guy was thinking about surgery and found this metric

Akiva Weinberger

Sphere times ball?

But not with the usual metric, with a freaky metric

0celo7

2:25 AM

g_2 and g_3 are the usual metrics

B^q is flat, so that's nice

Akiva Weinberger

Wait, the g I know takes two indices

0celo7

yes?

Akiva Weinberger

Oh wait I see

So $(g_1)_{ij}$ is the metric of $S^p\times B^q$ you're defining, whereas $(g_2)_{ij}$ and $(g_3)_{ij}$ are the usual metrics of $S^p$ and $B^q$ respectively

and $g_1$ is defined by that equation from earlier

0celo7

yes

did you not learn product metrics in do carmo

Akiva Weinberger

I did, but, like, recently

and it wasn't defined with an $f$ in there

0celo7

2:28 AM

well I'm taking the product of f^2g_2 and g_3

and then multiplying the whole thing by that other f

of course there's some abuse of notation going on

Akiva Weinberger

And finding $R$ does not look fun.

0celo7

there's formulas for the Riemann tensor

and even the Ricci tensor

Akiva Weinberger

Obviously

But not fun ones

0celo7

but not for the scalar curvature, which is what I need

>:(

and I don't want to have to think about how exactly one is supposed to contract things here

Akiva Weinberger

Isn't it the trace of a thingy?

0celo7

2:32 AM

yes, but there's multiple metrics floating around

ams.org/journals/tran/1987-303-01/S0002-9947-1987-0896013-4/…

NICE

Akiva Weinberger

So is this still for your thesis or just homework

0celo7

thesis

Akiva Weinberger

I mean I dunno what happens in grad school

0celo7

I just want a reference for this computation

I'm not in grad school

Akiva Weinberger

Oh

Sorry, I don't really know the structure of higher education

What even is an honors thesis

0celo7

2:34 AM

undergrad thesis

Secret

3:17 AM

users.mai.liu.se/hanlu09/complex

This is less colorful, but it at least free up a degree of freedom (blue) to plot 6 dimensional functions on a piece of paper

Ideas on how coloring should be assigned:

1. Hue = Argument (Red -> Green)

0celo7

lol, my notes

Secret

2. Brightness = Modulus (0-> infinity, Black -> white)

3. Saturation = contour lines

0celo7

$F'(0)$ is ~~injective~~ ~~surjective~~ bijective

how did I conclude such a thing...hmmm

Xander Henderson

magic

and hope

and duct tape

Secret

frustrates me that there are no single word for "neither injective nor surjective functions"

In general, all mathematical objects of the form "neither A nor B" received so little love to be named

Secret

3:23 AM

(cont.) Then pole order and multiplicity of roots will show themselves by regions where clors cycles around

user21820

@Secret There is a word describing functions that do not satisfy any nice properties.

Typical function.

=D

Secret

so by that logic, we will call "neither open nor closed sets" typical sets ? :P

user21820

For geometry, it is general configuration, like in "points in general position".

@Secret Nope. "do not satisfy any nice properties".

A typical set in a typical topological space is neither open nor closed.

Typically, a typical object in a typical structure does not satisfy any typical mathematical property that typical mathematicians care about.

"typical" is an atypical property. Just to be clear.

Secret

well that's odd cause I often found them interesting, especially the pathological ones

Secret

(cont.) and thus, we still have one leftover degrees of freedom (blueness) to plot something of higher dimension. making a grand total of 6 dimensions that can be cramped onto a piece of paper

Perhaps someday I should code a quaternion explorer

user21820

3:29 AM

@Secret But the pathological ones are not typical either.

Secret

well true...

user21820

More accurately, the pathological ones that people define are not typical.

Such as the nowhere differentiable continuous function.

0celo7

gah, what a garbage proof

Secret

by that logic, open sets are atypical unless $\text{pathological} \subset \text{atypical}$ :P

(or whatever type theoric analogue of $\subset$ is)

user21820

@Secret Open sets are indeed atypical, in some informal sense. The thing is that of all the possible topologies on a set, typical ones have far 'more' sets that are neither open nor closed, and so a typical set in that topology is neither open nor closed.

Closure under union and finite intersection is nowhere near enough to make open sets typical.

Secret

3:35 AM

hmm interesting

Secret

We can't know the properties of the "rogue elements", because they don't, in fact, exist. The point is that the first four axioms don't exclude the possibility of something unexpected being in $\mathbb{N}$. Maybe $i$ is in there, or $\omega_\omega$, or a woolly mammoth. There's no way to talk about the "properties" of completely unspecified things! But we don't want woolly mammoths in $\mathbb{N}$ so we add an axiom to the effect that we only allow the right things in there. — Kundor Dec 23 '14 at 4:51

lololololol

user21820

@Secret That remark is wrong.

Secret

Well, I don't see how we can say anything about these elements, other than they don't begin at zero cause by Axiom 2.2, we have if n is a natural number, then Succ(n) is also a natural number, so n can in principle be anything

user21820

3:51 AM

See the remark I made on KSmarts' answer, which is wrong for the reason I stated. You can see for yourself that 19 clueless upvoters do not realise it. Hurkyl's is also wrong, because he conflates properties on N (namely sentences with one parameter from N) with subsets of N. He does know better, but his answer as written is simply wrong. hardmath and Ross Millikan gave answers that are technically correct but fail to successfully address the asker's misconceptions.

Dan's answer is not only wrong but is a self-advertisement.

0celo7

ok come on latex

Secret

Zee: why? I am not gonna write $\sum$ everywhere

0celo7

@EricSilva case in point ^

Zee

It’s just such a dirty thing to do , what? You gonna save 5 cents worth of ink

0celo7

that's such an ignorant comment

Zee

3:53 AM

Just write things properly

Secret

imagine the insane number of $\sum$ you need to write in every line if Einstein never invented his summation convention

Zee

Beats having to decipher what you intended to say

0celo7

I don't think Zee is a real person

Zee

This is math not mystical physics , you have to be explicit

0celo7

lol looks like you're not smart enough for diff geo. sucks

Xander Henderson

3:55 AM

I don't think that we should be advocating capital punishment for the use of a particular notation

you might want to consider moderating your tone, @Zee

Zee

Using that notation is akin to mathematical terrorisim

So am sticking to it

Xander Henderson

Yeah, I'm forced to conclude that you don't know what you are talking about... equating notation to terrorism? Get a grip, guy.

0celo7

I concluded that long ago, @XanderHenderson. He only knows buzzwords and inflammatory statements.

Secret

tbh, in my rough works, I sometimes use einstein summation notion so as to save me all that truoble of writing out the $\sums$. Only when it is an infinite sum I cannot do that because you cannot always interchange summation signs

0celo7

@Secret if you're Wald, you can do infinite sums with Einstein notation.

Secret

3:59 AM

but how, you will need some symbol to denote you are summing to infinity without making unjustified swapping of sumation signs?

(No I have not read Wald yet)

Semiclassical

mathematical notation has one purpose: to communicate mathematical ideas clearly to your audience. If Einstein notation is sufficient for that purpose, then it's fine. If it's not, then it's not.

Secret

$$\sum_{k=0}^{\infty}a_k \sum_{m=0}^{\infty} b_mc_ke_m$$

which omitting the sums, it has to be summed this way: $a_k(b_mc_ke_m)$ and you cannot pull anything out the bracket otherwise you get a wrong answer

Semiclassical

of course, I'm the sorta guy who appreciates diagrammatic notation....

Secret

uh wait a sec... I think I wrote the wrong thing...

$$\sum_{k=0}^{\infty}\sum_{m=0}^{\infty}a_kc_kb_me_m \text{ vs } \sum_{m=0}^{\infty}\sum_{k=0}^{\infty}a_kc_kb_me_m$$

is what I am trying to talk about

Semiclassical

@0celo7 that seems the exception rather than the rule, though.

Secret

4:06 AM

what does Wald summation look like?

Semiclassical

for the most part, einstein summation is confined to finite sums.

Konformist Liberal

4:19 AM

does a "mathematician to be" need to know (for example) the proof of splitting lemma from heart?

Secret

To be dealt with later: Check why we cannot have a nonstandard PA model where e.g. there's a natural 0<c<1 and thus the successor of c forms another increasing well order that intercalate with $\omega$

Niing

5:35 AM

I have trouble understand this

why there is a $x$?

when I concatenate w1w2

0celo7

5:49 AM

Thesis draft link

comments welcome

@Adeek

Niing

6:08 AM

This answer saved me!

To append $w1, w2$, since $w2$ can be formed by $\lambda x_1 x_2 x_3 \dots$, and from the basis step $w\lambda=w$: $w1\cdot(\lambda x_1 x_2 x_3 \dots)\\
(\dots(w1\cdot\lambda)x_1)x_2)x_3 \dots)$.

Zee

That looks nice @0celo7

Niing

7:02 AM

Why the word 'disjoint' is emphasized?

ohhh

I got it, $T_1,T_2$ should not be the same!

Leaky Nun

7:39 AM

@BalarkaSen in Hatcher (P.138 on print, P.147 of pdf), it is claimed that an infinite dimensional CW-complex $X$ is homotopy equivalent to $T := \bigcup_{n \in \Bbb N} X^n \times [n,\infty)$

I don't believe him

in fact, the homotopy that he tried to create, forgot to specify what happens at $t=1$

Kappie001

Hi, is someone here knowledgeable on sparse symmetric eigenvalue problems?

I'm trying to find 200-300 lowest-lying eigenvectors of a 3M x 3M square symmetric matrix, and I am not sure if the method provided by python (eigsh, which interfaces with implicitly restarted lanczos method implemented in arpack) is the correct algorithm to use, given that k (number of eigenvectors) is so large.

Leaky Nun

8:07 AM

just solve the characteristic polynomial :P

Kappie001

wait, that is not serious right?

Kris Walker

8:23 AM

Hi. Quick question. Say you have two identical right triangles whose shortest sides coincide and whose hypotenuses are perpendicular. Is there any way to find the angle between the two bases?

abenthy

Can you think of an example for a vector bundle $E \to B$ where you can visualize the $H^*(B)$-module structure on $H^*(E)$ given by the cup product?

Or in some way get an idea of what its doing

Kris Walker

Was that for me? If it was, I don't really know what that means, so no lol. Sorry

abenthy

8:39 AM

@KrisWalker Was not supposed to be a reply to you :)

Kris Walker

I figured haha

Balarka Sen

8:57 AM

@LeakyNun It works, see proposition 0.16 in page 15

It's a standard thing

I can't be arsed to explain it right now in more detail though

Leaky Nun

@BalarkaSen so what is the map from $X$ to $T$?

Balarka Sen

I have 0 time to explain anything to you in any more detail right now

If you haven't figured it out by 6th of May, ask me then

Alessandro Codenotti

How many exams do you have to do? @Balarka

Mary Star

9:14 AM

Hello!!

Does someone of you have an idea about my question: https://math.stackexchange.com/questions/2708442/one-sided-significance-test-essay-key-question ?

Mudassir Malik

10:40 AM

Howdy... anybody here with 50 reputations?

Rick

How do I prove this?

Mudassir Malik

?

i cant use comments to ask for more information, because i don't have 50 reputations. will anyone do a favor for me by asking a valid question in the comment section?

Tobias Kildetoft

@MudassirMalik on what question?

Mudassir Malik

10:58 AM

@TobiasKildetoft I am weak in English.Its difficult for me to imagine the figure by writing. I need a figure of hoop in the accepted answer of math.stackexchange.com/questions/664/…

@TobiasKildetoft ?

Rick

12:03 PM

Someone please answer, I posted a similar question on the main site, but all I'm getting is downvotes

Secret

12:31 PM

Answer my goddamn question you GR lunatic!

nitsua60

@MaryStar @Rick @MudassirMalik you've entered the room at one of the quietest times of the week. Within a few hours it should pick up in activity. (See the chart at chat.stackexchange.com/rooms/info/36/mathematics)

@Secret (was that intended for this room, or the hbar? I.e. these may not be the GR lunatics you're looking for?)

Secret

obviously, my frustration is delocalised

and btw, only the CTC guy, 0celo has nothing to do with it

the three people you ping are not GR people, so that naturally rules them out

nitsua60

@Secret The triple-ping has nothing to do with you. (Sorry, it was a badly executed Star Wars reference.)

Secret

noted

Sha Vuklia

1:04 PM

https://math.stackexchange.com/questions/2708676/show-that-mathbb-z-20-mathbb-z-cong-mathbb-z-2-mathbb-z-times-mathbb-z-4-m
anyone here time to look at this question?

Sha Vuklia

1:16 PM

(nvm)

Astyx

1:53 PM

@Sha first notice that $7$ has order 4 and its powers are $1,7,9,3$ and $11$ has order 2 and its powers are 1 and 11

Now all the primes under 20 that are coprime with 20 (ie 3, 7 and 11) are in the image of your morphism, thus the image is $\Bbb Z /20\Bbb Z$

The orders of $7$ and $11$ tell you your map is compatible with projection of $\Bbb Z^2$ on $\Bbb Z/2\Bbb Z\times \Bbb Z / 4\Bbb Z$

The cardinality of $\Bbb Z/2\Bbb Z\times \Bbb Z / 4\Bbb Z$ is 8, which is also that of $\Bbb Z/20\Bbb Z$, meaning it's an isomorphism

(assuming you know it's a morphism already)

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$Mathematics$ Mathematics