Mathematics - 2020-04-29 (page 2 of 2)

abhas_RewCie

16:00

@Knight That's a JEE question, probably, don't remember where

Knight

I told him “it all depends on what the question makers thinks”

Archer

@robjohn Are you there?

robjohn

@Archer You use symmetry and integrate over $(-\infty,\infty)$ and divide by $2$

Archer

@robjohn But I am not allowed to use that method.

robjohn

@Archer which method?

abhas_RewCie

16:03

@Knight JEE Advanced question, confirmed..

doubtnut.com/question-answer/…

robjohn

@Archer symmetry?

Archer

abhas_RewCie

Who gives you such kinds of questions...

@Knight You know mean value theorem?

Archer

@robjohn See, the question has restricted me to use the rectangle ^

Knight

@abhas_RewCie Yep

abhas_RewCie

16:04

5

Q: Which function satisfies $𝑓: \mathbb R → [−2, 2]$ with $(𝑓(0))^ 2 + (𝑓 ′ (0))^ 2 = 85$,also $f$ is twice differentiable?

Let $f:\mathbb R\to[−2,2]\,$ be a twice differentiable function with $$\big(𝑓(0)\big)^2+\big(𝑓′(0)\big)^2=85.$$ Which of the following statements are necessarily TRUE? (A) There exist 𝑟, 𝑠 ∈ ℝ, where 𝑟 < 𝑠, such that 𝑓 is one-one on the open interval (𝑟, 𝑠) (B) There...

See the answer by Calvin Khor

Knight

Wow @abhas_RewCie Thanks

abhas_RewCie

@Knight That's just a trick question

Knight

But you see, you cannot solve it

You got to assume some function and check it

Hahahahaha

robjohn

@Archer Have you tried simply integrating $e^{-z^2}$ around that contour?

Archer

@robjohn Yes

robjohn

16:08

I will write it out. I am sure it works.

abhas_RewCie

@Knight No, you can solve it, the Math SE doesn't permits full answer, until the asker asks for it, so everyone was giving hints

Archer

@robjohn I am stuck on the integration on the imaginary axis i.e. from z = ic to z = 0

It doesnt go to 0.

abhas_RewCie

@Knight You can use MVT to solve it.

Archer

But the answer says it should.

abhas_RewCie

See the doubtnut link, I sent, it contains the solution...

That's a straightforward trick question... don't get struck, it's easy.

Knight

16:12

@abhas_RewCie No, you cannot solve it. Solving something means proving that option (A) is correct but you cannot prove it as full information is not given and all you can is to think of some function behaving like that

Thorgott

but you can prove it

robjohn

@Archer $$ \begin{align} \int_0^{ic}e^{-z^2}\,\mathrm{d}z &=i\int_0^ce^{x^2}\,\mathrm{d}x \end{align} $$

@Archer It is not supposed to vanish, but it is imaginary

@Archer The imaginary part gives the answer to the integral of $\int_0^\infty e^{-x^2}\sin(ax)\,\mathrm{d}x$

Archer

16:35

@robjohn But the answer says its 0

40 mins ago, by Archer

2

A: Gaussian-like integral : $\int_0^{\infty} e^{-x^2} \cos( a x) \ \mathrm{d}x$

You can integrate $$f(z)=e^{-z^2},z\in\mathbb{C}$$ On the boundary of the rectangle $$\{z\in\mathbb{C}:\Re(z)\in[-R,R]\wedge \Im(z)\in[0,a/2]\}$$ Given $a>0,R>a/2.$ Being $f$ entire, this integral is equal to zero. Using the fact that $$\int_0^\infty{e^{-x^2}dx}=\frac{\sqrt\pi}{2}$$ And observin...

This answer^

Oh it's from $-\infty$ to $+\infty$

Knight

@Thorgott With only that much information?

Can we prove the correctness of option (A)

Thorgott

yes

Akiva Weinberger

Is there an arrangement of five vertices and edges in 3D space such that there's a 120 degree angle at each vertex?

I want a (not necessarily flat) 120-degree pentagon, in other words

abhas_RewCie

@Knight Everything can be proved You don't need to show them with examples.

Akiva Weinberger

(Cont'd) If all the edges are the same length, I'm gonna guess no

Even if they're not, actually

Yes, this is true - consider projecting it onto a plane

Knight

16:55

@Thorgott Would you please guide me?

abhas_RewCie

@Knight Proving (A) and (C) wrong is very easy.

Knight

@abhas_RewCie (A) is correct buddy

@abhas_RewCie I couldn’t see what you removed

abhas_RewCie

Assume, $f: \mathbb R \rightarrow [-2,2]$,

Thorgott

there's an answer on main

abhas_RewCie

Let, $x_1, x_2 \in \mathbb R$

Knight

17:00

@abhas_RewCie It’s given, we need not to assume it :) LOL

@Thorgott main ?

What is main?

abhas_RewCie

such that $f(x_1) = f(x_2) = C$

Thorgott

the main site

one of you linked the question earlier and there was an answer already

Knight

Yes

@abhas_RewCie Okay, then?

abhas_RewCie

@Knight then let $x_1 = r$ and $x_2 = s$

and $x_1 < x_2$

so, if $f$ isn't one-one in $(x_1, x_2)$, then it must be a constant function.

Knight

When I thought of the problem for first time, I imagined if the Range is restricted to $[-2,2]$ then it must be something like periodic function

abhas_RewCie

17:03

if it's a constant function between $(x_1, x_2)$ then, $f''(x)|_{k} = 0, k \in (x_1, x_2)$

also, $f'(x)|_{k} = 0; k \in [x_1, x_2]$

Thorgott

that argument makes little sense

same goes for that notation

abhas_RewCie

so, $(f(0))^2 = 85$

which isn't possble

as $f(x) \in [-2, 2]$ for $x \in \mathbb R$

$\blacksquare$

robjohn

@Archer The integral over the whole contour is $0$.

abhas_RewCie

So, there is an contradiction and (A) get eliminated

Knight

@Thorgott I couldn’t understand this line:

In either case, by continuity of $f'$, $f'(x)$ has a fixed sign on a small neighbourhood of $0$, and therefore $f$ is injective restricted to this neighbourhood.

abhas_RewCie

17:07

@Thorgott when I write $f(x)|_{k}$ I mean $f(x)|_{k=x}$

feynhat

@Knight $f'(0)$ cant be zero. Without loss of generality, assume it is positive. Since $f$ is twice differentiable, $f' > 0$ in a neighborhood of $0$. Done.

Thorgott

@Knight If a continuous function is strictly positive or strictly negative at a point, it is also that in a neighborhood of that point. This is a good exercise in case you don't know it. Now, if $f^{\prime}$ is strictly positive or negative on an interval, it is monotonic there, hence injective.

abhas_RewCie

@feynhat yes, exactly... Good

Thorgott

@abhas_RewCie that notation still doesn't make sense, nor does the argument

abhas_RewCie

@Thorgott Hints != Arguments. I made some additional assumptions

@Thorgott That's simple notation, btw

geocalc33

17:10

how do you integrate $\int_0^1 \exp(7/\log(x))\log^3(x)(x^4-x^3+3x-1)dx$

abhas_RewCie

7 mins ago, by abhas_RewCie

so, if $f$ isn't one-one in $(x_1, x_2)$, then it must be a constant function.

Thorgott

that's not an additional assumption, you are stating an implication "if... then..", which is wrong

abhas_RewCie

@Thorgott method of contradiction.

Thorgott

the notation is pointless and contrived, just write $f(k)$

abhas_RewCie

@Thorgott give a second,...

@Thorgott agree

To disprove, (A) we start with assumption that such $r, s$ exists. then, $f(v) = C \forall v \in (r, s)$, so, $f'(v) = 0 \forall v \in (r, s)$. Putting this in the given equation, we get, $(f(0))^2 = 85$ which clearly isn't possible, as it violates the range of $f$ which is $[-2, 2]$

@Thorgott Now better?

Thorgott

17:16

No, why on earth would $f$ be constant?

The assumption is that $f$ is one-to-one in that interval, which is more than opposite of being constant.

abhas_RewCie

@Thorgott Ah, that's a very terrible mistake...

XD :P

@Thorgott So, I need to prove that there are two points for which they are equal? (to disprove it)

Thorgott

yes, but you won't be able to

geocalc33

how do you resolve 0 times infinity?

abhas_RewCie

@Knight Solution - ABD

sarthaks.com/61523/…

geocalc33

L'Hospital infinite limit Indeterminate Form zero*infinity

►

this is funny

"so then this is 1 over x overrrrrrrr!"

CowperKettle

17:31

5

geocalc33

HAHAHAHHAHAHAHHAHAHAHHAHAHA

that really tickles my funny bone

Does anyone know how to integrate something?

5

$$\lim_{n \to \infty} \log(n) \int_0^1 \bigg(\exp(\frac{1}{\log(x)}\log(x)x\bigg)^n ~dx $$

Archer

@robjohn You still here?

Mike Miller

What is it @ÉricoMeloSilva

robjohn

I am away from my computer for a bit. I cannot reply in depth because I don't like using my phone and MathJax.

Archer

@robjohn oh, I just wanted to ask how do I go about solving $\displaystyle \int_R^0 e^{-(t+ic)^2} dt$

user193319

17:49

0

Q: Induced Group Representation

Let $D_{\infty} = \langle a,t \mid a^2 = t^2 = 1 \rangle$ be the infinite dihedral group, and let $H = \langle at \rangle$. Given $\theta \in [0,2 \pi)$, let $f_{\theta} : H \to \Bbb{T}$ be defined via $f_{\theta} ((at)^n) = e^{i n \theta}$. I am trying to find the induced representation ${\rm In...

group-theory representation-theory group-homomorphism group-presentation dihedral-groups

robjohn

$e^{c^2}\int_R^0e^{-t^2}e^{-2ict}\,\mathrm{d}t$ Take the real part.

Archer

Oh then we'll again split it into imaginary and real parts

Thanks a lot!

Mike Miller

@BalarkaSen Unfortuantely the argument of spheres having 2-fields is pure algebra

Pichi Wuana

18:10

Is there a point in trying to isolate a set in a set equation?

I feel it is not the correct way

Érico Melo Silva

19:01

@BalarkaSen apparently cheeger has some notes on critical points of distance functions if youre interested

havent looked at em yet

Balarka Sen

Oh cool

found the notes from shmuel's site

Thanks! I will have a look

Érico Melo Silva

nice

let me know if this is interesting cause im intrigued but wanna know if there's something cool going on before i invest lol

Balarka Sen

Yeah definitely

I will have a thorough look tomorrow

Ted Shifrin

19:19

howdy, @Erico and a @Balarka

Balarka Sen

Hi @Ted!

Érico Melo Silva

hi

Ted Shifrin

Sure is quieettttt in here

Mike Miller

@TedShifrin You may have seen above but the homotopy theory argument doesn't even simplify much for the case of 2-fields. I am very unsatisfied.

Érico Melo Silva

what is a 2 field

Ted Shifrin

19:30

This stuff was in Mosher-Tangora, which I took third quarter my first year of grad school (and dropped in the middle). I've never learned it ... since it seemed to be all algebra.

Mike Miller

I am being too lazy to say two linearly indep vector fields.

Érico Melo Silva

lol word

Ted Shifrin

I'm fine with $k$-vectors just as with $k$-forms.

Érico Melo Silva

i actually only believe in k-vectors

Mike Miller

@TedShifrin The argument for 2-fields extends immediately to a statement for $2^m$-fields. That's crazy. It should be much simpler for the case m=1, because the actual answer is much simpler.

Balarka Sen

19:32

@TedShifrin I am stuck doing point set topology assignment

Mike Miller

I just don't believe that one can't get a geometric handle on 2-fields

Balarka Sen

Free thinking: If you take a small patch of a sphere in H^n and push it along geodesics radially emanating from a certain point it will grow flatter and flatter until it becomes a horosphere in finite time, when it becomes completely flat

This is Riccati comparison theorem in general yeah

If you have a very negatively curved space and you push a hypersurface parallely it becomes exponentially flatter

something like that

I haven't actually read the statement of Riccati but this feels true

Ted Shifrin

@Mike: So should the Stiefel manifold of $2$-frames be inherently simpler than that of $4$-frames? I dunno.

Mike Miller

Yeah? The guy of 2-frames is a 2-step sphere bundle, whereas the 4-frame guy is a 4-step sphere bundle.

Spheres are pretty simple.

Ted Shifrin

Well, I'm just thinking of $O(n)/O(n-k)$.

JBis

19:41

@TedShifrin I'm back. I'm trying to do the same problem as before but vertically instead of horizontally. Here's what I tried

Ted Shifrin

@Balarka, I've never thought about that. I suppose this should be a consequence of the fact that the geodesics spread apart predictably.

Balarka Sen

Yeah

Ted Shifrin

It sounds basically like the Rauch comparison theorem, which I've also never studied.

Alessandro Codenotti

Hi Ted

JBis

 Math.max(
        Math.min(
            largeWidth / numDevices,
             16 / (9 * largeHeight)
        ),
        Math.min(
            largeHeight / numDevices,
            16 / (9 * largeWidth)
        ),
    )

Balarka Sen

19:43

A friend was trying to explain Riccati but he was going off about taking covariant derivatives of covariant derivatives and I zoned out. I gather this must be what it is

I'll work it out at some point

First I have to figure out how countable $\omega_1$ is

Érico Melo Silva

@BalarkaSen classic riemannian geometry

Thorgott

not

Alessandro Codenotti

@BalarkaSen not very countable

Érico Melo Silva

i cant listen to what anyone tells me

just have to do it

Balarka Sen

lmao yeah @Erico

JBis

19:44

any ideas?

geocalc33

that's galaxy brain facts

Thorgott

formal definition of "very countable" required

Alessandro Codenotti

It's just countable, but countabler

Balarka Sen

its first countable tho

Mike Miller

@TedShifrin To make a point, certainly O(n) is more complicated than O(n)/O(n-1). That's all I'm saying.

I can't decide if I should ask about a geometric proof on MO or if it's just too obviously hopeless.

Ted Shifrin

19:45

Yes, sure. Anyhow, this is not stuff I have every studied or know.

Alessandro Codenotti

@BalarkaSen It is

Until you close it at the end

Balarka Sen

yup

Ted Shifrin

@JBis: I'm not going to think about this any more. Literally all you do is switch the two variables.

Alessandro Codenotti

What about second countable?

Balarka Sen

nah

geocalc33

19:48

via the starchart what is Kolmogorov (T0)?

Thorgott

it means "not nice enough"

Alessandro Codenotti

do you have to do separation axioms too?

geocalc33

ah

Balarka Sen

Nope, just 1st, 2nd, Lindelof and separable

lol

Alessandro Codenotti

Meh

It's not like separation axioms are very hard here to be fair

Balarka Sen

19:49

yeah

ordinals are locally compact hausdorff

Thorgott

oh btw Balarka

turns out we're gonna cover Nullstellensatz in my commutative algebra lecture this semester

so I will actually get around to it sooner rather than later

Balarka Sen

Niiice

Alessandro Codenotti

@BalarkaSen Yeah

Pretty sure $\omega_1$ is more than Hausdorff but I don't want to think about it

Balarka Sen

locally compact hausdorff are always regular but definitely $\omega_1$ is normal lol

Mike Miller

Nullstellensatz is the reason they're allowed to call it algebraic geometry

Alessandro Codenotti

19:52

All the way to $T_5$ according to the book

Balarka Sen

COMPLETELY NORMAL

boi

Mike Miller

It seems clear that algebraic geometry and geometric algebra should swap names

Balarka Sen

No way wtf

Oh I guess

I got confused

Alessandro Codenotti

I think algebraic geometry is an accurate name, there's only algebra and no geometry

Mike Miller

How is the latter part of your sentence related to the name

If it's algebra but done with some amount of geometric insight it's still fundamentally algebra, whence geometric algebra

But if you're really doing geometry with algebraic tools that's be algebraic geometry

geocalc33

19:54

I like geometry done in an algebraic way

Balarka Sen

they should just call algebraic geometry as "uncivilized language"

2

because thats what it is

Alessandro Codenotti

hmm fair

geocalc33

lol (+1)

Alessandro Codenotti

Just call it higher category theory, instead of hiding and pretending to be doing geometry

Balarka Sen

at least higher category theory is a meme

deformation theory of perverse D-modules is not even a meme

geocalc33

19:56

Deformation theory sounds like deformation is happening on some object. Maybe deforming a manifold?

Balarka Sen

deformation of character variety

thats what happens to algebraic geometers

Ted Shifrin

You can't fool me, a @Balarka. You have no character.

geocalc33

that's out of my wheelhouse

Stan Shunpike

@TedShifrin hey ted!

Ted Shifrin

hi @Stan

Balarka Sen

19:58

@TedShifrin My character isn't reducible to trivialities

geocalc33

deforming a manifold is actually called a diffeomorphism if I understand correctly

Archer

Does anyone here know how to evaluate complex line integrals using Matlab?

geocalc33

unless you're doing random geometry

Balarka Sen

random geometry is literally very random lmao

they build models of random simplicial complexes all day

and take expectations

its so weird

Alessandro Codenotti

A classmate of mine did a random surfaces seminar

He had some beautiful 3d printed models

Balarka Sen

20:00

I think it's damn cool

Thorgott

should the study of algebras be called algebraic algebra

geocalc33

imagine hitting minkowski space with random geometry

like fibrate that shit and make it crack

Ted Shifrin

@Archer: I gave up learning Matlab. I don't think Matlab knows the residue theorem. So you're meaning to parametrize the curve? Just give it the line integral to do, having parametrized it yourself.

Balarka Sen

There should be a proper model of a random 3-manifold and there should be a proper statement for "a random 3-manifold is hyperbolic with probability 1"

I don't know enough to know the proper statements though

geocalc33

ah

Archer

20:02

@TedShifrin I am not learning it..I just need to verify some of the answers that arent available anywhere

geocalc33

like fibrate the lines of const. time in the light cone, and then add a tension gauge on em until the universe spills its guts

Balarka Sen

Honestly any computation is already hard at the level of graphs. What is the expected first betti number of a random graph on $n$ simplices (easy)? What is the deviation (more complicated) ? What about a scaled limit as $n \to \infty$ (yikes)?

I tried these out once before, they require nontrivial probability

Archer

@TedShifrin How to give it the line integral?

geocalc33

@BalarkaSen you're forgetting I have a background in stats

Ted Shifrin

Parametrize the curve and compute the integral the way you learned in multivariable calculus.

geocalc33

20:05

I have calculated p-values so many times

and confidence intervals

Archer

@TedShifrin I want the exact value of $\int_C 1/(1+z^2)$ where C is the segment from 2 to 2+i.

Ted Shifrin

Well, that's easy enough to do in your head. What's an antiderivative of $1/(1+z^2)$? I assume you meant $dz$ even though you left it out.

Alessandro Codenotti

@Balarka some GGT people also think a lot about random groups

Balarka Sen

yeah its gromov's theorem that in an appropriate sense random groups are hyperbolic with probability 1

Érico Melo Silva

didnt gromov invent random groups

Archer

20:09

@TedShifrin arctan(z) ?

Ted Shifrin

Right.

Presumably Matlab can do complex arctan. Mathematica certainly can.

Alessandro Codenotti

@ÉricoMeloSilva The real question is what wasn't invented by Gromov

Balarka Sen

Lol

Alessandro Codenotti

To be fair he did invent pretty much everything in geometric group theory

Érico Melo Silva

correct

Balarka Sen

20:12

gromov works in mysterious ways

geocalc33

who is this gromov?

Érico Melo Silva

oh my god

Alessandro Codenotti

There's an heretic among us

Balarka Sen

@ÉricoMeloSilva thats him yes

5

Alessandro Codenotti

lmao

Archer

20:13

Ted Shifrin

What's worse, I've heard him lecture and met him ...

Archer

@TedShifrin So the examiner just expects me to plug in the values?

Érico Melo Silva

@BalarkaSen yeah i meean to type "oh, my god"

Ted Shifrin

Don't ask me, @Archer.

Archer

I think there's some other way to do this...I tried with triangle inequality, but that gave 1/3 as the bound.

Ted Shifrin

20:14

Triangle inequality will never give an answer.

What course is this and what is the question?

Alessandro Codenotti

Last Wednesday we had the first lecture of the geometric measure theory course and the professor just did an overview of the stuff we'll cover with some motivation, but at some point he was like "ok let's see one proof, every lecture should have a proof, even better if it is by Gromov"

Stan Shunpike

anyone here familiar with sinks in dynamical systems?

Archer

@TedShifrin I mean Triangle inequality then ML inequality

@TedShifrin I have uploaded the pic of the question. It's a course on Complex Analysis

Ted Shifrin

ML inequality is good for estimating integrals (typically as $R\to\infty$). It has no relevance here.

Alessandro Codenotti

@TedShifrin What was he lecturing about?

Archer

20:15

@TedShifrin Oh, then is there any other method?

Ted Shifrin

@Alessandro: There were several lectures over the years. I don't remember.

@Archer: IF this is an exam question, I've already told you too much.

geocalc33

Mikhail Leonidovich Gromov

Mikhail Leonidovich Gromov (also Mikhael Gromov, Michael Gromov or Mischa Gromov; Russian: Михаи́л Леони́дович Гро́мов; born 23 December 1943) is a Russian-French mathematician known for his work in geometry, analysis and group theory. He is a permanent member of IHÉS in France and a Professor of Mathematics at New York University. Gromov has won several prizes, including the Abel Prize in 2009 "for his revolutionary contributions to geometry". == Biography == Mikhail Gromov was born on 23 December 1943 in Boksitogorsk, Soviet Union. His father Leonid Gromov and his Jewish mother Lea Rabinovitz...

JohnnyApplesauce

Archer

@TedShifrin At least a hint? You've just told me ML inequality won't work and I should see the antiderivative for the exact value.

Ted Shifrin

I'm not telling you any more, @Archer. Go use your brain. I guess it's a little late to tell you to study calculus.

Archer

20:19

Ok never mind, let me try more.

Ted Shifrin

Good idea.

geocalc33

I'm not convinced that gromov has thought of everything though. What about Curtis Mcmullen

Érico Melo Silva

@AlessandroCodenotti what is in this course

geocalc33

what about integration Chad

Mike Miller

@StanShunpike What are you asking about them

Alessandro Codenotti

20:22

@ÉricoMeloSilva Sets of finite perimeter following Maggi's book of the same name

Érico Melo Silva

oh i like that book

Balarka Sen

Ok really about time I get done with topology

I mean set theory

Alessandro Codenotti

The professor said that he wants to reach at least De Giorgi's structure theorem, and we will do more if time allows it

Noice @Balarka

Any cool exercise in your pset?

Balarka Sen

all routine Munkres exercises i assure you

geocalc33

can we lift the ban on number theory while you're gone? @Balarka

Ted Shifrin

20:26

Oh, I forgot to ask @Stan what was sinking him.

Balarka Sen

number theory is banned until EnjoysMath solves twin prime conjecture

Alessandro Codenotti

@BalarkaSen unfortunate

geocalc33

do you think Enjoys can do it?

Balarka Sen

@AlessandroCodenotti Every course should be a year long graduate course. If I'm going to do point set topology I should do it from Engelking and in-depth, no half-assed Munkres crap

I can be a graduate point set topology student for a year

If they pay me to be one

Alessandro Codenotti

Fair enough

Ted Shifrin

20:28

I think that might be overdoing it.

Alessandro Codenotti

Year long because you should do Engelking's general topology in the first semester and Engelking's dimension theory in the second one

Balarka Sen

Exactly

Ted Shifrin

crawls in a hole

Balarka Sen

I mean otherwise it's too hard to get in the mode. I can't think about manifolds all day and then sit down and prove a fact about a separable Lindelof space which is T_3 but not T_4

/rant

Alessandro Codenotti

lol

geocalc33

20:33

Would Gromov say that?^^

Alessandro Codenotti

I do want to go through Engelking's general topology properly at some point

Mike Miller

@BalarkaSen I have a solution to this problem

LAD

Recently, I have asked some questions about continuous-time Markov chain and received no attention. 1, 2 ,

3 , 4 , 5. Could you please have a look at them and give me some suggestions?

Stan Shunpike

20:51

@MikeMiller @TedShifrin I am interested in systems of nonlinear differential equations where the components of the vector $\mathbf{x}(t)$ will have either logistic growths and exponential decays. I'm looking for a book that might discuss this.

Ted Shifrin

@Stan: But ordinarily you linearize around the fixed points of the dynamical system to get started. Try Hirsch-Smale's beautiful book on linear algebra and dynamical systems for starters. (The later edition with Devaney I don't know, but it may incorporate more applied things of interest to you. Check out both editions if you can.)

Golubitsky & Dellnitz might be a good starting place, too, although I don't know that book much.

manooooh

21:07

Hello all!

Is the following true?: $a+b>a$ with $b>0$

Érico Melo Silva

@TedShifrin maybe i should read hirsch smale since the world ended and i have free time

always wanted to

i always thought it was really bad that im a brazilian who doesnt know any dynamics

Mike Miller

Yeah Hirsch Smale is a good book

Érico Melo Silva

ill add it to my too long project list

@Mike did you give up on reading jost w me

Mike Miller

If p is a fixed point of your dynamical system on R^n and (Dx)_p: R^n -> R^n the linearization of the vector field at p then so long as none of (Dx)_p's eigenvalues are purely imaginary you should get exponential decay to p

@ÉricoMeloSilva No I just got busy

Érico Melo Silva

21:26

no problemo

robjohn

21:53

@manooooh for real $a$ and $b$, yes.

@TedShifrin Is it a nasty, dirty, wet hole, filled with the ends of worms and an oozy smell, or a dry, bare, sandy hole with nothing in it to sit down on or to eat?

Alessandro Codenotti

Love this reference

skullpatrol

22:34

@robjohn the way the death toll is climbing there is no hole deep enough to escape.

robjohn

@skullpatrol I wear a mask even if I am in a hole, if there are other people in the hole with me.

skullpatrol

Perhaps, if they found a black hole we could all jump into, we may not suffer as much.

masks are now mandatory in all of Germany @robjohn

skullpatrol

22:55

with a $5,000 fine for not wearing a mask

Thorgott

the (at most) $5000 are for store owners that don't make sure their staff are wearing proper masks

fines for individuals are like $100 at most, i think

there's also large variation between this in the individual federal states

skullpatrol

Thanks for the info pal

dsillman2000

23:17

If an $n\times n$ matrix, $\mathbf{A_n}$, is populated with integers randomly distributed on {-1, 0, 1}, what is the resulting probabilistic distribution of the determinant $\det(\mathbf{A_n})$? Essentially, if $X ~ \det(\mathbf{A_n})$ as described above, what is the pdf of $X$?

* $X \sim \det(\mathbf{A_n})$

Ted Shifrin

@robjohn is this an allusion I’m missing? Good to see you, robjohn.

Alessandro Codenotti

@TedShifrin It's the beginning of Tolkien's The Hobbit

"In a hole in the ground there lived a hobbit. Not a nasty, dirty, wet hole, filled with the ends of worms and an oozy smell, nor yet a dry, bare, sandy hole with nothing in it to sit down on or to eat: it was a hobbit-hole, and that means comfort."

skullpatrol

23:48

What's so special about Hobbits that they only live in comfort?

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$Mathematics$ Mathematics