Mathematics - 2022-05-31

Ted Shifrin

00:22

$\Bbb R^3$ with cross product is isomorphic to $\mathfrak o(3)$. Are they equal?

DanielSank

00:51

I have some notes that say that given two linear transformations $a$ and $b$ with commutator $[a, b] = \eta$, any transformation written in terms of those, i.e. $T = b^n a^m$ has the property $[a, T] = \eta (\partial T / \partial b)$.

Anyone know how to prove that?

ehhh maybe induction

Ted Shifrin

Just write it out?

DanielSank

yeah fair enough

What is it with commutators and derivatives showing up together so much?

Dumb question. Forget it.

Ted Shifrin

01:08

I don’t know that it’s dumb, but I question your “so much.” Lie bracket is a derivative, so maybe it is “so much.”

one potato two potato

02:39

I suddenly wonder: If $G$ is a group such that $G/Z(G)$ is cyclic then $G$ is abelian. But if $G$ is abelian, $Z(G) = G$ so $G/Z(G)$ is in fact trivial. Isn't this a contradiction if $G/Z(G)$ is nontrivial cyclic group?

Calvin Khor

question- I know that $C^\infty_*:=C^\infty_c(\mathbb R^n\setminus \{0\})$ is dense in $W^{k,p}(\mathbb R^n)$ iff $kp≤n$ and $p>1$ or $k<n$ and $p=1$. A text I'm reading says it should follow that if $kp≤n$ and $p≥2$, then $C^\infty_*$ is dense in $W^{k,p}(\mathbb R^n\setminus\{0\})$. Is this obvious?

I'm referred to a 1965 french out-of-print-and-not-scanned-anywhere book of Lions...

hmm, not even on ebay.

Peter Balabanov

02:57

@onepotatotwopotato I don't seem to get what contradiction you are talking about.

leslie townes

potato: there's no nontriviality. it's fine.

copper.hat

all these double negatives

Ted Shifrin

@onepotatotwopotato give me an example of a $G$ with nontrivial cyclic $G/Z$.

Peter Balabanov

03:13

seems awkward that formulas don't translate. it can't be fixed, right?

Ted Shifrin

@PeterBalabanov What are you talking about?

Peter Balabanov

03:54

@TedShifrin I don't see formulas convert in chat, just symbols surrounded by $'s

Calvin Khor

@PeterBalabanov there is a bookmarklet you can use here math.ucla.edu/~robjohn/math/mathjax.html (from the sidebar --->)

Peter Balabanov

oh thanks, i will try

Calvin Khor

yw. unfortunately you need to turn it on each time you enter chat

Peter Balabanov

ok, got it. don't you happen to know if i can use some tag to insert a list in stack question? enumerate/itemize don't work

Calvin Khor

On SE sites, you can use markdown instead

Peter Balabanov

03:58

I believe it's aren't, not don't? sorry, i'm not native

Calvin Khor

it should be "do you happen"

but dont worry i understood

let me double check the notation for lists and get back to you

Peter Balabanov

oh, no, i think i figured it out myself

thanks. i found "editing-help" page, it says about it

Calvin Khor

ok. Its one space on a new line followed by 1. for un-numbered its one space followed by -

@PeterBalabanov ok good

Peter Balabanov

i decided to remove this part of question at all. but now i know what markdown is

Calvin Khor

:)

Peter Balabanov

04:09

it's like doing things in a dumb way but for latex users

Calvin Khor

haha. if only latex could make websites

Ted Shifrin

I’ve done Latex for 30+ years and I don’t know markdown.

Calvin Khor

@Ted how painful was making a mistake on a typewriter?

Ted Shifrin

LOL

Calvin Khor

and how much did it cost to ship the latex code to the professional typesetters?

Ted Shifrin

04:11

I typed my thesis on a Hermes portable typewriter. However, if there had been no,Latex, probably instead of writing 4 textbooks I might have done a bit more research.

one potato two potato

@KarlKroningfeld Why do you think so?

Calvin Khor

i feel like i dont know how good we have it with latex and keep trying to break it

copper.hat

i had to use a Latex precursor called troff. awful.

Peter Balabanov

is it ok that i, like, spam questions? it's not like i'm trying to collect some easy answers on problems before the test. but these questions are different, really :)

but i got a test anyway. sometimes it's just a lot easier to get an understanding by asking questions. of course it could be a waste of time, but it's like choosing between two alternatives, spending time asking, or looking through books

Ted Shifrin

04:29

You need to show efforts to solve them. Just posting questions with no effort may incline us to think you want us to do your homework or exam for you.

Peter Balabanov

yeah, i understand that. asking without making effort is useless if you want to know something, it's always like that, so an easy detection

Ted Shifrin

In addition, for many questions, it’s hard for us to know what knowledge/tools you have to approach them.

Peter Balabanov

i think about it too when i ask. even when talking to an advisor or teacher i feel it

that it can be hard to get an answer because of it

Koro

05:19

@leslietownes: I think this is why normal operators are called normal. If $T\in L(v)$ is normal, then eigenvectors corresponding to distinct eigenvalues are orthogonal (normal).

Abstract Space Crack

06:24

0

Q: Attempt at twin prime conjecture based upon "CRT pigeon-holing" and concept of encoding. If this conjecture is true, then twin primes is.

Definition 1. $$ r(n) =\#\text{rad}(\{x^2 + 2x : x \in \Bbb{Z}, p_{\pi(\sqrt{n+2}) } \lt x \leq n\}) $$ or essentially $r(n)$ equals the number of different radicals that can occur in numbers of the form $x^2 +2x$ for $x$ in said range. And where $\text{rad}$ of the set builder is just applying ...

Asinomás

troff is cool

you are wrong sir copper

graffe

08:20

sanity check: if I toss 10 coins and count the number of heads and tails, is the prob of getting each possible outcome 1/11?

I see the answer is no

Calvin Khor

09:34

@graffe the coins are iid bernoulli(1/2) rvs. The sum of 10 of them is a Bin(10,1/2) rv.

graffe

10:16

@CalvinKhor Yes thank you . I had a momentary brain failure :)

Prithu Biswas

10:45

Are there theories about Continuous Functions $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ ?

A plate of momos

isn't that just MV calculus?

Prithu Biswas

@Aplateofmomos Idk anything about MV =P

A plate of momos

Ig you should learn it then, it does go over how to define continuity in that case too

or you can do topology where you can generalize continuity beyond the distance / open ball definitions

Prithu Biswas

@Aplateofmomos I have yet to master single variable calculus lol.

A plate of momos

lmao my studies were all over the place, ultimately now that I look back, I realize that it don't matter what order you do it. The only thing is whatever you are doing, make sure you understand that very well, because it will be useful later

Prithu Biswas

10:55

@Aplateofmomos it seems like $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is just a vector field. right?

A plate of momos

I mean

You'd think that it is but then you realize that is not how people consider vector fields in DG

It is a real shock when you realize it

3

Q: Relating the traditional definition of a vector field in terms of function with the differential geometry definition involving fibers and bundles

My first exposure to the concept of vector fields was in highschool physics courses, which had a simple intuitive idea of being a function which aassociates a point in a given region in space (domain where function is defined) with an arrow pointing in some direction. For example, I can give the...

differential-geometry vector-bundles vector-fields

this is me experiencing the shock

Prithu Biswas

@Aplateofmomos By the way it seems like imgur is blocked in my country XD.

Thats why the images are all broken.

A plate of momos

I guess for visualization purpose you could treat it as a vector field you wanted. You could take any two function, and form the naive vector field out of it (x,y)-> (f(x,y) , g(x,y) ) Or you could just ditch the idea of vector field and oyou can think of it as mapping point and outputting as pair of function

Bruh

why would they ban that

Ig you can use vpn

Prithu Biswas

This video is sponsored by NORDVPN.

A plate of momos

there is a browser named "Brave" in that you have VPN inbuilt into (in form of Tor tho, so idk if that is same)

Prithu Biswas

11:01

@Aplateofmomos Is this vector fields on surfaces?

A plate of momos

I suppose one would say generalization of surfaces, but apparently the old idea of the vector field as pairs/ triplets of functions also is contained inside it

I'm not sure of the exact connections and relations between the engineering idea of thinking of vector field as triplet and the idea mentioned in post I mentioned, but apparently the way in the post is more fundamental to what I understand

Prithu Biswas

@Aplateofmomos vector fields on manifolds?

A plate of momos

yes

It gets even more interesting when one finds out that differential equations can be thought of as "flows on surfaces"

1

Q: Are differential equations beyond coordinates?

In Physics when we write down Newton's second law, the differential equation we have is coordinate agnostic. Meaning, we can put in any coordinates into the equation and get a second order DE which models the motion of the object. Now, an interesting point here is with the coordinate agnosticism ...

ordinary-differential-equations differential-geometry mathematical-physics

Prithu Biswas

@Aplateofmomos By the way, there is a reason why I am thinking about $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$.

Remember the question on multiplicity you edited?

A plate of momos

yea

wat about it

dayum u got 13 upvotes now

I think that ur best question yet

likr in terms of score I mean

Prithu Biswas

11:10

@Aplateofmomos What if we generalize the problem from $f: \mathbb{R} \rightarrow \mathbb{R}$ to $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$?

A plate of momos

I guess that would be an MO level question

Prithu Biswas

@Aplateofmomos idk. I haven't yet worked on infinite subsets on ℕ.

So that generalization is for the future prithu.

Jakobian

L-semi-inner product

In mathematics, there are two different notions of semi-inner-product. The first, and more common, is that of an inner product which is not required to be strictly positive. This article will deal with the second, called a L-semi-inner product or semi-inner product in the sense of Lumer, which is an inner product not required to be conjugate symmetric. It was formulated by Günter Lumer, for the purpose of extending Hilbert space type arguments to Banach spaces in functional analysis. Fundamental properties were later explored by Giles. == Definition == We mention again that the definitio...

A plate of momos

Sorry but isn't the question in regard to R

Jakobian

How do you prove that such a product always exist on a Banach space?

A plate of momos

11:13

oh oops

Prithu Biswas

@Aplateofmomos One hypothesis I have is that every finite subset of ℕ with an odd maximum is Constructable.

Like I have a general method for it.

A plate of momos

I think your problem on some level can be thought of as a topological one. I don't think I fully understand it yet, but there is certainly some sort of topological aspect to it

I have a question

how did you come up with this constructive set formalism for understanding the spivak theorem ? @PrithuBiswas

I would also suggest you post your partial work as answer / inside the question so other peopl can build on it

Prithu Biswas

@Aplateofmomos As far as I remember, while my discussion with user21820 they started using the "{}" notation for indicating multiplicity for convenience.

It was a while ago, so I don't even remember.

geocalc33

13:05

Is anyone familiar with the partition function?

Partition function (mathematics)

The partition function or configuration integral, as used in probability theory, information theory and dynamical systems, is a generalization of the definition of a partition function in statistical mechanics. It is a special case of a normalizing constant in probability theory, for the Boltzmann distribution. The partition function occurs in many problems of probability theory because, in situations where there is a natural symmetry, its associated probability measure, the Gibbs measure, has the Markov property. This means that the partition function occurs not only in physical systems wi...

raspiduino

14:24

I have a quick question (might be stupid), but is sqrt(a^3) equal to cube root of a^2? Thanks!

Sorry I am not really familiar with latex :)

leslie townes

in general no, although there are values of a for which the two are the same.

raspiduino

Ok thank you so much!

ChocolateOverflow

15:32

Imagine 2 race tracks, 1 like a 0 and 1 like an 8. Cars driving on the 0 track make a 360deg turn every trip, while cars on the 8 track make a net 0deg turn. How come? I'm not good at maths but I think it has to do with basic geomery/topology

Ted Shifrin

16:18

Once you trivialize the tangent bundle locally, it amounts to the same thing, @Aplate. It's only for global concerns that you have to think more abstractly, really.

Leaky Nun

@ChocolateOverflow font fail XD

@ChocolateOverflow this shows that loops have a so-called "winding number", and you cannot deform smoothly between loops with different winding numbers

Ted Shifrin

@ChocolateOverflow Yes, it's because the track crossed over itself and the two pieces have opposite orientation.

Leaky Nun

imagine driving one loop vs driving two loops; you make 360deg in the former and 720deg in the latter

Ted Shifrin

Huh? @Leaky

Leaky Nun

@TedShifrin this is related to the argument principle thing from complex analysis right

winding number of loops

0 and 8 are two loops with different winding numbers

Ted Shifrin

16:21

Yes, the change in the argument of $z$ is $2\pi$ times the winding number, of course.

You have to be careful. It's turning number, not winding number.

You have to talk about winding number around a given point. What point are you using for the 8?

So what you're saying is just wrong.

Leaky Nun

I see

thanks for correcting me

Ted Shifrin

The total angle through which the tangent vector turns is $2\pi$ times the turning number. This is what shows up with holonomy, Gauss Bonnet, etc.

A plate of momos