Mathematics

12:15 AM

🥳👍

Grammar question which I'm sure is already posted somewhere on grammar stack exchange...but is it grammatically correct to answer a question with a single contraction?

For example: "Is it sunny out today?" Answer: "It's."

noballpointpen

I am not a native speaker, but it sounds not natural for me.

shintuku

@geocalc33 'it is' is okay, 'it's' is not

geocalc33

I think it's not grammatically correct as well but I'm curious to the reason

noballpointpen

Then I don't think you visited the right place to ask... we are not language experts here.

geocalc33

12:29 AM

I'm going to check grammar stack exchange

noballpointpen

Ted, if you don't want to answer/discuss, just let me know and I will be ok with it.

Since I am somewhat awaiting for an answer. Not sure if you're just busy.

shintuku

12:51 AM

what's $\mathbb F_p$, is it always the field of elements of a field $F$ modulo $p$?

Silly Goose

1:08 AM

in Dummit and Foote group actions are introduced and then it is found that any group action induces a representation of said group in the symmetric group over the set acted upon. But other sources say that it is Cayley's theorem which tells us that any group can be represented as a subset of the symmetric group; which is correct?

Cayley's theorem seems to prove more than is wanted since it proves that every group is isomorphic to a subgroup of a symmetric group

noballpointpen

I didn't study algebra yet, but can't these be just different ways of proving the same? Just a general observation from me.

shintuku

1:35 AM

@shintuku no it is an alternative notation for $\mathbb Z_p$ that emphasizes the fact it is a field

robjohn

2:14 AM

It’s good that someone could answer your question

one potato two potato

How can I reply to my own message?

shintuku

2:28 AM

@onepotatotwopotato you can get the permalink number of your comment by clicking on the arrow that appears on the left when you hover over your message. then begin your message with :(permalink number), e.g., :30976263

one potato two potato

@onepotatotwopotato go

shintuku

gj you did it

one potato two potato

oh neat

Silly Goose

what theorem(s) tell us that "Every finite-dimensional unitary representation on a Hilbert space V is the direct sum of irreducible representations."?

one potato two potato

2:59 AM

Given a group presentation, I think it's quite a hard problem to ask what are the normal generators of that group?

Sourav Ghosh

3:29 AM

Thank you :)

porridgemathematics

4:50 AM

im missing a simple argument here, can anyone help fill it in? $u$ is subharmonic on the unit disk, and $v$ is defined as $u(z^2)$ on the upper half of the unit disk, and $0$ on the lower half. Then $M_v(r) = M_u(r^2)$, where $M_f(r) = \frac{1}{\pi r^2} \int_{\Delta(0,r)} f dA$, $\Delta(0,r)$ is the open disk center $0$ of radius $r$

it should follow from a simple change of variables but im not able to make it so

porridgemathematics

5:05 AM

instead I get $M_v(r) = \frac{r^2}{4} M_u(r^2)$..

oh crap, sorry im being a moron

ignore my question

it turned out $M_u(r)$ really meant $\sup_{|z| = r} u$

in which case it makes total sense

nickbros123

6:43 AM

is the motivation for defining matrix multiplication the idea of creating a system of linear equations / row vectors $C_{m \times p}$ which is a matrix with m equations p unknowns, whose equations/rows/row vectors are each a linear combination of the rows of another system $B_{n\times p}$ which is a matri with n equations, p unknowns

Utkarsh

7:14 AM

Hii I want to know what is the direction ratio of the line $\frac{x-2}{2} = \frac{2y - 5}{-3} = z+1$. The component values along x and y axis are 2 and -3. What is for z? 0 or 1?

Mr. Feynman

I have a question about the connection 1-form. As I define it, it is as a *matrix& of one forms. As we said yesterday, when we have a $G$-structure we also require that the connection 1-form is $\mathfrak{g}$-valued. How does that make sense in general? Not all Lie groups are matrix Lie groups, so I have no reason to expect that the Lie algebra is made up of matrices...

porridgemathematics

7:39 AM

@Mr.Feynman connection one forms are only locally defined anyway, in which case they are $\text{End}(TG)$-valued one forms

so you can think of it as $\mathfrak{g}$-valued

for finite dimensional lie groups, the matrix viewpoint works

Mr. Feynman

8:14 AM

@porridgemathematics I'm not very grounded on the matter and I have never encountered a definition involving endomorphisms of the tangent bundle, nor I know how we get to the Lie algebra from that... I suppose this comes from some consideration about left translations (?)

@porridgemathematics For the time being I want to focus on this piece. Why does this work? There are non matrix Lie groups that are finite dim

porridgemathematics

8:58 AM

@Mr.Feynman for the first question, you could think about the christoffel symbols , so if I write (for an affine connection), $\nabla_{\partial_i} \partial_j = \Gamma_{ij}^k \partial_k$, then I could equally write in local coordinates, $\nabla = (\Gamma_{ij}^k) (\partial_k \otimes dx^j) \otimes dx^i$, which is a covector field taking values in $TG \otimes T^{\ast}G = End(TG)$

Ajay

Are Darboux and Riemann sums the same thing?

porridgemathematics

@Mr.Feynman currently what im saying just applies on any smooth manifold,

and for any affine connection

im rusty on connections of principal $G$-bundles, but the logic should be very similar

the lie algebra should come out of $T_{e}G$ and left translations yes

the details someone else will have to help you out with or you will have to fill in yourself, since im pretty far removed from DG right now

I just said finite dimenisonal lie group because then $End(TG)$ is a finite dimensional vector bundle, so matrices correspond to endomorphisms, i cant remember if something like ados theorem matters here

oh yeah, dont you have that the connection one form should be invariant under left or right translations (which one I forget)

leslie townes

ajay: i am not sure that they are consistently defined across all sources, but in the cases i've seen, they are not the same thing, although they give rise to the same notion of integrability and the same values of the integral

porridgemathematics

so that should explain why $End(TG)$ coincides with $End(\mathfrak{g})$ @Mr.Feynman

darboux sums are just extremal riemann sums

which is why they give the same integral

but anyway im pretty sure the connection one form for you will be an End(g) valued one form, not a g-valued one form...

you could say its a matrix of g-valued one forms though

one potato two potato

9:38 AM

I need to choose one of two classes: TDS (topological data analysis) and Probability theory.

robjohn

@Utkarsh exactly what do you mean by "component values"?

Mr. Feynman

@porridgemathematics Ooh, this is basically the definition of connections as functions taking sections of a bundle into 1-forms valued in such bundle [...] and then you just played with tensor products

@porridgemathematics That's something about which I got confused yesterday, damn it

porridgemathematics

9:53 AM

yeah

Mr. Feynman

In that case I'm getting confused on why in the case of vector bundles the connection form is a matrix of real valued one forms and for a principal bundle we have a matrix of $\mathfrak{g}$-valued one forms

What is the reason why for a principal bundle we wouldn't be content with a connection form made up of real valued forms? I mean, even without technicalities...

porridgemathematics

i think there is confusion in using terms here, to me a connection is locally as I wrote

terminology aside, it looks like $\nabla = (\Gamma_{ij}^k) (\partial_k \otimes dx^j) \otimes dx^i$

the matrices in those 'blah is a matrix of blah valued whatevers' is really always referring to $\Gamma_i$ with components $(\Gamma_{i})_{jk} = \Gamma_{ij}^k$

Mr. Feynman

What I call connection form (at least for vector bundles) is the matrix whose entries are $\omega^j{}_{k}:=\Gamma^j_{ik}dx^i$

porridgemathematics

right, but there is some abuse going on there

because we are presupposing a fixed coordinate frame, and the effect of applying the connection to a vector field, in those coordinates, is the same as writing the vector field in those coordinates, and matrix multiplying

the $\partial_k \otimes dx^j$ is thus being hidden

so sure, its a matrix of scalars if you want

but to say so you are smuggling away those tensorial components

in both the principal G bundle case or the usual case

so thats why you can always say a matrix of $TM$ valued one forms

because the components of the matrix really are $\Gamma_{ij}^k \partial_k \otimes dx^j$

so this way there is no special cases involved

the principal G bundle is only different because you can replace $TM$-valued with $\mathfrak{g}$-valued

actually I think matrices are going to just cause confusion when thinking about why you can go between some description and another description

Mr. Feynman

At this point I think it is really about terminology as you say, because for a vector bundle with a local frame $\{e_i\}$ (so we get rid of any confusion with $\partial_i$), a connection is such that $\nabla e_k= \Gamma^j_{ik}dx^i\otimes e_j$, so the point is my definition only takes into account the first factor while in the other case we are considering the whole thing

porridgemathematics

10:04 AM

I would always use the pure tensor notation to see why you can do that

I believe it really is down to terminology yes

theres nothing special going on in the principal bundle case

besides the additional constraints

Mr. Feynman

I'm using multiple references and they might not agree on these conventions so I'll take my time to check things a little bit more and I'll be back on this either here or directly asking a question on the site. For the time being, thank you for your help @porridgemathematics

porridgemathematics

no worries, and yeah I should have used $e_i$, the $\partial_i$ only refer to coordinate vector fields

Utkarsh

11:05 AM

@robjohn Direction ratios along axis.

robjohn

11:30 AM

@Utkarsh then $2:-\frac32:1$

note that $4:-3:2$ works, too

Thomas Finley

12:36 PM

Hey everyone! I have a question regarding the definition of a field and the operations within it. In the definition, it states that the operations of addition and multiplication should be well-defined. I'm a bit confused about whether these operations refer to the usual sense of addition and multiplication or if they can be defined arbitrarily.

Specifically, I'm wondering if it's possible to define addition as simply a name given to an operation that has no connection to the conventional notion of addition. Could addition, in this case, represent any operation unrelated to the traditional …

shintuku

2:28 PM

any comprehensible nonhausdorff spaces other than the line with two origins?

on an unrelated note: jstor.org/stable/25099190

Utkarsh

4:01 PM

@robjohn Oh yeah. Thanks.

Mr. Feynman

4:32 PM

In the definition of the Maurer-Cartan form, Frankel's book does the following:

Can someone help me understand why do we need $dg$ at all in the way he arrives to it? It maps a vector to itself, so why would I compose $(L_{g^{-1}})_*$ with it instead of just having $\Omega=(L_{g^{-1}})_*$?

> Thus $dg$ takes $Y$ at $g$ into $Y$ and $g^{-1}$ left translates [...]

In fact, seeing the intrinsic construction on Wikipedia, I wouldn't even think of putting $dg$ there

user 858770

5:02 PM

> Only the clearest of minds are the first to think of something nobody would think of.

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$Mathematics$ Mathematics