serious-discussion

[inner finch]

:yamin:

[neat lintel]

@thorn brook it's really easy to prove if you replace "homeomorphic" with "diffeomorphic." otherwise, it's a little bit non trivial. the proof im thinking of uses facts about de rham cohomology (i believe it uses the topological invariance)

(to respond to your question in #point-set-topology

i didn't want to interrupt the ongoing convo

[thorn brook]

Well frick, I don’t know what diffeomorphic is lmao but I will look into that! I don’t even know what the rham Cohomology is. Heck, I don’t even know what Cohomology is lmao! But I will certainly look into it, thank you so much!

I hope Hatcher talks about cohomology

He must, right?

[neat lintel]

diffeomorphic means there's a smooth map with smooth inverse

if you differentiate it you get a linear isomorphism on tangent spaces which implies they have the same dimension qed

[thorn brook]

Yeah sorry but that’s a little over my head. I will try to learn it tho, thank you!

[last oxide]

oh

thats the one about open maps

doesnt it use brouwer fixed point or some meme, too

[thorn brook]

Well I found some kind of PDF that uses that theorem

But the later parts were alien to me lmao

[fringe needle]

@sullen valley might check it out later thnx!

[last oxide]

inb4 some algebraic top memer comes in complaining this is much easier with homology

[modest rune]

you can prove it fairly easily with homology if you just mean the R^n case yes

[thorn brook]

Okay great! I will come back once I have learned more about homology!

[deep mango]

Mirzas here!

[sharp mulch]

Old news...

[deep mango]

Finally, this server is worth frequenting again

And totally not because I just finished my road trip or anything

I will be in berkeley soon. It's almost time to get on the train

If only ange were here.

[sharp mulch]

...

[deep mango]

Oh shit mirza is old news, I guess I didn't click on discussion-2 today

[sharp mulch]

[mortal oasis]

so uh how much of a math background do i need to read up on Lie Algebra

[sharp mulch]

A fair amount

[mortal oasis]

fuck.

im assuming a fair amount isnt just like Calc 3

lmao

[sharp mulch]

You should know linear algebra + groups + diff geo maybe?

Because when you learn about lie algebras you also want to learn about lie groups

[mortal oasis]

i uh know linear algebra

[vagrant kestrel]

e^matrix is a group

[mortal oasis]

mostly i guess cause im interested in how the DARTEL algorithim works

[sharp mulch]

Yeah the linear algebra involved isn't too complicated

Mostly orthogonal matrices and stuff

Because lots of lie groups are groups of matrices

[mortal oasis]

hm maybe ill take a look and see how confused i am

i like orthogonal matrices make everything clean

[sharp mulch]

Is it http://cslras.pbworks.com/f/DARTEL.pdf ?

[azure kettle]

e^ just converts additive groups to multiplicative groups

[sharp mulch]

I briefly skimmed this and I didn't see anything that seemed to be lie algebraic

[vagrant kestrel]

the paper tries to explain lie groups and algebras nice

[mortal oasis]

yep thats is

[sharp mulch]

It's just some numerical tomfoolery

[mortal oasis]

its for nueroscience which is why im hoping it keeps things at a reasonable level

ok

cool

[sharp mulch]

The lie algebra/lie group thing is completely unnecessary

I'm not even sure why they bring it up tbh

[mortal oasis]

to sound cool

probably

[azure kettle]

I mean sometimes it's more than just sounding cool

[mortal oasis]

i memeing

im probably read it tomorrow I have beentraveling all day glad it doesnt seem unreasonable

[sharp mulch]

"lie algebra" and "lie group" are both mentioned only twice

[vagrant kestrel]

it's cited a lot which means it's readable

[mortal oasis]

oh its like a gold standard for VBM

i believe

[azure kettle]

like even if it's technically unnecessary, might be a clearer way of communicating a concept sometimes

not saying it's the case here

[vagrant kestrel]

oh is this like the most famous neuroimaging algorithm or something

[mortal oasis]

yea

def for MRI

its the way i believe we currently id different tissues in the brain

from an MRI

[sharp mulch]

This paper is kind of weirdly written

A page and a half to talk about multigrid in vague terms

[mortal oasis]

i believe one of the big deals in VBM is that it standardize to a template image to help with comparisions and DARTEL is how that is done

[sharp mulch]

There is an appendix where they compute derivatives...

[azure kettle]

tell me that's not true

[sharp mulch]

Appendix A: deriving derivatives

They even provide a source where the derivatives are computed before literally computing them again

[azure kettle]

lmfaoooo

[mortal oasis]

hahahah im assuming there are a lot of biology people who dont care about how stuff is calculated

[sharp mulch]

This paper hurts me

Can I write papers like this and get 6000+ citations in biology?

Should I become a biologist?

[azure kettle]

dont let your dreams be memes

get your citations

however you can do it

[mortal oasis]

KEKW

[azure kettle]

since a biologist already published a paper on how to approximate areas under curves using partial sums

maybe you can develop on that

[mortal oasis]

hmm

[azure kettle]

and find a way to get the exact area under a curve

[mortal oasis]

yea wasnt there a doctor who did that

like 200 years later or something

[rose dock]

Lmfao

[azure kettle]

https://care.diabetesjournals.org/content/17/2/152.full-text.pdf

[mortal oasis]

https://care.diabetesjournals.org/content/17/2/152.abstract

[sharp mulch]

[mortal oasis]

i linked late

[rose dock]

Tai's method

[mortal oasis]

wasnt first

[rose dock]

We should start calling it that in calculus class

[azure kettle]

it's just simultaneously such an impressive and entirely unimpressive thing

like I have to give it to them, figuring out Reimann sums all by themselves is pretty impressive

[sharp mulch]

I think it's pretty impressive

Both being able to figure out Riemann sums

[mortal oasis]

I agree

[sharp mulch]

As well as getting to that point in your career without having seen integrals

[mortal oasis]

also how sad would you be when you realised

[light needle]

oh lmao is that the guy who rediscovered trapazoid rule

[mortal oasis]

like you think you make this wonderful math discovery

1993

[vagrant kestrel]

it's called mathematics outreach

[azure kettle]

the unimpressive thing is that they have such a shallow appreciation of what they discovered

[light needle]

This is symptomatic of a broader problem btw, the ineffecient communications between different fields

[cold needle]

i hope someone explained to them what happened

[azure kettle]

their only application of it is to apply it to glucose curves or something

kekw

[cold needle]

like this whole

pain

[rose dock]

Is this why calc is required for med school

[mortal oasis]

KEKW

[light needle]

someone pitched to me once that a stem librarian would be nice, someone who translates(jargon and such) between different fields

always thought that would be a nice job for me

[mortal oasis]

when someone says why do i need calc link them this

[cold needle]

h m m m

You don't wanna be this guy.

like impressive as it is its just pain

[light needle]

yeah imagine if you came across a problem while doing your research

and you wanna know if someone else has solved it

[vagrant kestrel]

do you want the MCAT to have calc problems

[light needle]

but you do not understand jargon from the other field or how to search in it or w/e

(mathmeticians with cs for example, maybe u spend a few hrs rediscovering something basic for an algorithm)

[vagrant kestrel]

I have no idea how you search papers to check if you've rediscovered something

[rose dock]

Paper: A novel method to search a sorted array in O(log(n)) time

[light needle]

ideally if you want a subresult that is already proofed you will just find that proof

or whatever analogues in different field

which is why i think a STEM librarian might be super useful

[azure kettle]

the best part is that the paper rediscovering calculus thanked a bunch of people who were in literal math-adjacent fields who should've immediately realized this was a meme paper

I think one was an electrical engineer or something

[vagrant kestrel]

the EE just nodded along

[azure kettle]

funny way to get exposed for not having actually read it

[cold needle]

cursed

[azure kettle]

in unrelated news, did you guys know that a person born in NY identifying as non-binary is leading to the downfall of Japan

[cold needle]

i love twitter!!!

im gonna delete my account soon

[azure kettle]

it's obviously well known that Japan has never had any LGBT people, and Japanese media definitely has no "leftist LGBT propaganda"

[bronze pelican]

Utada Hikaru ❤️

Lover her music

[cold needle]

not her i suppose

their

[azure kettle]

smh how can you like her music if she's going to destroy Japan

actually mb

like them*

[cold needle]

😌

now listening to simple and clean

letting autoplay do the rest

i should play KH sometime in the future

hmm

[vagrant kestrel]

https://www.youtube.com/watch?v=UPdlfIhzPEo

[hollow ginkgo]

Why is this from a math server?

[cold needle]

no idea

they are artefacts of the past

[vagrant kestrel]

KH from group theory pls

[cold needle]

lol

H, K subgroups of G of course

[azure kettle]

< G

[leaden torrent]

is before the current mod teams time too lmao

i dont think any of the current mods were around when it was added

maybe dami?

its still there because no reason to delete it

but its certainly weird

[narrow rock]

[leaden torrent]

yes we were talking about

thanks for confirming

[blazing pawn]

I did it

I destroyed japan

[cold needle]

finally

we got rid of one landmass

there are more to go

[summer nest]

downfall of japan?

[summer nest]

wake island theme plays

[pale orchid]

groudon is not happy about this

[deep mango]

Based moth. Weebs in shambles.

[vast surge]

Where are you going to destroy next?

[deep mango]

Please destroy texas please destroy texas please destroy texas

Buncho in shambles.

[vast surge]

A friend of mine is planning on burning down all of Florida at some point.

[deep mango]

Wow.

[vast surge]

Just, all of it.

[deep mango]

Florida isn't so terrible

[neat lintel]

please destroy toronto

[cold needle]

what

[deep mango]

So is texas, but texas needs help with it

[cold needle]

wait

im still in florida

what

no don't

[neat lintel]

do

[deep mango]

Metal in shambles

I'm going to eat Florida.

[cold needle]

packing my bags rn

[vast surge]

Yeah Toronto's close enough to where I am. Would you prefer something quick or something fun?

[deep mango]

[neat lintel]

quick, please

[cold needle]

true

[vast surge]

Upstate NY enough that it's basically the same thing

[deep mango]

Toronto is upstate NY lite

[vast surge]

No, Toronto's a city. We're super fucking rural.

[deep mango]

"""city"""

NY has a real city.

Albany.

[vast surge]

Yeah but Toronto's closer.

[deep mango]

Also Troy.

Troy is so pretty

[vast surge]

Like, the only NY cities between where I am and Toronto are Rochester and Buffalo.

[deep mango]

Ooh, Rochester.

How about Ithaca. Surely that's not too far.

[vast surge]

Yeah but it's the other direction

[deep mango]

The direction one should go

New york state has so many nice cities and towns

I bet Toronto's got more of an Erie Pennsylvania vibe.

Erie is the worst place I have ever been

[neat lintel]

the only good things about toronto are the parks/trails/beaches

[deep mango]

[vast surge]

Honestly so many people only know NYC; there's so many cool things here. Appalachians, catskills, finger lakes, hudson river valley, etc.

[deep mango]

"only"

What city has beaches? Come on

I want beaches

I hope you just mispelled benches

[vast surge]

Miami, but then you'd have to go to Florida

Some places in California

[deep mango]

I went to school right near SF

SF has a really gorgeous beach

[vast surge]

I briefly lived in Myrtle Beach, SC, and there were beaches there.

[deep mango]

I could have gone to UCLA for my phd but I didn't

The only part of that that makes me so sad is I wont have nice beaches

[cold needle]

mine has beaches

and it is not miami

[deep mango]

NYC doesn't have nice beaches, have to drive down to sandy hook and I've been there so many times already

[vast surge]

Well you're not missing much. The sand is rough and course and it gets everywhere

Not like you, padme

[deep mango]

I love rough coarse sand that gets everywhere

[vast surge]

I don't like sand

[deep mango]

I just love the ocean and water, that's all

[vast surge]

I like the kinds of beaches with pebbles and rocks instead of sand

And lakes

[deep mango]

What am I going to do. Swim in the east river, huh?

[vast surge]

But fuck sand

[deep mango]

I like those things too

[vast surge]

This sounds safe and hygenic.

[deep mango]

Yes

I wish I had a nice river nearby to swim in

And paddleboard in

And kayak in

[vast surge]

And get washed away by the current in, and swept to an undersea kingdom where you'll go on all sorts of magical adventures

[deep mango]

I would like to get washed away in the current one time

That sounds great

Hopefully someday I will be stranded out at sea.

[vast surge]

In all seriousness it's not fun when there's a current so strong that you're having a hard time getting/keeping your head above water, I've done that before

Don't swim alone, and don't swim out in open water if you're not a strong swimmer.

[deep mango]

On a boat that is just a little too complicated to manage as one person, and I'm the only one there.

A sailboat of course

And then I will have to learn how to control it and rig up solutions to do more than I should be able at once

[vast surge]

Trust me it's safer to just get a crew

[deep mango]

And get myself to shore

While catching fish, like in life of pi

They all died

Of

Uh

[vast surge]

Life of tau/2

[deep mango]

Scurvy? Idk

I'm not going to eat them

[vast surge]

(I'm kidding about this please don't ban me)

[deep mango]

What a romantic survival fantasy this is

[vast surge]

Also a good opportunity to get away from the stresses of academia and just do some math

[deep mango]

Math is not allowed on my stranded boat. I need to spend all my energy on crisis response and moment to moment problem solving!

Math would be so much more fun if it really got the heart pumping like a true crisis does

That happens during exams

That's why I like exams

[cold needle]

i dont like exams

cus im bad

[vast surge]

I love exams

I wish exams were 100% of the grade

[deep mango]

Ok I don't love them that much

[cold needle]

i wish graded homeworks and exams were about the same

[deep mango]

I want to take exams and have them barely matter

[cold needle]

some people just work better with more time

[deep mango]

But still feel like they have big stakes

[vast surge]

I find that the best way to do exams is to drink way too much beforehand, throw up a bit then show up to them 30 minutes late and leave like an hour to 90 minutes early

[deep mango]

The feeling of big stakes is so wonderful

[vast surge]

I've consistently gotten As when I did that

[cold needle]

that

sounds really not good

lol

[vast surge]

I mean, if it works it works

[deep mango]

Bananachair is a god.

[vast surge]

No, I'm just a mortal

There is only one God you should be concerned about, and he is currently dreaming in his city of R'lyeh

ph'nglui mglw'nafh Cthulhu R'lyeh wgah'nagl fhtagn

[timid portal]

Would anyone happen to know about that trick involving sequences where you can easily derive closed forms for the sum of natural numbers raised to some integer power?

I'm trying stuff out involving another cool thing I derived

$\sum_{k_m=1}^n\left(\sum_{k_{m-1}=1}^{k_m}\left(\cdots\sum_{k_1=1}^{k_2}k_1\right)\cdots\right)={n+m\choose 1+m}$

[fathom swallow]

Flip

[timid portal]

You could use this to help derive a closed form for $\sum_{k=1}^n k^p$ for $p\in\mathbb{N}$, but I want another easier reference to make comparisons

[fathom swallow]

Flip

[timid portal]

Sequence trick is something like: take a well-known sequence a_n with a closed form, and somehow b_n = a_(n+1) - a_n = n, or something

[worldly solar]

@timid portal you can get it for k^p recursively by summing the quantity (k+1)^p - k^p from 1 to n. One side telescopes, and by expanding and distributing the other side you can get the k^p sum in terms of the k^r sums with r<p

[edgy atlas]

A big reach but.. Does anybody here know how to do a correctness proof for the Fast Fourier Transform (FFT) algorithm? (In the simplest case, where the input vector is divisible by 2) Nobody in my study group could figure it out

[inner finch]

great

[timid plover]

Yeah

[toxic schooner]

Oh so the other channels arent affected?

[nimble shuttle]

which channel is that

[rancid meadow]

Ted muted #discussion (mod abuse)

[nimble shuttle]

owned

[deep mango]

Dead server?

[inner finch]

soon dead

[toxic schooner]

[light needle]

dead me

[devout nacelle]

This is Velleman, right?

[wicked ore]

Feelings of existential dread

[devout nacelle]

Or some similar proofs book

I remember having read this page before

[inner finch]

nice lighting

[wicked ore]

“Book of proof” by Hammack

[inner finch]

yellow lights are based

[wicked ore]

Perhaps

[devout nacelle]

Ah yes, Hammack.

[inner finch]

i thought they were named hammock

all this time

[compact tartan]

sodium light moment

[cold needle]

hammack

this is flashbacks to intro proofs class :|

[inner finch]

my condolonces

[cold needle]

yeah

[mortal oasis]

I hate matlab I hate matlab I hate matlab

please someone make the pain stop

please

[sharp mulch]

[lost surge]

Anyone have any recommendations of websites with banks of higher-level math problems? Like vector calculus, diff eq, linear algebra? All the math problem websites I've seen are like regular calculus or below.

[leaden torrent]

resist urge

not familiar with a unified resource but you can find a bunch of free textbooks for those subjects

(and some not-free textbooks...)

which tend to have plenty of problems

nlab has higher-level problems but not of this sort.

[lost surge]

Yeah, Nlab isn't really what I'm looking for. I have textbooks. But I want more problems. Guess I'll just have to use the textbooks.

[inner finch]

nlab is a problem in it of itself :nice:

[leaden torrent]

congrats mirza

you got the joke

[lost surge]

I thought Nlab was like a math wiki.

[leaden torrent]

...kind of...

[nimble shuttle]

[inner finch]

thank u thank u 🤝

[leaden torrent]

imagine if a wiki on sports was written by people who exclusively played badminton and also thought all other sports should be analyzed like badminton

using the terms from badminton

[zealous skiff]

namington

[leaden torrent]

and these people considered themselves superior to the non-badminton-ers

thats nlab

[zealous skiff]

are you the deez nuts addiction guy

or who was that

[leaden torrent]

???

[inner finch]

[nimble shuttle]

nadmington

[inner finch]

Society

[zealous skiff]

who is the deez nuts jokes guy

[leaden torrent]

no clue.

[zealous skiff]

wait i think thats daminark

[nimble shuttle]

[inner finch]

Since you're talking about someone unfunny you're probably thinking of ryc :nice

[neat lintel]

[inner finch]

These people sound pretty based

[crystal stream]

I accidentally forgot to write the hypotheses for stoke's theorem, had to look it up on wikipedia

finally understood what the heck the pullback is

[sharp mulch]

#❓how-to-get-help

[lost valley]

Oh, sorry wrong room 😅 😌

[hallow wasp]

what is pullback and pushouts

sound like duals

[bronze pelican]

universal property blah blah blah

[modest rune]

your dislike of category theory is cringe pt

pushouts and pullbacks are super important

they are in fact dual to eachother

[inner finch]

naheen yar sach batao

[bronze pelican]

cronge

[neat lintel]

category theory is okay

[bronze pelican]

category theory more like cateBORING theory

[astral marsh]

gotem

[vivid halo]

update: I have computed the integrals

[neat lintel]

what

[sharp mulch]

Oh nice

Zeros

Cool block structure

[narrow rock]

[meager sonnet]

Can't lie that looks like line noise to me

Mathematical equivalent of perl coding

[inner finch]

mathematical equivalent of cringe

[meager sonnet]

That's just cringe

[neat lintel]

on a connected symplectic manifold (M, ω) a vector field X is hamiltonian iff it's symplectic and \int_c ω(X, .) = 0 for all loops c in M

something something cohomology

really makes you thonk

nevermind this is a general fact that always holds i thought it was some cool symplectic thing

math contunues to disappoint

[sacred geyser]

sadness

[neat lintel]

i realized it as soon as i started thinking of the proof

lmao

[mortal oasis]

figued out my matlab issue!!

all i had to do was partition my drive install Linux and run the API on the Linux

[inner finch]

oh great, so you're switching to mathematica now

[mortal oasis]

if they have an SPM equivlanent yea

[toxic schooner]

mirza acting like people can understand her hindi/urdu

[toxic schooner]

[fast ivy]

I just thought about a topology-analysis problem

But idk if it might be worth discussing

[devout nacelle]

Go ahead

[neat lintel]

[toxic schooner]

go ahead, i would like to see the discussion of something i for sure cant understand

[neat lintel]

still typing ..

[hallow wasp]

⏰

🪥

[fast ivy]

Fix $V$ a real vector space, consider the set $C(V \times V, \mathbb{R})$ under the compact-open topology and $\mathcal{M}(V) \subset C(V \times V, \mathbb{R})$ the set of all metrics on $V$ under the subspace topology.
\
\
Now, quotient $\mathcal{M}(V)$ by the equivalence relation of equivalence of metrics, let's still keep calling it $\mathcal{M}(V)$ just for abuse of notation.
\
\
My question is, consider $\mathcal{N}(V) \subset \mathcal{M}(V)$ the set of all metrics of $V$ induced by a norm.
\
\
My question is, what can we say about $\mathcal{N}(V)$ topologically for a general vector space $V$? Is it dense in $\mathcal{M}(V)$? I know that for finite dimensional vector spaces it consists only of a single point, but that still doesn't say much about its closure, for example.

[fathom swallow]

MisterSystem

[toxic schooner]

[fast ivy]

I got motivated by this problem on my last analysis class, that was today.

We discussed Stone-Weierestrass

I prolly won't be able to solve it because it's too general, but I like to formulate my own questions lmao

I was wondering how well one can approximate general metrics on a real vector space by metrics induced by a norm in some sense.

One thing tho

When we studied Stone Weierstrass, we considered C(X,R) with the topology of uniform convergence.

Idk which one is more natural

If the compact open topology or this one to formulate the problem

But whatever

[hallow wasp]

What is C(VxV,R) supposed to be

[fast ivy]

OH

[hallow wasp]

compact open topology you said

[fast ivy]

That doesn't make sense

We would need V to have some topology in the first place

In order to the problem to make sense

[hallow wasp]

i thought thats for metric spaces

[fast ivy]

So we need V to be a topological vector space

[hallow wasp]

ig its for toplogies

oh no

topological vector space is just a vector space equipped with a topology?

[fast ivy]

Yeah

But also

[hallow wasp]

i only got lost in your second paragr

aph

[fast ivy]

The operations on the vector space are all continuous with respect to the topology

That's what a topological vector space is

[hallow wasp]

so scalar multiplication and addition are continuous?

[fast ivy]

Yup

[hallow wasp]

lemme come up with a definition

[fast ivy]

That's the requirement to be a topological vector space

[torn willow]

So,given a vector a and scalar c,f(v)=v+a and f(v)=ca are continuous functions?

[hallow wasp]

Ok i cant really think of anything clean but is it just saying that +:V x V -> V is continuous and *:V->V v->vf such that f is in the underlying field, is continuous?

[fast ivy]

$+ : \mathbb{V} \times \mathbb{V} \rightarrow \mathbb{R}$ where $(u,v) \mapsto u+v$
\
\
and,
\
$\cdot : \mathbb{R} \times V \rightarrow \mathbb{V}$ where $(\alpha, u) \mapsto \alpha \cdot u$
\
\
Are continuous

[fathom swallow]

MisterSystem

[fast ivy]

Yeah

[hallow wasp]

looks cleaner

[fast ivy]

I mean, we are working with real vector spaces

Because in a general field

We don't necessarily have a topology

In order to ask a function from F × V -> V be continuous

[hallow wasp]

where im confused is you say you quotient the subspace topology of M(V) with something i never heard of

[fast ivy]

Equivalence of metrics

Oh yeah

[hallow wasp]

what are those

i mightve seen it once

[fast ivy]

https://en.m.wikipedia.org/wiki/Equivalence_of_metrics

Here

But I will also type out the definition

[hallow wasp]

so if their induced topologies are the same they are equivalent

and u induce topologies by taking balls

at every point i think

so if the set of epsilon balls of two metrics is equal then their topologically equivalent

or do you have to check more than their basis?

[fast ivy]

Let $X$ be a non-empty set, $d_{1} : X \times X \rightarrow \mathbb{R}$ and $d_{2} : X \times X \rightarrow \mathbb{R}$, then what I mean by these two being equivalent is that $\exists C,K \in \mathbb{R}$ positive constants such for all $x,y \in X$:
\
\
$C d_{1}(x,y) \leq d_{2}(x,y) \leq K d_{1}(x,y)$

[hallow wasp]

oh

super concose

[fast ivy]

Btw

It's the opposite lmao

Lemme see

[fathom swallow]

MisterSystem

[fast ivy]

There we go

And yeah

When this happens

This means that d_{1} and d_{2} induce the same topology

[hallow wasp]

idk have hard time conceptualizing quotienting with this relationship

[fast ivy]

This is an exercise I had when I first studied it

If I find it again

I will take a screenshot

There's a cute fact about R^n

You can also talk about equivalence of nroms

And the definition is pretty much the same, with small changes

And in R^n, two norms are always equivalent!

In fact for finite dimensional vector spaces in general

[hallow wasp]

like W in M(V)/~ are distinguished by their different topologies?

[fast ivy]

So that's why I said before that N(V) consists of a single point for finite dimensional vector spaces

Because two norms are always equivalent

This is also an exercise

[fast ivy]

Yeah, you can think of this equivalence relation as being

"Oh, we consider two metrics the same if their induced topologies match up$

[fast ivy]

The weird thing is that I have no idea how the topology of M(V) goes lmao

It prolly isn't Hausdorff

So like, I couldn't guarantee just by knowing that for finite dimensional vector spaces N(V) is a single point that its closure doesn't behave like crazy

[hallow wasp]

wait

ok

finite dimensional vector space means M(V)/~ is one element because finite dimensional vector spaces have one topology?

[fast ivy]

No

[hallow wasp]

or all the metrics are topologically equivalent

[fast ivy]

I mean, the set of all metrics induced by a norm is a single element

Because two norms are always equivalent for finite dimensional vector spaces

Idk about general metrics we could induce on these guys tho

[hallow wasp]

why is that true?

[fast ivy]

The book I have used to study analysis is written in portuguese

So lemme MSE real quick to see if someone has a proof of this

https://math.stackexchange.com/questions/57686/understanding-of-the-theorem-that-all-norms-are-equivalent-in-finite-dimensional

[neat lintel]

"first pick a basis"

[fast ivy]

This is a nice exercise tho

[neat lintel]

[fast ivy]

You could try for yourself

[neat lintel]

the sully was supposed to come immediately after the basis message

but mistersystem is too fast

[fast ivy]

Lmao

Yeah, I still have no idea why I thought about this question

But I prolly won't be able to solve it too soon, but being able to formulate my own questions is a good way to learn stuff

I love doing it

It always happned to me that I was studying a topic, then I thought about a question that only after learning something else I could do it on my own

I will try to keep this one on the back of my head lmao

[forest jackal]

I like to think of that theorem of f.d. norm equivalence in terms of compactness. The unit sphere in the standard norm is compact, so any other norm attains a positive max and min on the standard sphere, and with homogeneity this gives you the equivalence. All the proofs are basically this idea though, sometimes disguised.

[fast ivy]

Never really thought about it this way

[forest jackal]

It makes the reliance on f.d. very clear

[hallow wasp]

yea whats homogeneity

[fast ivy]

I will reread the proof I know and see if I can highlight this idea

Anyways Gomez

[forest jackal]

homgeneity is the fact that the norms scale by |c| if you scale a vector by c. this is true of any norm by definition.

as a result it suffices to work on the sphere

[fast ivy]

What do you think about this question?

[neat lintel]

[fast ivy]

V is a real topological vector space btw

[hallow wasp]

oh ok that makes sense

[heady estuary]

Hey guys! Does Courant's book "Introduction to Calculus and Analysis" cover Real Analysis as well?

[forest jackal]

Initial impression: I would be quite surprised if you got anything like density, but idk would have to play around with simple examples to build intuition before I actually made firm conjectures on the behaviour.

(And am probably not going to play around with simple examples, as I have my own work to do :p). But if you wanted to attack this I would suggest first studying the finite dimensional case, even dimension 1.

It may be trivial.

[fast ivy]

I know N(V) consists of a single point because of equivalence of norms

Then

I would have to somehow

Construct a metric on a finite dimensional vector space

Say R^n

That is the uniform limit (on compact sets) of metrics induced by a norm

But that this limit is not itself induced by a norm

I don't really know how could I construct something like this

[hallow wasp]

also im assuming standard norm is euclidean norm

[forest jackal]

yea or if working with an abstract f.d. vector space V, choose an arbitrary basis, and use the corresponding isomorphism with R^n to port the Euclidean norm over to V.

[hallow wasp]

wish i had an example

[forest jackal]

or really you can just work in R^n itself once you have that isom

example of what?

[hallow wasp]

an abstract fd vector space

ye

i only think in Rn

[forest jackal]

polynomials of degree less than 420

[fast ivy]

Lmao

[hallow wasp]

ig thr problem with my thinking is you said there is an isomorphism between it and Rn

I can see it with the polynomials of degree 420 because the coefficients correspond to each entry of the vector

but what are not so straightforward examples

[forest jackal]

Any vector space has a basis, the basis elements play the role that 1,x,x^2,...,x^(419) play in the polynomial example.

any element of the vsp can be written as a linear combination of these uniquely

[hallow wasp]

yes

[fast ivy]

Real n×n matrices, polynomials with degree ≤ n for some natural number, the tangent space of a real manifold at a point

[forest jackal]

and you can send this element to the tuple of real numbers corresponding to these coefficients.

[hallow wasp]

I can see it with polynomials but i was wondering a less straightforward example

Real nxn matrices isn’t straightforward to me

or do you take the basis for the matricies

wait nvm

[forest jackal]

smooth solutions to the ODE x''(t)+x(t)=0 over R.

[fast ivy]

You just send each entry of the matrix to a an n×n tuple

[fast ivy]

Ok that was a really cool example

[hallow wasp]

oh so its isomorphic to R^n*n

also idk about that example gomez

[fast ivy]

I was really impressed when I learned that solutions to linear ODEs form a vector space at first

[forest jackal]

its nice because sometimes you will want to work with exponentials, sometimes with trig functions, etc

[fast ivy]

It's a really non trivial example of finite dimensional vector space indeed

[hallow wasp]

i dont see it

[fast ivy]

Suppose you have a certain number of solutions to that ODE

Say x_1, ..., x_{k} solutions to the ODE

Now

Take any linear combination

Between these solutions

You can see

That this linear combination is still a solution to the ODE!

And moreover

[hallow wasp]

oh yea

[fast ivy]

When the ODE is linear

You get more

That in fact

[hallow wasp]

because solutions are linear for homogeneous diffeqs

[fast ivy]

The space of all the solutions is actually finite dimensional

I.e

You need a finite number of solutions

In order to generate all the others

So that's why it's a finite dimensional real vector space

[fast ivy]

Yup

[hallow wasp]

oh ok

thats really cool

i never thought of that

[fast ivy]

That's prolly the best non trivial example of a finite dimensional vector space, can't really think of something better

[forest jackal]

same with linear recurrences, eg u(n)=u(n-1)+u(n-2). the theory is very similar from a linear algebra perspective.

these are all just kernels of a linear operator on a vector space. whenever you have a space with a linear structure, and you are looking at a linear equation/system of equations on this space, the solution space will be a vector space itself (a subspace of your original one).

[hallow wasp]

useful advice for the future, ty

[blazing pawn]

the cooler version of this is that local solutions also form vector spaces

in ways that can be patched together and stuff

its neat

[fast ivy]

Yeah

We touched a bit of ODEs in this last analysis course I took

[blazing pawn]

the cooler cooler version is that local solutions form a sheaf and let you realize ODEs as representations of the fundamental group

But i dont know if you have the background to get anything out of that statement

Or see what it means

[fast ivy]

I know what a sheaf is, it's a presheaf with additional properties and it let's you talk about "local structure", for example ringed spaces, locally ringed spaces, complex manifolds aswell as real manifolds.

But like

What I mean is that in the context of motivating someone to learn linear algebra

The example of smooth solutions of a linear ODE is really neat

And is simple enough to motivate someone to study the subject

[blazing pawn]

Oh yeah no dont use sheaves to motivate lin alg

its just a neat fact

[fast ivy]

Sheaves are useful to study vector bundles, and we all know that vector bundles are the natural next step to linear algebra lmao

Just kidding

[blazing pawn]

No no hes right

[hallow wasp]

wait

what are presheaves

or not worth going into?

[forest jackal]

lol

not yet, not worth learning about till you have covered more in general

[fast ivy]

We would have to talk about functors

And category theory

So like

Uhh

[forest jackal]

then you will realise you have been secretly working with them all along

[fast ivy]

The definition is simple enough

It's just a contravariant functor from your category to something else, like Set or the category of rings.

But like

[hallow wasp]

when do i reach thst point?

[fast ivy]

What is a functor?? What is a category???

[hallow wasp]

i know those i think

functor is a map between objects of categories

right?

or a map between categories

[fast ivy]

A functor is a map F : C -> D between two categories that maps objects of C to objects of D, maps morphisms of C to morphisms of D and also has two neat properties.

If your functor is covariant

Then they are as follows

If you have two morphism f : X -> Y, g : Y -> Z

Then F(g ° f) = F(g) ° F(f)

And

[blazing pawn]

I think that this is a kind of bad way to introduce them because the prototypical example of presheaves and sheaves everyone cares about is just topological spaces anyway

[fast ivy]

F(id _ {X}) = id _ {F(X)}

[blazing pawn]

I think its easier to explain the usual definition and then categorify the space afterwards

[hallow wasp]

please do

i like concreteness where i can get it

[fast ivy]

Do you know what a manifold or an algebraic variety is ?

[hallow wasp]

yea i think so

[fast ivy]

That's a nice motivation for sheaves

Nice

Which one of those are you more familiar with?

[hallow wasp]

n manifold is a where every neighborhood of a point is homeomorphic to Rn i think

i think idk tho

[fast ivy]

Yeah, it's a topological space that is Hausdorff, second countable and locally euclidean (i.e basically what you have said)

That's good enough

The interesting thing about manifolds

Is that for each open set

[hallow wasp]

algebraic variety has something to do with solutions to equations but idk

[fast ivy]

You somehow have some kind of linear structure

But only locally

It's that as if each open set

[hallow wasp]

second countable

i forget

[fast ivy]

Has the structure of the real algebra of continuous functions on that open set

[hallow wasp]

but it has something to do with its bases

[fast ivy]

We will make that precise

[fast ivy]

Doesn't matter to motivate sheaves

Only that each open set has some kind of structure of a real algebra

[blazing pawn]

I think the easy definition is sections of a map

For sheaves

[fast ivy]

That's basically what a sheaf does

You have a topological space

And you want to somehow give to each open set

The structure of something else (in the case of algebraic varieties would be of the prime ideal of a commutative ring)

In this case

Each open set has the structure of a real algebra

But not globally

[blazing pawn]

Ok heres how id explain this: fix a topological space X. for each open set U of X a presheaf associates a set F(U), which you can think of as "local information on U" and given an inclusion map V -> U of open sets you have a corresponding restriction map F(U) -> F(V), that is, restricting local information on U to local information on V

a sheaf is a presheaf in which this local data can be glued together

[fast ivy]

Yup

[hallow wasp]

shat is V

what is V

[blazing pawn]

Open set contained in U

[fast ivy]

That's a way better way to explain maybe, because is more explicit and direct I guess

But doesn't give an example

There's the classical one

[hallow wasp]

also F(U) is how you denote a presheaf attatched to an openset?

[fast ivy]

The sheaf of germs of continuous real valued functions on a topological space

[blazing pawn]

visual description

F is the presheaf

it associates a set F(U) to each open set U of our space

one presheaf F has an F(U) for all open subsets U of X

[hallow wasp]

so F(U) isnt apart of the topology?

[blazing pawn]

F(U) is a set

[hallow wasp]

its just a set that is possibly in X?

[fast ivy]

A presheaf is a map that associates a set to each open set of your topological space

And this is really powerful

[blazing pawn]

Its just a set

[hallow wasp]

oh ok

[blazing pawn]

not in X

[fast ivy]

Because gives you stuff like manifolds and algebraic varieties as I said

[blazing pawn]

probably anyway im sure you could come up with some convoluted example

[hallow wasp]

so there exist many presheaves of any topology

[blazing pawn]

the prototypical example here is sections of a map

[bronze pelican]

Hi

[blazing pawn]

Suppose we have a function p: R -> X

continuous

a section of p on an open set U is another function f: U -> R where (p circ f) = id

basically its a local inverse

[blazing pawn]

ok now im bored make pty explain it

namington lmfao

[bronze pelican]

What

Explain what

[hallow wasp]

sheaf

[blazing pawn]

i did all the set up

[bronze pelican]

A sheaf is an upside down covering space

[blazing pawn]

its actually sideways

[fast ivy]

Lmao

[hallow wasp]

wait what

[bronze pelican]

Just watch my video on Reimann Hilbert correspondence

Part 1

[hallow wasp]

p composed f is identity map?

[nimble shuttle]

Reimann

[bronze pelican]

Skip to where I explain sheaves

[hallow wasp]

what does that have to do with sheaves

[blazing pawn]

theyre the usual example

you send each set U to F(U) = set of sections of a fixed map p on U

[bronze pelican]

https://youtu.be/usnQ2BuBM6o
59:00

[fast ivy]

A lot, F(U) is usually denoted as Γ(U,F) which means sections of F over U.

And this comes from this example

Studying sections of, for example, a bundle map is one of the motivations for sheaves.

Not the only one

But surely a really important one

This comes a lot in manifold theory and algebraic geometry

For example, the tangent bundle with its bundle map

I really find the idea hard to grasp

But it's ok

You usually have to have a lot of examples to appreciate this I guess

It really is abstract

I mean, at least I work better with examples.

In the case of smooth manifolds for instance, we have the sheaf of germs of smooth functions.

Say we have a smooth manifold M

So it makes sense

To think about the set C^{\infty}(M,R) of all smooth functions between M and R

This is in fact a real algebra

Since sum, multiplication by scalar and multiplication of smooth real valued functions are still smooth.

One thing tho

When we talk about smoothness

It's a local property

So in order to compare two smooth functions around a point

It's usually very useful

To say that they are equivalent if there exists some small open set around this point where these coincide.

Equivalence class of functions under this relation are called germs

[hallow wasp]

what is they?

two points?

[fast ivy]

I am talking about two real valued smooth functions from a manifold M to R.

So we want to define an equivalence relation on this set of smooth real valued functions from M to R.

And say two functions are equivalent around a point iff there exists open sets U,V that contain this point and there exists a small neighborhood in the intersection where they attain the same value.

This equivalence relation defines what we call germs of smooth functions.

[hallow wasp]

hold

let f,g be two smooth real functions from M to R, f~g if there is U,V where x in U and x in V and x in B subseteq U intersect V where f(B)=g(B)?

sound wrong

[fast ivy]

Yeah, that's right, U, V and B open.

Prove this defines an equivalence relation

You are going to see

[hallow wasp]

and M is the manifold, the second countable locally euclidean, hausdroff topology?

[fast ivy]

That this has something to do with that "restriction map" in the definition of a presheaf.

[fast ivy]

Yup

[hallow wasp]

wait

this feels similar to bases of a topology

[fast ivy]

The idea is that to understand smoothness you only need to know the behaviour of the function around small open neighborhoods.

[hallow wasp]

so where would the restriction map come in?

from [f]:B->R? where [f] is an equivalence class

because originally it was M->R

[fast ivy]

It just takes a functions f : U -> R and restricts it to a smaller open set

That's why it's called restriction map, because of this original example

[hallow wasp]

this is a lot to retain

where would i come across this topic naturally?

all i got is munkres

[fast ivy]

Differential geometry, mostly complex differential geometry

And also

Algebraic geometry

Btw

I have a book here

[hallow wasp]

wym

[fast ivy]

That is a nice reference on the subject

[hallow wasp]

oh ty

[fast ivy]

It is called

"Sheaves in Geometry and Logic"

This is by the man Mac Lane himself lmao

[hallow wasp]

oh scary yellow book

[fast ivy]

Of the founders of Category Theory

Btw

I am using this example of manifolds

Because

I am currently reading this book

And this was the most interesting example he gave in my opinion

[hallow wasp]

ok so my main takeaway is that sheves give important information locally

[fast ivy]

that's a really good take

that's the common intuition everyone has for a sheaf

so you are halfway there

Now you need to spend just a small time to understand how the intuition connects to the formal definition

[hallow wasp]

yes this is where i need to learn more as gomez has said

this is a big reason why I see no resson for me to study category theory

[fast ivy]

and also understand the basic examples like the sheaf of sections of a map and the sheaf of germs of continuous/differentiable/smooth functions

[hallow wasp]

ill miss a lot of intuition if i dont knownenough

[fast ivy]

Category theory is good when you already have worked with a lot of examples and has built some intuition on concrete examples

It serves as tool to bind common similarities between different fields of math

such as common constructions and etc

Like, you see limits and colimits A LOT

[hallow wasp]

yea

i heard of those but idk their formal definition

[fast ivy]

But if you don't know what a product topology is, or a quotient space is or any of that

you won't get why the definition makes sense and is important

[hallow wasp]

are limits and colimits different ways of constructing new objects or something?

[fast ivy]

Basically

like

direct sum of modules, quotient module, tensor product are examples of limits and colimits

constructions in topology like the product space and the quotient topology

and others

[hallow wasp]

what are limits and colomits om category of Sets

[fast ivy]

cartesian product, disjoint union,

quotient sets

when you quotient something by an equivalence relation

[hallow wasp]

oh

[fast ivy]

pullback

[static jacinth]

how do you study math?

[blazing pawn]

it depends a lot on what you want to study

Like "math" is quite broad

[devout nacelle]

It will depend on your background with math, what you want to learn, how much time and effort you're willing to spend, and what you want to get out of it.

[static jacinth]

From the basics until currently. Because I’ve always been the memorization type and i excel at those subjects except mathematics. I’ve pretty much done anything to try & get better. Does your mind have to be open minded or something?

[cold needle]

need to move away from memorizing unless you need to

like yeah there are a few things to just have memorized

[static jacinth]

How do you approach it?

[cold needle]

but ye

hmm

im not sure

i like to read something and try to explain to myself what i understood from it

like

[devout nacelle]

Hmm, I'd say it's about getting used to the techniques and ideas with math rather than explicit memorisation of facts. The two are linked but not the same. In particular, with math you're supposed to be able to use what you already know even in unfamiliar contexts.

[cold needle]

the relationships and techniques and ideas

which will eventually lead to the result

[devout nacelle]

This becomes more and more natural as you practice and learn

[static jacinth]

So i have to go back to the basics of math if I have to?

[devout nacelle]

Also, explaining what you've learnt to someone else, or helping others with problems when you can is a good way of learning too

[devout nacelle]

Sure, it never hurts

What do you mean by basics here?

[static jacinth]

Like primary, middle school things

[devout nacelle]

Okay, you could use Khan Academy for review.

[static jacinth]

Are there others than Khan Academy?

[devout nacelle]

You might be able to find bits and pieces of content on YouTube or the internet, I'm just familiar with KA.

The interface is nice, you can keep a track of your progress so KA is good

[delicate rivet]

Organic chemistry is one if the channel in youtube

[devout nacelle]

Does the OC tutor guy on into that basic content?

[static jacinth]

I’ve tried KA many times but it seems I’m not able to understand it

[delicate rivet]

I think so

[static jacinth]

English isn’t my first language, might be the reason

[devout nacelle]

Ah

You could use subtitles, and slow down the playback speed if that helps?

I also think they give subtitles in different languages for a lot of content?

[static jacinth]

Yeah although I also want to try different resources

[devout nacelle]

You could try looking at Expii, I think they have some content at that level.

Other than that I'm mostly clueless, you could ask for help here in accordance with #❓how-to-get-help if you're facing difficulty with some topic.

It's completely natural to not understand something the first time you see it.

You get a hang of things with practice.

[static jacinth]

Alright, thank you very much for answering!

[devout nacelle]

No worries; goodluck!

[neat lintel]

Does anyone know where one could use a server to do some simulations/calculations on for free?

[toxic schooner]

u can go to #bots and use texit there

[neat lintel]

I should give mor econtext

I was messing around with python and some maths: and stumbled upon this beautiful pattern ```py
import matplotlib.pyplot as plt
import math

i = 0;
v = 1
for n in range(0, 360*v):
i += 1
plt.scatter(math.cos(n) + math.sin(n), i)
print(math.cos(n) + math.sin(n), i)

plt.show()

For v = 1, it gives off, unsure of the english word, "square diamond" like shape

larger values for v seem ot result in shapes similiar to parallelograms (again, unsure of the english word)

for larger and larger values, the shapes seem to get curved. I want to test really larger values for v without wiating an hour or 30 min

[toxic schooner]

that seems kinda cool

But idk which software would let u do it

[neat lintel]

Wtf the stickers are sus on phone

[neat lintel]

if you reply on phone it replies to a different message

random one

[toxic schooner]

oh

so its not only drakes discord thats fucked

[neat lintel]

its for everyone like that

[toxic schooner]

[torn willow]

Oh right,shas doesn't have a phone

[toxic schooner]

yea, i use discord on my lappy so idk about this much

[rose dock]

Perhaps google colab?

[vast surge]

cocalc.com is a thing that exists, though I've always found it to be a bit slow.

[neat lintel]

Thank you both! I found Colab to be more suitable for my intentions

[neat lintel]

: d

[sacred geyser]

when you mess up solving an ODE and accidentally make a Levy flight

the units on both axes are in radius of the earth

[sharp mulch]

[sacred geyser]

the ode is for the equations of motion of a rocket free falling to earth but

not quite there

[sharp mulch]

Indeed

[crystal stream]

I wonder how would it look irl

[sacred geyser]

my friend john sent the code to us and my buddy plotted this, his response was

[hallow wasp]

hihi

what is a levy flight

oh nvm

wikipedia

[neat lintel]

maths is really beautiful when visulized

[rancid meadow]

nGroupoidThesis.png

[narrow rock]

[terse flax]

cat11cat12cat13
cat21cat22cat23
cat31cat32cat33

[summer nest]

Cattrix

Why do weighted blankets cost so much

[vague lion]

just sleep under a mattress

easy

[toxic schooner]

[iron osprey]

https://cdn.discordapp.com/emojis/855875274473865258.png?v=1

[cold needle]

hello friends

i am working on the first few problems from CoM

so far it seems like review from linear algebra cus i remember the inner product space chapter

fun inequalities

[devout nacelle]

[cold needle]

so is this the book that most multivariable calculus class will use after linear algebra?

[neat lintel]

hi

tell me about one of the problems you did

[cold needle]

i did literally the first three problems

just now

so the first one was starred

i think that means its one that will be used

later

[neat lintel]

yes

[cold needle]

i had to show [\norm{x} \leq \sum_{i=1}^n \abs{x_i}]

[fathom swallow]

melia antiqua

[neat lintel]

[cold needle]

so i squared both sides

and removed some terms

and got a bunch of just

absolute value terms

greater equal 0

so i think thats done

[neat lintel]

alright

[cold needle]

its nice that squaring gets rid of any minus

😌

[neat lintel]

so this is saying l^2 norm <= l^1 norm

[cold needle]

o

what is those norms

[neat lintel]

$\ell^p$ norm on $\bR^n$ is $$|x|p = \left(\sum{k=1}^n |x_k|^p\right)^{1/p}$$

[fathom swallow]

TTerra

[neat lintel]

useful in analysis

[cold needle]

ah

okay

[neat lintel]

knowing how to compare these norms is important

[cold needle]

so this is intuitive since square rooting

i guess if i think cube root and so on

it just get smaller

how far can i go

wait

[neat lintel]

[cold needle]

that p is in the exponent

🤔

okay not immediately obvious i cant read lol

interesting

so then if i compare l^3 norm with l^2

ok i guess

i should try this on my own

kind of ugly nvm

wtf

[neat lintel]

[cold needle]

tterra this is scary

[neat lintel]

restrict to p >= 1 for simplicity

for p < 1 these things get weird

[cold needle]

ok