#serious-discussion

1 messages · Page 297 of 1

neat lintel
#

you see ai has gone so advanced they can deliver both food and airstrikes to your house

outer crag
#

Would you prefer food or an airstrike?

#

The choice is yours.

neat lintel
#

I prefer neither

outer crag
#

Interesting.

neat lintel
#

you're probably going to do airstrike if I choose one

outer crag
neat lintel
#

I don't judge i'd do the same

#

UAV would first be deployed within your area about 100 meters out to avoid electrical interference

outer crag
outer crag
#

No promises lil bro.

neat lintel
#

then a ballistic airstrike could target your nearest or exact location and do the job

outer crag
#

You know a lot about this topic.

#

I have my suspicions about you.

neat lintel
#

I refuse to talk further for I believe that you're

#

something

#

hey

#

@outer crag

#

would you like a quick recipe on a fun device???

outer crag
#

No.

#

I don't advocate or support hooliganism.

zealous garden
tawny mortar
#

if you train it hard enough. it's possible

fresh comet
#

@tawny mortar welcome to the mathcord! nachoWaves

tawny mortar
ocean harbor
outer crag
ocean harbor
#

good 4 u

outer crag
#

🫶

sonic prairie
#

morning guys

neat lintel
#

not screaming the keyboard's caps lock was on

#

xd

livid night
#

Its 0400

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Well 0415

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Why am I up

bold carbon
livid night
#

Quite the opposite, I fixed it... although im not enjoying it 12

neat lintel
#

I just woke up good morning to y’all and evening and night to others

ocean robin
#

good morning everyone :]]]]

serene vault
#

evening folks! (and morning/afternoon to ppl in other parts of the world)

latent edge
serene vault
#

wot

latent edge
serene vault
#

cute but also kek

latent edge
serene vault
#

returned from a day of depresso

latent edge
serene vault
#

currently doing it hahah

#

but also waiting to snipe amann and escher

latent edge
#

no baby gurl rudin?

serene vault
#

rudin's functional analysis?

latent edge
#

real analysis

serene vault
#

i didn't know about this book

latent edge
#

now u do

#

:3

serene vault
#

would it be necessary if i already am aiming for amann?

latent edge
#

idk amann

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i only read rudin

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some people try abbott

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i think it doesnt matter

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as long as u are learning the same stuff

serene vault
#

that's fair

latent edge
#

seiya :3

serene vault
#

:3

rocky shuttle
#

seyana

serene vault
#

sorena

rocky shuttle
#

soyana

latent edge
#

wakaruuuu

rocky shuttle
#

shirankedo~~

compact surge
#

nyanpasu

ocean robin
latent edge
#

its u

ocean robin
#

true!!!!

latent edge
#

up to homeo

#

:3

ocean robin
#

yuh‼️‼️

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speaking of homeomorphisms i need to study more topology EB_EeveeDizzy

livid night
#

I need

vivid quest
#

You need

ocean robin
#

i need

serene vault
#

need

latent edge
#

need

pure hollow
latent edge
#

:3

pure hollow
#

:3

ocean robin
#

:3

gaunt harbor
#

Is there any explaination for how did the ILATE priority order for by parts integration come from?

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Is there any significance at all for it?

solid yarrow
#

Hello

faint light
#

and since they're hard to integrate you dont want them as your v term

gaunt harbor
#

so technically

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if i messed up the order it'd be correct

faint light
#

also, ones which have a nice derivative take priority since you have to integrate u'v

gaunt harbor
#

Basically, what makes calculations easier, that has more priority?

faint light
#

i cant think of an example rn

gaunt harbor
#

true

#

arctan?

faint light
#

yeah smthn like that maybe

glacial lark
#

i was reading through some Oxford Maths admissions stats and it talks about calculating a students PSI score, what is this and how is this used?

clever adder
#

we are friends

novel apex
#

man i should frequent discussy 2 more often. vibes are immaculate

old oak
#

It's the superior discussy, yes

solid yarrow
old oak
solid yarrow
old oak
solid yarrow
old oak
solid yarrow
faint light
clever adder
#

yeah...

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I am your friend

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Could you help me?

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where is johnny?

old oak
#

Just reacting from afar 👻

junior veldt
#

oh wait owlcider do you know about distributions?

pure hollow
#

~~ you mean better functions ~~

junior veldt
old oak
junior veldt
#

ah ok, i meant the analysis kind

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i've been reading about them recently

old oak
#

I figured

junior veldt
#

they're very cool :)

pure hollow
old oak
pure hollow
junior veldt
#

mhm, right

old oak
#

The language of measures has served me well enough

junior veldt
pure hollow
#

Woah that's new

junior veldt
#

-# is it??

#

yeah distributions have helped me understand the yoneda lemma even better

pure hollow
junior veldt
#

oh i misinterpreted

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i thought you meant that me doing category theory was new opencry

pure hollow
pure hollow
junior veldt
#

they're not mutually exclusive~

#

but yeah it turns out distributions are to functions as presheaves are to objects

barren wind
#

what do you guys think of the 369 nikola tesla theory

vivid quest
#

I think that i don't know what the 369 nikola tesla theory is

barren wind
#

this guy

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is nikola tesla

vivid quest
#

This is a fail

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Also i still don't know tnwhat the 369 theory is

granite hound
#

Just got kicked off of one of the only picnic tables in my part of my city because it's apparently on private property

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I've been using that table for over a year and nobody has ever gotten upset about it before, hell I don't think I've ever even seen someone use it before

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(Its at the edge of a park-like area that I always assumed was public)

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But this old guy threw a fit about me sitting there reading a book, so I had to leave and find somewhere else to sit :(

vernal token
#

This book uses so much mathfrak..

Screenshot_2025-07-29-23-49-04-457.jpeg
granite hound
#

Anyway, venting complete, back to book

vernal token
#

It feels like there is not a page without there being some line full of mathfrak stuff bleakkekw

topaz trout
#

When does math become meth

granite hound
#

I lied I'm still pissed off

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Like there's nowhere to sit outside in this part of town because people like him hate the idea of homeless people existing

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So sorry I didn't notice the tiny private property sign that's facing the other way from where I came from

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And I left as soon as he told me! He didn't have to keep going on and on about it

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And frankly that table is closer to the sidewalk than it is to the paths inside their private little park (With no fence btw), so part of me is pretty sure that they put that table there for people to use

verbal quest
#

why do they have a picnic table on their private property next to a park
or is it like museum or something

granite hound
#

No it's an old people condo next to a church

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So I mistakenly though that the park, like the church next to it, was open to the public

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But I guess it really is more connected to the condo than it is to the church

verbal quest
#

you should donate a table to the church to set up right next to theirs
as an F U
because you know what will happen🤡

granite hound
#

That'd be funny, but I don't think I could really make that work

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And I'm literally moving in 2 days anyway

verbal quest
#

ok, if you're leaving anyways
new plan
put up signs for a potluck picnic at that table on the day you're leaving

granite hound
#

Lmao maybe

#

But again, this is not the first time I've hung out at this table and this is the first time anyone's made a stink about it so I'm pretty sure it's the minority of people there with sticks up their asses

verbal quest
#

yea, or someone left a mess recently
or both

manic shell
ocean harbor
jaunty ibex
errant zenith
#

Hello

solid yarrow
jaunty ibex
jaunty ibex
junior veldt
elfin smelt
undone pike
vast wraith
rocky shuttle
#

mathfrak for integration variables bleakcat

errant zenith
#

Cat

vocal ibex
#

Its math?

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Gng who is still learning

mint canopy
vocal ibex
#

Idk

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Nothin

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Im boutta go to 10nth

serene vault
#

welcome to the mathcord!

errant zenith
#

welcome to the mathcord!

vocal ibex
solid yarrow
solid yarrow
vast wraith
#

Stillwell is awesome

dreamy ore
#

the hyperbola the worst quadratic curve

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actually everything other than the ellipse (and circle) sucks

burnt ledge
#

No!! No!!!!!!

dreamy ore
#

yes

#

ellipses are awesome

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hyperbolas blegh

serene vault
burnt ledge
#

Hyperbolae are fine :-(

dreamy ore
#

theyre fine i guess

#

not as cool as the ellipse tho

vast wraith
burnt ledge
#

circles are good but ellipses can go right to hell

dreamy ore
#

what!!

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ellipses are just circles but more fun

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theyre like circles if circles were a pain to deal with

burnt ledge
vast wraith
#

i mean… true

burnt ledge
#

theyre like if a circle had nothing good about it

dreamy ore
#

hyperbolae look boring, feel boring, and have boring properties

vast wraith
#

tfw arclength of an ellipse

dreamy ore
#

ellipses on the other hand

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are awesome

#

-# also im biased cuz ive done far more work with ellipses than hyperbolae

#

-# like ive done a lot of stuff dealing with the earth

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-# and the earth is an ellipse

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-# ...oid

vast wraith
#

ellipseoid

burnt ledge
#

ellipsoids at least sometimes have something going for them

dreamy ore
#

:3

#

if e = 0 then π_0 = 3.14159....

vast wraith
dreamy ore
#

if e is not 0 then well

#

we can figure that out

burnt ledge
#

Circles vinidacted once a-maybe-gain

vast wraith
#

ok but what if i want the arclength between any two given points on the ellipse

dreamy ore
vast wraith
#

imagine living in the real world pff

dreamy ore
#

im naming my daughter ellipse

#

if i ever have a daughter

vast wraith
#

also guh i was supposed to practice byee

dreamy ore
#

byee

foggy meadow
burnt ledge
#

True index: over 9000

burnt ledge
dreamy ore
#

anyway hyperbolae are just wannabe ellipses

latent edge
#

circle > hyperbola > ellipse

burnt ledge
#

circle > parabola > hyperbola > ellipse i think is the complete ranking

latent edge
#

i still agree

dreamy ore
#

circle > ellipse > parabola > hyperbola

raw cedar
dreamy ore
#

grrr

raw cedar
#

i wouldve ranked it this way

dreamy ore
#

my ranking makes most sense cuz its in increasing eccentricity order

latent edge
#

Hyperbolas are the closest to my heart

dreamy ore
#

ellipses are just fun

#

hyperbolae are unbounded and unfun

latent edge
ocean harbor
plain seal
vivid halo
#

Hyperbolic 3-manifolds are friend shaped

heady pagoda
#

I am taking refuge here until the rage bait tsunami abates

rocky shuttle
#

I believe in 鹿乃ism, the Deer God

heady pagoda
rocky shuttle
heady pagoda
pure hollow
#

nG I gotta know why does number theory require a lot of more complex math?

vivid halo
#

that said number theory is a very old subject and almost every other area of mathematics turns out to be useful in one way or another to problems in number theory

pure hollow
desert breach
#

bruh

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i cant send gif?

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awh hell nah

vivid halo
#

algebraic geometry, symplectic geometry, mathematical physics...

#

these also use a lot of heavy machinery

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these fields overlap with number theory a lot but there is plenty of stuff in these fields which is more separate from number theoretic motivations

pure hollow
#

Makes sense

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Math is indeed vastnozoomi

vivid halo
#

but yeah number theory in particular benefits both from really heavy algebraic geometry machinery and also lots of hard analysis input

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and often you can get the two to play off each other

pure hollow
vivid halo
#

yeah absolutely

#

maybe not "hard" analysis but the foundations of analytic geometry are being completely rewritten as of late, mostly motivated and informed by number theory and arithmetic geometry

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one reason why a lot of modern number theory relies on such heavy machinery is that people have become quite comfortable with the strategy of translating problems in number theory into more geometric problems, and a lot of the spaces that arise most naturally by doing this end up being pretty exotic and require rather strong geometric foundations to work with properly

#

this strategy has been incredibly successful historically it makes pretty heavy demands on formalism

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Grothendieck and his school developed the modern approach to algebraic geometry through schemes for this sort of reason, there were explicit problems in number theory (the Weil conjectures) which exceeded classical methods

pure hollow
vivid halo
#

depends on background lol

pure hollow
#

Like?

vivid halo
#

I mean if you want to learn the basics of modern algebraic geometry you have to go through something like Hartshorne or Vakil

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and you would probably want to learn that material before learning any other more exotic flavors of geometry that show up in number theory these days

pure hollow
vivid halo
#

this is a completely psychotic recommendation if you haven't gone through algebraic geometry or been exposed to more classical stuff on analytic geometry before, but I would suggest the various notes and lectures from Clausen and Scholze

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this is the most modern and least shitty way to approach analytic geometry and it fixes almost everything that was wrong with all the previous approaches

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if you want something like p-adic geometry specifically then the Berkeley lectures on p-adic geometry from Scholze-Weinstein are really good (required reading for people who later want to understand this Fargues-Scholze paper geometrizing p-adic local Langlands)

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the adic space/perfectoid space approach to p-adic geometry in those notes has been around for a lot longer; that formalism still has some issues, but it's more than expressive enough to do a ton of geometry and number theory

junior veldt
#

hiiii nG

pure hollow
junior veldt
#

and hi convergence!

pure hollow
junior veldt
#

hru

vivid halo
pure hollow
junior veldt
#

yayyyyy

#

same

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i made a new thread :3

pure hollow
junior veldt
pure hollow
junior veldt
#

i think it's accurate

pure hollow
#

Perhaps

junior veldt
#

why wouldn’t it be?

pure hollow
junior veldt
pure hollow
junior veldt
#

and monic!

pure hollow
junior veldt
#

:3

fathom swallowBOT
#

Gave you the Role that does nothing selfrole.

gentle palm
#

Hello, i am new to this server but i am looking forward to lurking around and reading, and maybe on the occasion, actually saying something :)

#

Also, i wanted to ask since this seems like a good place for it, does anyone have any advice for doing math in terms of ergonomics? Like how to do a lot of math without hurting your wrist or eyes.

#

I personally struggle with writing math, especially for long times or quickly, so i use latex and typically type my notes in lectures. Its the only way i can keep up with the pace of the lecturer.

But recently i discovered that my drawing tablet stand (for digital art) fits a notebook on it perfectly and it also collapses in my bag very nicely. It does unfortunately mean that my notes are horizontal and you have to use graph paper instead of lined, but i find it minimizes neck and back strain since i often hover over my math with my eyes very close to my paper. I imagine a nicely-shaped wood block could serve the same purpose, so id recommend it for anybody who wants to do math comfortably but does not have giant blackboards at perfect standing eye-level for doing math on at home :)

bright hill
#

is it that different from riemannian geometry?

#

you essentially just study smooth manifolds endowed with a special type of 2-form, right?

vivid halo
#

in the Riemannian situation you study smooth manifolds equipped with nondegenerate symmetric 2-tensors, these give you metric tensors on tangent bundles and then you can do all the usual differential geometry stuff with curvature and so on

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in the symplectic situation you don't have local invariants like curvature by Darboux's theorem, so you really only have topological invariants in the end

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symplectic geometry is much more rigid as a consequence

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not quite as rigid as complex analytic geometry but close!

bright hill
#

what're the usual motivations for symplectic geometry? just physics?

#

all I know tbh is that you it's related to hamiltonian systems or smth

vivid halo
#

not just physics

bright hill
#

whatever that might be

vivid halo
#

although physics is a big reason why these things are important, phase spaces of classical mechanical systems are symplectic manifolds

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a lot of the techniques originally motivated by physics are useful in other areas though

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a large part of symplectic geometry is related to stuff like mirror symmetry and enumerative invariants in algebraic geometry

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like here's a really nice enumerative result that really kickstarted things like Gromov-Witten theory

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compute the number N_d of rational curves of degree d passing through 3d-1 points in general position in CP^2

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N_1=1 (one line passes through two points) and N_2=1 (one conic passes through five points) are ancient results

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N_3=12 (twelve cubics pass through eight points) is also pretty old but much harder

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N_4=620 is like, borderline impossible with classical methods

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all the way up through the 1980s N_5=87304 was the best result on this particular enumerative problem

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well then in the 1990s Kontsevich just computed the whole generating function all at once

image.png
bright hill
#

ok this is lowk goated

vivid halo
#

the result follows by the associativity of the cup product in the quantum cohomology ring, which is an extension of the cohomology ring of a symplectic manifold

#

just as the usual cup product encodes intersection numbers of submanifolds, the quantum cup product encodes "quantum intersection numbers" of submanifolds (connected by pseudoholomorphic curves, counted by Gromov-Witten invariants)

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in particular even though symplectic manifolds have no local invariants and only topological invariants, these invariants are really rich and often interact rather deeply with quantization

haughty lion
vivid halo
haughty lion
#

Wait every manifold has a metric

vivid halo
#

Kahler manifolds are quite well studied, they simultaneously carry complex, Riemannian, and symplectic structure

vivid halo
haughty lion
vivid halo
#

Kahler is one of the handful of special holonomy structures a manifold can have

#

special among Kahlers are Calabi Yaus

haughty lion
vivid halo
#

you can often study moduli spaces of metrics and say nice things about these

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the case of conformal/complex metrics on Riemann surfaces is quite well studied in particular

velvet dagger
#

Especially closed hyperbolic n-manifolds (as long as n = 2)

haughty lion
latent edge
#

b...but n=3

vivid halo
#

sure

#

I mean people also study all the distinct smooth structures you can put on topological manifolds

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S^7 has 28 distinct smooth structures

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R^n has a unique smooth structure unless n=4 in which case there are uncountably many

haughty lion
vivid halo
#

4-dimensional topology is fucked up lmao

#

like uniquely so for a lot of reasons

haughty lion
#

I thought it was just some weird 4-manifolds which have lots

velvet dagger
# haughty lion Even R^n?

The fact that there seems to be a "God-given" one is less a function of uniqueness and more, there's one we interact with. But you could in principle just start writing down a metric and it can work

haughty lion
vivid halo
#

Milnor is great

#

I remember working through the hypersurface singularities stuff for a course once

haughty lion
#

I'm just scarred from even approaching papers after peeking into kervaire's paper on the 10-manifold with no smooth structure

junior veldt
#

the what

haughty lion
#

this paragraph

image.png
junior veldt
vivid halo
junior veldt
#

ok

#

yay :)

vivid halo
#

there's plenty of physics that can only exist in the case d=4 :^)

haughty lion
#

me and a friend thought it was a forgotten assumption

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maybe I can try again after another semester's worth of topology

junior veldt
jaunty ibex
neat lintel
#

Any one plz give me best Organic chemistry doubt asking server

jaunty ibex
serene vault
latent edge
#

Seiya :3

serene vault
#

:3

latent edge
warm lance
#

organic chemistry tutor

neat lintel
iron nimbus
neat lintel
alpine copper
undone pike
#

is our universe really 4d though

undone pike
vivid halo
vivid halo
#

and certain consistency conditions/anomaly cancellations only hold in this dimension

undone pike
#

hmmm ok

#

what do you mean by consistency conditions? jc

vivid halo
#

when you try to define various field theories and try to perform quantization one thing that can stop you from doing this are various anomalies

#

without certain anomaly cancellations you can't produce well-defined quantum field theories and you can't extract well-defined numbers from them

#

Bosonic string theory for example has a conformal anomaly if you try to run this construction on an arbitrary spacetime manifold, but the anomaly cancels in 25+1 dimensions

flat harbor
#

i dont think lorentz invariance is real

vivid halo
#

Yang Mills is special to dimension 4 because that's the dimension where curvature 2-forms have Hodge duals which are also 2-forms, otherwise it doesn't even make sense to write down the Yang-Mills functional

solid snow
#

when the hell do i get to this fancy physics

vivid halo
#

nlab

solid snow
#

Oh hell naw

vivid halo
#

nlab is quite good for learning physics if you are mathematically inclined

flat harbor
#

any decent qft book

solid snow
#

zoo wee mama

flat harbor
#

amazon book store, perhaps

undone pike
undone pike
pastel blade
undone pike
#

someone made it for me

pastel blade
undone pike
foggy meadow
livid night
#

Goodmorning chat

restive salmon
#

Gmm

jaunty ibex
pure hollow
silk hearth
#

Hello, how’s it going?

#

Hope everyone is having a nice day!

cold patrol
cold patrol
silk hearth
#

Likewise friend!

cold patrol
silk hearth
#

Sorry no, why do you ask?

cold patrol
silk hearth
#

Oh, Its just a common name, I suck at football

cold patrol
#

really?

silk hearth
#

Anyways, just wanted to say hi

cold patrol
#

btw, nice to meet u

silk hearth
#

Nice to meet you too, sky

cold patrol
silk hearth
#

I’m boring, don’t have many

#

How about you?

jaunty ibex
cold patrol
silk hearth
#

It’s okay friend, math is interesting enough right?

cold patrol
#

i think you are injo

silk hearth
#

Injo? What does that mean?

cold patrol
silk hearth
#

México

outer crag
#

Welcome to the server @silk hearth

silk hearth
#

Thank you! Excited to be here!

cold patrol
silk hearth
#

How are you doing ysst?

#

I’m confused sky, why do you think we are bots?

cold patrol
#

arent you?

#

let you prove you r human...........

silk hearth
#

I don’t think so, at least

rocky shuttle
#

the only bot I see here is higher!

cold patrol
fresh comet
silk hearth
#

A vocaloid fan!!

#

Are you not?

#

(Zen I mean, not higher!)

rocky shuttle
#

She covers a lot of vocaloid songs, though

silk hearth
#

My apologies, I’m not really familiarized with her, I’m just aware of her existence

#

So…

#

May I ask why is higher! a bot or…

#

I haven’t unlocked that server lore yet

rocky shuttle
vocal vessel
#

Higher is a benevolent android

silk hearth
#

Boredom

#

Like android 16? He protects nature and gives pep talks to teens and stuff?

vocal vessel
#

And kills squirrels.

#

...cant be too perfect

silk hearth
#

The squirrels had it coming man

#

They are turning carnivorous

rocky shuttle
silk hearth
#

Hippos are evil, so that’s okay too

vocal vessel
#

They spin their tails when they shit so it sprays everywhere.

old oak
silk hearth
#

Well, I was referring to them being extremely violent

vocal vessel
silk hearth
#

But yeah, that too

vocal vessel
#

The ultimate evil

silk hearth
#

You know a creature is bad when someone named MalevolentAntichrist refers to it as “The Ultimate Evil”

vocal vessel
old oak
old oak
vocal vessel
#

Oh perhaps she does. But so does higher. They even have photographic proof

vocal vessel
silk hearth
#

Omelette du fromage

#

🤌

#

Found It

cold patrol
silk hearth
#

Busy.

spice ginkgo
cold patrol
#

let me introduce myself

spice ginkgo
cold patrol
#

iam 26 years old and live in San Francisco

spice ginkgo
#

go on

cold patrol
#

man

#

how about you?

cold patrol
glass gull
#

can someone explain function spaces to a 17 yo like me?

#

and then explain what it means for a function to be dense

cold patrol
glass gull
#

yeah

#

is that it?

cold patrol
#

Just like we group numbers into sets (like integers, fractions, real numbers), we can group functions into spaces based on rules they follow. For example:

#

All functions that are continuous on the interval [0,1]

glass gull
#

oh okay

#

that sounds straightforward enough

#

so what does it mean for a function to be dense

deep mango
#

a function space specifically is a set of functions where the functions can be added and multiplied by numbers without going out of the space

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so its a vector space of functions

#

usually function spaces are sets of functions that share a certain property

#

like continuity, or integrability, or something like that

deep mango
#

a subset of a function space can be dense in the function space

#

that means that any function in the space is the limit of a sequence of functions just from your subset

cold patrol
#

individual functions are not dense on their own.

deep mango
#

so for example, power series give you sequences that converge to lots of functions

#

it turns out that the stone-weierstrass theorem says that the set of polynomials is dense in the space of continuous functions

cold patrol
#

"Dense" applies to sets of functions, not single functions.

deep mango
#

on a closed interval

#

similarly, fourier analysis tells us that the set of finite fourier series (finite sums of cos(nx) and sin(nx) are dense in the space of L^2 functions on [0, 2pi]

#

(those are the functions whose square is integrable)

#

in particular, bounded functions

silent junco
#

source? sotrue

deep mango
#

no

glass gull
deep mango
#

if V is a function space, and U is a subset of V, then U is dense in V if...

#

whenever f is in V, theres a sequence of functions fn in U such that fn converges to f

#

this is what density always means for subsets

glass gull
#

oh okay i think im getting it

#

what does it specifically mean for a subset

silent junco
#

do you know how the real numbers are constructed from the rational numbers?

deep mango
#

yeah rays has a good point, the rationals are dense in R

glass gull
#

oh

#

i do not

glass gull
deep mango
#

oh

#

just that every element of U is also an element of V

glass gull
deep mango
#

like U is smaller than V

glass gull
#

oh i see

#

and it must contain the same elements

#

or no

deep mango
deep mango
glass gull
#

okay that makes sense

#

thank you!

deep mango
#

so you mention universal approximation theorems in your bio

#

thats another density theorem

glass gull
#

yeah thats why i was learning about it

deep mango
#

universal approximation theorems say which function spaces neural networks are dense in

#

neural networks are special functions

#

they form a subset of different function spaces

glass gull
#

like the output of a nn?

#

okay..

deep mango
#

the question is, how well can we approximate different functions with a neural network?

glass gull
#

well theoretically to any desired degree of accuracy

deep mango
#

a. is there a sequence of nns (that you could get from, say, training) that converge to a function you want?

#

b. how do we find that sequence (how do we train)

glass gull
#

UAT only guarantees existence therefore cannot answer b

deep mango
#

part b isnt answered by density. density just says there is some sequence, not how to find it

glass gull
#

or am i wrong

deep mango
#

yes

#

thats right

glass gull
#

okay

#

so all density theoresm are existence theorems?

deep mango
#

ummmm

#

yeah i think so

#

maybe

#

yeah cause density says "for every function in V, there exists a sequence in U..."

#

so yes

glass gull
#

do you study UAT in math.. or did you do deep learning too

deep mango
#

not really, i just started learning about it recently from someone else im working with who does ML

glass gull
#

interesting

#

i know what the theorem states im just trying to learn the math behind it

#

and why the proof works

#

the original proof was proved on sigmoidal functions using contradiction

glass gull
deep mango
#

yes if you allow general nns

glass gull
#

whats a general nn?

deep mango
#

like any neural network

#

i dont know if you specify to specific types

glass gull
#

okay i get what youre trying to say

#

so uh

#

question 2

#

whats a discriminatory function

#

why cant i paste images

#

bruh

deep mango
#

not sure

solid yarrow
#

why are infinte products so hard to calculate ._.

#

welp

#

series were not easier

latent edge
#

Just take logs :3

solid yarrow
#

why am i even complaining ._.

#

well.

#

not in thtis case

latent edge
#

then it becomes a series problem

solid yarrow
#

it was a smart move

latent edge
#

Now you cry cuz it's hard

solid yarrow
#

witht factorization

latent edge
#

can you show me?

#

I'm curious

solid yarrow
#

i didn't see that

#

i am scared of having to evaluate more of those

#

not sure i wanna continue reading that ca book

#

:(

#

i hate calculating stuff like this

latent edge
#

Reminds me of Euler product calculations

solid yarrow
#

welp

#

the chapter is about some euler product

#

i thought it sounded fun

#

yeah

#

...

#

fun

latent edge
#

It is fun but yeah it's messy calculation

solid yarrow
#

though i also don't just wanna skip the exercises

#

i am trapp

latent edge
#

would take it any day instead of this

image.png
solid yarrow
#

wtf

#

this looks even worse

#

._.

latent edge
#

It is

#

I am trying to find a way to simplify the product

solid yarrow
#

:(

#

good luck

#

i am thinking of skipping some parts of the book

#

not sure how i'll do that

#

usually i just start and go linearly

#

._.

#

i could go to the section that sound the most fun

#

though i will most likely have to backtrack lots of stuff

#

studying with book hard

vivid halo
solid yarrow
#

oh hi nG

#

how's the espresso addiction game going?

latent edge
vivid halo
solid yarrow
# vivid halo what

didn't you tell me a while ago, that your workplace had an espresso machine

#

and that you

latent edge
#

just noticed something actually

solid yarrow
#

used it

#

too often

vivid halo
#

yes but I'm not in my workplace these days since I'm on vacation

solid yarrow
#

oh

#

i didn't know that

#

sowwy

#

._.

latent edge
#

how did the 1-X^(-alpha) in the denominator go away

image.png
#

I never questioned that

#

I just took what my prof did for granted

vivid halo
#

lmao

#

I dislike the tildes over everything

#

otherwise it would be so clean

nocturne glacier
#

The room you are in suddenly grows dark, and a figure appears before you. Before you can react or ask a question, the figure extends his own query. There are two pills presented to you. One is a strong Blue color, and the other a deep Red.

"I stand before you, and every human being on your planet. I present them the same question, and that is this.

“If you you take the Blue Pill, and over fifty percent of the human race chooses as you do, then all will live."

“If you take the Red Pill, then you are guaranteed to live, but if more than fifty percent of the human race chooses as you do, those who chose the blue pill will perish."

Which pill do you reach for?

Alternative question: Which choice saves the highest estimate amount of lives assuming 8 billion people choose randomly?

Bonus question, what does the average probability of people taking the blue pill have to be for it to be the better choice for you to save as many people as possible on average?

iron nimbus
#

@nocturne glacier welcome to the mathcord!

iron nimbus
latent edge
#

Then they were like "well I put the tildes to indicate it's different and might conflict with the normal hecke"

latent edge
#

You mentioned Euler products

jaunty ibex
jaunty ibex
vivid quest
#

automorphism

ornate bridge
#

@vivid halo recently you said you usually google when you think about a problem a little and don't find a solution; what do you do when you can't find anything? Or does that usually not happen

vivid halo
#

Use noggin

latent edge
#

exactly

ornate bridge
# vivid halo Use noggin

so after you google because you thought about it for a little (you said it can be as little as 5 minutes) and don't find anything, that motivates you to think about it more yourself

vivid halo
#

Uh yes lol

ornate bridge
#

Are there some searching techniques you picked up over the years?

vivid halo
#

Mostly just know how to use Google properly a lot of people don’t know how to Google properly

ornate bridge
latent edge
#

tfw I was whining 3 hours ago

#

Now I get it

#

I didn't use noggin

junior veldt
#

classic mistake

latent edge
steel blade
marsh blaze
latent edge
#

@vernal token

#

Remember the arctan sum over fibonacci?

#

I think I figured out a messy way to solve it

haughty lion
solid yarrow
#

Vol. 2

latent edge
solid yarrow
#

oh good?

#

or oh bad

latent edge
solid yarrow
#

lmao

latent edge
#

@solid yarrow i figured out something in my research

#

Some progress eeveekawaii

junior veldt
#

nice!!

solid yarrow
#

proud of you

#

what did you find o.O

next quest
#

"analysis" is the most nondescript name of a math field i've ever seen, who named this

solid yarrow
next quest
#

also "groups" "rings" "fields" "magma" ts pulled out of a hat

glass gull
#

whats a measure in measure theory? i dont understand it

solid yarrow
glass gull
#

like a way to quantify things

cold patrol
#

is the way to assign a size

glass gull
#

wdym?

solid yarrow
glass gull
#

i do not

solid yarrow
#

here you go

#

for future reference... just look it up ._.

glass gull
#

i already saw the definitions

solid yarrow
#

yeah

glass gull
#

i just didnt understand it

solid yarrow
#

so what are you struggeling with then?

glass gull
#

so i came to discord

solid yarrow
#

what did you not understand

glass gull
#

wait

#

what does it mean to generalise

#

measures

#

like length

#

etc

solid yarrow
#

i am not sure what you mean?

glass gull
#

the reason i dont understand is prolly bc im pregrad

solid yarrow
#

i am also pre-grad

#

._.

#

if pre-grad means pre-grad-school

glass gull
#

i cant paste images

#

bruh

solid yarrow
#

no image perms

glass gull
#

In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events.

solid yarrow
glass gull
#

what does generalisation mean here

glass gull
#

no i mean like

#

im literally not in uni

solid yarrow
#

oh

glass gull
#

i need this bc im trying to understand the proof behind

#

the universal approximation theorem

#

for a project

#

so after understanding measures im probably going to ask you what a signed measure is

solid yarrow
#

so you want to know why a measure is a generalization of geometrical lengths?

#

or what is the question

#

i don't fully understand i believe

glass gull
#

do you want to know like the bigger question

solid yarrow
#

sure?

vivid quest
#

A fox in the chat

solid yarrow
vernal token
vernal token
#

It's funny that you continued trying it, I thought you found it dull

errant zenith
iron nimbus
vivid quest
#

They do that all the time to hunt in the snow

#

So that the prey can't go away

delicate kernel
#

they do it to catch mice/voles running around in tunnels in the snow that they can hear

vocal vessel
#

Mice + voles = moles

delicate kernel
#

oh I can't send gifs apparently 🥲

split marten
#

When you think about it mathematically, do you think God exists? According to Gödel's approach, it seems like He does, but of course that's not enough; it would be strange to attribute existence to Him.

solid snow
#

lol

split marten
cold lantern
split marten
cold lantern
eternal crow
#

i believe in God

vivid topaz
#

what is Mathematics even about

split marten
#

And what are you basing your opinion on? I'm curious about the views of theists (I'm not an atheist).

solid snow
#

the answer is that nothing godel did proves anything about god

cold lantern
eternal crow
#

I've made a conclusion that the most fundamental things of existence are axioms. Like even physical things were defined into existence perhaps by words

solid snow
#

well no

eternal crow
#

and in the bible, God brought things into existence with his words "let there be … (something)"

solid snow
#

bro forgor

eternal crow
#

so ya, I've concluded maybe that's just it

cold lantern
#

You have to differentiate logical/mathematical axioms and "axioms" in other fields like Physics or Biology, which more often than not would be called laws.

eternal crow
#

idk the godel stuff. i think ive seen a short about it once but i forgor

cold lantern
cold lantern
split marten
#

Personally, due to the limitations of our logic and reasoning, I try to think of the thing called God as something beyond us and beyond that cycle. Personally, I don't think such a thing can be perceived, but I don't want to attribute any qualities to it in the full sense.

#

Don't hesitate to inspire me where I am lacking.

cold lantern
#

I agree and share the same point of view, as I wrote above "I don't believe in the typical gods, e.g. Christian, but I'm not ruling out the existence of a higher power."

#

Are you religous / do you believe in one specific god?

split marten
#

And what I wonder is, if this is something like the limits of human thought, what exactly should humanity strive to develop? To be honest, the main reason I turned to mathematics was my instinct to try to understand the absolute in some way. I may be younger than the average age here, but my desire to be certain about things is very strong.

split marten
cold lantern
split marten
cold lantern
cold lantern
split marten
#

If there is an absolute limit (not being able to comprehend God), or if we can say that it is not possible to perceive the third dimension from the second dimension, then in your opinion, in which direction should mathematical and abstract research go?

cold lantern
split marten
lapis rock
#

1+1=?

split marten
cold lantern
lapis rock
#

you smart

split marten
lapis rock
#

11+11=?

split marten
#

its hard

lapis rock
#

no so ew

#

ez

split marten
#

i does not want to respond

lapis rock
#

=1111

split marten
#

Actually, I was going to say that, you know 😕

cold lantern
lapis rock
#

Don't worry if you are a beginner, come to my house and I will teach you for free.

#

😂

#

No, brother, I teach mathematics for free.

split marten
cold lantern
#

Your talking points are vague. What do you by "third dimension calculated by the second dimension"?

lapis rock
#

My friend, I am a mathematician, don't worry, I will explain to you.

#

When the first dimension meets the second, a break occurs, resulting in a third dimension, which returns to the first dimension.

cold lantern
#

If you mean comprehend visually then no. If observer and space are 2D the observer cannot "see" the third dimension.

lapis rock
cold lantern
split marten
chrome sapphire
#

💩

lapis rock
chrome sapphire
#

valorant ?

lapis rock
#

do you have roblox

cold lantern
#

We have to differentiate between our notions of limitations. 1) We cannot physically comprehend or do something at all no matter what we do. Example seeing 4d like 3d. Write a number out with 10^10000000 figures because we don't have enough ink or in that case atoms. 2) Things we didn't prove yet like the Riemann Hypothesis or other open theorems. -> We can't do anything about the first one but we try to push boundaries and solve them in the second case

split marten
cold lantern
#

Thinking about this is nice, but that's not what mathematicians do. We leave that to the philoophers. Mathematicians have different goals and it is the second point in my earlier answer.

#

Also you have to learn to be precsise and ask precise question. That's the cement of all of mathematics. In my opinion this discussion is very vague, I don't even really understand your points/questions.

split marten
cold lantern
#

This is going to be my final message, I have to leave. 1. If you are asking about things which have nothing to do with math and seem impossible e.g. proving existence of God, flying etc. you can philosophize about that, but mathematics isn't going to help you directly and it's not the aim of mathematicians. 2. If you're asking about things which in the context and realm of mathematics seem impossible, i.e. to solve or prove then that's the entire point of mathematical research - there is no general answer on how to approach these problems.

split marten
valid hornet
#

Godel proposed an argument for God based on the concept of a positive property, but the specific version he presented leads to model collapse, I think.

#

There are other versions that avoid it, though. I'm not as knowledgeable on that type of argument.

valid hornet
zealous garden
#

Either way Newton was off his rocker

#

Smart guy but out of it

foggy meadow
#

I mean that fits the middle ages man.

#

There were soooo many people like that.

valid hornet
fresh comet
#

@split marten welcome to the mathcord!

odd swift
#

does it matter if i take the sat or act

lavish bobcat
latent edge
vivid quest
#

Oinc?

long lark
fresh comet
#

@lavish bobcat welcome to the mathcord!

long lark
#

i know im being nitpicky here and its not the main point but i dont think thats true

vivid quest
lavish bobcat
vivid quest
cobalt barn
#

Guys

#

Should I begin with algebra

fiery nest
# cobalt barn Should I begin with algebra

i’m starting calc 1 in a few weeks.
i took precalc last fall.
i took trig spring semester.
i never actually finished algebra, never took geometry, but i’ll fill in the gaps as i go.

#

algebra is kinda crucial

#

unfortunately they don’t teach it at my college—passed precalc with good marks anyway

neon garnet
#

ASpikaThink she did say that that was challenging for her

neon garnet
fiery nest
#

precalc and trig both did use a lot of algebra

neon garnet
#

mhm algebra feels kinda important to me lowk

fiery nest
#

i only ever passed prealgebra, in my early teens, then just never really used math again until last year

neon garnet
#

i'm glad you're using it!!

#

i gtg for now. good luck learning ^^

dusky locust
dusky thistle
dusky locust
jaunty ibex
split marten
split marten
neon garnet
dusky locust
neon garnet
neat lintel
dusky locust
neat lintel
neon garnet
#

i'm considering finding a nice textbook to read soon

foggy meadow
#

It's not that far away.

dusky locust
neat lintel
#

He was born at 1643, about 190 years after the middle ages was over. I do think that's a lot of time tbh

neon garnet
neon garnet
neat lintel
foggy meadow
neat lintel
dusky locust
# neon garnet for computers ^^ probably

Ooooooo I see I see, I couldn't do all the computer stuff, my brain would probably pop before I could understand the basics (which is already happening to me with physics and mathematics in itself)

foggy meadow
#

I guess not, it was commone for the 1500-1600 to have gnoticism.
Also some sort of magic.

neat lintel
#

in 476 AD, certain barbarians called ostrogoths conquered Rome

#

That's when antiquity ended

#

And the middle ages started

#

In 1453, the turks conquered Constantinople

#

That's when the middle ages ended

#

And the modern period started.

foggy meadow
#

Cool, neat.

foggy meadow
neat lintel
#

But gnosticism exists since the beginning of christianity

foggy meadow
#

I would recommend the channel Esoterica to learn more about the texts from the time period.

foggy meadow
neat lintel
#

the apostles's disciples were actively opposing gnosticism in the 2nd century AD. Heck, you have a whole book in the 180's against the gnosticis

foggy meadow
#

Yep.

neat lintel
foggy meadow
#

Your still avoiding the point.

neat lintel
#

Wait there was a point? I thought this was just a chill talk

foggy meadow
#

My point was that Newton was like everybody else back then.

neat lintel
foggy meadow
#

Yeah I was watching stuff on different medical people from the time.
Where he was considered a heretic later, and was killed or maybe not.

neat lintel
#

Very different

foggy meadow
#

Na.

neat lintel
#

As different as you can be

#

Otherwise he wouldnt do anything

#

He would just be drinking with his buddies in late 17th century london or whatever

#

But ik what you mean

foggy meadow
#

I mean different to us maybe, but to other people at the time.
Idk man.

neat lintel
#

Bro was 84 yrs old when he died

foggy meadow
#

Apprently he was late middle ages.

neat lintel
#

What's his name

foggy meadow
neat lintel
#

What's his name bro

foggy meadow
#

Pietro d'Abano

neat lintel
#

Maybe i can pull it off from memory

#

Yh no clue who that guy is

foggy meadow
#

Yeah apprently he wrote a l o t.
And was influentual for the middle ages, but it often forgotten.

neat lintel
#

Well if he was a medieval sorcerer then he got what he deserved

#

But he probably wasnt a sorcerer was he

foggy meadow
#

No, yes???

#

I mean like 3/4

neat lintel
#

Just some very smart guy i think

#

An alchemist

#

Yk what i'll give you a chance

#

I'll watch that video

foggy meadow
#

Okay, nice!

neat lintel
#

40 minutes?

#

Why does that guy deserve a 40 minute long video

foggy meadow
foggy meadow
neat lintel
#

You do know there were non-sorcerers who were much better than him right

#

non-sorcerers were the true geniuses of the middle ages actually

#

Avicenna

#

Averrois

#

Aquinas

foggy meadow
#

Yeah, but that's not interesting to me. Ew, Aquinas (if that's the same one).

neat lintel
#

(though Aquinas only made theological and philosophical contributions tbh)

foggy meadow
#

Yeah, I didn't like his stuff even though I probably believe some of it.

neat lintel
foggy meadow
#

Probably, I just don't like what he says.

neat lintel
#

But that's where his contributions end

#

If you want the actual best then Ibn Sina is your guy

#

(Avicenna)