serious-discussion

[solar hawk]

It depends on how far you wanna go

For the portion of ITM we’re reading

[latent zodiac]

i wanna read the entire book

not like 2 chapters

tf

[solar hawk]

Only real analysis upto contour

[solar hawk]

We’re doing 4 but ok

[latent zodiac]

mb gvng

[solar hawk]

Then the pre reqs r listed at the front of the book 😭

It’s a lot

[storm sage]

rudin doesn't deal with complex numbers lol

he has a followup book real and complex analysis

[craggy parcel]

lee says in the intro analysis up to baby rudin is enough

[brittle socket]

ISM explictly states that knowledge of the Implicit and Inverse Function Theorems are required, iirc.

but yeah for ITM you don't need that iirc

[bright hill]

What do you wanna do?

Do you wanna hop into manifolds as fast as possible? Do you want to do like

Calc 3 but rigorous?

[jade anchor]

hi anyone here

[north topaz]

don't ask to ask

[graceful walrus]

@tight plinth hi

[tight plinth]

hello!

[latent zodiac]

ye

[bright hill]

well my (purely theatrical) hate for munkres urges me to stir you away

and recc shifirin

but I have never read analysis on manifolds so I can't say

[tropic mica]

can i assume 3=pi=e

[old oak]

And large values of e

[old oak]

You can do what you want, it's a free country. You might not be correct, but that's a different matter.

[prisma swallow]

How do you deal with forgetting stuff? Spaced repetition?

Literally forgot alot of multivariable calculus stuff

[old oak]

Also pi=4, as is well-established:

[long matrix]

ur the one whos wrong

[old oak]

Frequently, yes

[prisma swallow]

QED

[safe ruin]

... what

[arctic zephyr]

does 26% = 0.26

[ocean harbor]

,w does 26% = 0.26

[fathom swallow]

Wolfram Alpha doesn't understand your query!
Perhaps try rephrasing your question?
Click here to refine your query online

Requested by kensukeken

[ocean harbor]

[vast wraith]

yes, % is just an abbreviation for 1/100

[south kelp]

l^infinity norm goes brrr

[magic meteor]

i just realized that "but obviously the square is longer" is just as attractive as an argument

[shadow aspen]

this is not the best way to prove pie=4, you could use inverse squares. Even approaching infinity there still would be small edges that make up the difference.

[long matrix]

the irony is the perimeter of this thing is indeed 4r

[tawdry pecan]

lmfaoo

[old oak]

One of so many reasons why I don't do geometry

[vast wraith]

true

[tame whale]

yo anyone wanna do a sec 5 situation problem rn?

[pastel tusk]

even if pi was 4 it would be 8r not 4r

[long matrix]

sh

[brazen mulch]

hehe doing geometry >.>

[neat lintel]

Geometry is easy

but the real thing is calculus

[fresh comet]

nlab...

[visual breach]

Wdym, this is rather just technical

[neat lintel]

can anyone help me solve a circuit by mesh analysis?

[loud snow]

Ping me in help channel

[neat lintel]

can i send you in dms?

[loud snow]

Ok

[narrow carbon]

Is it category theory?

[fresh comet]

seems more like algebraic geometry to me?

then again, the two are closely related afaik, and I'm no expert anyhow

[raven plaza]

Schemes are algebraic geometry

[narrow carbon]

Oh seems far beyond than my level
Real analysis introductory course ---- AG

[loud snow]

Polynomials are algebraic geometry

@raven plaza u there

?

[raven plaza]

Hello

Indeed polynomial in algebraic geometry

[loud snow]

I always curious

Algebraic geometry stuff you know

When I learned a little I was always confused

[raven plaza]

I have also only learned a little

[loud snow]

Like I think I get the scheme things but I really dont

Hopefully when I older I get more info

[raven plaza]

Schemes are just manifolded rings

[latent edge]

I learned until schemes

[loud snow]

U know that doesn’t improve understanding of nothing

[latent edge]

Schemes were my final goal

[loud snow]

I know some defnitions n stuff

But then the real problems are

What are questions someone would normally have ya know?

Like idk purpose of divisors

[latent edge]

Idk I might need to know what is a stack and weird stuff like M_0,g bar

[loud snow]

But I know where they show up

Idk how they show up in classical ag

Idk natural questions to ask there either rly

Where the algebra?? I ask sometimes

[raven plaza]

You can do intersection theory with divisors

[loud snow]

That not helpful

Thats not a question right?

I cant give you list of questions commonly asked about things in classical ag

I just know random factoids

Like riemann roch idk motivation rly but is important!

Im too young to know answers

[neat lintel]

Merhaba.

[flint forum]

hello

[visual breach]

Ono

I mean it's fancy, but it is also stuff.

[loud snow]

Probably depends on context

And you definitely dont need to know any of this

[tame whale]

can anyone help me solve a sec 5 trig problem its for marks pls

[mental stratus]

are you in a test?

[tame whale]

nah he let me bring it home

can you help me with it ?

[vast surge]

Like, homework or a takehome test?

[tame whale]

homework

i worded wong mb

[quasi jetty]

To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.

[tame whale]

yea i did that it dont work

@vast surge can you help

[vast surge]

I probably can, but I don't want to.

[onyx slate]

what is the equation in standard form of a parabola that contains the following points (-2,-11.5) (0,-2.5) (4,-20.5)

[vernal token]

Ask this in a help channel

[tacit magnet]

Thats solving 3 equations with 3 variables

[surreal bison]

vandermonde!

now do hermite interpolation

[rocky shuttle]

why standard form when you can do lagrange or newton interpolation

[surreal bison]

what is "standard form"

[rocky shuttle]

i assume with respect to the monomial basis

[surreal bison]

oh okay

but you can just

add

the lagrange ones

using

addition

[rocky shuttle]

to get the representation in terms of the monomial basis? sure

but you can also just leave it as it is

[visual breach]

How do you add

Ohno I forgot addition

[neat lintel]

alright lets do this @tender tulip

me: chaotic evil

[tender tulip]

LMAO

[neat lintel]

you?

amukh: neutral good

[tender tulip]

Probably chaotic neutral

[neat lintel]

ok

all the mods: lawful good

[tender tulip]

Absolutely the fuck NOT

[neat lintel]

all the admins: lawful good

[tender tulip]

The only lawful good mod is Eric

Most I’d say are lawful neutral

[neat lintel]

ok

[tender tulip]

Wew is chaotic neutral

[neat lintel]

fr

[tender tulip]

Most of us are probably chaotic neutral

[neat lintel]

uh what about the honorables

[tender tulip]

Almost all true neutral

[neat lintel]

hyzae is neutral evil

[tender tulip]

She’s lawful evil

[neat lintel]

nah

stabbing

[bronze pelican]

Hyzae is chaotic good

[tender tulip]

I catalyzed that when she joined the server by saying “hello chat” ad absurdum

[tender tulip]

My main bit is going schizo doing horrible problems in advanced channels

[neat lintel]

@bronze pelican

[bronze pelican]

I meant chaotic evil

[fresh comet]

am I chaotic evil

[tender tulip]

Lawful neutral

[bronze pelican]

No you're lawful good

[rocky shuttle]

Lawful good

[fresh comet]

this sounds off

[neat lintel]

ok who else

[fresh comet]

I'd have thought I'd be in the chaotic category

[neat lintel]

ok xela is next in /top

and boot

[fresh comet]

are we making a chart?

[neat lintel]

yeah

[fresh comet]

[neat lintel]

idk what site to make it on

but

[fresh comet]

what do you have rn?

[neat lintel]

bc none of them support just text inputs

i should make a git repo

https://github.com/speedydelete/mathcord-alignments

[neat lintel]

@fresh comet @tender tulip

[surreal bison]

hwy

just do roles

are you asking an llm that you run on all my messages?

[tender tulip]

What am I chat

You chose

[neat lintel]

against discord tos according to fiona

sadly

i should code that

i need to make easyai.py

[surreal bison]

what does it break

the "can't read messages" rule??

[neat lintel]

idk

ask her

[surreal bison]

no you

you didn't...ask her...why??

[neat lintel]

no

[surreal bison]

how uncurious are you

[neat lintel]

dami also said no

i mean

it makes sense

[surreal bison]

you didn't ask "what specifically is the thing that is prohibited"

[neat lintel]

the tos probably says "you waive all rights to your msgs by making an account" or smth

[surreal bison]

wait

what?

are you crazy?

if it said that then the bot would be more likely to be fine

(of course they could later prohibit using the api for that)

[neat lintel]

ah here

#discussion message

[surreal bison]

also i think someone could surely put protected I.P. in th emessages and still be able to legally enforce such IP

pulling messages for training data is also against ToS and is pretty easy to get banned doing

[neat lintel]

oh, you still own it according to the tos

[surreal bison]

i wish discord was FOSS

[neat lintel]

actually i could make a bot

that would export weights to you

not against tos right?!

@surreal bison

[surreal bison]

read

of course

somebody could totally do this anyways

[neat lintel]

and get banned

according to fiona

[surreal bison]

well i mean

if one is allowed to pull messages

[neat lintel]

yeah

[tall halo]

Is there a shortcut to memorise the derivatives of arccos,arcsin,artan and argcossintan?

[static loom]

I would just rederive them when I needed them

for instance y=arccos(x) you write as cos(y)=x, differentiate to make -sin(y)y' = 1 so y' = 1/sin(y) and now cause you know cos(y)=x then with sin^2(y)+cos^2(y)=1 you have sin(y) = sqrt(1-x^2) and so y' = 1/sqrt(1-x^2)

pretend I put the -1 in there correctly

[storm sage]

you also can't use bots to run accounts on discord

any botted account has to be specially designated as a bot account

[neon silo]

For me there is no shortcut to memorization, things that you use more frequently will just become easier to recall with time. Because I've had to use it a lot I now know that arctan is 1/(1+x^2), the others I'd rederive as Merosity prescribes if I needed them

[tall halo]

I heard what you said by 3+ Proffessors.

[surreal bison]

...there is obviously a shortcut to memorization, what?

[wispy socket]

In general using mnemonics can help memorise stuff. Not sure if it would be practical for derivatives but whatever floats ur boat

I can’t really think of any other way except just doing questions that involve these derivatives until u basically ‘memorised’ them

Proving them, as mentioned, may help

[neon silo]

That may very well be the case. I'm guessing you're thinking of something along the lines of flashcards, for example. I reject this because whenever I've seen my peers do something like this they typically forget the things they were studying 2-3 months after the exam if it wasn't directly relevant to the next course, so to me this type of memorization is empty and bears nothing in the long term.

Maybe not what you were going for but to me the shortcut to memorization, if anything, is understanding since knowing why the formula is true could be said to be better than just knowing that it is true (and could be seen as memorizing serval things at once if the form of argument can be applied elsewhere). Although this is not really a shortcut since understanding can take significant time sometimes, and once you're there you may be familiar with the result from multiple different perspectives.

However if OP's goal is just to pass a test perhaps the flashcards are worth a shot.

p.s. mnemonics are cool, I just have a hard time implementing them.

[surreal bison]

...if one forgets 2-3 months after the exam, one probably doesn't need it. but also one should of course still be flashcarding

understanding hells a whole bunch yes. with this specific thing i wouldn't bother memorizing (i in fact don't have them fully memorized)

[surreal bison]

I guess my point was, if you actually really wanted to memorize, there are obvious shortcuts that are effective. If you forgot 2-3 months after the exam (realistically for this kind of thing it'd be more like 1 month) then you aren't still using the flashcard mechanism

It definitely bears something. It is useful to, say, memorize a bunch of definitions when you are learning something new

[neon silo]

I'd probably agree most of the time that they didn't need it then, though it can be tough to know in advance if or when something will be needed again. Of course there's nothing wrong with having to brush up of things, we all need to from time to time, but I feel like if one finds themselves needing to relearn a whole area essentially from scratch then they have wasted their time at some point (or they've just been out of it for a long time).

[neon silo]

I suppose my root problem is that I see memorization on the whole as a crutch to actual learning if it's forced in some way and not something that coalesces naturally. That is to say it's a tool that one can use in school to get marks, not to learn a subject more deeply. I'm having some trouble with this now though as the example of memorizing def's seems like a good one, so maybe it's fine as long as the scope of memorization is kept to the most fundamental things.

[narrow talon]

anyone good in programming?

[still slate]

Is wolfram alpha the only good program that can help me with complex separable differential equations? Any other recommendations?

[south warren]

yapping

[still slate]

Thought so. Not sure if I'm ready to pay up for that extra compute. Thanks.

[grizzled perch]

I feel like asking the question directly is more productive

but i do know some programming

[pulsar finch]

What is the smallest positive integer that has exactly eight distinct positive factors?

[fair mural]

probably the product of the first 8 primes

[deep mango]

I was thinking product of the first 3

[fair mural]

[neat lintel]

!help

[quasi jetty]

To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.

[deep mango]

2*3*5 has 1, 2, 3, 5, 2*3, 2*5, 3*5, 2*3*5

[fair mural]

oh that’s what they meant

i guess the way i interpreted it does have an obvious answer

[deep mango]

But idk if you can do something smaller

By doubling up on a prime

[fair mural]

i don’t think so since 8 = 2^3

if you chose the same prime twice you would be counting the same number more than once in that count

[deep mango]

Yeah

[pulsar finch]

I cannot send picture

[split oak]

24=2³×3

[pulsar finch]

I was about to send number of distinct factors formula

[split oak]

24 has 8 distinct positive factors

[fair mural]

oh that’s interesting

[deep mango]

Grah

[long matrix]

reported for doing peoples homework for them

[deep mango]

Fun patrol says no fun allowed

[long matrix]

im fun yes

ur not fun

[deep mango]

Functorial

[deep mango]

Youre funky

[long matrix]

funky

[deep mango]

Hahaha

👃

[neat lintel]

chat

why is zero considered a number

[long matrix]

its not

[dark chasm]

hi i keep losing this occupied math channels and it started to happen yesterday. i edited my roles and it worked and now its the same. is there a problem with the server or if my account is the problem how can i fix it?

[pulsar finch]

is this enough justification? took me a while to get started

let the prime factorization of n be:

n = a^x * b^y * c^z . . .

number of divisors = (x + 1)(y + 1)...(z + 1)

we need to find the minimal one for which (x+1)(y+1)(z+1).= 8

divisors of 8 is 1, 2, 4, 8
8 = (x + 1)(y + 1). . .

2^x . 4^y
y = 1
x = 3

naturally you would want to use 1 for the power of 4

this problem is way out of my league but still wanted to try tbh

[icy locust]

N = Z^+ so tru

[long matrix]

ill consider 0 a number when someone tells me convincingly what a number is

[icy locust]

number is someone who makes your head numb

[white temple]

i can tell you very unconvincingly that a number is an element of a ring (not very good answer but an answer nonetheless)

[long matrix]

then everything is trivially a number

[bitter ember]

What does it mean when I say 2!

[rocky shuttle]

is everything an element of a ring

[bitter ember]

Mathematically speaking

2! = What?

[white temple]

for the sake of defending an admittedly poor answer, yeah i suppose so

[vast wraith]

selfie (unfortunately my roses got in the way)

oops that’s the wrong one of the two

better

also yes it’s totally on topic, uh, something something mathematical model for generating something akin to roses

[fair mural]

but this is #serious-discussion, not #math-discussion

[vast wraith]

i like having both at the same time

[static loom]

http://algorithmicbotany.org/papers/ specifically http://algorithmicbotany.org/papers/abop/abop.pdf is relatively popular if you're into that kinda thing

[vast wraith]

yayyyyy thanku

[vast wraith]

fun fun

[static loom]

yup yup

[vast wraith]

just like me

[tulip void]

Help

Homeless people often approach me and ask for food or money does this mean I look like a kind person or a person who can be easily taken advantage of

Or someone who has money

How to avoid this

[rocky shuttle]

Stay at home

[zealous garden]

||The contents of this post have been removed at the discretion of the FBI||

[deep mango]

I rarely get approached and i think its cause i have a confident nyc walk

Like im a resident

If you look worried about being somewhere then i think people notice and imagine youre a tourist

[jagged forge]

i definitely do notice i've been approached many times at hotels and airports

[flat harbor]

lovely

[ocean harbor]

you might get kidnapped by them so be careful with it lol

if they're still begging you then just tell them that I made money to teach my students or smth

and move on

[craggy parcel]

its most likely because its visible in your body language that you notice them. most people actively pretend homeless people dont exist when they walk by them, so if you give eye contact or even change your pace a little bit, you are signaling that you notice them

[dreamy ore]

i just ignore them

[wanton coyote]

God this whole conversation makes me sad

[dark chasm]

i am having issues with the server where am i supposed to text the problem can anyone help?

[hybrid blade]

does anyone have a recommendation for a good laptop for a grad student? price limit about £500

[ocean harbor]

do you mean where to get help in general?

[night grotto]

is (tan)(sin) the same thing as sin^2/cos

[wet mortar]

yes

[vast wraith]

tfw 16th century europe

(from oystein ore’s cardano: the gambling scholar)

[tropic mica]

pi=3=e

is this correct

[vocal vessel]

No pi is 4

[tropic mica]

oh ok

[vast wraith]

from a translation of cardano's Liber de ludo aleae (1564)

i feel personally attacked 😭

[loud snow]

Chromebook

[white temple]

get an old lenovo thinkpad, replace the crappy old 5200rpm drive in there with a SATA SSD, and if you feel so inclined, upgrade the ram. that thing'll run linux like a charm and leave you with plenty of money from your £500 budget still in your hands

[thick yacht]

Was this put into practice?

[vast wraith]

thank you for reminding me! i was meaning to check... will note down a reminder, am about to head to sleep now

i would be willing to bet quite a lot against it though, hehe

that it ever was, that is

[fair mural]

it’s as correct as it is funny

[gentle crane]

how valid is the tabular method for intergration by parts

[tropic mica]

So it’s pretty funny

[next schooner]

what do you mean by "how valid"?

[gentle crane]

For yr13 maths

Cuz for emx^3sinx it's fine but something like e^x sinx it may actually make it harder it visualise (for me anyway(

[next schooner]

yeah you typically only use the tabular method when one of the derivatives goes to zero

so for x^3 you differentiate it to 3x^2 then 6x then 0, so you can use the table

if you can't do this (like you can't for e^x sin x) then the method doesn't really work

[gentle crane]

I know that instead you wait for a loop

Or realistically until it's simple enought to Intergrate

Then u intergrste the line

Which gives u the same thing

[surreal bison]

...what is this "tabular method"

do you just write iterates in a table

but like why

[gentle crane]

@surreal bison u know u have a u and V

And you derive one and Intergrate the other

Yeah you write that out in a table

[surreal bison]

what the hell are you talking about

[gentle crane]

Its more for depth ones

Like x^4sinx

[surreal bison]

i am talking about recursively applying integration by parts

[gentle crane]

?

When I said depth?

Or what

Its a standard method just Google it

[surreal bison]

depth? what?

[gentle crane]

I'm just talking about using it for stuff that doesn't terminate

[surreal bison]

like depth of the recursion

[surreal bison]

if it cycles then you are fine

you can just use a very complicated technique known as addition

if it is not cyclic and doesn't terminate, you are screwed. but not really since it's probably still a neat identitiy

take for example the gamma function

[gentle crane]

Yeah if it's cyclic

So as I said for e^xsinx as an example

[surreal bison]

\Gamma(z) := \int_0^\infty t^{z-1} e^{-t} dt

one shows the recurrence relation by parts.

[surreal bison]

okay but this is distinct from an example that doesn't terminate

take \Gamma(1/2) for example

[gentle crane]

?

[surreal bison]

look at the example i gave

[gentle crane]

Idk what gamma is so hence I can't really imagine it

[surreal bison]

i just fucking defined it

[gentle crane]

Well I do I see it's a function

[surreal bison]

by parts you immediately get \Gamma(z+1) = z \Gamma(z)

you can just evaluate \Gamma(1) and see that it is 1

thus $\forall n \in \mbb{N}_{> 0}, \Gamma(n+1)=n!$

[fathom swallow]

Xela

[surreal bison]

now technically to define \Gamma outside of \Re(z)>0 you need to do analytic continuation

because that integral doesn't work there

but you can just use the recurrence relation

\Gamma(z+1)=z\Gamma(z) so we want \Gamma(z)=\Gamma(z+1)/z

and if you pray to legendre's spirit, this ends up being meromorphic with poles at the negative integers

the point is that one can't just terminate

i mean for analytically continuing the function that suffices

e.g. jump from -1/2 to +1/2

but for, say, actually computing \Gamma(-1/2), it doesn't

so in fact

\Gamma(1/2) = \sqrt(\pi)

there's actually a form for all rational ones?

\Gamma(n + \frac{p}{q}) = \Gamma(\frac{p}{q}) \frac{1}{q^n} \prod_{k=1}^n (kq+p-q)

[ocean harbor]

,, \Gamma(n + \frac{p}{q}) = \Gamma(\frac{p}{q}) \frac{1}{q^n} \prod_{k=1}^n (kq+p-q)

[fathom swallow]

K²en [𝔸𝔨𝔦𝔯𝔞]

[ocean harbor]

,, \Gamma \left(n + \frac{p}{q}\right) = \Gamma \left(\frac{p}{q}\right) \frac{1}{q^n} \prod_{k=1}^n (kq+p-q)

[fathom swallow]

K²en [𝔸𝔨𝔦𝔯𝔞]

[ocean harbor]

https://cdn.discordapp.com/emojis/1112022027847606383.gif?size=48&quality=lossless&name=catcombing

[ocean harbor]

this is funny

[surreal bison]

this happens regularly with pre mathematically mature pepple

post ones are more likely to have simply forgotten the definition

[me]

[ocean harbor]

https://tenor.com/view/calculus-cat-cat-math-studying-cat-gif-23164559

[ocean harbor]

lmfao relatable

[loud snow]

[tropic mica]

Why is discussion 2 so rejected compared to discussion 1

[fresh comet]

it's just overflow

however, if you want slower conversations, this channel is great

[tropic mica]

Btw that help channel you just saw

They’re talking about ghandi

What

[fresh comet]

dunno, this kind of stuff happens from time to time

[tropic mica]

O_O

[fresh comet]

as long as it gets back under control (and it seems like it has), it's fine

[tropic mica]

https://tenor.com/view/super-gif-15396035736016205249

[fresh comet]

no image perms

you might get active soon though

[tropic mica]

WHAT

They added image perms

[fresh comet]

in the discussion channels, it's always been that way

except for #chill

[tropic mica]

https://tenor.com/view/honest-reaction-gun-side-eye-my-honest-reaction-true-reaction-gif-3139670107205599383

No way

[fresh comet]

I personally never felt it though, because I didn't really use these channels until I was Active already

[tropic mica]

Time to spam 300 emojis in advanced

[fresh comet]

don't do that...

[tropic mica]

Oh I can’t even change my server name anymore

Okay good thing I didn’t put some dumb stuff 😭

[fresh comet]

issok

just keep using help channels

that's how I got Active

[tropic mica]

Oh witj the amount of question I ask I’m almost there

[wild hedge]

When was the last time they had mod apps?

[tropic mica]

Considering mods respond under 20seconds if you ping them

Prob long time ago

[wild hedge]

I wanna apply lowkey

[frigid dust]

Is a constant function always a contractive function?

[river moon]

check the definition

d(f(x), f(y)) <= k d(x,y)

[frigid dust]

Okay! UwU

[queen hornet]

who here is familiar with JNF

jordan normal/canonical form

[visual breach]

Likely most math majors

[raven plaza]

Jordan natural form

[odd narwhal]

You can dm modmail to have it changed

[lapis hull]

When are you available shiny?

This Thursday perchance?

[odd narwhal]

Good question

Thursday is busy, actually. I think sometime next week would be best

[lapis hull]

That's fine!

Sunday and Thursday work best for me, I can probably also do Monday though

[deep mango]

Good morning derpus

[lapis hull]

Good morning rycie bycie

[deep mango]

Dont worry i know how it feels to provide your availability and get left on read

[ocean harbor]

they aren't looking tho

[rocky shuttle]

[lapis hull]

LOL

[deep mango]

Grahhh

[lapis hull]

any updates wrt d?

[deep mango]

My hands are all sparkly this morning

No...

[lapis hull]

damn....

im sorry that sucks

[deep mango]

So annoying

[rocky shuttle]

D

[deep mango]

Idk why im bothering

[lapis hull]

why

[deep mango]

Idk you know how when you get numb / a limb falls asleep then when it wakes up it feels all sparkly

like you're drinking soda

[lapis hull]

oh pins and needles

[deep mango]

yes

i have that mildly in my hands

but they werent asleep

[flat harbor]

gevening chat

im watching my kawaii kouhaitachi giving presentations on classical mechanics

[rocky shuttle]

meowning bloe

[flat harbor]

helohelo zan

[deep mango]

like whatever, the more he insists hes interested the less bad im gonna feel about just double texting / reminding

[rocky shuttle]

how do your kawaii kouhaitachi perform?

[flat harbor]

uhh

well last weeks presentation was pretty nice

[deep mango]

lol

was last weeks by the same person

[flat harbor]

oh this week just started

im NOT slighting them!

[flat harbor]

different kids, last week were the new guys and this week are the ones ive seen since last yr

[deep mango]

oh...

ok

[flat harbor]

but cmech presentations can be very calculation heavy

[deep mango]

right

[flat harbor]

which end up very boring if the presentor doesnt have good presenting skills so i dont blame them

[flat harbor]

ok the prof just started an impromptu cmech lecture time to sleep

[ocean harbor]

https://cdn.discordapp.com/emojis/1155009924141678602.gif?size=240&quality=lossless

[urban nexus]

bing chilling

[fresh comet]

Aki in a discussion channel

[neat lintel]

:3

[rocky shuttle]

:3

[jaunty ibex]

hello everyone

I think I became analytic number theory pilled in no time

[latent edge]

based

keep it up

[fresh comet]

[unreal sky]

who here can help me with a complex number problem ? 😭🙏

[fresh comet]

!help

[quasi jetty]

To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.

[unreal sky]

ive asked

but we are stuck

https://discord.com/channels/268882317391429632/1018700692027883620

[ocean harbor]

https://cdn.discordapp.com/attachments/695514882216427540/1231377074648780830/attachment.gif

[unreal sky]

https://tenor.com/view/cat-thumbs-up-cat-thumbs-up-meme-stock-image-gif-27117830

aw

[jaunty ibex]

$\sum_{n_1,n_2,n_3} \Lambda(n_1) \Lambda(n_2) \Lambda(n_3)$

[fathom swallow]

veronica

[jaunty ibex]

Very cool

[ocean harbor]

https://cdn.discordapp.com/attachments/937568987565404170/1241587158226305165/caption.gif

[latent edge]

https://tenor.com/view/geometry-dash-dash-spider-part-gif-6950215967713207161

I died at 94% in sirius

[ocean harbor]

https://tenor.com/view/gd-geometry-dash-sakupen-circles-stfu-stfu-i'm-playing-gd-gif-2682408120184002311

[ocean harbor]

HAHAHAHA IMAGINE

[latent edge]

damn I wanna play gd now

[ocean harbor]

go play

[dreamy ore]

i cant believe people still play that game

[ocean harbor]

it's goated

[dreamy ore]

i regret the years ive wasted on that game

[ocean harbor]

it's amazing

[latent edge]

why

[ocean harbor]

without it my life is shit

[dreamy ore]

cuz its a shit game 💀

with a shit community

[latent edge]

[dreamy ore]

at least

it was shit when i left the game

idk i quit like at least 4 years before 2.1 dropped

[ocean harbor]

since when

[latent edge]

I don't care about the community

[ocean harbor]

ah that's understandable lol

[dreamy ore]

2.0 had been out like

17 millenia by that point

[latent edge]

2.1 is cool tho

[dreamy ore]

and the community was just toxic bs

[dreamy ore]

uninterested

but yall do you

[ocean harbor]

cuz those are kids

just ignore it

[latent edge]

yeah just play with sane people

[dreamy ore]

yeah i happen to remember a large number of just plain assholes in the community lol

i remember the topic of generic featured levels was a big thing when i quit

tbh i dont remember

i dont think sonic wave was verified when i quit

phobos was though

[latent edge]

tbh gd changed a lot

[dreamy ore]

idk if that gives a time frame ¯_(ツ)_/¯

[latent edge]

generic featured levels are still a thing but not very favored now

and creativity of some creators is just insane

[dreamy ore]

do people still make nine circles levels

[latent edge]

not that I know off

[dreamy ore]

lol

[latent edge]

they do some parts of it

it's a common theme in collab levels I think

I do see them sometimes

[dreamy ore]

lol

[fresh comet]

nc levels still exist, but are far less common than in 1.9/2.0

[latent edge]

yeah

[dreamy ore]

i still maintain that its not very fun at least for me

[latent edge]

i freaking died at 50 in sonic then I just quitted

[fresh comet]

some of the recent nc style levels are very well done too

[dreamy ore]

its just annoying after a while

eyekilling visuals to bad music

clicking till u get carpal tunnel

[latent edge]

lmao

[dreamy ore]

not my type

[fresh comet]

tbh, I don’t hear of top players struggling with carpal tunnel anymore

what changed

[latent edge]

tbh cant be worse than freaking blade of justice

[dreamy ore]

they accepted it as an inevitability

[latent edge]

this level is cursed af

[dreamy ore]

or its like idk fucking human evolution

gd players got stronger tendons

[fresh comet]

idts?

[dreamy ore]

no other explanation

unless u want to accept the evolution theory

guys

[fresh comet]

oh actually

I know why

[dreamy ore]

can we talk about

[fresh comet]

It’s cause people use up arrow now

[dreamy ore]

how good of a fuckin album mbdtf is

😩

[fresh comet]

what is mbdtf

[dreamy ore]

cant believe i have not listened to the full album in order until now

[dreamy ore]

my beautiful dark twisted fantasy

kanye album

goated album

[fresh comet]

hm

[dreamy ore]

cant decide if this or graduation is better

its defo about.. at least tree(3) * grahams number cubed + 7 times better than the fucking dubstep i was listening to when i was wasting away at gd

[ocean harbor]

map packers

blade of justice is fun

https://tenor.com/view/blade-of-justice-geometry-dash-gif-20201672

[latent edge]

no

the dual part is cancer

[thin kiln]

You're a genius

the sine* sorry, It's because in french we say ' le sinus'

What kind of software is that ?

[ocean harbor]

ibis paint

[magic anchor]

hello

[ocean harbor]

henlo

[plush zenith]

Hi

[broken zephyr]

hi, if i have college credit for calc 2 and 3, but not calc 1 will i have to retake calc 1

[plush zenith]

How, isn't calc 1 a requirement to take 2 and 3

[broken zephyr]

because i took them through my highschool

so i had the highschool credit

[plush zenith]

Yep so you already have the credit, if your uni thinks that credit is enough for calc 2 then I don't think you have to retake

[broken zephyr]

ok thanks

[latent edge]

that's a question for your university

[ebon pelican]

hello, id like to ask something
im currently studying as a natural-sciences major and usually focusing more towards applied math
sadly my course does not introduce any pure math, so i'd like to learn it but not in a school setting

what order of topics would you recommend so that one can be somewhat knowledgeable in general topics in pure math

[latent edge]

what topics are you interested in?

[ebon pelican]

probably real and complex analysis

but im open to other fields

[latent edge]

For real, either go with Tao or baby rudin. I personally used Tao

For complex, maybe Lang

I used Howie first then went with Lang for more exposure

Howie is fine to be fair and it doesn't require that much real analysis knowledge

So you can start right away if you want with complex

[ebon pelican]

what prior knowledge do i need to have before analysis

[latent edge]

basic understanding of calculus is prefered

be familiar with derivatives, integrals and series

[tulip tapir]

baby rudin has a (not sure if its totally complete) solution manual which may be helpful if you're self studying

[latent edge]

but still real analysis will be difficult if it's your first time

[ebon pelican]

what's next after real analysis

i mean complex

[latent edge]

well that depends

I guess you could do harmonic analysis/measure theory if you are interested

just depends on what you want

[jaunty ibex]

My road was like
real -> complex -> elementary number theory-> analytic number theory

[tulip tapir]

you also don't need to do a specific progression of things that are based on each other

[jaunty ibex]

Seems like I need harmonic analysis now

[ebon pelican]

maybe i'll get back to what fields i may do after doing real and complex analysis

[tulip tapir]

you could just do algebra

[latent edge]

that's hella based

[ebon pelican]

how will i know that i have learned the material

that's something i struggled with in self-studying

[latent edge]

you can do abstract algebra to get a feeling of what algebra feels like

and a bit of topology

[ebon pelican]

i have only done the typical calc 1-3 + ODEs

i will try learning analysis since summer break is near

thank you all

[jaunty ibex]

I was biased before that I liked analysis so I did algebra and topology

Now I'm back to analysis because I enjoy it more

I thought maybe I would like algebra or topology more but I was wrong

for me it's analysis>topology>algebra

[ebon pelican]

what do i expect to learn in topology

is it like more extreme geometry

[vast wraith]

i don’t think you typically do any geometry at all in general topology

[worthy stream]

does anyone here can code in vs code?

[static loom]

it's arguably one of the easiest things to code in

[latent edge]

as a vim user I agree

[vast wraith]

i use way too obscure of a text editor to participate in editor discussions :S

[final bluff]

Coding on stone >>>>>

[neat lintel]

i do

[worthy stream]

nothin guys i had small problem some pal fixd it for me

[fresh comet]

every so often, I get this video on my youtube recommended page
https://youtube.com/watch?v=LLCF7vPanrY

it's pretty interesting, and somewhat scary too

[scarlet jasper]

Those are rookie numbers, gotta pump those numbers up

[rocky shuttle]

Is this an incremental game?

[fresh comet]

I can't tell if your being serious

[rocky shuttle]

[fresh comet]

would maybe be a fun incremental game, but maybe bad for the humans on (the fictional) earth

[jaunty ibex]

Ask deltoid maybe he can come up with a formula for occurrence/frequency of these numbers

[fresh comet]

oh hi succ

er, veronica?

what am I supposed to call you now

[jaunty ibex]

vero from now on

[fresh comet]

okay okay

@rocky shuttle would the objective of such a game be to make and detonate as many nukes as possible?

[rocky shuttle]

and then predict the next one

[jagged forge]

an incremental game about nukes would go hard

of course the end of the game would be nuclear winter or something

[fresh comet]

world tension +50

[jagged forge]

lmao

oh you could have an actual doomsday clock as a game mechanic

[rocky shuttle]

I just want to make numbers go up!

[fresh comet]

higher!

[rocky shuttle]

higher!

[latent edge]

Am I the local number theorist now in mathcord

[jaunty ibex]

For some reason I can't see advanced channel or access math channel anymore

[rocky shuttle]

Local time series analyst

[latent edge]

Maybe because you have the dash role

idk tho

[worthy prism]

maybe

[fresh comet]

you do have the em dash role...

[latent edge]

just ping mods i guess

[fresh comet]

I've only ever seen it assigned by moderators to ban people from the adv channels

[latent edge]

Oh

Let me check adv

[fresh comet]

it could be possible that you have it for a different reason though

[latent edge]

Oh that makes sense

[fresh comet]

so I wouldn't count that out just yet

[latent edge]

vero said something kinda inappropriate

[fresh comet]

oh

[latent edge]

hence nuked

[fresh comet]

well uh, yeah

that's gonna happen

[jaunty ibex]

uhh that's fine but why I can't access math channel

[fresh comet]

which channel? math discussy?

[jaunty ibex]

or even #advanced-number-theory

[fresh comet]

that role bans you from them

[jaunty ibex]

I can only react

huh wtf

[fresh comet]

mhm

[jagged forge]

destroyed

[fresh comet]

you might have to just wait it out, or you can contact modmail if you have issues

[jaunty ibex]

I had a question to ask in adv NT rip

[fresh comet]

just ask Deltoid outright ig

[jaunty ibex]

sotrue

[tulip tapir]

guessing its bc of the epsilon delta thing lol

[jaunty ibex]

Oh but even the other person said something equally bad lol

[fresh comet]

possibly, yes

[fresh comet]

Zorn is better known amongst the users here I think

[jaunty ibex]

Ohh nepotism moment

[fresh comet]

you're also on an alt account, so ppl may not know you

[jaunty ibex]

[tulip tapir]

lol

[fresh comet]

in any case, you can DM modmail if you want to appeal ig

[jaunty ibex]

How far can you go with $\sum_{n\le x, (n,k)=1} 1$

[fathom swallow]

veronica

[latent edge]

how far you went?

[jaunty ibex]

$x \sum_{d|k} \frac{\mu(d)}{d} + O(\sum_{d|k} \mu(d))$

[fathom swallow]

veronica

[jaunty ibex]

whatever the sum is idk what it is

the O term is probably something x^{1-epsilon} can't go that far

[latent edge]

I think you can go further with sieves

[jaunty ibex]

ofc u would say that

[latent edge]

i mean sieves are cool

they are really powerful to the extend sometimes you shoot yourself in the foot

[jagged forge]

i don’t know anything about sieves but that looks like a sieve to me

[latent edge]

with brun sieve you can get a lower/upper bound for the sum you want ig something like $\frac{C_{higher} \phi(k)}{k} x + O(x^{1-\epsilon})$

[fathom swallow]

Steiner

[latent edge]

same thing for lower

[jaunty ibex]

huh that's pretty close to my intuition

why the phi tho

[latent edge]

for every interval mod k there are phi(k) such integers

so you get that constant for free

the only issue is how uniform the distribution is

that you can go wild with sieves

[jaunty ibex]

ic

[latent edge]

yeah it is a sieve-like sum

it screams for sieve

[neat lintel]

https://media.discordapp.net/attachments/1019458299525091448/1245135807023546389/TLiltI3.png?ex=6657a695&is=66565515&hm=9c3e953f34ecc73506b97b7f36fffcd8eb1b623f318a178f57ef6b09c793534e&

[fresh comet]

I cannot tell if this is crankery

[craggy parcel]

i think its a joke

"scuttlebug" "mario bump" "jamboree"

[fair mural]

no way they seriously said 4*65536 and not just 2^18

[craggy parcel]

2^18 feels like it should be bigger

[sacred arch]

In times of struggle, remember the wisdom of the ancients: When putting pigeons in holes, if you have more pigeons than holes, then some of the holes will need to pull double duty

[spiral spoke]

I have learned about LCD methods of integration first order differential equations and other basic stuff in high school. I learn further. I am stuck now. What should I do next

[north topaz]

I mean it's 8^6 so less than 10^6

[craggy parcel]

makes sense

[north topaz]

https://tutorial.math.lamar.edu/Classes/CalcII/CalcII.aspx check out this

there's also calc III and DEs

[spiral spoke]

@north topaz Thanks bro but I have covered these topic already. I want to dive deeper

[north topaz]

it depends on whether you want to continue with pure mathematics, applied, or stats

if you want to continue with pure I suggest you do real analysis

Stephen Abbott's Understanding Analysis is a good textbook for that

Or Tao's Analysis, books I and Ii

for a more comprehensive coverage of the topic

Also you didn't say if you've learnt multivariable calculus already (which is what calc III is)

[spiral spoke]

Yup I haven't studied multi variable calculus

[north topaz]

okay then

[north topaz]

I'll DM you my notes for multivar calc, hope it helps

(typed notes by the uni)

[spiral spoke]

Thanks bro

[stray lily]

yo

how do i get started here

[old oak]

By presenting your thoughts on whether 0.999..... = 1

[stray lily]

im scared to answer

[deft rock]

Does this server help with 7th grade geometry??

[stray lily]

probably yeah

maybe i can help you

what is the question

[deft rock]

Can i send in dms?

[stray lily]

yeah sure

i am new here so idk how that works

[deft rock]

Alright thank you

[ocean harbor]

these 06 kids won't shut the fuck up about graduation and prom

https://cdn.discordapp.com/emojis/1072870091965874237.gif?size=48&quality=lossless&name=panch

[north topaz]

the other movements of Pomp and Circumstance sound very dissonant btw

(that's the famous graduation song)

also known as "land of hope and glory

yes it's a British imperial tune

[ocean harbor]

I added never gonna give you up song on graduation day

I hope they add it

[old oak]

Surely that's the hymn of the people who don't graduate and stay for another year

[ocean harbor]

im a 05er

taking another semester

[brittle socket]

Hi fisherman

[bronze stream]

he

[brazen hazel]

help-8

[vast surge]

I just looked this up and this does not sound like a song that's appropriate for a graduation ceremony per se. It sounds like the propaganda song that would play at a military parade.

[north topaz]

exactly

it was a British imperial tune

just look at the lyrics to Land of Hope and Glory

[jagged forge]

i didn't even know it had lyrics

[sonic sentinel]

an idea ive had for some time to reduce LLM hallucinations: let the LLM prompt itself

so for example and if i ask it "how many cars were sold globally in 2019" and the AI suspects it might hallucinate an answer

it starts a new chat with itself

and it asks (as the user) "how many cars were sold globally in 2019"

it makes the assistant reply in the format "The number of cars sold globally in 2019 was"

then it generates an answer

it then messes with the weights of this "inside" AI a bit to get a second prediction for what comes after "The number of cars sold globally in 2019 was"

it can repeat this some number of times

if it keeps generating the same answer every time (say 74.9m, 74.9m, 74.9m), then it can be fairly confident its not hallucinating

if it keeps giving answers around the same ballpark (say 70m, 75m, 73.2m), then the answer is probably in that ballpark and the "outside" AI can respond to the user accordingly

if it keeps giving wildly different answers (say 40m, 100m, 8m), then it can just tell the user it doesn't know

[north topaz]

well now you know

[sonic sentinel]

caveats i can come up with: (1) the LLM needs to be aware whether there is a high chance to produce a hallucination in response to a particular prompt (2) the "inside" LLM needs to be a simplified version of the original LLM, otherwise it could start doing this "prompt itself" procedure recursively forever (3) would it take a lot of time to mess with the "inside" LLM's weights? if yes you could probably counteract this by messing with the weights in advance.

[ocean harbor]

https://cdn.discordapp.com/attachments/695514882216427540/1231377074648780830/attachment.gif

[neat lintel]

): I told someone id learn latex and never did

[ocean harbor]

I learned latex here

very useful

[neat lintel]

nice

i need to but i keep putting it off

well, more like want to

[ocean harbor]

are you a beginner?

👀

the most horrible thing in latex are tikz

very annoying and hard also

[brazen hazel]

im sorry, i googled it while i waited, and i found my answer on the internet, again, i am sorry

[ocean harbor]

you're fine :3

[neat lintel]

worse

havent started

[vale plover]

@crystal stone applied for part-time positions to a couple of mathnasiums (k-12 math tutoring center) and kumon learning centers (k-12 math and reading/writing after school program) in my area. any advice on interview questions i should expect?

not sure if you applied to these places specifically but what are some generic questions i should anticipate?

[ocean harbor]

check pinned messages in #latex-help

they're really helpful

[neat lintel]

thx

In an AP. Sp = q, Sq = p and Sr denotes the sum of first r terms. Then, Sp+q is equal to

[tame egret]

🛑 stop

[vast wraith]

fangirling

[ocean harbor]

twelve essays damn

[sonic sentinel]

it looks like this method (or something close to it) does exist

selfcheckgpt from what i can understand of the diagrams is something like that

i am officially smart enough to have come up with something invented a year ago, no photographs please

[stray lily]

a little more and you can invent on ur own

good job

[dense ridge]

my mathematician advisor told me that one is not a “serious” mathematician if one does not accept infinity in their math, or accept proof by contradiction always. What is your opinions on that? to me it seems extremely philosophically dogmatic. Its not that i hold such positions, but rather the display of such aversion to other ideas of optimal mathematics. He likened the idea of not accepting infinity to flat earthers. Which i just find distasteful really.

[old oak]

That's a very judgmental way to put it, but I think it's fair to say that in most areas of mathematics if you refuse to accept ZFC, the notion of infinite sets, or LEM, you'll be putting yourself on the fringe and are likely not to be taken seriously.

[vast wraith]

it is possible to work without these things without making any drama about it

[alpine kindle]

i think finitism ends up being the more extreme of the two

not accepting lem is probably fine

in some sense

most things don't require it

[dense ridge]

yes i agree, but its a whole other problem of wheather it is good that such people not taken seriously, i think it isnt, and it seems like he is fine with not taking them seriously, personally if people try their best to argue for a point in good faith i would either take it seriously or be apathetic. i also think fringe ideas are good if it is in good faith which i think most people who would propose such ideas are.

[dense ridge]

who do you think are making drama?

[vast wraith]

historically there have been some mathematicians who claimed all maths using nonconstructive methods as basically invalid

[dense ridge]

sure, but thats not a judgement on people, which i feel this is more directed towards. he is basically not only saying this, but that those who do it are nutjobs.

in the other direction i mean

[vast wraith]

well, if someone claims your life's effort is invalid, i am fairly certain they'd take that personally

that said, this sort of attack is not a thing that's been happening since the early 20th century afaik

[dense ridge]

well i would be even more angry at the assertion i wasnt being serious or good eneough or really trying. fields require going down roads which may be unproductive i think.

[rocky shuttle]

what does invalid even mean when you're allowed to set up your own axioms in math

[neat lintel]

@vivid halo heres my cas so far (small part)

[neat lintel]

violates the rules of logical deduction

[rocky shuttle]

how are non-constructive methods invalid then

[dense ridge]

well not everyone is willing to accept just any axioms. the entire problem is what math is or shoulde be

[neat lintel]

wtf

non constructive proofs are valid

btw there are 9 mukhs now

[wispy cosmos]

If nothing is impossible, is it impossible for something to be impossible?

[old oak]

there's a fairly good argument to make that at this level most mathematics is unproductive, it's just abstract art and arguments over the preferred style of abstract art

[dense ridge]

maybe, but i just find it worrying and disheartening to have the very top leaders of the discipline be dogmatic in this way, preferably i would want a wide array of ideas and philosophies to be worked on without having to resort to personal attacks of the rest, it almost makes me lose faith in the value of our current acedemic system of mathematics when paired with all its other problems. do you get what i mean?

[jagged forge]

now i want to know what started this discussion in the first place

[dense ridge]

not so far up an anecdote of mine

[old oak]

i think most mathematicians don't really care about the philosophy, they just want to do mathematics in a way that's commonly accepted, and I agree that hostile reactions to stances such as rejection of infinity are unpleasant and unnecessary

but also I don't see how I could collaborate with a mathematician who doesn't accept the ZFC axioms, we just wouldn't be compatible

my useless art would not mesh with theirs in a pleasing way

[dense ridge]

yeah i definitely see how that could not mesh well, yet as you said respect still needs to be shown to different yet still possibly valid methodologies. i agree, i just wanted to know if i was totally off base. Thank you for your thoughtfull opinion and argument and willingness to engage with me, good night or whatever time it is for you.

[vast surge]

I probably could if it's just something like, "yeah, philosophically something's off with ZFC. Whatever. That's totally seperate and unrelated to the math that we do that assumes ZFC".

[glass escarp]

Hey guys I made this video for my Calculus class. It was a project. Please like, comment, and subscribe. My grade depends on it 🙏 🙏
https://youtu.be/ScPp2nhowgA

[hazy island]

Hello

[tiny cloud]

Bruh

[stray lily]

hello

[strong sluice]

Great video. I love your word choices.

But does your grade actually depend on it?

[bronze stream]

can anyone suggest me best book for linear algerba for super begginers please

[narrow talon]

methematics

[rocky shuttle]

mothemotics

[neat lintel]

Elon musk views this

That if something doesn’t break the current laws of physics he thinks it is possible and pursues it

Otherwise he will say it’s impossible if it doesn’t make sense according to the laws of physics

The next level of people are those who think anything that has already been done by someone means it’s possible and anything yet to be done is likely impossible

This chain of thinking is poor because whenever someone breaks a world record usually many other achieve the same thing quickly but before then no one could

An example is track running time like 4 minute mile

[narrow talon]

guys how would I turn a linear scale into a logarithmic scale

if I have a function f(x) would it be enough if I just replaced the argument with 10^x ?

along the linear scale it'd just be 10, 100, 1000,... no

[crystal stone]

Mathnasium will give you some test on arithmetic and algebra

You have to score above about a 98% on it for them to hire you?

I remember when I had just graduated UCLA, and I was just married, I needed some amount of income. I applied all over the place, and I was rejected from mathnasium because I got a 94% on their placement test

When I tried talking to the manager he was like "Well you didn't get the score you needed"

[north topaz]

there's so many tutoring opportunities though, like just go somewhere else lol

[crystal stone]

Thanks for the advice, let me go back in time to 2018 and get a job

[oblique copper]

I need help to solve an exercise

It's about trigonometry

[sturdy gulch]

wsp

[sturdy gulch]

what's the problem

[oblique copper]

I'll send to you

[vast wraith]

i have started looking into this, will let you know what i find in time

it seems to just be a passing example for Cardano, not much elaborated on… but in the age of sail, a method like this would be a very big deal, and there had been (failed) attempts in the 15th century by different methods… it seems many of them inspired by Archimedes… i should mention Tartaglia also gets involved as a riposte to Cardano

oops but it’s 3 am gn

[thick zealot]

Does anyone here teach real analysis and linear algebra privately?

[fresh comet]

is Dami typing up linear algebra

[velvet dagger]

@fresh comet you'll see me stop typing but don't worry, a Damiocrat always pays his debts

I'm just moving it to a Google Doc

Since I think it's gonna get... long lmfao

[fresh comet]

mhm

[velvet dagger]

There's a clear part 1

[fresh comet]

I'm not worried about that lol

[fresh comet]

you seem very trustable

and even if you didn't go through with it, I'd still be fine with that

[velvet dagger]

This is part 1. I think this order is fairly unambiguous (in principle you could try to do the stuff about basis before linear maps but I think this is better since in principle you can speak about linear maps not in the finite rank setting)

Right now my logic is that part 2 will be some multilinear algebra (mostly centered around bilinear/quadratic forms, and then a proper presentation of the determinant)

Part 3 will be structure theory of linear operators (so here's where the heavy lifting comes in, stuff like min poly, eigenshenanigans, diagonalization/triangularization results, and canonical forms)

Then finally norms, inner products, and such

Some logic/contentions about this order

• Everything in the first three parts is over a general field k (or at least maybe with general conditions like the min poly splits, or some condition on characteristic). Part 4 makes it clear that, okay now we're using the order and possibly some topology, so this is no longer raw algebra. This choice I'm least happy about since there's a lot of geometry of linear maps over R and C that I'd like to feature throughout ideally, but it feels trickier to work it in "smoothly"
• Some of the "structure theory" can be done sooner, and arguably it feels weird to talk about tensor products before mf eigenvalues, but it does feel like it's kinda split up: unless you're gonna pull some weird Axler shit, you can't really get the full force of the structure theory without determinants. And I don't wanna just mutter volume of tetrahedron or just write down a long formula, I want to present it in a good conceptual way

@vast surge curious about your thoughts on this lol

[vast surge]

Hmm

This is for an intro LA course?

[static loom]

silly take: only mention the fields Q and F_p, then handwave away the other fields as just field extensions of these

[vast surge]

Proof/theoretical based, for math majors - I think that this is mostly good. I'd shift quotients and first iso to after bases though, but that's just a personal thing.

[velvet dagger]

Yeah. The idea is that it's for people who are moderately clever, so they know what a proof is and don't need to be spoonfed the fact that cubic polynomials over R form an R-vector space but also cubic polynomials over C form a C-vector space zomg?!??!1?one

[velvet dagger]

Yeah that I guess was informed somewhat by aesthetics. Like oh quotients are a thing with or without finite dimensionality! :0

[vast surge]

I think the dictionary between linear maps and matrices is the correct way to motivate matrices personally. Matrix-vector multiplication is applying the linear transform to our vector. Matrix multiplication is composition of linear maps.

The way that it's often introduced, it feels like you're just given a complicated formula and told "here's how this works, have fun!"

So I'd honestly shift that a little bit further up, maybe even right after linear maps are introduced

[velvet dagger]

Yeah, I was thinking there might be a roundabout way to say oh encode systems of linear equations as matrices, wait hmm Ax=b kinda thinks of A as a map and asks for a pre image of b... That map happens to be linear... Oh linearity is a good property!

But it feels weird and you definitely wouldn't wanna define matrix multiplication at that point

So it felt oddly split up so I just kinda said screw it

[vast surge]

You can encode systems as matrices, sure, but you're not doing systems until the end

[velvet dagger]

Yeah I'm saying one alternate approach I considered was starting with linear equations and using that to kinda suggest linearity but it still didn't really suggest abstract vector spaces

And I was like eh they'll have to take a leap of faith anyway

So I'll just start with that

And then reward their faith

[vast surge]

Oh I see

First iso is basically right after you do kernel and image

Is the goal to draw parallels between rank-nullity and first iso?

[velvet dagger]

Yeah my idea was, first half is just vector space and linear map stuff, no coordinates

Now let's introduce coordinates and see how everything we did falls out

[vast surge]

How deep into the spectral theory are we planning to get here?

[velvet dagger]

So this is where the decisions are tricky

I haven't defined inner product spaces yet, which I know feels screwy but aesthetically part of me kinda wants to do all the norms/inner product stuff in one go

[vast surge]

I feel like - is this a course structure or a textbook idea? If it's a textbook, a lot of stuff could be added as "optional chapters"

[velvet dagger]

Corollary no spectral theorem

[vast surge]

Yeah the other thing is I'd have shifted those to being part 2.

[velvet dagger]

Yet

[vast surge]

Norms are just length, and inner products - especially the dot products in finite dimensions - are a fairly intuitive notion in a way that determinants aren't

[velvet dagger]

But yeah I mean, under this flow I'd do eigenvalues, eigenvectors, characterstic polynomial, Cayley-Hamilton, diagonalization = eigenbasis

[velvet dagger]

What do you think in particular of this. This is kinda the main stuff I'm wrestling with rn

[vast surge]

Will you be doing invariant factors and JCF stuff here?

[velvet dagger]

Yeah

[vast surge]

I think a way around this would be to sort of interlace the general treatment with applications specifically to R and C, as needed. Pedagogically it helps the earlier sections (R^n is easier to visualize than an arbitrary vector space, and students need toy examples), without necessarily sacrificing generality.

[velvet dagger]

I guess I'm less worried about sacrificing generality so much as it being a cognitive thing

Like let's say I talk about norms for a while and then go to determinants

Determinants have a geometric interpretation that only really works in R

[vast surge]

It also makes it so that looking at R^n and C^n doesn't feel "tacked on at the end" but rather integral or even motivating ideas throughout the thing

[velvet dagger]

But as a definition they make sense no matter what

And I feel like it's this weird "Wait hold on a second is this over a general field or is this R or what's going on?"