lmao
#serious-discussion
inner finch
fathom swallowBOT
The current time for Brandon7716 is 12:04 AM (EDT) on Wed, 13/10/2021.
clever knot
Have yet to unlock the no sleep hack
wonder how society would function if we didn't have to sleep
More working? More free time? Or both?
inner finch
unfortunately more working
clever knot
Would that imply we would be a more advanced society?
uncut tusk
Ireland before they discovered whiskey
gentle bay
I bet it would be much more.
halcyon basin
if bosses knew people didnt require sleep i bet my left kidney it would be hard abused
hard abused
sleep is a bit of relief from a terrible day, if you dont have it then there is no reset point basically
neat lintel
if bosses knew people didnt require sleep i bet my left kidney it would be hard ...
you don't let the boss abuse you
you become the boss 😤 😎
neat lintel
just dont close ur eyes
halcyon basin
Yo whats wrong with math twitter
never knew math concerned itself with political issues
limber thunder
twitter is cancer
steep mountain
highly recommend staying off twitter
its a toxic enviroment where the only concern is debating with no real purpose for the debate itself
dense drift
This server may be one of the best things on the internet. Never in my life have I been able to get so many math answers so quickly
I was wading through the dark, just stumbling along whatever math I was doing and ignoring all of the questions I had since getting help took too long or nobody could answer.
I am infinitely grateful
vague lion
cheers lol
dense drift
Also if anyone has a similar discord or forum or whatever for physics and/or computer science/python, I'd be happy 
vague lion
try #old-network
dense drift
oo
vague lion
the other servers are kinda stinky tho ngl
vivid halo
sick burrow
This server may be one of the best things on the internet. Never in my life have...
this server is one of the best things on the internet just because of the quality of shitposters here
there are few places that can make me laugh as consistently as this server
steep mountain
sometimes im searching logs for some book and i get a good laugh
surreal sapphire
this server may be one of the best things on the internet because i cannot tell if this was sarcasm or not
sick burrow
it was not
surreal sapphire
sick burrow
shitposting is a solid 90% of the reason I'm here
dense drift
the other servers are kinda stinky tho ngl
stinky xd lmao
sick burrow
uh oh
dense drift
Where are the shitposters
dense drift
sick burrow
steep mountain
was searching for LA books and got a good laugh
dense drift
Imagine trying to be wholesome but you are all jaded as hell
But seriously though, there's people here who spend multiple hours a day just like... helping other people?
Out of their own free time, for free, for nothing in return?
How nice is that
swift sinew
It’s called addiction
vague lion
it's called perfectionism
steep mountain
for nothing in return
vague lion
unanswered questions bug me
dense drift
That's nice
dense drift
*for nothing in return*
You don't even get likes or updoots here
surreal sapphire
you get dms you didnt ask for
fast ivy
~~you get dms you didnt ask for~~
latent forge
brazen bough
crystal stone
was searching for LA books and got a good laugh
Axler is such a good book
Everyone that disagrees is wrong
If you don't believe me, your entire point of view is written in a book titled "linear algebra done wrong"
So Axler agrees that it's the right way, and the book that treats determinants as central
is titled linear algebra done wrong
checkm8 atheists
neat lintel
I was messing around more with the split-complex numbers and found a nice property for reducing root-finding of polynomials when x is a factor
So if p(x)=xg(x) where g(x) is another polynomial over the split-complex bois, then p has 0 as a root and all of gs roots, but also any roots you get from the polynomial $\sum_{k=0}^n 2^{k-1} a_k x^k$ where $a_k$ are the coefficients in $g(x)$. This comes from the fact that division is equivalent to multiplication by the conjugate and division by the split-complex modulus. But that modulus is 0 if we are on the diagonals $|x|=|y|$. Checking the polynomial with the powers of 2 arising is sufficient for that root check before you can throw a factor of x away
fathom swallowBOT
zd
neat lintel
You can prove it by looking at (t+tj)^n and (t-tj)^n (encodes all of |x|=|y|) and using the binomial theorem
maybe this is a question for #groups-rings-fields but anyone know of general root-finding strategies in commutative rings?
split-complex numbers don't form a field so we can't treat it as a subfield of C
... I will cross post 😈
actually nah its busy
crystal stream
split-complex numbers don't form a field so we can't treat it as a subfield of C
they don't?
oh right
no inverses
agile cliff
lmao
thick nymph
mods slowmode pls
hollow obsidian
hi babes
quick pulsar
Wtf happens
lyric lynx
thank god
inner finch
did they @ everyone or something lol
lyric lynx
i had like 10 pings before they finally locked down the channel
inner finch
oh lmao based
fair valve
poopy pant
thick nymph
i got 3 pings yeah
finite lily
@everyone
orchid zinc
it b like that sometimes
shell pier
Well that happened lol
narrow vector
anyway now I'm here @compact tartan is a qtpi
storm drift
lol ive never posted here, the 4 pings got me worried
inner finch
lmao
uncut breach
ping storm
kindred crag
I was wondering why I got 6 pings here
shell pier
I got pinged 7 times
fast ivy
That was intense.
inner finch
damn i got 0 
digital dust
lmao that was intresting
kindred crag
yeah same
narrow vector
I love how Texit was spamming with them
valid flare
wild totem
bruh
fading badge
I didn't wanna see that
flat garnet
i hate having a username beginningof the alphabet now
sonic ether
o_O
polar wadi
i hate having a username beginningof the alphabet now
hehe...
flat garnet
rip a-c gang
blazing mortar
Lmao
hollow obsidian
cause tis funny
sonic ether
heres one @inner finch :))))
blazing mortar
What happens?
inner finch
so truE
flat garnet
yo but whats the link tehy spammed i wanna see
blazing mortar
This server is so active all of a sudden
uncut breach
lels in #discussion
flat garnet
lol
sick badger
🤣
sonic ether
he probably has been caught sending an exam question
wintry grove
Hey
grave ivy
NOO the reactions 😦
wintry grove
POV you haven’t opened this server in years
digital dust
@leaden torrent are bot commands disabled in #discussion
you might have to change a setting
leaden torrent
they shouldnt be
sonic ether
users?
fast ivy
odd narwhal
Just ban everyone
mossy cargo
,purge 300 would just clear the last 300 messages in the chat, yea
fathom swallowBOT
mossy cargo
oops
sonic ether
uh ok
mossy cargo
seems purgebot isn't working in discussion tho
digital dust
@leaden torrent use @fathom swallow?
mossy cargo
Lol it had an error in discussion lmaooo
leaden torrent
i am
raw drift
wth
mossy cargo
HAHA
eternal bronze
lmfaoo
sonic ether
big purge
leaden torrent
okay
mossy cargo
very helpful texit
leaden torrent
we gonna have to do this piecemeal
digital dust
ahh ok lmo
sonic ether
maybe just summon The Owner
mossy cargo
try 99
odd narwhal
Owner won't be able to do anything nami can't
old herald
wtf happened
sonic ether
ww3
dapper oriole
huh
inner finch
deez nuts
dapper oriole
who tagged me
inner finch
hi yami
brisk skiff
what happened
brisk skiff
why did i get @
royal canopy
needed to know what 2+2 is
dapper oriole
who tf knows
sonic ether
so like my question is
brisk skiff
i think a bot @ everyone
sonic ether
what is x
inner finch
we got botted 
sonic ether
perms level
inner finch
it'd become a ghost ping then
old herald
yes by the English majors 🙀
fast ivy
Spam bot attack
neat lintel
He was trying to purge either way
blissful lichen
this server is literally always getting attacked
sonic ether
how many bots were there
neat lintel
First time I've seen it get raided actually
fast ivy
Nami, one of the server's moderators, is about to write an explanation for what happened.
blazing pawn
7
digital dust
Nami, you could try using mee6
blazing pawn
theres really not that much to it some bots joined they spam pinged everyone
blissful lichen
First time I've seen it get raided actually
well sometimes people just randomly ping roles
neat lintel
It was just a script kiddy pinging names
blazing pawn
i mean like
neat lintel
Only people with high alphabetic names got pinged
sonic ether
i see
neat lintel
Down to like C I think
sonic ether
makes sense
nimble parrot
there aren't many roles to ping 😛
blazing pawn
u have to send messages pretty fast to get rate limited
hollow obsidian
makes sense i got pingged cause im A
blazing pawn
They werent pinging roles they were pinging people
swift sinew
makes sense i got pingged cause im A
I didn’t 🧐
nimble parrot
the question is why didn't @stiff bobcat catch it? it has mass mentions detection, unless disabled, or unless it was offline at the time
unkempt nacelle
pings?
hollow obsidian
I didn’t 🧐
interesting
hollow obsidian
the question is why didn't <@!155149108183695360> catch it? it has mass mentions...
they hecked it
nimble parrot
what does that even mean
neat lintel
🤔
unkempt nacelle
what is a heck
inner finch
deez nuts
neat lintel
i dont have permissions to send messages in #discussion anymore is it just me or is that everybody
neat lintel
the question is why didn't <@!155149108183695360> catch it? it has mass mentions...
Maybe it's disabled
leaden torrent
it shouldve been caught
unkempt nacelle
pings?
leaden torrent
maybe its misconfigured
leaden torrent
we'll have to look into it
leaden torrent
pings?
read #discussion or #changelog
swift sinew
I’m interested. What drastic measures were taken
sly vale
Well you cant post in the main discussion channel so...
neat lintel
don't now 
rugged field
just mass report thar server
I caught their ids and also them admitting to it in that server
neat lintel
lol
neat lintel
just mass report thar server
Good idea
rugged field
their only reasoing being "Fuck maths" and "I cant do maths"
swift sinew
their only reasoing being "Fuck maths" and "I cant do maths"
Check the engineering servers
neat lintel
Wait, how to report without filling out long ass form?
rugged field
Message Link: #896691846921719828 message
User ID: 892546948152840263
Server ID: 896682217948975135
light lily
if its any consolation i don't think its just them... this is like the third big server i've been in thats gotten raided with random mass pings tonight
devout veldt
sly vale
vibes in not pinged
nimble parrot
maybe its misconfigured
the configuration for it can be a bit confusing. you go to modules -> automod then:
make sure this is set like this (a bit counterintuitively)
and mass mentions to something like
nimble parrot
by default it's all disabled
royal canopy
why not delete the messages that have invites to random servers? That seems like the safest thing to me
leaden torrent
@compact tartan maybe look into this?
nimble parrot
any mod with admin perms can look at it
leaden torrent
thanks for the help btw
nimble parrot
👍
leaden torrent
why not delete the messages that have invites to random servers? That seems like...
just about to do that
there are so many messages its hard to find them all
we did tell discord to delete when they banned
but discord doesnt always listen
royal canopy
search in: discussion has: link
solar raft
btw theres like 100 more messages left till completely purging the bot's messages. how come u cant purge it?
royal canopy
should pull up all the ones that have links to discord servers
nimble parrot
oh yea, @stiff bobcat also has discord invite detection, if you want to simply delete those
compact tartan
<@!207092805644845057> maybe look into this?
look into what
compact tartan
eh
leaden torrent
might help prevent this
nimble parrot
compact tartan
I don't like the idea of auto deleting invite links
because I'd say most of the time people use them for ok things?
leaden torrent
its not invite links
compact tartan
mass mentions yes
compact tartan
I guess we could do that temporarily
leaden torrent
is there a reason to not disallow mass mentions?
compact tartan
I did
compact tartan
yeah
leaden torrent
anyway i flashed message perms in #discussion
seems we got all the bots
sometimes they hide themselves with renames
but not in this case
neat lintel
Oh you're in discord.py server too
Apparently Danny (the maintainer) is giving up on the library last I checked
neat lintel
I think the main framework for python's gonna be discord-interactions in the future
nimble shuttle
neat lintel
Discord hardly cares breaking changes lol
compact tartan
led
neat lintel
I'm sure they'll break the new API in like 2 years too
compact tartan
that's a strong way to put it
but I did pressure a lot of discord staff about it
because there's a lot of underhanded shit going on
odd narwhal
Not surprised
neat lintel
They hardly seem to care about feedback
real sigil
neat lintel
The video link copying problem has been around for so long
real sigil
Smh bots
proper sapphire
lol
cold needle
lmfao
real sigil
Someone has a media. version of that gif and it's hilarious
neon iron
Clever guys do you recommend I join their discord they seem pretty funny
inner finch
deez nuts
lofty robin
goteem
pale orchid
looking at the fourier method?
mint patio
Stein and Shakarchi's first lecture, the one on Fourier analysis
A lot of it is flying over my head tbh but as I continue to do some more passes I'm sure it'll get in there
I definitely appreciate seeing how naturally it comes up though
pale orchid
you know eigenvectors?
mint patio
Yes. Our LA class was behind too so we had to rush through eigenvectors and eigenvalues
pale orchid
like Av = lambda v
mint patio
But I understand the idea and how to find them
pale orchid
the nice thing about complex exps is that for many differential operators
D exp (i x) = lambda exp ( i x)
they are so called eigenfunctions
mint patio
yeah I remember this. So the complex exponential is the eigenfunction of a lot of differential operators
pale orchid
ye
mint patio
why would it not be i*lambda exp(ix)?
mint patio
So for some operators the i might cancel out?
Or some other phenomenon that causes it to disappear
pale orchid
sure, the operator could be i d/dx
mint patio
Interesting
pale orchid
in diff eq the i is usually just absorbed into the coefficients whose value you find using boundary conds
so they kinda vanish if not needed
alternatively you can treat exp as the analytic representation of sinusoids and end the procedure by taking the real or imag part
mint patio
I haven't done anything with boundary conditions, only initial conditions. Dunno what boundary conditions are :P
So this has some relation to the complex Fourier series. Because we can write that completely in terms of e then when using complex Fourier series in the solution of differential equations (like heat, wave, where the solutions are the same up to a constant) then since complex exponential is an eigenfunction for lots of differential operators we can find that constant...? something like that? just trying to figure out the relevance
Oh wait
Is a boundary condition just an initial condition but in space lol
So the solution (assuming it's multivariable) has to intersect some certain coordinates? Or it has to look like a certain curve at the boundary of the domain?
pale orchid
p much
mint patio
to which question :p
pale orchid
Is a boundary condition just an initial condition but in space lol
this
mint patio
sadge
pale orchid
I haven't done anything with boundary conditions, only initial conditions. Dunno...
and here, the idea is diagonalization
oof i qupted the wr9ng one
mint patio
AH I wanted to mention that when you said eigenvectors...we completely skipped diagonalization 💀 too behind
mint patio
I think it's a process where you decompose a matrix into a triangular matrix? So its eigenvalues are just the pivot
pale orchid
nu
mint patio
eek
pale orchid
you decompose it into QDQ^-1
Q has eigenvectors as columns and D is diagonal with the eigenvalues as its elements
it basically says that a diagonalizable linear transformation is simply a scaling of coordinates if you look at it in the basis of the eigenvectors
some differential operators are diagonalizable in the fourier basis of complex exps
so the operator's action can be understood by scaling complex exponentials and then combining them back
so you take a function, express it in a basis of complex exps, scale them according to your transformation, and chnage back to the original basis
the change of basis is done via projections onto the eigenvectors
and well, as rice told u earlier, you do that with dot products in euclidean space, and more generally with inner prods
like the fourier transform for squarw integrable functs
mint patio
so you take a function, express it in a basis of complex exps, scale them accord...
what's the purpose of this?
oh wait
to understand the operator's action (I think I kinda understand what that means)
did you all learn this as part of your engineering study edd? or did you learn it on your own
like not just during your phd but over masters too
or whatever else
pale orchid
a bit of both
i saw fourier and frobenius series as part of my diff eq course
and then signals and systems later on in my bsc
from then on it's like bread and butter for anything. since we deal with nice signals, it's always valid to ask if stuff would be nicer in the transformed domain
also stuff like control theory / stability theory with fourier and laplace
mint patio
is signals and systems and control or whatveer all that bad? I always hear everyone complaining about it lol
pale orchid
and in complex analysis and modern discrete control as well as sampling theory, the z transform pops up a lot
people find it difficult
mint patio
i saw fourier and frobenius series as part of my diff eq course
I might see the method of frobenius maybe?? we're covering power series solutions eventually
pale orchid
presumably because it has a ton of statistics that is usually explained poorly
mint patio
I did really well with stats in Hs and I think I'll do good when I take it next sem too
so hopefully it won't be too bad :D
pale orchid
stuff like
autocorrelation functions for deterministic and stochastic signals
parseval theorem, wiener khinchin theorem
and transfer functions of different flavors
like so
(wide sense) stationarity, etc
and once you discretize everything, same deal with vectors and matrices
dat toeplitz structure
i always keep this one around
mint patio
that looks very nice
yeah LOL
visually satisfying is my favorite type of satisfying
pale orchid
a correlation matrix with a two-level block toeplitz structure
comes out naturally when dealing with random matrices
ah
and as for the purpose of diagonalization
sometimes the original differential equation is super nasty
but solving it for one complex exponential is easy
and then you just add those complex exponentials up
that's the so-called fourier method or separation of variables method for diff eq
inner finch
toeplitz 😳
pale orchid
2 level 
mint patio
this is a personal attack
compact tartan
parseval theorem, wiener khinchin theorem
is that the one about existence of spectral density
upbeat fiber
how hard is linear algebra
is it gonna kill meh
I've heard is mad hard but some people say it's easy idk 
Right
Gonna tryhard on that shit
You don’t need much before getting into linear algebra right?
Just basic stuff no?
ALSO PLS tell me is there going to be any trigonometry in LA?
🤢
toxic schooner
pale orchid
<:blobseensomeshit:830896128505544704>
yeah
frosty zinc
ALSO PLS tell me is there going to be any trigonometry in LA?
Trigonometry
Wdym? Trignometery is basically manipulation of complex exponentials. Nothing much to be scared about.
But yes - Lin.Alg. is beyond those things
frosty zinc
It can, yes. But it's not the point/focus of Lin.Alg. IMO
pale orchid
no, it isn't
toxic schooner
steep mountain
frosty zinc
This is going to be fun <:sadcat:547124024585486359>
I fail to see the problem here? Those are just complex exponentials
upbeat fiber
I don’t like them
No, it’s not that I don’t like them. It’s that i am not good at solving them
steep mountain
have you considered studying them again from scratch
steep mountain
usually when im really bad at a certain concept i go back to how its taken initially
and build up the diff
upbeat fiber
I feel like i should put more time into complex numbers instead
steep mountain
assume you know nothing kinda mentality
pale orchid
you need trig for complex numbers
upbeat fiber
I know some stuff in trig but I’m really bad at halve angles and such
Not sure what its caled
steep mountain
trig rules?
pale orchid
oh those identities. you can derive them with complex numbers tho
but you need to understand the basics well
unit circle and stuff
steep mountain
there is a lot of identities you cant memorize them all
pale orchid
that's just memorization, yea
upbeat fiber
I suck at memorization 😞 ig ill do my best in la if i fail i take it next year
I actually look forward to it. I think it’s gonna be more fun than discreet math
steep mountain
discrete is great wym
upbeat fiber
Caught in 4k
steep mountain
upbeat fiber
Discreet math is awesome
Fr people look at me like a psychopath when i say discreet math sucks lmao
Number theory and stuff is fun
But induction
…
That thing scares me
Even worse
pale orchid
how do you know any of those things, but not trig and linalg?
upbeat fiber
Structural induction using haskell
upbeat fiber
how do you know any of those things, but not trig and linalg?
Wdym?
steep mountain
how do you know any of those things, but not trig and linalg?
cant be good at everything heh
pale orchid
those are super basic tho
steep mountain
i sucked hard at integrals till end of my freshman i was taking calc 3 and topology
upbeat fiber
Sheesh
steep mountain
there is always that 1 basic thing you miss in hs
upbeat fiber
Integrals are fun af
Calculus is hella lit
Well
I didn’t study in hs
I started studying since covid started
Which sucks
steep mountain
eh same here didnt realize i like math till college
upbeat fiber
I wish I took math seriously in hs but didn’t
steep mountain
did well in hs but never got invested into it you know?
upbeat fiber
IRGHT LMAO
F
I didn’t eve do well
But i never failed math in hs
Different breed
steep mountain
i think what you do rn is all that matters
upbeat fiber
Ig
pale orchid
spoken like a true anime protagonist
upbeat fiber
I’m more invested in math rn than programming
But I’m doing well in discreet math which is fine
steep mountain
there was this nice argument against regrets which says
how can you expect things to go well even if you did invest time in that one thing even if you were a better person that would possibly lead you to a worst you today
poorly said by me but you get the point
steep mountain
its like you know what maybe i was a retard in hs but im enjoying myself rn
upbeat fiber
I just feel like. I could’ve learned more if I started earlier
steep mountain
there is always that
i mean newton was inventing calculus at age 18
makes you wonder if its worth the effort lmao
just take it easy
steep mountain
ye is admirable really
worn juniper
say f(x) = blah blah, is x the parameter of f?
is it said like that or is the semantics wrong
steep mountain
the variable you feed the fn is the parameter so yes
technically it would be b l a and h
you get the point
worn juniper
ahh kk thanks
frosty zinc
I don’t like them
This is the first time I am hearing someone dislikes working with exponentials
frosty zinc
Oh. These.
Never used them all that much
upbeat fiber
i find it hard*
same, they told us you have to learn about them but we never got them on the exam lol
frosty zinc
I mean, leave the exams behind but anything relevant to actual problem solving - These identities are never useful to me. Instead, I just use the complex exponential form of it -
$\ sin(x) = \frac{e^{ix} - e^{-ix}}{2i}$
$\ cos(x) = \frac{e^{ix} + e^{-ix}}{2}$
$\$ And even these you can easily relate or derive by keeping Euler form of complex number in mind
upbeat fiber
This is literally the first time i see these
fathom swallowBOT
HarshlyDOOM
upbeat fiber
I don't think they go through these in hs here unless in advanced math in hs
yeah this is college level math here
Is this the stuff that pops up in linear algebra?
frosty zinc
I don't think they go through these in hs here unless in advanced math in hs
yeah, it's rare to see these in HS (I did but that's a different story). But this is how most people treat trigonometric functions as .
frosty zinc
Is this the stuff that pops up in linear algebra?
You first see this when you do complex numbers and you encounter the polar form of complex numbers as -
$$\ z = r(cos(\theta) + isin(\theta)) $$
Where magnitude of complex number,
$$\ r = |z| = \sqrt{r^{2}cos^{2}(\theta) + r^{2}sin^{2}(\theta)$$
and, argument of complex number is -
$$\ arg(z) = tan(\theta})$$
What you may not have learnt is that this polar form is equivalent to writing -
$$z = re^{i\theta} = r(cos(\theta) + isin(\theta))$$
fathom swallowBOT
HarshlyDOOM
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
high rivet
leaden torrent
is there a reason youre posting this or do you just think its interesting
it does confirm my priors a bit
burnt dune
hi
upbeat fiber
If i have a Haskell/ functional programming question where do i ask ? 🤔
crystal stone
we don't really do a whole lot of non-computational programming
you might try #computing-software
upbeat fiber
ooou
crystal stone
But I think for that it's best to go to the #old-network and look for a CS server
Since we mainly specialize in mathematics
upbeat fiber
yeah i'm on the programming channel but they are busy sadly :(
neat lintel
no inverses
you can get inverses 1/(a+bj) = (a-bj)/(a^2-b^2)
as long as you don't have |a|=|b|
basically points on the main diagonals don't have inverses
it's an inconvenient ring to work in cause there are 4 separate orbits of the action by elements of the form re^(tj) (the right-facing hyperbola)
namely the sets of elements of the form re^(tj), rje^(tj), and the positive and negative versions of each (always making r strictly nonnegative), and then the diagonals don't even have polar decompositions
at the same time it's nice cause it's mostly units, and the points living on a diagonal follow a multiplication rule (t+tj)*(w+wj) = 2(tw+twj)
and for opposite diagonals
(t-tj)(w+wj) = 0, which is nuts
crystal stream
is there an analog for all the cool theorems from complex analysis?
neat lintel
unfortunately not many, and where there are it's only partial
You can show e^(tj) = cosh(t)+jsinh(t)
but that's only the right facing hyberbola as I mentioned
so multiplying by j gives you a flip over to the top facing one, or by negative j to the bottom facing one, or just by -1 to the left one
and then scaling by a positive real r lets you cover the entirety of R^2 minus the diagonals
I was playing around with trying to do calculus on it but I couldn't get some things to work, apparently even z^2 isn't differentiable (z is cause it's already linear so ofc it will be linear locally) so that basically means bye bye calculus since there's only one kind of polynomial that's differentiable
But I might be wrong cause I was just going off of what wolframalpha spat out
You can look at the Jacobian of a function and force it into the form of a split-complex element in matrix form to get an analogue of the Cauchy-Riemann equations, but I think it breaks because the sufficient condition proof relies on topological properties of R and C that don't occur in the split-complex numbers
namely neighborhoods not getting fucked up by those diagonals
crystal stream
i'm guessing we don't use the usual R^2 metric
neat lintel
You can because the split-complex modulus doesn't even follow the triangle inequality 😂
there's nothing better really
I think it might work when only considering a single hyperbolic region though
like if you look at the right facing region you could do re^(tj) -> r and it's fine I would imagine, maybe? haven't proven it
idk what the metric would look like for 2 nonzero elements
wait wtf am I doing lol
uh
maybe there is a fun metric you could make out of it (only one region) I am curious
also curious as to whether calculus works when you restrict domain and image to only one region
you will lose a lot of functions
leaden torrent
Sure, but you can get some unintuitive behaviour near boundaries
neat lintel
pog
leaden torrent
But like as an example
If f: R -> R is given by f(x) = |x|, then f restricted to [0, 1] is infinitely differentiable everywhere
Since it's identical to g(x) = x on that region
neat lintel
right ok
also the other big issue with calculus was using the traditional definition you end up with a factor of 1/(h^2-k^2) and it's a limit as (h, k) -> (0, 0) with no saving graces from simplifying so it is just doodoowater, limit doesn't end up existing (at least according to wolfram I hope it's wrong so bad 😭 )
so maybe even picking and choosing regions won't work
the only reason I am clinging to hope that Wolfram is wrong is because I tested out writing differentiable functions as their linear transformation near the point like
f(y) ~ f(x) + Df(x) (y-x) and it seemed to be pretty close no matter in which direction I slightly perturbed y-x
but that doesn't mean much
neat lintel
Ok I got something cool. Triangle inequality works for sure if you only consider linearly dependent points that are off the diagonals, it definitely works in some other niche cases but this is a nice property
Any line through, but excluding, the origin is a metric space in the split complex numbers under abs(||z||)=|x^2-y^2|
wait frick
I had some variables switched nvm
sad
but because of that switch it means you can compare any point and the REFLECTION BY j of a point linearly dependent to it using the metric
you can actually choose 2 lines to compare your point with since you can reflect again by making that second point negative
but you can't necessarily (afaik) compare a point to points on its own line or the negative of that line
crystal stream
is there a split-complex logarithm?
neat lintel
you could definitely make one yeah but probably only for the points that have polar decomposition
So the action of the logarithm on a hyperbolic section would be like
transforming hyperbolas to vertical lines
unlike how i can be written as e^(pi/2 i), j doesn't have a polar form, it's just 1je^(0j)
Because if there were then you could move smoothly from the right facing unit hyperbola to the top facing one, and that's impossible
and same thing with -1
So you just have to define 4 different split-complex logarithms 😂
to make things nicer I guess you could define log(-1)=-1 so that you only have two and the transformation on two additive inverse hyperbolas is a plane
you could make it all one function with log(j)=1 if you don't care about it being injective
oo or even more logical would be log(rje^(tj)) = (t+log(r)j), so they swap to keep the upward/downward facing sections facing those ways even after they transform into planes
crystal stream
so the squaring function maps the hyperbolic plane to the first "quadrant"
neat lintel
so the squaring function maps the hyperbolic plane to the first "quadrant"
not sure what you mean, could you clarify?
neat lintel
this is a funny math.se post because I wonder if the poster playing around with Laurent series even tried to find a nontrivial differentiable function on the split-complex numbers :P
ugh I was thinking of something else to do too but it's inconvenient that split-complex Cauchy-Riemann isn't sufficient :/
I can't imagine them not being necessary though so if you want to play around with split-complex differentiability then the Jacobian being in the form of a split-complex linear transformation gives you $j\frac{\partial f}{\partial x} = \frac{\partial f}{\partial y}$
fathom swallowBOT
zd
neat lintel
just sad that it's not iff :(
mint patio
If I say "Let $k = n\pm m$ and I have $\int \cos(nx+mx)dx + \int \cos(nx-mx)dx$, would it be valid to equate this to $\int 2\cos(kx)dx$?
fathom swallowBOT
feather
mint patio
Never mind that's pretty stupid huh 
I can't do that because writing each as int cos(kx) + int cos(kx) doesn't assure me that each term is the same
crystal stream
afaik cauchy riemann isn't iff in C either
no analog for morera's theorem i'm guessing
oh wow an argentinian did some work on split complex numbers
noice
Isotropic rectangles play a fundamental role in this theory since they form the domains of existence for holomorphic functions, domains of convergence of power series, and domains of convergence of functional series.
neat lintel
it's iff in C as long as u and v are differentiable both ways I think
woah whats all this, looks cool
ahh huh they use a Wirtinger derivative
never heard of these but holy shit cool to see that people decades ago did exactly what I did xD
"yeah it's not actually differentiable so we'll just pretend it is up to what the Jacobian tells us (i.e. analogue of Cauchy-Riemann)"
I wish I knew Spanish damn, this source looks awesome (Vignaux, J.C. & A. Durañona y Vedia (1935) "Sobre la teoría de las funciones de una variable compleja hiperbólica", Contribución al Estudio de las Ciencias Físicas y Matemáticas, pp. 139–184)
ok another thing to examine with split-complex bois: analogue of Mobius transformations
although I don't know much about them 😂
onyx idol
neat lintel
wait right they're inverses of stereographic projections after transformations on a Riemann sphere
https://en.wikipedia.org/wiki/Möbius_transformation#Projective_matrix_representations this is exactly what one would want to pay attention to when applying this I think
trim dagger
@uncut steppe I don't know ed systems in other countries but calc usually sees precalc as a prerequisite
Which is often an extension to algebra II, covering unit circle and a lot more trig functions
fair mural
isn’t precalculus just trig
trim dagger
Yeah
uncut steppe
I live in America
fair mural
or is it something else too
trim dagger
It is
fair mural
ok
trim dagger
Oh I assumed you didn't because different places usually change what they call classes quite a bit
fair mural
where are you in calculus right now?
uncut steppe
ooh! I am from Cali
trim dagger
I'm used to alg I > geo > alg II > precalc > calc
trim dagger
I'm in calculus I atm, currently around logarithmic diff
I've touched on integrals on my own but honestly forget everything I ever learned
fair mural
that stuff is interesting but i honestly never use it lol
neat lintel
tell me something cool about logarithmic differentiation
uncut steppe
I'm used to alg I > geo > alg II > precalc > calc
Awhh! Well I don't know what to take next year. Since we only need to go up to Math 3. But no specific "class name".
fair mural
integrals are really cool
trim dagger
It's a little cooler than implicit differentiation
fair mural
you like math so you’ll love them when you get to them
trim dagger
I love math
trim dagger
I do more homework for the fun of it
junior and senior years of high school I made the homework for my classes and didn't have to do any of it but got full credit fo rit
fair mural
lol
that’s cool
i like math but i’m taking statistics right now and it’s not my favorite
although it’s probably just the teacher
trim dagger
I'm personally less of a fan of applied maths so it could be that too
uncut steppe
I am in APUSH & it's horrible due to the teacher </3
trim dagger
I'm pure track
trim dagger
I mean there are applications for me
I'm a math ed major, the application is teaching other people stuff they'll never use
fair mural
bruh
trim dagger
lmfao
fair mural
do you have any other math you’ve been wanting to try to learn?
after i get done with calculus i plan on doing linear algebra
trim dagger
I have a bunch of required classes for my majors and fortunately some that I've always wanted to take are in there
I'm hoping to hit real analysis at the very least but my uni does have a math ed graduate program and I don't even know what that covers yet
fair mural
i haven’t really done anything with analysis, is it just like the study of some kind of math?
i’ve heard of it definitely
trim dagger
I would think so but I'm not in any yet
Abstract analysis is a requirement for me but it's a bit far out; real analy is an elective that I justr want to take
trim dagger
It revisits calculus I/II but with proofs and shit
fair mural
oh
trim dagger
I don't have my list my advisor does and didn't save it for me to get online
meager sonnet
analysis generalises calc
trim dagger
I get to do computer science too which I've wanted to all through high school but they were never able to hold the class
meager sonnet
it rigorously presents limits and continuity and the like
trim dagger
Yeah my prof makes it sound like a blast
fair mural
well hopefully you get lots more classes you like
trim dagger
At this point I have space for one elective that can be anything I want
Everything for the double major also satisfy all of my elective spots but one
fathom swallowBOT
feather
mint patio
Is that not an elementary antiderivative of e^(-x^2)
neat lintel
"elementary" probably includes "not in terms of any limits"
timid spindle
What if we solve non-linear DEs numerically?
JK
Unless!
crimson mica
Need help. What's that animal that looks like a velvet worm except it lives in the sea and it's transparent?
mint patio
"elementary" probably includes "not in terms of any limits"
ah yeah I just checked the wikipedia page and it has to be a finite number of terms :(
frosty zinc
Need help. What's that animal that looks like a velvet worm except it lives in t...
I read that as wavelet worm.....
Why am I like this....
brave hollow
this proves my old theory, physicists are self centered
neat lintel
is there a name for R[x]/(x^3+1)?
toxic schooner
neat lintel
to be clear I mean R is the reals not just some general ring
bronze pelican
is there a name for R[x]/(x^3+1)?
its isomorphic to C
toxic schooner
It is?
vivid halo
yes
toxic schooner
vivid halo
since x^3+1 has a conjugate pair of complex roots
bronze pelican
The only nontrivial finite extension of R is C.
vivid halo
as soon as you adjoin a single complex root to R it's C
neat lintel
Oh because this quotient is the same as adjoining a root of x^3+1 to R right?
vivid halo
yes
well
it's the same as adjoining all the roots of x^3+1 to R
obviously if you adjoin the real root of x^3+1 to R you get R again
bronze pelican
notice that x^3 + 1 factors as (x^3 + 1) = (x-1)(x^2 + x + 1).
So the quotient R[x]/(x^3 + 1) is isomorphic to the tensor product R[x]/(x-1) \otimes_R R[x]/(x^2 + x + 1)
The first factor R[x]/(x-1) is isomorphic to R.
The second factor R[x]/(x^2 +x + 1) is isomorphic to C
toxic schooner
bronze pelican
So you get R[x]/(x^3 + 1) is isomorphic to R \otimes_R C = C
vivid halo
you're adjoining all the roots of x^3+1, one real root doesn't contribute anything new, the conjugate pair of complex roots turns this into C
bronze pelican
neat lintel
I don't know why that's true but if it is, that's very interesting because then R[x]/(x^3-1) would also be C, right?
or wait maybe that's what you were thinking of
bronze pelican
If you do something like R[x]/(x^5 - 1) then you get C^2
neat lintel
cause (x^3+1) does not factor as (x-1)(x^2+x+1) but rather (x-1)(x^2-x+1)
neat lintel
oo and that has 2 complex roots as well lul so either way you get gamered, it's C
bronze pelican
it has a factoization over R as a bunch of linear factors times a bunch of quadratic factors
the linear factors contribute nothing
Each distinct quadratic factor contributes a copy of C
I'm not sure what happens when you have repeated quadratic factors
toxic schooner
This stuff is all in ring theory?
neat lintel
you can have higher degree factors irreducible over R right tho
bronze pelican
no
neat lintel
x^4+x+1 doesn't work? no real roots
bronze pelican
if it has no real roots then it factors as a product of quadratic polynomials over R
neat lintel
ahhh ok
that's awesome, how do you show that?
probably some shit I forgot from my 2nd course in algebra
bronze pelican
i forgor ☠️
neat lintel
ok I don't know how to do an in depth proof but I thought of one that relies on other stuff
minimal polynomial has to be degree 2, maximum for those complex roots since they can have at most 1 other conjugate
over R
bronze pelican
this is sort of using that C is algebraically closed
neat lintel
so any polynomial after getting rid of linear factors has to be a product of those irreducible quadratics
distant vortex
All real polynomials can be factorized as linear factors over C. If it has a linear factor (x-z) where z is a complex number then (x-z*) is also a linear factor. So factorize it as linear factors over C, then multiply all the complex linear factors with their conjugates which gives quadratic factors over R. Then you can factorize every polynomials as a product of real linear and quadratic factors.
neat lintel
I don't mind 😂
oo that's nice awesome
I would have still remembered this stuff if I hadn't taken such a busy semester when I took that class
iirc you can generalize that fact to Galois conjugates looking at elements in the extension field as elements of a vector space over the smaller field
bronze pelican
I thought there was a way to conclude it without using FTA
neat lintel
everything is hard until you find it and then it's trivial 
bronze pelican
No nvm it's not
how do you do it
distant vortex
No, lol
I thought my previous proof was wrong for like a second
That's what the no nvm it's not was for
bronze pelican
oh ok
distant vortex
I'm looking for proofs but they all use FTA, cry
bronze pelican
I thought Dummit and Foote proves FTA using this fact, but maybe im mistaken
neat lintel
All real polynomials can be factorized as linear factors over C. If it has a lin...
this proof doesn't use fta tho I think right? https://en.wikipedia.org/wiki/Complex_conjugate_root_theorem#Proof
distant vortex
No that part is fine
FTA gives us that it can be factorized as linear factors over C
neat lintel
right FTA is just equivalent to "C is algebraically closed"
bronze wren
Can somebody give me an upper intermediate compnitorics question
Fucj I meant to send this in math discussion
neat lintel
hm it's really weird that K=R[x]/(x^3+1) and C are isomorphic tho cause the modulus you would define naively in K doesn't even map down to R but is a vector field and setting it to 0 gives you a fancy surface (z=-x^2/2y) rather than having a single point as a solution like you would have in C
also the "unit curve" in K is not a circle either, but an unbounded curve that kinda loops around to 1 on its way through the unit region and then yeets out to infinity again
that means you can make some really really unintuitive spanning sets for C over R that do wacky fun things
me likey
neat lintel
ahah actually all of this is ignoring a huge issue I just realized
you're basically pretending the entire time that even though your new element k is a root of x^3+1, that it is neither a root of x-1 (defined not to be 1) nor x^2-x+1
but it has to be the second one making 1, k, and k^2 not linearly independent
and you get your dim=2
and so really the spanning set by x+yk+zk^2 can instead be written as (a+b)+(b+c)k^2, and just substitute k^2 with e^(2pi/3 i) and you're in the complex world
that is funny
ahah ok I just wrote it out, so the explicit mapping from K to C using a span of a set of 3 is $a+bk+ck^2 \mapsto (a+\frac{b-c}{2}) + \frac{\sqrt{3}}{2}(b+c)i$
And setting any of a b or c to 0 gives us an isomorphism between K and C
fathom swallowBOT
zd
neat lintel
so if we call this mapping T, plotting the vector field of T(p)-p actually does look like all the vectors hop onto the xy plane
Ok now to play with R[x]/(x^3) since x^3+1 and x^3-1 give you nice maximal ideals, I want shitty ideals only
wait im buggin how are x^3+1 and x^3-1 maximal they're not right? but they still quotient with a PID to make a field (C)
(x^3-1) is a subset of (x-1) right?
or do I have it switched
simple raven
it's the same as adjoining all the roots of x^3+1 to R
But (X^3+1) isn't irreducible over R so R[X]/(X^3+1) can't be a field (it's the product of R and C)
neat lintel
there we go so this is all fucked LOL
neat lintel
what about this? @simple raven
simple raven
what about this? <@!280791755300012032>
I don't understand why there is a tensor product, for me it's just a product of rings
By the Chinese theorem
neat lintel
wow ok
well that mapping I found (fixing one boi to 0) can't be an isomorphism then
should have noticed it earlier since (x^3+1) and (x^3-1) are noooot maximal and cant give you a field by quotienting but I just ate up what folks said 😭
simple raven
You have an explicit zero divisor, [the class of] X-1 multiplied by [the class of] (X^3+1)/(X-1)
vivid halo
yup
neat lintel
wow yeah that too
so I'm very happy now thank you
I still should write a+bk+ck^2 with the correct basis of 2 tho LOL
a+bk^2 suffices
simple raven
(x^3-1) is a subset of (x-1) right?
Yes, it's a strict subset so (X^3+1) can't be maximal (because it's not the ideal (0) )
neat lintel
fuck yeah
bronze pelican
I don't understand why there is a tensor product, for me it's just a product of ...
Ohhh ok
simple raven
If you do something like R[x]/(x^5 - 1) then you get C^2
For this one you get R x C x C
vivid halo
algebra leaving my brain as soon as I passed my qualifying exam
simple raven
@neat lintel if you want to make sure you've understood, try to compute $\mathbb{R}[X]/(X^n - 1)$, for an integer $n ≥ 0$
fathom swallowBOT
Adrien
neat lintel
ok just refreshed myself on CRT for rings
oh god is this gonna be a bunch of cyclotomic stuff 
since you're basically factoring x^n-1
simple raven
oh god is this gonna be a bunch of cyclotomic stuff <:sad:532803514892812288>
Yeah
But it's not that hard don't worry
simple raven
since you're basically factoring x^n-1
Yes, over R[X]
You don't need to give an explicit factorization
neat lintel
oh ok then it's just R[x]/I_1 \times ... \times R[x]/I_k where the ideals are principle and generated by irreduccible polynomial factors of x^n-1 right
But R[x]/irreducible is gonna be a field since R[x] is a UFD so it's a product of a bunch of fields
you're not going to have any elements in each of these quotient rings with degree above 2, and you won't have elements in the quotients by ideals of linear elements above degree 1, and that should determine the dimension of each
so the quotients by linear terms give you R and the ones by quadratic terms give you C?
and it's just a product of all that right
simple raven
oh ok then it's just R[x]/I_1 \times ... \times R[x]/I_k where the ideals are pr...
Yes it's a product of fields
simple raven
so the quotients by linear terms give you R and the ones by quadratic terms give...
Yes ! So you just need to count the number of reals roots
neat lintel
ooo and that gives you exactly how it decomposes holy crap
k real roots and it's Rx...(k times)...xRxCx...(n-k times)...xC
idk the direct sum symbol in latex LOL
simple raven
Yeah
neat lintel
each takes 2 complex roots
simple raven
Your irreducibles factors need to be in R[X]
simple raven
oh ok then it's just R[x]/I_1 \times ... \times R[x]/I_k where the ideals are pr...
Also, it's a product of fields because you don't have powers of the same irreducible factors
X^n - 1 has no repeated roots in C
simple raven
that is awesome, thanks
Np !
So if n is even, you have 2 reals roots (1 and -1)
If n is odd, you have 1 real root (1)
neat lintel
also I guess you would get that R[x]/(linear) or R[x]/(irr. quadratic) give you R and C specifically is just because the cardinality is <= when taking quotients?
simple raven
Cardinality ?
neat lintel
X^n - 1 has no repeated roots in C
that is good to know
simple raven
If you quotient by a polynomial of degree 1, you get R (not hard)
If the polynomial is of degree 2, you get a finite extension of R, of dimension > 1 so it has to be C because it's the only non trivial extension of R
It's not the same over Q
neat lintel
yeah I would know how to do the degree 1 case since the isomorphism is basically right there for you already
ok got it
simple raven
Great 
neat lintel
gaming
wow, the unit curve for R[x]/(x^3+1) looks like this
fractal moment
So one of the reasons I'm messing with this stuff is to discover "unit curves" (even if there's not a definable norm or modulus) from the sum formula for e^(t?) whatever ? happens to be, basically just forcing it to cyclically generate the basis for that ring (1, ?, ..., ?^n)
And whatever happens, because of the nature of that sum you will get a wacky set of functions that is actually multiplicative in the world of a_0+a_1?+...+a_n?^n
basically generalizing the unit circle, hyperbola, and line you get in the case of complex, split-complex, and dual numbers
neat lintel
huh I definitely screwed something up
since it's not a field we can have (k-1)(k^2-k+1)=0 without either being 0 and making k constructible from k^2 and 1, just means that it's not an integral domain
RxC is dim 3 as a vector space so that basis I had way way earlier was probably fine
timid spindle
can someone help me in #latex-help ?
neat lintel
check 📌s on #latex-help
timid spindle
check 📌s on <#840667252793802752>
sorry. I always forget pins. I'll see if I can find something in that book.
mint patio
tell me why my calc 1 class is unironically harder than my diffeqs class
mfs giving some dumbass questions
obsidian jolt
tell me why my calc 1 class is unironically harder than my diffeqs class
Don’t you need to have calc knowledge to take a diff eq class
obsidian jolt
like how are you supposed to Laplace transform without knowing calculus
mint patio
I've taken up to calc 3 and am currently taking diffeqs, but I'm retaking calc 1 since I didn't do well the first time I took it
mint patio
but the professors make it 200x harder than it needs to be
neat lintel
$\dv{x^2}{x}=?$
fathom swallowBOT
Tesseract
steep mountain
thats cute
obsidian jolt
I’m finding calc 3 difficult to visualise
mint patio
nah but like avg of 65 is wtv it happens for hard classes but this is a freshman calc class
neat lintel
nah but like avg of 65 is wtv it happens for hard classes but this is a freshman...
How are they running it?
obsidian jolt
nah but like avg of 65 is wtv it happens for hard classes but this is a freshman...
Maybe a lot of people uninterested in math are being forced to take it
are you at uni?
Or school
neat lintel
am I supposed to say x tess <:sully:651816820122189834>
As you are well aware, that is a very, very difficult calculus problem.
steep mountain
we had averages of <30 on calc 3, real analysis, integral analysis ,prog in C , intro to top
cause hard=good
mint patio
Nothing abnormal besides the questions. They just ask a lot more challenging questions than they need to
obsidian jolt
we had averages of <30 on calc 3, real analysis, integral analysis ,prog in C , ...
wow
neat lintel
What's actually interesting is how Archimedes managed to compute the area under a parabola.
obsidian jolt
also what’s integral analysis
mint patio
do they separate analysis into differential and integral analysis 
wat
obsidian jolt
yeah but what does it cover that doesn’t happen in a calc class
neat lintel
Nothing abnormal besides the questions. They just ask a lot more challenging que...
Like what? Obtuse word problems? Overly complicated derivatives?
steep mountain
my uni has this great idea to compact the curriculum to 2.5 years
steep mountain
so on freshman 21% passed
neat lintel
do they separate analysis into differential and integral analysis <:sully:651816...
I've heard people divide calculus into integral and differential calculus, but not analysis.
mint patio
I never struggled this much with calc 3 either
mint patio
yeah same Tess that's why I sullied
mint patio
It's not
steep mountain
too bad it was the 1st course lmao
mint patio
That's why we're confused 💀
neat lintel
The major divisions of analysis for undergrads are real, complex, and functional analysis, if memory serves.
steep mountain
professors really think its a flex to make the course as hard and notorious as possible
mint patio
Like why is this important
why not just use induction
like a normal fucking person
lol
instead there's all this bullshit
and u know why?
so we can do limit definition of integral
obsidian jolt
How is this calculus though
mint patio
The limit definition of the integral involves sums like this
(Calc 1 here is both differential and integral calculus)
neat lintel
That's a rather lovely problem, actually!
obsidian jolt
I’ve seen that before
mint patio
It's pointless imo
obsidian jolt
while doing summation things
mint patio
I hated doing shit like this when I took my intro analysis course
Where you had to use axioms of real numbers to prove the most basic shit
So obnoxious
obsidian jolt
It feels like there’s literally an endless amount of stuff you can learn
neat lintel
It introduces you to the idea of a telescoping series, gives you an opportunity to develop facility for series manipulations, and proves a very useful result; I think it's a rather nice problem, actualy.
mint patio
obsidian jolt
In which class does one learn about lebesque integration?
mint patio
analysis
obsidian jolt
Feels like there’s an endless amount of stuff to learn
mint patio
It feels like there’s literally an endless amount of stuff you can learn
you don't say :p
obsidian jolt
After calc 3 I’ll do linear alg, diff eq, then analysis and abstract algebra then other shit
steep mountain
yeah but what does it cover that doesn’t happen in a calc class
just proofs
neat lintel
As far as I know, Lebesgue integration just gives me an excuse to integrate over pathological spaces.
tall badge
there's a side to these formulae that isn't shown too often by exercises just saying "prove this"
mint patio
Like if you see some of the advanced people here talking about math you think they're just making words up