Hi can someone tell me in what conditions is lim( f(x)*g(x)) = lim(f(x)) * lim (g(x)) ? The limits must exist and be finite?

anyone have a mnemonic for the difference between principle and principal? some clever wordplay that will help me remember?

certainly valer if they exist and they're finite then it's true

@valer if they exist then they're finite :)

it's also true if one of them is $\pm \infty$ and the other limit exists and isn't $0$

if you get $0 \times \infty$ you can't draw a conclusion yet

@GFauxPas where $\infty * k$ is interpreted as $\infty$ if $k>0$

and $-\infty$ if $k < 0$ yes

@Fargle and to moreover replace the actual founders by a convenient mythology of their own creation

To the extent that it's understood as fictional, I don't mind that. But that's not how they want to use it.

@Semiclassical Yep. It's honestly distressing how much people refuse to be aware of the actual history.

yeah

it's why I get exercised (to use an example that showed up in h-bar) about movies like 'The Imitation Game'

whatever artistic merits they may have, the degree to which they create their own story and pass it off as 'history' irritates the hell out of me

One can't be entirely historically accurate, to be sure---it's a movie

But there's limits to that.

@Semiclassical good news for u it doesn't have any merit

hah

I didn't want to jump to any conclusions there

Soo basically i can't distribute the limit ( if one of them or both don't exist) and if after i distibute the limit i get an indeterminate form. Thanks

right

I am sure that the dimension of symmetric complex square matrices over $\mathbb{R}$ is $n(n+1)$?

is there an inverse gradient "function"?

i saw this but the problem is that I don't actually have a function, just a data set of the gradient at various points.

so i want to be able to take in a data point and spit out the original, pre-grad datta point.

@Fargle i think it's important to note that all of us fail to properly historicize even when we have good intentions and we have to be constantly aware lest we become the stupid

cf: me in the hbar at some point getting mad at psychologists for being bad at critiquing freudian theory because they dont know how to historicize

context

otoh there are people who are sorta awful even within their historical context

Lovecraft being the example I have in mind.

dude what a dick

lol

also all his prose is actually garbage

Lovecraft is weird to me in that wrote stories of horror and the bizarre that remain influential and genre-defining

his trajectory kind of ruins the idea for me

but the things that horrified him were largely immigrants and minorities

the like "incomprehensibility of the universe and fragility of our perceived reality" is a dope sound bite

but to lovecraft that just means the fragility of his pompous white sensibilities

how much of the original Twilight Zone series have you seen

all

kk. then you definitely should remember the 'eye of the beholder' episode, with the woman undergoing reconstructive surgery

the fact of Lovecraft's racism is sorta like the conclusion of that episode, where the reveal is how different that society's notion of beauty/grotesque is from our own

@heather Here's the relevant principle. If $f$ is a $C^1$ (differentiable should be enough) function defined on a simply connected domain $\Omega \subset \Bbb R^2$ (eg an open disk) then $\int_\gamma \nabla f = f(\gamma(1)) - f(\gamma(0))$ for any path $\gamma$.

@Semiclassical esplain further pls

So if you know the value of your inverse-gradient at some point $x \in \mathscr{D}$ of your data set, say the value is $f(x) = p$, for any other point $y \in \mathscr{D}$, $"\int_x^y \nabla f = f(y) - p"$, from which you recover $f(y)$.

So what you have to do is to find out how to approximate this continuous object $\int_x^y \nabla f$ over your discrete data set $\mathscr{D}$. There are many efficient ways to approximate an integral discretely

Well, suppose you read Lovecraft without knowing how much of a racist git he was (and you somehow miss the parts that directly reference immigrants etc)

You just have to push it through using line integrals

From that you do get a certain notion of cosmic horror and terror at the unknown, which is all well and good

@Semiclassical when u ignore that one story that's literally about how scared he is that white people and black people can be related lol

lol

but then you read a bit more and realize that what horrified him in real life was not cosmic unknown aliens but minorities and immigrant populations

if ur lovecraft what's the difference

the effect, to my mind, is rather similar to the ending of that episode, where you realize your own notions of what is normal/grotesque are very different than what Lovecraft had in mind

I don't think the fact that that terror (of racial intermixing) came from the bottom of his stomach (proved by how jarring his stories are) is lol-worthy. It's proved that there is something seriously tribal about people

Tribalism in our guts, in one way or another

i remember reading him in high school (me: a minority (who is also racially mixed) and child of immigrants in america) and being like whaaaat the fuuuuuck why does the internet liiike this aaaasssssss

lol

lol

I am not a minority and neither am I a Caucasian, so I cannot connect with either points of view

I do think it's something to take more seriously

I don't think I have actually read anything from that guy

I think most of what the internet gets from Lovecraft is rather indirect, i.e. people influenced by Lovecraft rather than people reading him directly

@BalarkaSen maybe but this is reductive. There's definitely an effect that comes from not belonging to the "default tribe" that makes me question the notion that identification w a "tribe" is as simple as this

Everything is reductive.

My point still stands

hard disagree

Except reductionism. That's emergent. :P

i would say that if youre, for example a minority in america who is socialized in a highly mixed environment, the projective identification w the default is a much stronger effect than the "tribal" identification with people like you

i never felt very comfortable in either minority latino spaces or predominantly white spaces, and i dont think this is an uncommon experience

The way i'd put is that 'horror at the Other' is rather universal. Horror at contemporary minorities/black people/miscegenation, on the other hand, is very particular

If you believe in dialectical logic, every thesis has an antithesis. In that sense every thesis is more reductive than then antithesis. I wasn't debunking your perception of what triablism meant, I just felt labeling things as "reductive" is not a good argument :P

@BalarkaSen "this is reductive" is bad argumentation, "this is reductive because X phenomenon" is not necessarily bad argumentation

The former is interesting, the latter not so much

I'll continue this discussion after dinner

the mythology of an ancient eldritch being sleeping in a lost undersea city, that's interesting to me

@Semiclassical it'd be interesting if his prosaic recourse to making us think something is indescribably horrific wasnt just to say it's indescribably horrific

the fact that "The Shadow over Innsmouth" is basically about Lovecraft's horror at racial intermixing, on the other hand...

well, sure

like take a good example of a writer: borges, dude's not even writing horror but i kind of get this cosmic sense from his writing that feels like what lovecraft wishes he could do in his wildest wet dream but cant

eh. I think Lovecraft occasionally does pretty well

like you get this kind of dizzying feeling when you read something like the aleph (or at least i do) and it strikes that soundbite of lovecraftian horror better than anything any of lovecraft or his successors writings imo

though I think I've actually read more of Poe directly than Lovecraft

so im also not a fan of poe but i think he's better than lovecraft

It depends on the story, of course

I still dig The Raven, for instance (though that's poetry rather than prose)

he definitely did some impressive things i think, he has a kind of compelling way of writing neuroticism and he is like the originator of a lot of "genre"

but he harshes my vibes bro

I think Lovecraft is one of those writers who is more interesting as influence than as direct source

i think lovecraft made an aesthetic that works well in the video game era lol

im not really a fan of a lot of writers he influenced w his horror like king or smth

eh. you say that, but i'm not sure how many video games have really replicated it

i think if they faithfully replicated it i wouldve liked it less

hmm

i just like fighting tentacle bois in my games bro

lol

I'm trying to think of what some of the examples would be

Bloodborne is close

gud gam

i do think it doesnt live up to the "indescribable" sound bite, but smashing tentacle bois was fun

the closest it came was when (spoilers) u find out that there's like giant monsters that were invisible for like most of the game just hanging out over ur head

right

@EricSilva Borges is writing about paradoxes. I don't see how you're comparing him with Lovecraft at all.

isn't there that one story of Borges where a fictional city/society creeps into existence in the real world

It is cosmic thrill, but in a different sense, of temporal and metaphysical breakdown.

im saying lovecraft fails to live up to the hype of his own kind of horror, but the sentiment u get from reading borges is often closer even though he's writing in a completely different mode

I think that's a bit closer to the usual notion of 'horror' than typical Borges

@Semiclassical Tlon Uqbar Orbis Tertius, yes

@EricSilva I don't believe in universal sentiments from reading texts. What is yours?

I think the shortest story of Lovecraft's that I've read and can really recommend is The Music of Erich Zann

@BalarkaSen i described it as this "dizzying feeling", like recognizing that reality has fragility to it. that's the core of the idea of lovecraftian horror but idt lovecraft gets there. I think borges gets that feeling even though he's not writing "cosmic horror" whatever the hell that is.

@Semiclassical I don't think it's universal in the sense you're describing it. John Carpenter, according to whom "horror of the Other" is the right-wing horror, describes it as imagining a certain set of humans as a group, and the horror comes from other groups.

In that sense, it's exactly codifying tribalism

i guess 'universal' is the wrong word. pervasive, maybe

'primal', perhaps

Primal is much better

@EricSilva I see. I think that's a superficial comparison, but if that floats you boat...!

@BalarkaSen i mean this is precisely the reason i think just describing it as tribalism is banal, the horror at the other is like a one way mirror, it's reified by the default, but if youre otherized in the way say, latinos in the US are, it's like an incomprehensible mess to understand how you identify

I think you're coming at this from a much more politicized and social point of view than is good to analyze these pieces of literatures

@BalarkaSen it like, isnt a comparison, im saying borges does a thing and lovecraft doesnt do a thing, not comparing their works

The thing that's also hard to appreciate with Lovecraft is the extent to which Eugenics was a thing back then

which I'm not saying somehow 'explains' Lovecraft---I think they're distinct

Of course, I cannot comment on how the otherization has happened with immigrants or black people or other racial minorities in the US, as I haven't been in that situation. But I do not think that extrapolating white supremacy with the general notion of "tribalism" is correct approach

But it is really hard for me to fathom how respectable the subject of eugenics was during that era

@BalarkaSen i mean, the tribalism thing was to me, a completely separate point from the literature stuff, but i think to say that lovecraft isnt heavily politicized is a bad reading of lovecraft (his stories drip with his very socio-political fears)

borges is different, but his writing isnt apolitical no matter how much the dude insisted it was, he was kind of a political naif and his writing isnt free from it

I think there's at least some of his writings which can be read apolitically

I know, but many many many authors have emulated the Lovecraftian horror of otherworldy terrors without the socio-political commentary that Lovecraft provided

the library of babel for instance

Lovecraft was just one person. The Horror of The Other is a big ass genre

@BalarkaSen yeah, that's sorta what I was after when I said Lovecraft has been more significant as influence than as source

@Semiclassical reading apolitically != writing apolitically

fair

I just mean, tribalism can be of much general groups of humans than races

Humanitarianism, of identifying yourself as part of the process of human endeavors instead of as an individual, which many do, is also a form of tribalism

I feel like 'cosmopolitanism' is maybe the better word there

i.e. viewing yourself as a citizen of the world

@Semiclassical Agreed.

@BalarkaSen i think there was a misunderstanding of my point, i mean, as an aside i think this is kind of a weird point because i think dehistoricizing writing is not a very enlightening way of reading lit, but i was saying that lovecraftian horror isnt a thing that lives up to what it endeavors to be

Well, to be clear: Lovecraftian horror as "horror written by Lovecraft", or "horror sharing themes and motifs with what Lovecraft wrote"

my point of bringing in borges was to give an example of a writer who captures some of this breakdown of subjectivity (that is supposed to occur in lovecraft but doesnt) successfully

^

i wasnt comparing borges to lovecraftian horror

the second one

the first i would just call: lovecraft

eh, I think that's a bit dismissive

there's a lot of 'Lovecraftian' horror out there

I think you're giving much more importance to Lovecraft as an author than Lovecraft as a genre

i think i better say cosmic horror

From my cursory conversation with you, it seems you are at the polar opposite of the "death of the author" philosophy

I think it also helps that HP Lovecraft is a good sounding name :P

And I don't really care much for such extremist point of views tbh

@BalarkaSen this would definitely be wrong

authorial intent $\neq$ historicizing the author

@Semiclassical I do like that word.

@EricSilva again, though, this seems to conflate "how successful was Lovecraft as a horror writer" vs. "how successful cosmic horror can be in evoking that"

I feel like the second is too dismissive of the possibilities in the genre

@Semiclassical idt genre writers are generally successful at doing cosmic horror but maybe im burned by bad genre writers

sure, but that's again focusing on the actual and not the possible

Does IUT count as cosmic horror?

100% yes

inter universal horror

[Politics] I am getting really really frustrated with describing my political alignment to politicians recently

@BalarkaSen the thing is my viewpoint definitely isnt extreme though, im in for extremely wide critiques of literature and like "deauthorizing" but to dehistoricize a text completely will mean there are things in it you cant understand, that's all im saying.

borges is like, literally full of intertextuality

I do not agree with that, as understanding is completely subjective. If you want to understand it in context to history, of course dehistoricizing it is incorrect.

@EricSilva absolutely

What kind of political alignment am I when I want all parties to be optimised on solving each problem by all attacking the problem at different angles at the same time?

For example, I read Borges on my first read without following the intertextual references. I still understand the paradoxes he's trying to formulate.

If that's end-goal, I think that's a perfectly ok understanding

@BalarkaSen im not saying that understanding the text isnt subjective, it obviously is, what im saying is that there are things the subject itself cant extricate from the reading

there is no ahistorical reading if youre mired in the history

like as a latin american i cant not read borges as dripping with argentine sensibilities, it doesnt even make sense for me not to, it's there and it's something that's related to what i come to the text as a subject

I personally think Borges is better suited to write paradoxes completely mathematically than literarily or metaphysically

He has a great essay on the Zeno's paradox to that effect

i think what id say is that, i never even had the option of approaching borges without historicizing it, and i think that's why it strikes me as a weird reading, so i would modify my point

That is completely fine. In your subjective view of the world, it's impossible for you to dehistoricize most things. That is an acceptable statement.

I do not think the same is true of everybody

i mean ok, i disagree, what claim id make is that dehistoricizing and approaching a text ahistorically are different. The former id say is not really possible, while the latter is fine and even illuminating

If you broaden the definition of history to be integral to the existence of an individual, then you're right that it's not possible, but also to my view that claim loses... any punchline, really

It's like saying you cannot exist without existing

Something really tautological like that

what no, that's nonsense lol

to say that you cant exist free of your own intersubjectivity isnt a tautological point

you cannot exist that you does not exist

it's pretty substantive, most americans dont even believe it

IT does not exist that you exist that does not

I don't see why it isn't. That's what Berkeley's notion of existence is

For it to exist, it has to not exist, but before that it is nonexistent

@BalarkaSen idk what berkeley's notion of existence is

Nonexistence is an existence, and it does not matter how invisible this message is

@EricSilva One exists because one is percieved

Because the fact is that nothing can beat nonexistence. All that exists will eventually become nonexistent

We collectively exist because we perceive each other in simultaneity

We do not exist because it is not being perceived as so

Also, this chat does not exist

and dies not eixist

Also this wifi doesn't exist

@EricSilva lol I really doubt those two subsentences are correlated

is existence an asymmetric relation, that I have no idea

hello. Could someone help me to understand the solution of this exercise? math.stackexchange.com/questions/2824474/… Thank you

can it be that A exist because it is perceived by B, but B is not even though it is perceived by A

and that exercise is too hard

:( why

@BalarkaSen I thought God's perception of us was what was supposed to guarantee everyone and everything's existence

@BalarkaSen what i was arguing is definitely not that our intersubjectivity is the source of like our entire existence

more that it's a powerful force that we have to mediate and arent so powerful as to be able to fully overcome

@Fargle Rather, Berekely effectively defines God as the simultaneous perception.

It's about optimization

that's not such a simple proposition because i think where the line is drawn is actually confusing

Ah. Been a while since I've read it

The thing that "glues our existences", something like that.

I think his theistic interpretations are less important than that statement

The way I took that was that God's consistent and omnipresent perception is what guarantees that things don't cease to exist after we stop perceiving them

i.e. object permanence

but either way this is immaterial to the current discussion I think

That's right. But I don't think the right conclusion from that is "God exists"

I 100% agree

I don't like much of what I've read of Berkeley

though I will say he had a good point in that calculus was sitting on rocky justification in his time

@BalarkaSen this strikes me as kind of uncharitable to people who havent engaged w this kind of thought, i think there are probably reasons beyond like the inaccessibility of philosophy and historical situation that would render a normal person kind of unsympathetic to the claim

$$\lim _{x\to 0}\dfrac{(1+x)^{1/x}-e + \dfrac{ex}{2}}{x^2}$$

How to evaluate this? I have tried L Hospital but that didnt help.

@EricSilva You're right, I'll redact that.

@Fargle Have a look at Borges's "A new refutation of time"

He discusses the metaphysical content of Berkeley's thought

Which are more important than Berekely's thoughts :3

i think what id say is two fold: 1. berkeley's point isnt tautological, 2. my point is very much not berkeley's

Ok, then I clearly do not understand your point. Want to reiterate?

I might be missing something

cool, given $k$ random integers, the probability that they're all relatively prime to another is $1/\zeta(k)$

Euler's product formula ftw

i think my point is more freudian

(Side-note: XXXTentacion is our generation's Tupac now)

@Abcd try putting $x = 1/n$ and $n \to \infty$

mumble rap sux

our subjective experience arises from a mediation between something internal and something intersubjective, and in that sense we cant dehistoricize (or perhaps we can im not solid on this point, but maybe to dehistoricize is it's own kind of rehistoricizing, just not to the time and space of the writing, but rewriting from our own position)

I don't think you need to consider $x = -1/n$ seperately

but maybe you do

the mediation that happens isnt the same as the simultaneity you were talking about @Balarka but i think it's born out of that simultaneous perception but not the same as it

@EricSilva Is this something along the lines of "archetypal understanding of every story ever", but expanded and broadened to it's fullest potential instead of that narrow (and false) statement?

@BalarkaSen wasn't he shot in a robbery, entirely unrelated to east vs. west rap?

@BalarkaSen are you referencing something w the quotes

That is to say, we understand every narrative using an approximation by our collective understanding of "core narratives"

im brain dead rn cause i have a migraine

perhaps but id have to turn that statement over a lot

@anakhronizein yea lol

So how is he anything like Tupac?

He isn't; it's a meme now

@EricSilva Not anyone particular, no. I guess the importance of archetypes is a prevalent theme of Jung but I haven't read Jung

I am too old to be reading Jung :3

oh i dont really buy jung

kdot is the new tupac tbh

mortal man reified it

long live king kendrick

Kendrick is ok

nah dude he's the greatest boi

how much a dollar cost is my daily humbling track

@Semiclassical is this how physics works

No, it really isn't.

twitch

lmao

the pictures never show it, but pennyfarthings actually traveled backward for this reason

not really related: once in my complex analysis class the professor started talking about angles between contours

@EricSilva I can't connect with most of the songs he writes, but I do like his lyrical complexity and great sense of sonics

and I asked which inner product we were using and he didn't like the question

lol

i remember you talking about that

"the concept of angle goes back at least to the ancient egyptians, I mean in the sense of geometry"

whoa okay sorry boss

funnily enough, some notes I'm writing have at least three different inner products

what about ~full generality~ tho

He's right

Please don't defile geometry with nlab

one is the usual Euclidean dot product. another is $\langle X,Y\rangle = E[XY]$ where $X,Y$ are real random variables

what is nlab

and another is $\text{Tr}(A^\top B)$ between matrices

that one I haven't sen before Semi

the matrix one

@BalarkaSen idk if i really "connect" either

I had seen that before, but I hadn't appreciated it as such

if you do QM enough, you run into the concept of a density matrix $\rho$

at least like, not in the way i feel i obviously connect w artists like criolo who are writing from a perspective i connect more viscerally, but i think there's something about his music that vibes w me and idk if it's my material experience or growing up listening to rappers from compton

which satisfies $\operatorname{Tr}\rho = 1$ and has nonnegative eigenvalues

@EricSilva Makes sense

Does anyone know of a good diff geometry problem set (with solutions)? Ideally using the g metric notation instead of doing that EFG crap

main thing is that expectation values then arise as $\operatorname{Tr}[\rho \mathcal{O}}$

@DemmiLi do carmo doesnt have solutions but it has lots of good problems that people on this chat can definitely answer if you ask nicely

I have a rather narrow category of "things I immediately like", and the criterion for immediate liking is very much related to if it's elaborating on a perspective I have independently come upon and has been deeply moved by

and from there you can get to inner products in rather the same way as I said above with random variables

Eg this is why I am a big fan of the Def Jux era

if i know the prob density function and i know the standard deviation. Does $\int_{x-\sigma}^{x+\sigma} \rho(x) dx$ give me the probability of x being 1 standard deviation away from the average?

It feels like a group of people whose narratives are fundamentally related to some battery of thoughts that really haunts me

@EricSilva thanks, I already know about do Carmo. I was hoping for a more modern approach with the metric tensor instead of the EFG notation of Gauss.

there are 3 do carmo's my dude

@MohammadAreebSiddiqui You've got $x$ as both the integration variable and in the endpoints. Can't do that.

his riemannian geometry book doesnt use the EFG bois

(plus, the interval $(x-\sigma,x+\sigma)$ isn't centered at the average value $\mu$)

my bad, it should be $\int_{\langle h\rangle-\sigma}^{\langle h\rangle+\sigma} \rho(x) dx$ right?

<h> is the average of h

(Eg this, @Eric)

\langle \rangle looks better in $\LaTeX$ than < > btw

not that that answers the question but just saying

@BalarkaSen i jam w this

that seems to make sense on its own now

tbh

thanks XD

given that your integration variable is $x$, I think you mean $\langle X\rangle$

@EricSilva I really like that album and that song in particular

I would suggest not taking the random variable to be labelled as $h$ in any case; the general practice would be to denote it as $H$

@BalarkaSen i guess somethings are hard to arrive at independently, e.g. why i have trouble appreciating like country or something

so that you can distinguish between the random variable and a possible value of such

maybe a lot of it is also shit but idk im too dumb to judge

My appreciation for country is dependent on what genre of country

@EricSilva do u like classic country like johnny boi

who is johnny boi

the book didn't state any so i just went with my own small letter

johnny cash

oh i love cash

Folk-genre country, e.g. bluegrass or strings, I'm fine with

he's special

Good!

Me too

@Semiclassical i appreciate bluegrass on a technical level bc i play mandolin but i cannot listen to it for some reason

well, there's a difference between respecting a genre and enjoying it personally

i mean i enjoy to play it but not listen to it

by contrast, I have a pretty hard time respecting what one might call 'pop-country'

it's not so much respect as it is one kind of enjoyment vs another

i also like to play classical stuff but i dont listen to classical stuff and actually kind of have strong anti-classical music feelings

I more mean in the sense of: If someone likes bluegrass, that's not something that'd make you question their taste necessarily

ya i get that

by contrast, there are definitely country songs which strike me as tasteless

@Eric classical in the sense of baroque or more general

@BalarkaSen more general

i got burned on how elitist it is to move within classical circles

Like how a lot of people feel about disco

ahh

i see

@MohammadAreebSiddiqui there's no official rule saying what letters to use. but by convention, people use capital letters to denote variables and lowercase letters to denote particular values the variable may take

it's a convention

Right, so the average of variable G, $\langle G \rangle$, could be any $g$. Did i get it right?

sure

or something like $\operatorname{Pr}(G = g)$

What?

was the superscript there intended as an exponent or as a label

i just put it there before it felt wrong to leave it on its own but there wasnt a point to it

mmkay

Yeap! got it

I'd also say that your earlier remark was not quite right: the integral you wrote is not "the probability of x being 1 standard deviation away from the average" but rather "the probability of being at most 1 standard deviation away from the average"

for a continuous random variable, the probability of any particular real value occuring is zero

you always talk about the probability of the outcome being within an interval of finite length

Ohhh so the probability ranges from h-sigma to h+sigma

not only those 2 values

right i was confused about that thanks!

right