#serious-discussion
1 messages · Page 213 of 1
this also sounds very wrong lol
there is lots of different kinds of math in cryptography, most of it elementary
but if you want to see more advanced math, you can do that too
i mean just to define a standard notion of security, you need to know what a turing machine is, probability theory and (depending on how you want to look at it) game theory
meow
no.
definitely, im still getting exposed to the basics of it all so I can definitely see that I have a bias perspective
I became a math major and wanting to pursue academia in it because of cryptography
Outside of reading application papers before, im happy to being able to read more theory and mathy stuff now
Oh shoot i forgot i already said that, practically repeated myself soz
in that case my recommendation would be to take classes in algebra, number theory and elliptic curves and then take as many CS (and possibly even EE) classes as possible 
Yep! I have read through Silverman and tate, reading silverman AoEC again for a better understanding in a reading project, and hopefully will progress to some elementary papers on current problems with ec's nearing the end of sem and or even next
I also finished my ug math major, and added cs to get that side too as im only a 3rd year
but in generality it also makes me start thinking about other potential grad school programs since i can either go to the application/discrete side of it all, or do the more pure mathy side of it all since there are two ways to tackle the problem
A couple of options im considering is an Algo,Comb,Graph (discrete) phd, math phd, cs with crypto focus, and of the sort
Thank you for the recs tho!
I also took an AG course and also taking AT this semester
I planned on also sitting in a Hodge theory course but it was honestly too much for me haha
Hodge theory is great I love it
It seems soo cool, I wish I had a better understanding of the prereqs honestly
We are also approaching hodge theory from a complex geometry perspective rather than algebraic
I'm planning to read more about these after my course in representation theory
Just realizing very lately representation theory is not my thing lol
Maybe im missing something, but can you explain how rep theory is used in hodge theory?
im also not too familiar with it other than the general idea lol
Atleast from the approach I tried to study about hodge theory, it mainly needed a complex analysis, lin alg, and manifolds background
Idk the connections but I know they exist. I was just saying atm I'm focused on representation theory
Ohhhh gotcha my bad
@surreal sapphire what I meant by that is that going deeper than a couple of courses in the average uni
Oh also, Loch, are you interested in cryptology too?
He works in crypto
Thats cool af
Not as in NFT of course
Yes ofc lol
cryptographer is more common
cryptanalyst is also a thing but i dunno the difference
which one is better for a beginner C or python
either
Awaa did bad on my calc 3 exam
Chat how do we feel about mint ice cream
I hate it
I'm trying to learn both and so far I like C a little better but python has a simple syntax but because I'm a beginner I don't know the power of coding
I used to hate it but I like it for some reason
cryptographer works on encryption creation via implementation, cryptanalyst analyzes/breaks down the encryption, cryptologist does a bit of both iirc but i think more mathy
lovely
my brain exploded with bliss after talking to a grad student resulting in my realization that Data Science is just topology but with coding theory and different terminology (over generalization ofc)
this makes me feel less out of place
@versed drum
I got a question,
what should I study in math for the ACT or SAT exams?
the prep guides
there are also official practice exams available which gives you an idea of the kinds of questions that may be asked
oh really? i kinda hate it cause it reminds me of toothpaste, is it similar for you?

either are fine but i think for a beginner better to stick with one than do both
otherwise it can get especially easy to mix things up or get confused about differences
Oh we're talking about mint ice cream?
First of all toothpaste is delicious
So is mint ice cream
really? :O+
And there is no second point
i like drinking cold water with mint gum
I do spit out but don't rinse my mouth with water, i just swallow the rest
You think it's ok?
Must be
Just toothpaste lol
dont do that!!
its not gonna kill you, but it might make you stomach upset
with time
so we got Big Pharma telling us to not swallow toothpaste now
flips calendar to see 1984 January
Radioactive toothpaste used to be a thing
How's Schoolhouse (the tutoring service by Sal Khan)? What is it like to be a tutor on there?
integrals are the base of a lot of things tho like area and stuff
Nuh uh
I mean
area under a curve
The way you are expected to get the solution in a traditional way
antiderivative then
subsitute the x values
minus
done
except like the hard ones
Like an integral possibly has infinite number of solutions
ye
Like the same solution can be manifested into different functions
derivatives and or differentiation is still the goat
It can be logarithmic trigonometric or any hypothetical function f(x)
I have made few algorithms to rigorously deduce integrals
how
Like the same thing can be understood intuitively without that trig substitution or anything
how
the only way i know is trig sub
is it possible to use u sub
cuz it's like easier
Cuz it's 1/(1+x²)
You can look it geometrically
like on a graph?
1 be the base x be the perpendicular and √1+x² be the hypotenuse
base as in adjacent
Wait I can't send images in this channel?
That's pathetic
oh i know
Check dm
so like
post in #chill
Nvm we had a Convo in dm
i work in post quantum cryptography and formal verification
but uh, very few people work in actually implementing (new) methods of encryption
my work is more making small improvements to known methods
and somewhat focusing on "small" implementations, think embedded systems and the like
post quantum cryptography is quite ressource hungry (compared to classical methods anyway), so im trying to find improvements here
Can you give an idea of it in a simple way?
quantum computers break all classical encryption due to shors algorithm
so we need new ways to do public key cryptography
do you know how that algo works well? i want to understand it better
Yo people , quick question ❓
What's
ln(0)
no
undefined because e^x =/= 0 for all x
Thanks , and what's the result for ,,\int ln sinx dx
something bad
where Li_2 is a 'polylogarithm'
actually Li_2 is a known enough polylogarithm to get its own name: https://en.wikipedia.org/wiki/Dilogarithm
Why does a factoring algorithm actually help to break an encryption?
roughly because the big encryption/decryption scheme is based on knowing the prime factors of some specially chosen very very large integers
you can think of the prime factors as like a password
(extremely rough analogy)
all encryption is based on one-way/trapdoor functions, which are easy to compute in 1 direction but hard to compute in the other direction
it is very easy to compute the product of two large integers
it is very hard to factor a large integer though
this is the one-way function underlying RSA
it becomes easy to factor a large integer (that has two prime factors) once you know one of the prime factors
This is the best function
this fact makes it a trapdoor function
shors algorithm breaks the hardness assumption, so this is no bueno anymore
oboy. Tell me about how it's related to modular forms and spectral sequences or something 
well the first really nice property of this function is that it computes volumes of hyperbolic 3-simplices
I did see something about hyperbolic stuff and simplices on that wiki page. Sounds cool
the second really nice property of this function is that it computes algebraic K-theory groups K_2 and K_3 of fields

these two facts are related
this is kind of a miracle because K-theory is like, insanely hard to compute in general
there is some general story about how polylogarithms are related to algebraic K-theory it just gets very hard to write down after the dilogarithm
didn't somebody get a Fields medal just for giving a good definition for K_2 or something?
but the dilogarithm case is very clean
not that I know of?
hm, maybe it was only a PhD.. there's some story about this I recall
maybe Quillen for defining algebraic K-theory in general
that could be it
but yeah specifically you can write down a length 2 complex defined in terms of the 5-term functional relation satisfied by the single-valued dilogarithm
the cokernel of the differential computes K_2, the kernel computes K_3 up to mild torsion issues which can be fixed
this should be true for other polylogarithms it's just that the functional relations they satisfy get very complicated very quickly
we know this up to K_7 and up to the tetralogarithm
but yeah polylogarithms show up everywhere and they have very simple integral representations
oh yeah the other fun thing from this story is like
if you've seen the analytic class number formula before, it is a formula for the residue at s=1 of the Dedekind zeta function \zeta_F(s) for any number field F
one of the main terms involved in that formula is the regulator which is a determinant of logarithms of units of F
the polylogarithms appear similarly in formulas for \zeta_F(n) at integers n>1
so for example you can write the special value \zeta_F(2) for any number field in terms of (a determinant of a matrix of) special values of the dilogarithm
when F is imaginary quadratic you can prove this by computing the volume of an arithmetic hyperbolic manifold in two ways: first way involves a volume formula involving \zeta_F(2), second way involves decompsing this into ideal hyperbolic geodesic simplices and computing their volumes with the dilogarithm
Special values of zeta functions are arithmetically deep things and this gives you a nice way to study them: rather than considering infinitely many functions with the number field F varying, you can consider a single function but now with its evaluation points varying
Eh for you maybe
yes definitely
Woah that’s literally what I want to do
This is a talk delivered on April 5th, 2024 at the current developments in mathematics (CDM) Conference at Harvard University.
1.5 million views
i truly despise multiple choice questions in maths
now that i am able to take my studies seriously it's the fucking death of me
written questions are great because you can do a little positing and get some feedback on your suppositions but if you get a multiple choice question wrong they just give you the correct answer and tell you to kill yourself
completely unrelated to the above, is the product of periodic functions always periodic
it seems everyone on stack exchange only wishes to discuss the inverse
have you thought about it?
i have
even put it in desmos
wait
do you mean killing myself or the product of periodic functions
the latter.

right right
at the very least i know that one of my functions is either cos or sin (im trying to answer a question about fourier series) but im just thinking about how those behave when multiplied with some periodic function
the answer is "in general no, but when encountered in practice, it often is"
in particular, if you know the ratio of the two periods is rational, that implies the product is periodic as well
(why?)
many such cases
i gotta figure out the fourier series without knowing the function
so im thinking that i have to see what the period of the innerproducts would be - that way i know what terms will be vanishing by way of looking at the coefficients
Ye that sucks
Is this, I guess, a hard wall to computing K theory? Like...will we need better methods to "write down(?)" polylogs to compute it further?
well this is not the only way to compute algebraic K-theory (and the story I am mentioning is pretty specific to fields)
Ahhhh
but like most methods for computing algebraic K-theory blow up combinatorially and are hard to get a handle on
it's a similar problem to computing homotopy groups of spheres, even stably
oh also, fun thing related to that integral that kinda started this whole convo, about two years ago we walked into the math department and were like "ah what if I try to integrate...say....ln(tan(x))"
That was our first intro to polylogs and it still shows up in my nightmares sometimes
like you can do it by hand in low degree, or we can break up the problem into a sequence of easier problems and maybe understand the first couple of steps
I see
but I think these things are not meant to be simple in general
this just ended up being an exercise in annoying partial fraction decomp and like four change of variables
lmao yeah
oh we failed the exam the week after..which was just anticlimactic
rip
yeah
but yeah the dilogarithm is very beautiful and shows up everywhere, the trilogarithm is still quite beautiful but starts to get hard to understand
beyond that it's kind of a nightmare
wait I still have the pictures on my phone one sec
I've only ever seen the dilog in that one context, one time when fucking with the gamma function, and once in ryzhik's table of integrals book
one spooky thing is that there is a weird complex-valued variant of the real single-valued dilogarithm
oh boy
this is spooky because among other things it lets you define a "complex volume" for arithmetic hyperbolic 3-manifolds among other things
what's an "arithmetic" manifold?
they are defined from arithmetic subgroups of PGL_2(C) acting on hyperbolic 3-space
specifically ones obtained from quaternion algebras over certain number fields
when such a hyperbolic 3-manifold is arithmetic and defined over such a number field F like this, then the volume will be related to \zeta_F(2)
I feel troubled that I still have these mental walls that split "arithmetic" "algebra" "geometry", etc... even when the actual borders and topics starts to blur like crazy at some point
most knot complements are like this
is "discriminant" here at all related to discriminants of polynomials? AFAIK algebraic number fields are related to algebraic numbers and those are roots of nonzero polynomials
yeah they are related
at least when the ring of integers have a power integral basis (so it's of the form Z[a]) then the discriminant is equal to the discriminant of the minimal polynoimal of a


broke: wow what a beautiful formula math is so wonderful
woke: deriving this is a fucking nightmare
this is not so hard to prove actually
is it some polylogarithm calculation 
no
I mean one might want to compute the same volume in terms of dilogarithms later to prove this neat result about \zeta_F(2) but that's after you have this volume formula
the volume formula involving \zeta_F(2) is a volume computation for the Tamagawa measure
you can prove these sorts of formulas for any arithmetic locally symmetric space without any more work
it's really easy if you only care about the volume up to some nonzero rational factor, if you want the actual volume you just have to be a bit more careful about how things are normalized but this can be done very generally without so much issue
ah
is there some specific reason why we would want to know the volume of such spaces?
ik there's a similar kind of calculation when deriving the class number formula
as in it's the volume of some lattice
https://www.bbc.com/news/articles/clyz6e9edy3o
TL;DR A scientist worked years on a biology problem (not niche and had important real world impacts afaiu), asked a research-based AI to work on the same problem as him, and the AI made five plausible sounding hypotheses which the scientist thinks would have taken years for a Normal Human Person to make from the research the AI had access to.
so uh
does anyone have a plan for what to do with the inevitable silicon-based lifeforms which are smarter than us in almost every way
maybe we can tack "dont have any sort of negative effect on humans please" somewhere in the system prompt
that should work right
🥱
you could also
your way through the whole ai thing and wake up at the end
in which case, hope for soft landings, friend
overridden by the openai and intel/mil community system prompts
at least we got decent local models and cheapish cloud infrastructure
???
doesn't matter whether openai is malicious, they're reckless
also open source models make most forms of ai misuse more prominent
if open sourcing ai is a net positive its because its being used to do productive stuff
you'd run into basically the same problem here if you made everyone on earth more intelligent while keeping them at the same level of compassion: people would be more productive and have to rely less on external authority, but they would also be better able to cause harm to others
(this issue is less important in humans since most forms of increasing intelligence also increase compassion as a byproduct)
kid named probabilities:
oh yeah whats up with higher math and ai
yeah I more or less think of this about all tech advancements
people are treating it as obvious that llms would become good at coding, writing, etc and shit at math
but i havent seen much of an attempt at explanation
aside from like "theres less of it in the training data" or "math uses higher level abstraction", but from what i understand about llms this doesnt seem to explain it too well
I don't understand the technicalities either but I'm sure the key is in the fact that they're language models, they're great at tasks that involve generating coherent text, but the claim that they can reason logically is either outright false or they just can't do it at an advanced enough level yet with the constraints on inference and context window sizes
like I have a hunch this is why but I haven't read anything supporting this
Can you learn logic by just listening to people speak a language?
Can u learn reasoning?
a language is organized and hence has some inherent logic doesn't it
hence why things like reinforcement learning seem to be working fine I guess -- let the model explore the training data by themselves and with enough compute time they'll perform decently at other tasks
I guess the question is whether there's a more efficient way, whether this is actually reflective of how we humans learn, etc.
this is how learning language works
oh logic mb
i read it as language
they can reason logically, how else would they write functioning code of program ideas that doesn't exist in their training data
black magic?
people who don't have conversations for extended periods of time tend to end up worse cognitively afaiu
also almost all human reasoning is learned/taught through language
well yeah, but when people talk about what a language is they're usually including syntax, grammar, vocabulary, maybe some specific cultural nuances, and excluding the actual stuff that that language describes

funny thing is I'm sure there are authors (from cognitive sciences, philosophy etc.) who actually support the idea that "reasoning" is just a recombination of things much like what LLMs do
I don't wanna yap much bc I'm supposed to be busy
and ofc this is not my specialty and I prob should read more
if you trained gpt4 on random english sentences, only requiring grammatical correctness, you would have a model that speaks random english sentences only requiring grammatical correctness
for it to actually learn the meaning of what it is saying, there has to be some bias in the training data towards sentences with a certain type of meaning (such as sentences that are true)
sounds right, but i think of it as "its weird to call reasoning as pattern generalization" and not "its weird to call pattern generalization as reasoning"
i would guess, just intuitively, not a lot of support for this idea, that you see people (both in present time and historically) sort of having a u-shape curve of how intelligent they think animals actually are as they get more experience with them
like: first you're prone to anthropomorphizing the animal, then as you see the mistakes the animal makes you think its dumb, then as you learn more about how the animal experiences the world you become more forgiving of the animals mistakes
actually, i change my mind on this
from Claude:
However, [...] humans don't learn reasoning purely through language - we have embodied experiences, sensorimotor learning, and causal learning from interacting with the physical world from birth. Language helps us formalize and communicate reasoning, but isn't its sole source.
thinking about the topic more i think a big determining factor in whether an LLM ends up being able to do a certain type of reasoning is how often that reasoning is expressed in terms of language (or at least text that exists in the models training data)
spatial reasoning and object manipulation are good examples, we understand that stuff intuitively, from direct experience, and when we want to communicate that sort of reasoning to someone else you would normally use diagrams with language only playing a small role
I'm sure there is data about the impact of modern maths reforms at school on reasoning later
It's a similar problem llm's have to deal with
Ig
math ends up being an excellent candidate for this sort of blind spot because it relies so heavily on intuition (you know this if you tutor someone in math and start seeing yourself have to verbalize concepts you always "just understood")
also ive heard math people complain that math papers are good for proof verification but tend to be bad for understanding the process used to make that proof
yes because failed attempts are usually not included
I think "intuition" might be just reasoning in another language than formal maths
One that we're more familiar with
Bu i still think language is the major component to reasoning
Idk if it makes much sense (even to me)
from my experience exploration->solution->proof isn't just removing failed attempts, though that is a big part
theres also stuff like idk what motivated you to think in this way
wdym
The trend in certain countries to reform mathematical education from very early on in the 60's/70's
Which happened to be a very bad idea
im not sure what you mean, but for the purpose of llms being trained on a piece of language, it has to be published online and included in its training data. a lot of intuition is not.
Valid
But we also as humans aren't always taught our intuition, it just appears with experience, why not the same with llm's?
Like textbooks, they don't teach intuition
its not merely that mathematical reasoning is complex, its that its complex in a way that doesn't let llms learn it
"it appeared to me in a dream" 
chad ramanujan
there's also that one guy who was trying to prove his dead friend wrong
I'm not sure it bound to stay like that
Is it grammatically correct
this makes me wonder: would ramanujan have been remembered more or less greatly if he could actually describe what the fuck he was thinking?
i will say that a lot of mathematical eureka moments come up in unexpected times when doing other stuff

theres times where im not consciously thinking of steps or things but just abstractly pushing against a problem in my head
From my very little experience, someone very good at maths trying to explain his thought process is rarely easy to digest
i dont know how this could be translated to llms or if it even needs to be
not necessarily llms either tbf
@sonic sentinel
well yeah thats what happens
base LLMs are "taught" in the sense that the stuff its fed on is intentionally made to set the conditions for learning
but nobody's out here giving lectures to random weights
Some companies try to feed LLM's things that look as much as textbooks as possible to simulate education
And i think it works pretty well
I wish I was so smart that people's simulacrums of me started telling them insights in their dreams
fuck its weird to think that far more likely than not some of my classmates have had dreams including me
That's normal
I've had dreams with some classmates inside
Well...
Not exactly what you said but...
bro said well
lol something that i actually did in a level maths was pretend i was the british examiner reviewing my work and this would unironically give me some ideas on what to do on the problem
Is that forbidden?
if you're an llm you could literally summon another instance of yourself to check your work, i find it so weird this isn't really used
stand power
sand powder
It is used
But it can't be used to much times in a row before it becomes shit
Idrk i've heard it on a youtube video
no i meant like something integrated into the actual system
mmm maybe you're talking about best-of-n?
I need to review the 30min video to find the source smh
i was unfortunately born with a condition that made me unable to understand french
and i never really recovered from it
Ok so it's an interview of a researcher at hugging face. at some point, she says without specifying which model exactly that they trained models on their own outputs after filtering them (using other llm's or humans, or running the codes in the case of coding)
And it gives good results up to a certain point of at which it becomes dumb
yeah thats cool
i heard that was mainly what was happening with all the CoT models you see these days
Wait
She said deep mind and gpt4 did that
But "model collapse" happens if the chain is too long
That happens for humans too lol
hi i have a question, someone once told me a lot of advanced math is reducing questions in various fields to equivalent questions in linear algebra, which makes them easier to solve
for example the entire field of representation theory is based around this
how true is this statement in general
Pretty true
so much of the maths I'm doing right now feel like cheating, precisely because we're encouraged to reduce the problems
this exponent too big? throw it out, make it run towards zero and say byeeee
M.@sharp mulch m mm mm m mm m...,mmmmkm mm mmmmmm mm mmmm.mmm mm mm mm m.m mm mmm..mmmmkmmmmmm,m.mmmv h. Vvvvvvvvv.vvvv vv gvvvvbvb v hvvvvv.vvv v.vv .vv vbvvvh vcvvvg v gv gv v vvvvg cc vvvgvg g vc vvv.g glvvvvvgv g vvvvvv, m mm
<@&268886789983436800>
what
Just cuz you don’t speak Martian doesn’t give u a right to persecute those who do
I agree
Hey, I have some problem.
I took part in the programming contest in codechef site.
There was a difficult problem for me.
But interesting
Here is the url.
Who can explain about this.?
How to find Null Space of ?
Step 1: Row reduce the matrix
Step 2: Find independent columns (which do not have pivot elements) and dependent (pivot elements are there)
columns.
Step 3: Number of independent columns = Nullity of the matrix
Number of dependent columns = Dimension of the matrix
is this correct?
independent columns are pivot columns rght
very confused
pivot columns: columns with a pivot
free columns: columns without a pivot
rank of matrix: number of pivot columns
nullity if matrix: number of free columns
i would not use the phrases "dependent columns" and "independent columns", or "dimension of matrix", they are too ambiguous
Do you find a woman's softness, sweet submission, and vulnerability appealing—something that evokes your instinct to protect and care for her?
wtf
Yes
Isn't this what the female nature is about?
<@&268886789983436800> two bad posts back to back 
I don't think there is need to ping mods. I mean, the questions are weird ❓ but there's nothing wrong
Btw where is the math discussion channel? Cuz I thought these were math discussion, but they are not
#math-discussion is over there! 

No. I trust that she can take care of herself. I find her character something that might make me want to care for her
don’t discourage pinging the mods please
yeah if unsure it's probably best to ping (or DM @polar panther with a message link) and we'll determine whether it's worth moderating or not
No
No
Gender roles are made just to make women look inferior and submissive to men
There's no point of having gender roles in an ideal society
Is chess a game or something that makes your more intelegent
it only makes you better at chess
lord there's this question i encountered when i was teaching last week it's been haunting me
dang ion rmb how to use latex lmaooo
idk id say there some general transferability
it's question 4d
it irritated me severely that they just equated the root to 3 instead of expanding the brackets but i cant think of any reason why not besides "i wouldnt do it that way"
cause in my mind the rationalisation is that you "lose" a solution but none of the solutions you get by expanding are the ones they got
you could also raise 9 to the power of both sides
now recall your laws of indices
ah okay you still get to that step with the 9
no you could also use the difference of two squares
$(x + 1)^2 - (3(2x - 3))^2 = 0$
south
or $(x + 1 + 6x - 9)(x + 1 - (6x - 9)) = 0$
south
yeah then the left bracket gives 7x - 8 = 0 so they've missed a solution
,w 2log9 (x + 1) = 2 log9 (2x - 3) + 1
ah you have to sub into the original, wait
yeah turns out it doesn't work
,calc (8/7 + 1)/(2 * 8/7 - 3)
Result:
-3
this is negative sorry so you can't take the log of a negative number
ive gotten 8/5 and 10/7 tho, if it was a missing solutions thing wouldnt at least one answer be the same?
yeah that's very clever, recognising the inside of the log has to be positive
so it must equal 3
normally you get two solutions yes
to a^2 = b^2
they should justify it yes
but you need to remember the domain of any logarithm
is x>0 ouioui
No
but that doesnt necessarily mean that i cant solve it by expanding cause i could very well not get a negative value (as you see by my solutions)
I'd say you become a better thinker about strategy
but that doesn't mean you become more intelligent
Intelligence is just a small factory in it
if your steps aren't all reversible
It's mostly about super fast working memory
you need to check every solution you get by subbing back into the original
You need to have exceptional working memory inorder to be a chess prodigy
right right
like a^2 = b^2 is the non reversible step
cause it can come from a = b or from a = -b
everything else can be reversed
ah is that why my further math teacher liked it so much
Creating chess strategies require intelligence
that's their mistake then isnt it
it's not a mistake
sec
wtf is question 3d saying 💀

sksksksk that's a typo in the textbook it's written z sqrt y
real maffs moment
this textbook series has so many mistakes lmao when i was a student it was never a surprise for us to ask a teacher to clarify something and he was like no shit ur right the book is wrong
now i am the teacher who must do this
ok i accept that 2 is the only solution having substituted back but i have no clue why doing it the way i did it only yielded incorrect solutions
i couldnt tell you 😭 they just write anything

No need to enforce your feminist ideology onto me; i believe that's the qualities of the female nature and my beliefs will remain that way
Bring an argument and prove why that's bad
You wanna know why?
We first show that $\Phi$ is linear. Let $v, w \in \mathcal{D}_pM, k \in \R$, and let $f \in C_p^\infty(M)$. Then
\begin{align}
\Phi(v + kw)(f) = (v + kw)([f]_p) = v([f]_p) + kw([f]_p) = \Phi(v) + k \Phi(w)
\end{align}
so $\Phi$ is linear.
\medskip\newline
Now define $\Phi^{-1}: T_pM \to \mathcal{D}_pM$ by $\Phi^{-1}(v)([f]_p) = v(f)$. We need to show that $\Phi^{-1}(v)$ is well defined, that it really does send elements of $T_pM$ to elements of $\mathcal{D}_pM$, and that it really is the inverse of $\Phi$.
\medskip\newline
To show that $\Phi^{-1}(v)$ is well defined, suppose that $f$ and $g$ are smooth real functions such that $[f]_p = [g]_p$. Then by definition, $f \equiv g$ on some open neighbourhood $U$ of $p$. This implies that $v(f) = v(g)$, which in turn means that $\Phi^{-1}(v)([f]_p) = \Phi^{-1}(v)([g]_p)$. This proves that $\Phi^{-1}(v)$ is well defined.
\medskip\newline
To show that $\Phi^{-1}(v)$ is really an element of $\mathcal{D}_pM$, we need to show that it is linear and satisfies the product rule. Thus, let $[f]_p, [g]_p \in C_p^\infty(M)$ be arbitrary, let $k \in \R$, and suppose $v \in T_pM$. Then
\begin{align}
\Phi^{-1}(v)([f]_p + k[g]_p) & = \Phi^{-1}(v)([f + kg]_p)
& = v(f + kg)
& = v(f) + kv(g)
& = \Phi^{-1}(v)([f]_p) + k \Phi^{-1}(v)([g]_p)
\end{align}
and
\begin{align}
\Phi^{-1}(v)([fg]_p) & = v(fg)
& = f(p)v(g) + g(p)v(f)
& = f(p)\Phi^{-1}(v)([g]_p) + g(p)\Phi^{-1}(v)([f]_p)
\end{align}
which proves that $\Phi^{-1}(v)$ is indeed linear and satisfies the product rule. Thus, $\Phi^{-1}$ does in fact send elements of $T_pM$ to elements of $\mathcal{D}_pM$.
\medskip\newline
To prove that $\Phi^{-1}$ is the inverse of $\Phi$, let $v \in T_pM, w \in \mathcal{D}_pM$ and compute for any $f \in C^\infty(M)$ and $[g]_p \in C_p^\infty(M)$:
\begin{align}
\Phi(\Phi^{-1}(v))(f) & = \Phi^{-1}(v)([f]_p) = v(f)
\Phi^{-1}(\Phi(w))([g]_p) & = \Phi(w)(g) = w([g]_p)
\end{align}
That is why
ASda
LaTeX source sent via direct message.
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<@&268886789983436800> Free gift for all your moderating work
Someone is gonna get vanished
abdalghaffarさんの金玉は三時間後で消えます。
<@&268886789983436800>
Don't troll
Nobody asked you for your misinformed take on gender roles either.
Happy to read an argument :p
I'm not asking to argue with you about gender roles. I'm asking you to stay out of arguments that cause mod pings.
this is the most basic bitch shit I've ever heard
what do we live in
1950 lmao
"instinct to protect her"
lmao
you can't make this shit up
Doable
Thanks 🫡
nah. sounds to be like incompetence. and walking problem.
That's the reason why women lagged so behind throughout the history
Well your perspective of what a women should be will only work in a society of fools
An ideal human should attain his own individuality
Being enforced a gender identity is pathetic
The difference must only be biological
Not sociological
All the traits you have mentioned just to make them inferior to males that's what society has been doing since ages and it's pathetic
And even majority of women have nothing against it they just blindly follow the same
Women should be just like men ( excluding biological differences )
In simple sense an ideal women should abolish the feminine identity/nature
women were heros back then
men were caretakers of women and women were caretakers of men
with different rights
you still havent explaained why its paathetic though
argument?
from what ive seen the more power women get the more destructive and degenerate society is
simple observation; and thats a objective fact
@neat lintel what do you want a woman to be like?
married women should be house wives
you’ll have to elaborate on this a bit more, I fear
this is not an obvious fact, if true at all
and I highly doubt it’s true
Only a fool wanna live that life
So you mean women are dumb?
You really want your partner to be a housewife?
Well these roles only work in a society with low intelligence
no
I mean society is already so dumb where majority is religious
that’s also a dangerous perspective to have, imo
Nuh
The concept of God should be separated from these pre-existing religions
I'm new here
welcome to mathcord! 
Thx
I'm saying they should fulfill their duties, which include managing the household and family at a minimum. In summary, I don't believe women should have political power or be able to become leaders, aka patriarchy. I could give you examples of amazing women who were submissive housewives and at the same time scholars, and also their negative influences at a point where they had political power.
Amazing who?
Give examples
I am not talking about the political power
Umm Darda, Fatima bint Muhammad, Aisha bint Abu Bakr, Umm Salamah, Shuhda al-Baghdadiyya
I am talking about individuality
These were teachers and submissive wives at the same time
I agree
Wth
Elaborate
read about their biography
they were influential women in the islamic golden ages (and early islam)
What's islamic golden age is all about?
Wtf, are people still discussing femininsm in 2025?
all about sciences and knowledge and debates and Islam.
Governed by the sharia divine law
Ottoman and abbasid for example
Nuh i ain't calling them educated by that mean
alaot of historians complimented how they structured societies and gender roels
some of the muslims there are some of the smartest recorded intellectuals
All the women should be housewives?
yes, but not restricted to
Yes?
yes
What if your future wife denies to be a housewife
divorce
Don't marry bro @neat lintel
A married woman can be a house wife if she wishes but by no mean should she be locked down into that role only
my christian family are comfortable with it and my mother is a amazing teacher
@neat lintel what if you were a woman?
Would you have done the same?
Be a housewife?
Not purely a housewife
like I sid
we need women for more things
teachers and etc
Why can't a man do the households?
because a man by the masculine nature has the physical and mental capability to work and take care the wife or wives along with an potential family
So you mean men are mentally superior to women?
Physical capabilities have nothing to do much nowadays
We are not in stoneage anymore
in what aspects?
not true
That's what you said
there is a distinction between feminine and masculine aspects
thats just the coin toss nature we are born in
Most of the feminine nature the women develop is because of the surroundings and the way they are grown up
Guys I study in class 11 and I have discovered some theorems in mathematics so where can I publish them
this was common in 1800s and by mid 1900's lol
Like what theorems?
Most probably they have already been discovered by somebody else
Happened with me too
how physical capabilities matter today bro 😭
care to showcase ?
I have discovered tons of theorems and what I got to know they were already discovered by somebody else 100's of years back
Its all about presenting yourself
military?
Factronomials
Exponential theory
Fractional power series
if you talk about women filling these roles yes
I agree on this point
Yeah sure but they are still not complete I will complete them till 22 march
statistically there are men born w/ fked up hormones that cant lift a log and there women who are born to lift strong stuff, but yes like in MOST MOST cases men are stronger and they are more suited for the military
firefighters, police officers, farmer, athletes, waste management, construction workers, oil rig workers, ship/dock workers
That women and men shouldn't have different requirements for military just to give women extra rights
strength matters in the military
ye
You are literally describing working class
If anyone interested then dm me
We are talking about an ideal man or an ideal woman
They both should focus on their individuality and carry an intellectual pursuit
That's it
Im saying the men masculine nature is needed for these important aspects of a healthy and functioning society
I dont disagree that these positions must interest men and not women
yes but enforced in being housewives if married
if ur a woman and u want to be in these positions u must be born with extraordinary nature
dude it can be easily multi-tasked
Wtf?
like strength, physicality etc
No
not true
not even steroids could help you
how
@neat lintel there are exceptions
there are like, 0.3% of women who are stronger than the average men
idk the percentage actually but y get my point
im saying we also need women for feminine jobs aswell
Bruh yk what
career is a career bro why do you make distinctions
the ones on steroids lol ?
because its problematic
the youth wouldnt developed
political career, business career, they are all neutral
if both maan and women were aalways too busy the kid would have mental issues and that is proven
The naturally strongest woman in the world is definitely stronger than an average man
yes thats what im saying
i have no idea what average means anymore
You are in a mathematics server 
if both the man and woman gives the kid an iPad and ignores his development and neither teaches itabout femininity nor masculinity , the kid will have problems
It also depends upon the kids
For a typical kid
but thats like
Maybe yes
not just that
its an issue parents will have to manage
But yes being controlled by gender roles you are just making yourself dumb
if the kid doesnt get educated by his parents and doesnt spend time with his parents aswell
on how they want to nurture their kid
@neat lintel
Bro
It's about perspective what's more important to you
i just gave you examples of amazing house wives who are probably higher in status than all 3 of us combined
The society or the individual
You covered islamic golden age*
And we are not considering that superficial status
roman empire is even worse
maybe greeks
im not a historian
but gender roles and rights were established perfectly during the islamic golden age
@neat lintel what's important for the society isn't necessary a supirior way of life
Guys does anyone know how to publish a math theorem
@neat lintel let me give you an example
if you are so unsure then you should probably wait instead of rushing lol
but society isn't "optimized" if we assign women to the roles they had in the islamic golden age 😭
There's a restricted area where any individual steps in it gets detonated. There's a danger sign board in the border. There are 100 individuals each individual has assigned its own task and then there comes an animal who couldn't read that sign board and enters the area which gets detonated. Hence the rest of the people now can safely pass that area
And that was only possible cuz the animal was dumb enough to not be able to read and understand that signboard
So according to you being that dumb is an ideal trait?
the only thing married wives were assigned with is managing household and kids, but not restricted to house-hold managing
Pls reply guys
Dm me your theorems
Register an account on vixra and publish
Ok
oversimplifying what im saying ngl
Ok

R u saying to me?? To dm??
yes
And they were heavily criticized if they were to pursue a noble career
outside of the housewife thing
they werent allowed to have roles that allowed them to have access to political influence
but as far as I studied I dont think thats the case for the caliphates
so people or woman following the same roles are basically foolish and slaves
That's it
Nothing too complicated
Science fe
If a woman pursued science
she would be heavily criticized
Science does impact politics as far as I am concerned
why would anyone married a dumb women to raise kids though? Women were educated in their youth and had access to free education
If you dont allow women to become scientists thats dumb asf bro
even if they decide to get married and all
Being educated and being intelligent is a different thing
same thing
Why do you want a woman to raise kids that's the responsibility of you two
no
education helps express potential genetic intelligence
Not at all
learning algorithms and techniques for things doesnt help you think more efficiently ?
of course
a man is also responsible
but the housewife is also more responsible because shes not the one working all day to finincially protect them
Ideal couples shouldn't have kids tho
thats a messed up statement
It will just be a distraction from their intellectual journey
ideal couples would give birth to ideal and intelligent kids
No
An ideal person is not a slave of the society
Then the ideal child would need to be facilitated in order to maximize its intelligence 😭
Actually I retract this
w/ super professors and inhuman teaching
If you are talking about general masses
Sure they are fine to follow their surface level way of life
thats how it is in the capitalist world
.
.
It's just like how animals are useful for human beings in various ways
They are useful only because they are inferior to us
Same for the humans
And the society
"Muslim woman"
Lol
the ideal women is a muslim women
Ok
No
actually heres a lecture by one of the greatest modern islamic scholars
debateable
cant make up a opinion/stance if you dont know the roles and qualities of a muslim women
I already gave you an example
your example belongs somewhere that isnt found in the early islamic ages
It explains everything
again not true
islam teaches tht men and women are equal
but born with differen nature, different rights, and different qualities
Well I have explained everything
Now you if you still can't understand this
I have nothing to say
Ideal Muslim Woman fills her role perfectly according to the needs that an "Islamic" world needs
now I dont believe an Islamic world is an ideal world
If y do ok
@karmic spear wasn't my example sufficient?
You havent made a argument against my islamic stance on gender roles though
Ideal woman is not the ideal islamic woman based on all peoples' perspectives
only for the Islamic World
Yes
its not recommended to make a judgment based on something youre ignorant of
By the way that example covers a lot on the world of politics and how the elites manage us
but not related to the topic we are discussing
I am busy rn I will talk later
Im not ignorant
why do you think a ideal islamic state is not a ideal state in general?
Because the ideal Islamic state does not optimize liberalism
@neat lintel what about couples without any child ? According to u?
And I believe individualism, political equality etc are way more important in our society than anything else
Freedom is everything
If you take humans as work machines and you optimize them on a massive scale then life becomes dull
If you want to make an empire sure
Liberalism is where a country is ran by human-made laws that are always problematic
so youre a communist?
But if you are talking about life in general, living must be however y want to engineer it based on the potential y carry since the moment y were born
Wtf
no
how are the things I said associated by communism?
They are literally the opposite idea of communism
idk
never came across a scenario like that
Every political idea is problematic
Should they follow the same roles?
but a wife who isnt a mother has less responsibility with managing household
somewhat yes
Wth?
with cooking and etc
Cooking is a basic thing both men and women should do it
it is basic but not with a stressed mind
huh
like i said a man should be working and providing for his wife financially
men are caretakers of women and women are caretakers of men
You are again discussing about the average humans with surface level perspective of life
no
of course
Political ideas have flaws
because its the human mind
Systems have flaws
@neat lintel
I dont believe thats applied to the islamic law though
not even terriosts like hitler thought so aswell
Yes because you believe that a flawless being created these laws
I believe opposed to you, that a human mind made these laws
and that Islam is nothing more than an Idea
Every political idea is a religion if you view it objectively
you think like thaat because youre oversimplifing everything
Communism is a religion, Capitalism is a religion, every religion is a religion
life is too much to discuss every aspect
not true
Nah. I prefer the type of person that has ascended beyond that. I prefer to associate with people that are able to take care of themselves.
not the definition of a religion
humans cant really do that
at some point youre gonna need to be married
History defines it that way tho :p
Then I must be something beyond human if I'm currently doing that.
give me a university who defines religion like tht and Ill retract my commentary
you will see the effects if you havent and havent lived long enough
I don't know, its my perspective, called it fucked up or however you want but I am 100% sure that in the most historical university books you will come to see that religions are history
Or I've just ascended beyond ye mere mortals.
If you want to believe that Islamic laws are flawless because some perfect existence made them, ok go along w/ it
You advocate for freedom as if you want to take advantage of it and utilize it for learning
But in the end of the day if you analyze the system of Islam, politically you will come to see that its an ideology
Atelast on how it applies to politics
Its only just a system
You dont need to be a muslim to affirm that
With flaws that are ignored because you believe that these flaws are exceeded by obeying a higher god
who claims that your system doesnt have any flaws at all
you believe that this being exists, I believe that this is mythology
Its all about perspective
And thats when you start to shape your different opinions about men and women
again you claimed to not be ignorant of islamic law, so if you arent ignorant, im willing to hear a argument proving it being problematic
Not being ignorant of a religion doesnt mean I am accepting the claims of that religion
Not being ignorant means I know what this religion says and how this religion has evolved through out history
And therefore, I decide on what stance I want to hold against that
then what part of the islamic law is problematic?
Waiting for someone to bring up islam's approval of slaves
man
look
slaves werent approved
islam waas the first religion to free slaves
Its sections literally talk about how it is okay to have sexual relations with slaves I don't know what you are smoking.
Allah told slave owners to free slaves if they could because if the caliphates freed slaves all at once it would create disastors
specific rulings
Look
concubinage is a valid thing
lol
why dont you talk about the other secular empirs
how they established feudalism
and man made laws
abused their people
...maybe because the topic is about islam and how its laws can be problematic.......

and islam came and liberated these lower class people by giving them rights to education and health-care for free at a low tax rate of 2%?
yeah true
Islam did actually massively improve things for a lot of people
women were little more than cattle before islam, that much is true
It was fairly forward thinking in its age, but it is no longer that in this day and age
Evolution
in what aspect?
dont take extremeists as examples
If I explained why liberalism is problematic would you alter your beliefs?
No because like you, I have a perspective that I have justified on why I think it's the most "suited" for society and the individual
im more than happy to break down that tree of subjective validations lol
Lots of things in islam state that a woman needs a man's premission, sometimes for things a lot of people take for granted, take a look: https://islamqa.info/en/answers/69937/can-the-wife-go-out-without-husbands-permission-in-islam
Ok
what are ur thoughts
Subjectivity explains everything by the way you said it yourself
I just read it, and found no issues and everything seems pretty reasonable to me when you read the context of the hadiths
partially
I just can't see a better way of living
You are born
Then raised
whats a better idea than being able to do whatever the fk with every resource you have available to you , while operating like a human , inside of laws and rules
fortunately islam doesnt promote divisions
and these divisions are the catalysts for conflicts
sorry but
prophet muhammad for example
I think Islam does not have a good reputation for being peaceful
told us that it is obligatory to support a poor man or donate to charity or it would result in major sin if one refuses & capable of it
but you could say to me that if all people could turn to the idea of the islamic state then there would be no wars ok and no conflicts sure
prophet muhammaad also used to feed orphans and play with them, and authentic stories like the blind old man and prophet muhammad
again i said I would engage in discussions if you dont bring up extremist groups which 99% of the scholars condemn
they violated quranic rulings
alright
there will be conflicts no matter what, but not noticeable and not very significant
thats just something we can minimize but not eliminate
human-maade laws are the worst when it comes to reducing these types of issues
Sharia is made by human
even democracy is childish compared to islamic law if you truly understood both philosophically
not in my view
God made Sharia?
if you can show one flaw in the sharia I would leave islam because mistakes occur by humans
yes
thats why freedom exists 🤷♂️ I just hope you utilize it and take advantage to rectify your beliefs
Well, if that's what you believe I cant change your perspective
im very open minded
We could just agree that we disagree
How was god created?
and willing to hear arguments
Yk what
Ahan , are u religious btw?
No
Alr
@karmic spear you?
I am agnostic atheist
4 verses of clarification
yes agnostic atheist
anywaays ima go now I had a nice time discussing with you guys
But yes the idea of god what religions claim is definitely wrong
Have a great day
you too : )
wdym
Hmm
most religious texts are often laughably false if you're judging them at the level of literal factual truths, but that's also kind of not the point and value of those texts
"earth is 6000 years old" or whatever lol
Simplified I just believe that:, Humans had insecurities and tried to explain stuff. Humans then tried to explain stuff through out gods and religions. Then they made the idea of religion more sophisticated and "reasonable". Then they used these ideas in order to create political systems and justify both how they will operate and things they cannot discover through their perception (metaphysics.). Which as a thought is very flawed because it comes from a human. But the world is so random and absurd that you cant reject metaphysics, so I take my stance as being agnostic. Most religious texts (bible, quran) are very "human" if you read them and you could also say that its pure mythology. Thats why I dont reject the idea of something higher we cannot even imagine, but I completely reject the idea of religions. Because they are super flawed
and they defy truths yes as y said
I can only see how religion has benefited the political world even tho I believe political systems without any base on religion are better and I am not a communist
they are very human and there is a very big range of how you can pick and choose from the texts, like even if you are agnostic you can derive lots of profound lessons and meaning from a lot of these texts, and even if you are religious you can find plenty of ways to extract excuses to excuse one's own awful bigoted behavior 
I prefer that religion would have less of an influence on society but this is still a lot better overall than mass nihilism
I personally prefer myths as stories but I can see why people prefer them as religions
yeah
yeah I enjoy mythology a lot even as an atheist and I've got a lot of meaning from reading various religious texts from time to time in this way
thinking about metaphysics and forming religious helped a lot w/ evolution I think
and there is more to come
in the future
It's just there lack of understanding related to consciousness and human narure
Humans tend to relate everything to the other humans
Humans can have emotional attachment to a thing which isn't even alive
They can have emotional attachment to a stuffed toy
The reason?
Cuz the stuff toy tends to have a cuter features similar as a human child which triggers their instict to nurture it
That's why humans can somewhat get attached to these things
Same goes for the concept of god
The thing with humans is that
Most of the things that humans find obvious and evident is just a result of their natural instinct and unconscious influences
Or maybe everything is already pre determined
Your every action is determined
And who do you think is the one who determines?
It's just a domino effect
Like just take an example of basic physics
Two objects are comming together at a certain velocity with certain path

