im solving (-1)^(2/6)
#serious-discussion
leaden torrent
are you sure?
warm light
im making it 6root(-1^2)
leaden torrent
what facts are you using?
leaden torrent
if youre using $a^{b/c} = \sqrt[c]{a^b}$, that only works for positive $a$
fathom swallowBOT
Namington
leaden torrent
since you run into principality issues otherwise
warm light
oh
leaden torrent
for example, $-1 = (-1)^{1} = (-1)^{2/2} = \sqrt{(-1)^{2}} = \sqrt{-1} = 1$???
fathom swallowBOT
Namington
fathom swallowBOT
To many apples
leaden torrent
yes.
fair mural
would you take the 6th root first or square it first in that case?
or would you just simplify it
i imagine simplify
leaden torrent
formally you square it first and then pick a square root based on context clues
the better way is to just simplify
since then you dont have to deal with this ambiguity
for negative $a$ and $b, c$ coprime, $a^{b/c} = (-1)^{b}(-a)^{b/c}$
fathom swallowBOT
Namington
warm light
so does $\sqrt[3]{x} = \sqrt[6]{(x)^{2}}$
fathom swallowBOT
To many apples
leaden torrent
yes, since that notation indicates the principal root.
warm light
whats principle root
leaden torrent
er wait, thats a 3
no
lmao
principal root is the root with greatest real part, but to correct myself
those are not equal
for that exact reason
warm light
if $x^{1/3} = x^{2/6}$ is true, then shouldnt so does $\sqrt[3]{x} = \sqrt[6]{(x)^{2}}$?
leaden torrent
not necessarily
fathom swallowBOT
To many apples
leaden torrent
youre erroneously thinking those notations are equivalent
which is only the case for nonnegative bases
warm light
so positive bases and 0?
deep mango
branches 
leaden torrent
i dont think it has a name, its just... the definition of fractional exponents
warm light
ok, so can i simplify$\sqrt[6]{(x)^{2}}$ into $\sqrt[3]{x}$ when solving an equation?
fathom swallowBOT
To many apples
leaden torrent
youre potentially changing your solution set
so you have to watch for that
but that will only add false solutions, not remove them
so it just means you need to check your solutions at the end.
warm light
can i simplify $x^{2/6}$ into $x^{1/3}$when solving an equation without having to look for changing my solution set
fathom swallowBOT
To many apples
leaden torrent
as long as you know x is nonnegative, sure
defining fractional exponents for negative bases is hard in general
theyre usually left undefined, but you can define them in a not-very-well-behaved way
which is exactly what you saw here
and why you ran into issues when you didnt simplify
my general rule of thumb is "avoid fractional exponents of negative numbers in general"
they shouldnt be necessary for solving equations anyway
rather than having $x^{m/n} = y$, you can always have $x^{m} = y^{n}$
fathom swallowBOT
Namington
leaden torrent
which you can actually work with
devout pasture
nice but why
iron osprey
Math is tools u can use
To do things
But theres no known bound on what u can learn
And its generally all useful
rancid meadow
Not true
iron osprey
No known bound to me anyway
Like
The existence of a bound is known
But the least upper bound isnt known
Hence its unknown whether any1 has attained the maximum possible
sick kite
The idea of an upper bound for mathematics is an interesting idea, because it makes me wonder if there will ever come a time where mathematics is so complex that humans will not be biologically able to understand it. Theoretically it seems like a possibility but hey idk
iron osprey
Even if it does we'll just specialize
swift sinew
Aren’t we already specializing 
sonic sentinel
The idea of an upper bound for mathematics is an interesting idea, because it ma...
i don't think "being able to understand it" is the main concern
we can use the four color theorem for example without needing to understand the proof behind it, it would be unfortunate to have to skip like this in the future, but its still possible to expand math
the main concern imo, is that the previous math work in any field takes more than a human lifetime to understand the boundaries of, and so very few people could actually expand it
surreal sapphire
we can just blackbox more
and our methods of creating new mathematicians are always improving
sonic sentinel
the main concern imo, is that the previous math work in *any* field takes more t...
so like the amount of time you need to understand an unsolved problem and have the experience to solve it is more time than you have to live
clever knot
the main concern imo, is that the previous math work in *any* field takes more t...
yeah I mean does there become a point where you cant even learn enough in that specific field to find new things within your own life time? Like you spend your whole life just catching up
sonic sentinel
if this happens we will probably have to wait on biology and psychology, if those field face the same problem, it could end up being another form of great barrier
sick kite
Maybe we would have to improve our actual bodies using technology 
vale hare
Why do you not like the Aluffi exercises?
Curious, because I think they are quite nice. I've heard some people say they are all 100% trivial and others say they are difficult. And also I've heard people recommend the book "because it has tons of good exercises", while i've heard others say that aluffi sucks because the exercises are terrible and that if you wanna use it you should read it and then do the exercises in another book. However, I've not heard either side give much reasoning behind their assertions, so I'm curious why you think what you think about the exercises
jovial ember
Who’s the most beloved user on this server? React 
if it’s chmonkey
velvet dagger
@vale hare I didn't feel like there were that many exercises which were really interesting
And a number of them were just long. Like oh I don't feel like writing this part of the theory out fully so ima just have you do it as a guided exercise. Each step is very straightforward so it's just a lot of writing but not too much thought
Unless the exercises my friend had were not a representative sample in which case I might have to take it back lol
jovial ember
Who’s the most beloved user on this server? React <:chmonkey:773795952125411338>...
THE PEOPLE HAVE TURNED THEIR BACKS ON ME
vale hare
Unless the exercises my friend had were not a representative sample in which cas...
I can't say I've seen much of that, that the actual theory gets put in exercises. Generally the exercises are actually quite short. The only ones that are really long in my experience are the ones with a little negation sign, because those typically develop their own topics over time. But they aren't cited anywhere else in the writing of the book. I've only done the exercises in the first 4 chapters tho. Maybe in the later chapters theres more of that.
clever knot
how do i vote jan niku
swift sinew
Could someone please change my nickname to “ .close” I’m testing the bot
(That’s with a space in front of the period)
clever knot
Dont think that would break it or do anything but you can try
swift sinew
True true. I’m wondering if there’s a way for a robot to use my name and close a session
Unless robots are not able to use that command either
worn cradle
So...I have an interesting request, so I'm making a game that has what I've coined psionics ( science based magic) and I was looking to hire someone to help me calculate the energy requirements of each power to ensure that the costs of them are balanced perfectly.
There is alot of fantasy like ability and some we may just have to ignore or estimate (such as the time based ones, though to clarify there is no seeing the future or visiting the past.)
gilded spade
So...I have an interesting request, so I'm making a game that has what I've coin...
I would do that if I was smart
jaunty gate
funny number members
leaden torrent
So...I have an interesting request, so I'm making a game that has what I've coin...
i feel like thats more of a game design question than a mathematics one
just an fyi that we generally discourage these sorts of transactions
we dont disallow them, but its very easy to get scammed over the internet and we have very little way to prevent it
so... be careful if youre going through with it
molten cipher
I wonder how good it is printed as book http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf
thinking of printing it as B5 papersize book. But I feel like there's a little bit too much whitespace margin on the side.
the printing company will print whatever pdf is sent, so I'm a bit wary about arbitrarily cropping it. 
anyone has any opinion (is it good enough to print as it is at B5) or advice (need crop?) ?
limber thunder
yeah crop it a bit
I don't use the margins too much myself but I print chapters in half letter at home sometimes, a margin of 2-3cm is on the tight side but useable
B5 is bigger than half-letter so you can go for something slightly larger
molten cipher
ok, I guess I really should crop it, guess I need to look around how to crop a pdf
sick burrow
So...I have an interesting request, so I'm making a game that has what I've coin...
This sounds like a fun thing to do yourself
Or possibly something to hire someone for if you're serious enough, but I agree with Namington that this server isn't the right place for it
reminds me of a magic system I was designing for a fantasy setting I hopefully wanted to run a ttrpg campaign in
that was like super hard time magic
that I was very careful to make sure satisfied conservation of momentum which led to some really cool emergent properties
worn cradle
Ya, I'm thinking of hiring a tutor off of fiverr.
Ya time magic is a tricky subject, It took me a good month to figure out how it could function in my world 😅.
But no matter how hard I thought I couldnt reason the existence of traveling back or looking forward
sick burrow
but then I was trying to expand on it with like a "flow of temporal charge" and realized I would basically need to solve a heat flow equation to figure out how much power casters could draw 
it was actually a really cool system and I want to come back to it some day
worn cradle
but then I was trying to expand on it with like a "flow of temporal charge" and ...
Thats my issue right now, being a perfectionist, I won't allow anything in my game unless I have atleast a basic understanding of the laws that govern it, otherwise its just not happening.
sick burrow
since I managed to do time travel that didn't break causality while also being cool, and also time magic that didn't break kinematics
worn cradle
Especially since I never had the opportunity to attend College or university since I got kicked out of school for....making a bully fly....
sick burrow
obviously threw relativity out the window but that's a confusing and counterintuitive mechanic irl so I don't really care
worn cradle
It was only the first floor but apparently he broke his arm and a rib so I got in trouble for it.
sick burrow
all physicists after Newton were cranks
worn cradle
What is that reaction 😂?
sick burrow
all mathematicians after Cantor were cranks
note: the first case is not inclusive, the second is
worn cradle
Ya, any kind of math that doesn't include these symbols √¶∆ I'm great at. Also good at π since I've managed to memorize the first 5 numbers of 3.14159 😁
I can calculate Pythagorean and percentiles in my head, just the moment you throw algebra at me I become caveman
sick burrow
Ok compute $\pi_m(S^n)$ for all m,n
fathom swallowBOT
R is countable
worn cradle
I'm gonna guess the S is sin?
sick burrow
No, it's n-sphere
worn cradle
Ah, so my caveman wisdom is telling me this is to calculate the 3d circumference of a sphere?
Cause I see the m next to Pi which I'm going to guess means I need to (ironically)square that number?
sick burrow
no it's mth homotopy group
worn cradle
in general how do i find the point which increases the most rapidly?
Ask in the questiobs area mate, this spots for general discussion.
worn juniper
Ask in the questiobs area mate, this spots for general discussion.
i mean i thought the question quite general
sick burrow
Group of homotopy classes of mappings $S^m \to S^n$
fathom swallowBOT
R is countable
worn cradle
So many big words, why did I have to fail English so badly 🤣
Oh! Homotopy is that thing you can use to count the number of holes In any euclidean shape?
GAAHH BAD AUTO CORRECT!
sick burrow
number of girls in a Euclidean shape
worn cradle
I type one wrong letter and autocorrect decides it free real estate 😅
sick burrow
Homotopy groups roughly do count holes, yes
I think?
I know pi_1 does but I'm actually not sure if the higher homotopy groups do
I generally think of homology as being more of the hole counting thing
compact tartan
homology is shapes that are not boundaries
homotopy is spheres that cannot be shrunk
last oxide
no it's mth homotopy group
mth homotopy group thoi
sick burrow
Oh I bad
worn cradle
Yep...what?
sick burrow
I said the wrong thing
Said mth homology group instead of mth homotopy group
To be fair
last oxide
it also depends on the homology
its generally singular tho
if you are in this setting
worn cradle
homotopy is spheres that cannot be shrunk
Thats a very odd thing to need in math since anything can be shrunk when removing/displacing mass from its center, but I guess when you say can't it's for things of which mass and volume need to stay constant?
compact tartan
Thats a very odd thing to need in math since anything can be shrunk when removin...
that's the deal, not everything can be shrunk
sick burrow
Actually no I'm stupid
compact tartan
if I have a plane with a hole in it
I can draw a circle around the hole
the circle cannot be shrunk to a point because of the hole
sick burrow
They're usually homotopic but it would be a weird font that would have t homeomorphic to l
compact tartan
(the circle is a generator of the fundamental group (1st homotopy group) of the plane with a hole)
last oxide
Thats a very odd thing to need in math since anything can be shrunk when removin...
it depends on where you are is the point
if you remove a hole from in the middle of a circle, then you wont be able to shrink it if that hole is "not in your space"
last oxide
if you are in euclidean space, you can just shrink shit
sick burrow
pi_m(S^n) can be nontrivial for m>n
compact tartan
well
sick burrow
or why this would be so hard to compute in general
because I can't visualize it at all
compact tartan
the simplest example is pi_4(S_2)
sick burrow
yes but then I have to visualize 5 dimensional Euclidian space
compact tartan
yes
sick burrow
see I can't do that
swift sinew
sick burrow
it's painful how true this is
worn cradle
if you are in euclidean space, you can just shrink shit
In forth dimensional space you can just pass thru shit too!
last oxide
In forth dimensional space you can just pass thru shit too!
I think youre still thinking of euclidean space
compact tartan
I wonder if there's a syntactical way to see it
worn cradle
You know what's funny though, multidimensional worlds are stupidly easy to code into games.
sick burrow
you could show me a symbolic poof of pi_m(S^n) mniip
(well I'd be very impressed if you could do it in general)
but for a specific case
compact tartan
syntactical not symbolic
compact tartan
no?
sick burrow
math is when follow formulas
last oxide
I mean
you have the hopf fibration 
sick burrow
ok sure syntactical
compact tartan
by syntactical I mean by using homotopy type theory
sick burrow
wait
what
anyways
my point is
you could show me a proof
and I might even be able to follow it
compact tartan
you can syntactically show that $\pi_k(S^1) = \begin{cases} \bZ &, k = 1 \ 0, & k \ne 1\end{cases}$
fathom swallowBOT
mniip
sick burrow
but until I learn to visualize 5D space, and I won't, it will never feel morally correct
compact tartan
Syntactically, a circle is a space with a point $b : \mathbb{S}^1$, and a path $l : b = b$, and it is the universal space with these elements in the sense that it is initial. Given a fibration with fibers $C : \mathbb{S}^1 \to \U$ (classically you would talk about a fibration $\pi : \Sigma C \to \mathbb{S}^1$), a point $\beta : C(b)$, we can lift the homotopy $l : b = b$ up into $\Sigma C$, and and look at the other end, which will be $\transport^C_l(\beta) : C(b)$. If we're also given a homotopy $p : \beta = \transport^C_l(\beta)$, then we can define a section of $\pi$, $f : \prod_{x: \mathbb{S}^1} C(x)$, such that $f(b) = \beta$ and $\ap_f(l) = p$.
fathom swallowBOT
mniip
compact tartan
On the other hand, the integers $\bZ$ come with a pair of maps $(+1), (-1) : \bZ \to \bZ$ which can easily be shown to be inverses, so by univalence we have a homotopy $<ua>(+1) : \bZ = \bZ$ in the space of spaces
fathom swallowBOT
mniip
compact tartan
Next we use the aforementioned property of the circle (circle induction) to define a map. We pick a trivial fibration $C(-) = \U$, $\beta = \mathbb{Z}$, and $p = <ua>(+1)$. This gives us a map $f : \mathbb{S}^1 \to \U$, with the property that $\ap_f(l) = <ua>(+1)$
fathom swallowBOT
mniip
compact tartan
By functoriality we also have $\ap_f(l^z) = <ua>(+z)$ for any integer $z$. We can construct a function $<code>(a) = \transport_{\ap_f(a)}(0)$, $<code> : \Omega \mathbb{S}^1 \to \bZ$, and given the above it satisfies $<code>(l^z) = z$.
fathom swallowBOT
mniip
compact tartan
On the other hand we can trivially define $<decode>(z) = l^z$, $<decode> : \bZ \to \mathbb{S}^1$
fathom swallowBOT
mniip
compact tartan
we can show these two to be inverses, in the forward direction we use induction over z, in the backward direction we use induction over code(a)
therefore we have $\Omega \mathbb{S}^1 = \bZ$, and since $\bZ$ is a mere set, this implies that the first homotopy group of $\mathbb{S}^1$ is $\bZ$, and all the others are trivial
fathom swallowBOT
mniip
sick burrow
you say this like I actually have any idea what you're talking about
the extent of my knowledge of homotopy theory is chapter 1 of Hatcher
bronze pelican
visualize 5d space with a flat vector field in R^3
fallow kiln
wtf? 14-D airplanes exist?
leaden torrent
wtf? 14-D airplanes exist?
im assuming this isnt a joke
"plane" here is used in the sense of "coordinate plane"
you know, like the x-y plane has 2 dimensions
x and y
icy forge
I did a search on their post history, and I would assume it is a joke
leaden torrent
a 14 dimensional plane would have 14 dimensions/axes
fallow kiln
im assuming this isnt a joke
I thought he meant a 14-D designed airplane
mint patio
hand writing proofs is so much more satisfying than writing them with technology
but fuck hand writing
neat lintel
dan
sick kite
who is dan
mint patio
:dan:
why are we allowed to make this assumption
what if it turns out that u(x,t) \neq F(x)G(t)?
what if it's F(x) + G(t)
@deep mango just inc you have something
steep mountain
are you asking why we made it or why we can make assumptions in general?
neat lintel
mint patio
Why we can make that assumption (without actually knowing the solution yet)
should I really post this in questions
,
steep mountain
that's how hypothesis work in science in general
we assume something based on intuition or what we already know
and either arrive at a sound proof or its proven wrong/ modified
mint patio
so because we didn't arrive at a contradiction our initial assumption must have been correct?
similar to induction
what intuition/prior knowledge did we have that motivated this assumption?
steep mountain
this specific one idk 
but thats usually the main issue
"setting up a good hypothesis"
deep mango
<@!533879085542146059> just inc you have something
this is related to the fact that multivariable functions f(x, y) can be approximated by sums of functions g(x)h(y)
mint patio
this is related to the fact that multivariable functions f(x, y) can be approxim...
whattt
I did not know this
deep mango
since your equation is linear, if you find solutions of the form g(x)h(y), then you can add lots of these together
and get more solutions
this isn't a proof ofc
but it's the idea of why it's productive to make this assumption
think about this this way: if you have a continuous function f, then split up R^2 into a bunch of squares.
on each square with bottom left corner (x, y), pick g(u) to be equal to f(x) and h(v) to be equal to 1 for u and v just above x and y.
so g(u)h(v) is a little box function, which is 0 everywhere but on this square [x, x + delta x] x [y, y + delta y].
it's like a piece of a riemann sum.
its value on the square is f(x, y).
now you can add all these together (you might need to make the squares half-open)
narrow rock
stone-weierstrass
deep mango
and you get a sort of riemann sum for f(x, y). the terms of the riemann sum are all products of single variable functions.
swift sinew
Isn’t our assumption of separability also using the fact that we’re just hoping we can construct a separable solution to the boundary problem, which then must be the unique solution?
deep mango
in certain contexts (for example, continuous functions vanishing at infinity, or integrable functions) the linear span of these product functions will be dense.
when you restrict to ONLY solutions to the differential equation... well, you could just happen to ruin density
but you don't in a lot of important cases
as can be seen by how the separation of variables solutions give you fourier series in the time variable
or something like that
laplace transforms or whatever
idk
mint patio
on each square with bottom left corner (x, y), pick g(u) to be equal to f(x) and...
I'm still stuck here
Trying to visualize/draw it out
deep mango
Isn’t our assumption of separability also using the fact that we’re just hoping ...
well sums of separable solutions are not separable
so
mint patio
as can be seen by how the separation of variables solutions give you fourier ser...
this is what I thought too, it makes a lot of sense
deep mango
if you like
find a bunch of separable solutions
and then linearly combine
you can get solutions for any boundary data
(in the important cases)
my point being, you solve the equation assuming you can separate, THEN you see what boundary data makes sense for seperable solutions, (after all you'll get an equation in x which doesn't depend on t, so it has to hold at t = 0)
THEN you use the stuff that you now have solutions for, in order to construct new stuff
swift sinew
Ah you’re right thank you. I’ll definitely have to look over my PDE notes again to refresh myself, since I don’t remember it as well as I thought
deep mango
e.g. with wave, your separated equation in x looks like G(x)'' = -c^2 G(x) or something like that. The solutions are Asin(c x) + Bcos(c x), which tells you you found a solution (when you also solve for t and then multiply) which has that initial data.
maybe your equation is on [0, 2pi] and you have boundary values of 0, which would tell you that B = 0 and that c has to be an integer.
ok, NOW we have a bunch of solutions for A sin(nx) initial data, and so we can take linear combinations of these in order to get new initial data (and the solutions will just be linear combinations of our original separated solutions)
but these are just fourier series, and we know basically any function can be written as one.
(L^2 is the easiest place to do this, but ofc your boundary data should probably be differentiable because we're doing a differential equation, and it's on a compact interval so then it's bounded anyway. the fourier series will converge and you'll get a solution)
deep mango
I'm still stuck here
sorry. you should imagine a little skyscraper, located on the square [x, x + delta x] x [y, y + delta y].
delta x and delta y are fixed numbers
(not delta times x)
now you want the skyscraper to have height f(x, y) so that it's close to f.
but it will have a flat top, and everything outside the square will be 0.
it happens that any function like this is a product of functions: take g(u) to be equal to f(x, y) on [x, x + delta x] and 0 everywhere else.
and similarly, h(v) equal to 1 on [y, y + delta y] and 0 everywhere else.
i just need g(u)h(v) to be equal to f(x, y) whenever (u, v) is in the square.
and 0 otherwise.
mint patio
Oh wait I get it now
deep mango
you can come up with much nicer functions than this
mint patio
Ohhh yes this makes so much more sense now, skyscraper example is a lot better
deep mango
for example, multivariable fourier series let you represent functions as sums of products of complex exponentials.
but this is probably the more direct way to see that linear combinations of product functions are dense.
mint patio
We can make our delta x, y as small as we wish (hence the area of our skyscrapers becomes smaller) and then "squeeze in" another product function between the two smaller skyscrapers?
deep mango
yeah
like
it's just like a riemann sum but in 2 dimensions
you make your mesh of square smaller, and you tile R^2 by more and more squares
and the total function is the sum of the functions on each square
mint patio
yup yup
deep mango
(if you make sure to do some kind of half open intervals nonsense, at the moment you run into problems where the squares overlap)
mint patio
now I see where the fourier series naturally comes in
cold needle
today in fourier class he showed me a little thing about heisenberg uncertainty principle 
deep mango
is this the thing were a function and its fourier transform both can't be tiny?
mint patio
the solution of the wave equation is a multivariable function of x and t
so we can write it as a sum of product functions (one of x and one of t)
then using separation of variables, each product function has to satisfy the system of odes
and the solution to the system is a sum of sines and cosines bla bla, so each product function is a sum of sines and cosines
so the final solution is a sum of product functions and hence a sum of sines and cosines (i.e. a fourier series)
and then there's some specific stuff to get down the exact form
but that's the general idea yeah?
deep mango
is this the thing were a function and its fourier transform both can't be tiny?
wait this makes no sense the fourier transform of 0 is 0
whatever
i don't remember what it is
deep mango
the solution of the wave equation is a multivariable function of x and t
so we c...
yep!
it's kind of backwards, cause like
we know that sums of product functions approximate functions
but do we know that sums of product solutions approximate all solutions?
sure, there are less solutions than there are functions
but there are also less product solutions than there are product functions
so who knows
but the point is, it's a good guess to say "hey, let's try anyway and hope that product solutions do approximate all solutions"
cold needle
is this the thing were a function and its fourier transform both can't be tiny?
something like that yea theres this product of two integrals and one of them has the fourier transform of the function found in the first one and it was bounded below by a nonzero positive value
deep mango
and then you do it and you get product solutions which you can do things like fourier theory on
cold needle
there was a quadratic term in each integrals as well 
idk something to do with std dev or sth
whatever the physical interpretation was
deep mango
it won't always be sines and cosines. for example, the quantum harmonic oscillator is a PDE (sorta similar to the wave equation) but whose separable solutions have initial values which are the hermite polynomials.
instead of sines and cosines
ok, that seems bad, but then it turns out that hermite functions can also approximate any function if you linearly combine them.
so
it's fine there too
it's not fine super often though which is why no one does separation of variables anymore in an advanced pdes course lol
but it works for all the important examples (some would say the examples are important BECAUSE it works.)
mint patio
why do we care if it approximates all possible solutions? as long as it works out as a solution then why can't we just use it every time? 
well I guess probably initial/boundary condition stuff right?
since not every solution function is going to meet the ics and bcs (I think)
deep mango
something like that yea theres this product of two integrals and one of them has...
yeah these are called the "localizations" or something
i don't quite remember
you look at the square integrals of (x - a)f and (xi - a)fhat
deep mango
why do we care if it approximates all possible solutions? as long as it works ou...
yeah you want to be able to solve for any initial / boundary data
that's the idea
mint patio
mmm yeah okay that makes the problem a lot tougher
deep mango
these solutions have particular kinds of initial / boundary data
but then when you add them up
you can approximate whatever initial / boundary data you want
cause if you set t = 0 you get a fourier series in x
narrow rock
did someone say localization
deep mango
(or a hermite polynomial sum, etc)
and once you can get any initial / boundary data
you know what the solution was for that initial / boundary data : just the sum of the solutions corresponding to all the pieces you used to make it
(because the equation is linear)
it's just like how for ODEs, you find two solutions to your second order differential equation, and then you choose the coefficients so that the function and its derivative have the values you want at t = 0.
and then those coefficients are just the coefficients you plug into your general solution.
same idea but now there's a whole sequence of coefficients, not just a few.
(that corresponds to the fact that linear PDEs are infinite dimensional linear equations, and that linear ODEs were finite dimensional linear equations).
deep mango
this is the sort of thing that makes me like PDEs, not "uhhh the solution to laplaces equation in R^2 is log|x| and in R^3 is 1/|x| bluh bluh"
separation of variables is still very important
if you study quantum mechanics and schrodinger's equation, separation of variables can get pretty complicated!
and it's basically the main tool there (when you're in an intro class at least)
for example, you might remember from chemistry the weird kinds of orbitals that electrons can sit in around an atom
those are separation of variables solutions to schrodinger's equation in 3d with the coulomb potential (the force between an electron and a proton is proportional to 1/r^2 where r is the distance).
mint patio
what does it mean for an orbital to be a solution? I don't remember the specific orbitals besides the shapes lolll
deep mango
the equation looks something like $i\partial_t u(t, x) + \Delta_x u(t, x) = u(t, x)/|x|$
fathom swallowBOT
ryc
deep mango
i forget exactly
cause 1/|x| is the potential, and then when you differentiate you get the force -1/|x|^2. idk. whatever
deep mango
what does it mean for an orbital to be a solution? I don't remember the specific...
those orbital shapes are like clouds, and they are shaded corresponding to how big the function is at each point
basically the point is, an electron is way more likely to be in the shaded area than outside it (if it has a certain energy, corresponding to the constant you split things up with when separating variables)
the amount of shading is the value of the solution to the equation
or like the absolute value squared or something to make it positive and real
solutions to schrodingers equation tell you how likely your particle is to be near each point. they're probability distributions.
mint patio
ohh
deep mango
and that 1/|x| part changes to different functions depending on what forces are acting on your particle.
in this case, it corresponds to the force of attraction between two charges (an electron going around a nucleus for example)
mint patio
is all this background knowledge and physical context self contained in a pdes course
deep mango
no i took a quantum mechanics class
mint patio
ah
deep mango
and the whole time we just solved pdes with separation of variables and said cool things about the different solutions lol
it was very fun
mint patio
quantum mechanics is so cool, it was my fgavorite part of genchem 1 (obviously very generic and introductory stuff)
but I suckkkk at physics
idk maybe someday I get better and see if I can take the class (though there's probably a lot of prereqs lol... :sadge:)
we need a sadge emote @ moderators
doesn't do it
thank you ryc
brave hollow
if mods add sadge I will do some good deeds
swift sinew
This is probably a question moreso for a question-devoted channel
merry river
sry
vestal river
Open question for anyone who's ever actually finished a mathematics book cover-to-cover (doing a representative set of exercises): how do you do that? did you follow a schedule, how to you avoid burning out/getting bored, what if you want to read multiple texts at once? Looking for advice 🙂
crystal stone
Open question for anyone who's ever actually finished a mathematics book cover-t...
You uhh pick a small book
Or you go to grad school and to prep for PhD quals you work through the books outlined in the material to study
jovial ember
Open question for anyone who's ever actually finished a mathematics book cover-t...
I’m almost completely done with Matsumura and have 4 exercises left
In my personal experience I had traumatic thing happen to me and then studied the book like all day for a month to avoid thinking about my life and being sad
Idk how on earth you’re supposed to finish a book without focusing your whole life on it
vestal river
ok im glad it isnt just me that has this issue
forest jackal
its been a very long time since I read a math book cover to cover, and I am a working mathematician. you have to be more selective with what you read time & effortwise eventually
a schedule is a good idea, but don't be too rigid about it, because some things naturally take longer to learn than others.
jovial ember
How on earth do you learn all the things you want to learn though?
Maybe the answer is you never get to
But the doors are starting to open up and I feel like I ought to read 6 different dense books
And I don’t see how you’d ever have the time to do it
forest jackal
you accept that you will never know everything you need to. you absolutely master the basic machinery in your field and you get much better at picking up niche things as and when you need them
early grad school is a great time to read books though to tool up, that is before the proportion of your time spent actually attacking open problems really ramps up
jovial ember
I guess for me the issue is that it feels like there’s maybe 5 or so topics which all underpin the basic working person’s toolkit and it seems impossible to pick up the basics in each area
Deformation theory, descent theory, intersection theory,…
forest jackal
but in my experience you will never read quite as much as you would hope to in this time, there's simply too much
jovial ember
Each of the canonical books is like, something it feels you could spend a year doing a reading course to work through
But no one has the time to spend 4 years doing reading courses mastering all of 4 books lol
forest jackal
yeah its often like that unfortunately.
best you can hope for is to learn one or two really really well and at least have a vague idea of what is involved in the others
jovial ember
Right
I imagine what it comes down to is
You just ignore some of the really sticky details in most of them
forest jackal
exactly
jovial ember
And resign to knowing the ideas
tawny spoke
just learn all the buzzwords and pretend to know what you're doing 😎
forest jackal
well you will have plenty of time later in your career to explore these topics further, and by that point you will have so much more experience and context from other fields that you will pick things up a LOT faster
forest jackal
but it usually won't take the form of reading a book cover to cover anymore
jovial ember
Sure
Or maybe there’s an MSRI conference on it
Or you attend some summer school for a few weeks on the topic
Lol
forest jackal
again this probably varies between fields, since the body of knowledge is kinda stored in slightly different ways in different fields
yeah exactly, I was one of the people involving in the teaching of a summer school and I still learned things from the courses that were not in the centre of my ballpark haha.
jovial ember
I really liked having like
Encyclopedic expansive knowledge on things
Usually before like… needing it?
forest jackal
ikr
jovial ember
But it seems like it’ll have to end at some point
vestal river
This has been a more enlightening conversation than I could have hoped, thank you!
deep mango
keen void
Ok how do I use it or write it
keen void
K
deep mango
3blue1brown's videos do way way better than i ever could
neat lintel
3b1b is the place where I got my intuition for calc
toxic schooner
3b1b is the place where i got into math
sick burrow
eryc if you're so good at explaining
toxic schooner
lol
sick burrow
Explain how
neat lintel
3b1b is the place where i got into math
same lmao
deep mango
you are pretty good at explaining ryc
yes but it's 11:30 pm and i still have homework to do and i'm tired
toxic schooner
Oh right, i forgot it will be late night for you
deep mango
also thank you
toxic schooner
pale orchid
toxic schooner
pale orchid
cold needle
oops let me unpin in the other chat oo
patent sphinx
69420
neat lintel
I know this is a question, but it isn't one of those questions
Does anyone know a website with a readable font about all trig identities, including different ways of writing htem?
E.g sin(2v) = 2sin(v)*cos(v)
normal crane
I'd suggest Google
neat lintel
Does anyone know a website with a readable font about all trig identities, inclu...
symbolab
hollow pagoda
omfg, I actually clicked it.
distant reef
hi
bronze pelican
According to the internet, $80 in 1881 has “relative income worth” of $29,607 today.
fresh ginkgo
by any chance anyone know if there is a server for javascript help?
nimble shuttle
No.
rancid meadow
by any chance anyone know if there is a server for javascript help?
yeah whats Up bro
hollow jackal
these new help channels and the help bot are dope. good work mniip
sick burrow
how are there idiots with PhDs.
genuine quesition
or I guess what I'm really asking is how there are PhDs that don't know how to teach
like my sociology professor
has a PhD and has been teaching for like 20 years
but can't teach or grade based on any reasonable metric
vagrant kestrel
there's a reason #math-pedagogy is the furthest channel on the server
sick burrow
wouldn't a good PhD program filter out people like that? Or at the very least wouldn't someone learn to teach after 20 years of doing it?
plucky kayak
r = r+1
fair tinsel
I think what’s more concerning is that you’re studying sociology
pale orchid
unfortunately phds don't usually deal with pedagogy
and in some unis, no pedagogy skills are required in order to be able to teach
icy forge
but can't teach or grade based on any reasonable metric
Did you look at their research? I'd be surprised if they have no research (rare in any case) and no teaching (common among professors), but it really depends on how the uni values the professor
Valuation is typically done with some citation figures and whatever
surreal sapphire
and in some unis, no pedagogy skills are required in order to be able to teach
excuse i had a full 3 (three) hour seminar on outdated teaching methods
clever knot
didnt think to look in pins
surreal sapphire
i should use me fixing some typos as an excuse to re-pin and have it at the top
devout nacelle
Smh make a textbook out of it
clever knot
you could also pin in under proofs maybe?
surreal sapphire
yes, something like a 300 page one with exercises "consider an arbitrary family of sets ..."
devout nacelle
Send it there, I'll pin
surreal sapphire
you could also pin in under proofs maybe?
it is now done with a version that has less typos
bronze pelican
Each speaker will be assigned a group of graduate students to work on a project in the month leading up to the AWS, as well as during the workshop
oh so its not just a one week project
you have a month of preparation
vague lion
what exactly are the hidden channels?
bronze pelican
stray kite
what are some examples of "abstract" vector spaces which are commonly used? 'abstract' as in 'not using numbers'
compact tartan
well, the scalars are usually a number field
so the vector space usually needs to reference the number field somehow
icy forge
What if the scalars were a field not mappable to/from the reals (or any totally ordered ... equivalent)
compact tartan
I guess maybe your question is, what vector spaces don't "look" like they're just coordinate columns?
(deep down they all really are, this is the basis theorem, but)
consider the vector space of polynomials with coefficients in R
or the vector space of functions R -> R
or the vector space of three times continuously differentiable functions [0, 1] -> R
icy forge
(deep down they all really are, this is the basis theorem, but)
What's the basis theorem? o,o
compact tartan
or the vector space of sequences N -> R that converge to 0
stray kite
i think its "all vector spaces have a basis"
devout nacelle
what are some examples of "abstract" vector spaces which are commonly used? 'abs...
Space of functions over R, subspaces being space of continuous functions over R, space of differentiable functions over R; polynomials over a field
compact tartan
imagine thinking polynomials are functions
devout nacelle
compact tartan
I mean this unironically. Monomials are linearly independent on any infinite subset of R, but if you're working over finite fields this is an issue
devout nacelle
icy forge
compact tartan
in Z_5, any number to the fourth power is 1
and yet in Z_5[x], x^4 is not 1
though as functions Z_5 -> Z_5, x^4 is not distinguishable from 1
devout nacelle
Right
icy forge
Do you have cool spaces where some 'additive identity' is not 0-like (or does not use number 0) and some 'multiplicative identity' is not 1-like
e.g. vector space of matrices with elements in R, 0 is just 0 matrix, not cool
compact tartan
ok so there's a standard trick of considering R+ as a vector space over R
stray kite
"Give an example vector space which is not a subset of R^n, does not contain any functions or polynomials or matrices"
so this is what im trying to figure out
i thought of function spaces and polynomial spaces but acc to this even those arent allowed, so i was wondering what the most "abstract" example i could think of was
icy forge
ok so there's a standard trick of considering R+ as a vector space over R
Yeah this seems like the prototypical example, even in old books :|
compact tartan
with $1 \in \bR_+$ as the zero vector, addition of vectors given by $a \cdot b \in \bR_+$
fathom swallowBOT
mniip
compact tartan
and I'll let you figure out what multiplication by a scalar looks like
devout nacelle
"Give an example vector space which is not a subset of R^n, does not contain any...
Take an arbitrary field other than R
icy forge
It's a famous example o,o
stray kite
Take an arbitrary field other than R<:sotrue:826971355542192148>
i mean that works but i want to figure something out for this lol
icy forge
and I'll let you figure out what multiplication by a scalar looks like
I mean, I would wonder if I could figure this out without the book telling me
compact tartan
"Give an example vector space which is not a subset of R^n, does not contain any...
what does it mean to "contain" a function
leaden torrent
id assume he means
not defined as a space of functions of some other set
or a subset thereof
devout nacelle
Right
stray kite
yaeh
compact tartan
well in that case
compact tartan
C(R) x C(R)
leaden torrent
/ evil
icy forge
but it doesn't contain any (singular) functions
You mean this in the sense of, pairs of functions are not functions?
compact tartan
they are not
compact tartan
that's not a well defined notion then
devout nacelle
The pairs contain elements which are functions, not the space
stray kite
well im not really trying to figure out the question anymore lol i failed it anyway
so ig what im asking is like, i can think of 'abstract' examples for what kinds of groups we can form, such as the group of rubik's cube rotations, but im not able to think of anything which is as ""abstract"" when it comes to vector spaces
icy forge
Yes, I understand, it's the same as saying members of R^2 are not real numbers ([1,1] is not a real number, but 1 is)
compact tartan
ok so there's another example
which I think might fit the bill here
quotient vector spaces
vectors would be equivalence classes of vectors of some other vector space
stray kite
hmm
interesting
compact tartan
tensor products are a readily available example of this
deep mango
L^1 spaces
devout nacelle
Solution space of an autonomous linear ODE (I think I got it right) forms a VS I think?
But they forbid functions so :pain:
compact tartan
L^1 spaces <:sotrue:826971355542192148>
true!
L^p spaces aren't actually made of functions
deep mango
(if the measure is not discrete)
compact tartan
an equivalence class of a single function is still not a function
icy forge
an equivalence class of a single function is still not a function
I still don't get members of Equiv class vs equiv class :(
blazing pawn
well im not really trying to figure out the question anymore lol i failed it any...
does abstract mean like
vector spaces which are not just R^n (allowing for ones isomorphic to R^n)
oh i see
deep mango
I still don't get members of Equiv class vs equiv class :(
Thing in set is not equal to the set
{1} is different from 1
icy forge
Ah.....
That's indeed the simplest way to have the difference
It's just that {1} 'represents' 1 or 'acts like' 1? Or can you make a trivial map doing so?
blazing pawn
it depends on the situation
icy forge
By that you mean there is no trivial map, or there could exist no such map
deep mango
But I do feel like the spirit of this question is to give a set of random crap and turn into a vector space using the axioms. Like the vector space generated by {😺, 🐕} over R, or something
icy forge
But I do feel like the spirit of this question is to give a set of random crap a...
Abstract objects in math sometimes do feel like that
deep mango
This is literally what homology feels like too
icy forge
homology has visual intuition
You mean counting holes or
blazing pawn
Yeah but like the addition represents doing something
devout nacelle
I still don't get members of Equiv class vs equiv class :(
Equivalence class of an element is an entire set of elements equivalent to your element under the relation, while the members are just elements of the set, and can be taken as representatives of the class in some cases
compact tartan
pick a well-ordering of R, consider a monotonic map from R to the ordinal that represents R's well ordering class. Translate vector space structure along that map
deep mango
I havent thought about this as anything but an abstract sum of embeddings
blazing pawn
Visually the way I think about it is like the simplices give you a skeleton, chains are paths along that skeleton sort of, cycles are closed paths
blazing pawn
Singular is just fucky simplicial
blazing pawn
like you just let your skeleton deform
Idk i dont have good intuition for that
the proof of their equivalence just sucks
swift sinew
<@!507116799284740106> you should be able to type .reopen
Isn’t that feature turned off, it looks like that .reopen period is where channels are put into hidden and on read-only
(sry for the ping)
carmine pike
does mniip manage @quasi jetty ?
compact tartan
@carmine pike after a channel is closed there's a grace period before it's reopened or hidden
during that period you can type .reopen
you can't type because the channel was hidden (the grace period ran out)
carmine pike
hm I thought we had basically finished but ig it was only after the grace period I realised the actual solution to the problem
dusty compass
I found a great anagram of Banach-Tarski : Banach-Tarski Banach-Tarski
I'm so funny
deep mango
sick burrow
The Group in Logic and the Methodology of Science administers a program leading to the degree of Ph.D. in Logic and the Methodology of Science. The Group is not a part of the Department of Mathematics or of the Department of Philosophy; rather, it is an independent program staffed by faculty members from Mathematics and Philosophy, along with several from Electrical Engineering and Computer Science. Students who want to pursue the Ph.D. in Logic and the Methodology of Science should apply directly to the Graduate Program in Logic and the Methodology of Science, rather than to Mathematics or Philosophy.
Students in the L&M program are expected to study both mathematics and philosophy, though they need not meet all the breadth requirements for a Ph.D. in either field. They must pass one examination in the foundations of mathematics, one examination in philosophy (Area I), and a third in either mathematics or philosophy. Although there are no graduate instructorships in Logic and Methodology of Science, students in this program may, if qualified, hold graduate student instructorships in the Department of Mathematics or in the Department of Philosophy.
Anyone know anything about this program at Berkeley?
sick burrow
like how hard
swift sinew
Read #❓how-to-get-help
strong surge
ok
broken arch
@peak pilot Continuing here as to not reactivate the help channel. Finding the particular and minimal homogeneous solution in this case is very much feasible thanks to the Extended Euclidean Algorithm, but how it manages to do that isn't all that trivial.
hollow ginkgo
Is there a name for the kind of structure that is like a monoid but does not have to be closed under its operation?
jovial ember
An operation is definitionally closed
jovial ember
And I think you won’t get a satisfactory theory if you don’t know a•b in S
Like
This only really makes sense when trying to look at sub-objects
I feel
Anyway idk
hollow ginkgo
This only really makes sense when trying to look at sub-objects
That happens to be exactly what I’m doing!
sick kite
wdym by graph
sick kite
i would say no, because if you mean x-y plane (R^2), then there are certain values of R^2 which aren't possible in R^1 so its not the same
neat lintel
The numberline is the graph made from the function f:R->point
vivid halo
Was about to say this 
neat lintel
because people define graphs of functions as (x,f(x))
you can generalize for multiple variables
ancient flame
leaden torrent
f: S^n -> C
deep mango
I do a little \vec thing
fathom swallowBOT
ryc
hollow ginkgo
isn't the number line technically a graph?
How?
sick burrow
Each point is a vertex and there are edges connecting adjacent points
deep mango
sick burrow
Oh huh I meant to say each integer point is a vertex but this way is even funnier tbh
neat lintel
Today I learned about a-Torsion functor. I am not exactly sure how it works because it was defined as:
$\Gamma_A(M) = \bigcup_{n \in \mathbb{N}} (0:_M A^n)$
fathom swallowBOT
mMahael
neat lintel
I never saw the notation and im trying to solve a problem which is
Show : $\Gamma_A(\Gamma_B(M)= \Gamma_{A+B}(M)$
fathom swallowBOT
mMahael
neat lintel
Anyone have any insights, I now know the notation had to do with annihilator of A^n , but isnt annihilator of A contain anhillator of A^n?
A is an ideal of the ring R and M is an R-Mod
meager sonnet
A is an ideal of the ring R and M is an R-Mod
You are far more likely to get an answer in #groups-rings-fields
neat lintel
You are far more likely to get an answer in <#496784958430380033>
thanks
meager sonnet
np, most higher level questions will definitely get buried here
neat lintel
Is there a way to get pinged whenever someone asks questions in the algebra channel? I want to get better at algebra
meager sonnet
there is
you already have the role assigned
uhh, I don't think people use it a ton, unfortunately
neat lintel
Its all fine );
waxen stump
neat lintel
Is there a channel we can talk about social aspects of school like dealing with profs and advisors? Some of the social navigation seems not obvious
surreal sapphire
you can try the discussion channels
neat lintel
Im undergrad and im expecting this might happen in grad school. What is the best reaction to give when a Prof acts like you are stupid or asks “why are you in math?”
During office hours while asking for help. Is it something I should take to heart or do I dismiss prof as an asshole?
I will have them again for topics differential topology as a prof next semester and I was thinking of asking for letter of rec until they made this comment
surreal sapphire
that ... sounds super unprofessional
static loom
math people are not exactly known for their communication skills, you might want to be sure that this is actually what they actually meant
swift sinew
Some profs are assholes. But there might be a chance they’re asking to make sure you’re asking yourself that question. Obviously he could very well still be being an asshole tho
neat lintel
Has me a little sad an anxious because I am doing fine in the class and getting 90% on problem sets. So I wonder if I truly am not doing enough or if they are an ass.
swift sinew
In that case, I’m not sure why he would’ve asked that or behaved that way. Did something prompt him?
neat lintel
Some profs are assholes. But there might be a chance they’re asking to make sure...
Thats exactly what ive been doing for the entire week after the office hours
steep mountain
i only have 1 question for you
do you like math and would you want to keep doing it?
(well thats 2 but anw)
deep mango
When did mero lose honorable 
static loom
When did mero lose honorable <:hmmCat:831508801646755850>
I never lost it
I left the server and just haven't asked for it back yet
neat lintel
I was discussing how I felt in the class because we were discussing cohomology/homology and to summarize I stated that a lot of the applications felt as if I need another class to have motivation for learning cohomology/homology. I also made a small statement “I am not very comfortable with complex functions” which they interpreted as I dont know what complex numbers are. What I meant was that I was not comfortable visualizing how complex polynomials act on open sets. Neither have I graphed a complex function, with hue maps. This is in reference to talking about local degrees.
neat lintel
i only have 1 question for you
do you like math and would you want to keep doin...
After thinking about the question my conclusion is that my reason for wanting to go to grad school is to experience math research. I enjoy doing math too. Other than that I have no reason. So Im not sure if I am making the right decision since my reason isnt concrete or good enough.
I dont have any specific areas of math Im interested in besides algebra and topology
simple raven
After thinking about the question my conclusion is that my reason for wanting to...
for me it's a good enough reason, you like maths and you want to do more, I don't know what else you could desire
curiosity is a good reason
neat lintel
My question is, do ai readress this to him/ be confrontational about what he said or do I act as if it hasnt happened even though it made me feel bad. For the goal of grad applications/rectifying relations.
simple raven
can you ask to someone else ?
steep mountain
ask what the others think "preferable someone who knows your math skills"
the professor could just be an asshole
neat lintel
Ok Ill ask another prof
simple raven
I think, if you can, try to ask others professors for letters of recommandations
swift sinew
Speaking of, are letters of rec from very accomplished employers even worth adding to an application for grad school?
leaden torrent
my guess would be "no unless you were somehow doing very advanced research with them that you can make public"
which as a pre-grad student would be... quite the feat
that said, thats uncommon in mathematics in general so maybe it wouldnt be as bad as my first impression thinks
it just seems dubious
but itd at least make your app unique?
swift sinew
That’s true. I’ll use that as a backup if I’m unable to find a sufficient number of quality academic letters of recommendation
zinc torrent
After thinking about the question my conclusion is that my reason for wanting to...
I’m in the same boat. I love Math, so I will try my best to be paid to study and do math research. I figure that’s all the reason you need.
dawn osprey
Hey late workers-- what's your strategy for dinner when you are working late in the office?
About once a week there's usually a day where I have to stay late to finish up work. I often don't know in advance when it will be, so I can't just bring dinner from home that day. It's becoming a problem because it means that I won't be able to get home and eat until 8 or 9, and then it messes up the rest of my schedule... Do yall have any clever easy-dinners you can do at the office?
I have access to a microwave and a fridge. My first thought is to have canned dinners on hand but, then I need to have a can opener at work, and plates n stuff handy...
Some ideas I'm looking for are dinner options that can be stored long-term at the office without spoiling. I want to stray away from junk food because a) its unhealthy and b) I already have that kind of stuff there (slim jims and crackers, not ideal but they get me by)
deep mango
I just.... Buy dinner when that happens
Like something relatively good
I understand that this isnt always an option vut those are my priority days for buying takeout.
dawn osprey
yeah i'm not entirely against eating out i suppose. but that's a 20 minute excursion minimum, and to be honest its pretty pricey around here. i can justify it when i go with coworkers but its harder when its just me ya know?
i'll look into more options close by though, i haven't exactly explored this area in totality
The campus food options are practically nonexistent, most of the non-meal-plan stuff closes at 3pm 🤦♂️
vivid halo
buy some soylent I guess
alternatively buy some protein powder + trail mix or something
if it's just one meal once a week or so you can just stock up on whatever supplements that can get you through that
protein bars are good too
I'd personally go with protein powder and nuts, high protein and high fat
dawn osprey
Unfortunately my victims roommates don't exist; I live in a studio and my office mates never show up lol
neat lintel
in the following letter series some of the letters are missing a_ab_ab_bc_ba_cab , i am confused in this
anyone pls?
dawn osprey
if it's just one meal once a week or so you can just stock up on whatever supple...
i had some protein bars handy but esp when im like, really hungry, they just make me sick :((
Yeah okay, I'll see if I can find a combination of healthy snacks that will do. Thanks!
deep mango
buy some soylent I guess
proper tide
upper patrol
any reason why i have a "role that does have nothing" role
who went through the effort of giving me one
jovial ember
You got it when you reacted to a post
About the introduction of roles or something
It said like “click this for a role that does nothing”
upper patrol
oh
im deeply disturbed that former me clicked on that reaction for a worthless role
mint patio
ask your local moderator to take it off for you when they are active in chat
cold needle
or click on the react again
neat lintel
Hey, anyone here that know how to use Wolfram Notebook?
errant junco
Does anyone know how to get neat colors for the mandelbrot set? I am using even distribution of hues, but it doesn't seem to be working too well
nimble shuttle
wdym by even distribution of hues
iirc mandelbrot sets are often colored based on how "fast" the sequence associated to a point converges
errant junco
I am using HueSaturationBrightness color mode. In the code, I set the maximum number of iterations. The code then uses that number, to split all hues (0-255, well 230 since I don't want it to loop back to red) into n-equally sized slots. It then saves these colors and uses them after a point left the stable range
nimble shuttle
maybe adjust the weighting somehow
so that the color dropoff isn't so sharp
also personally I think it looks better with a black background so you could change that if you want
errant junco
maybe adjust the weighting somehow
I just did that and thank you for the suggestion.
I used a variable to determine the speed of the change independent of iterations.
So v and m
v[1] = v[0] * m
v[2] =v[1] * m
...
v[n] = v[n-1] * m
v[0] =1
m < 1
nimble shuttle
yah
errant junco
added a solid red background. I can't get black theme to work without hardcoding the background
Final product:
Now I just need to add a zoom function and I will be done
compact tartan
@errant junco try coloring the inside based on how close the point got to |z|=2 at any point during the iteration
(a proxy metric for being close to the outside of the set)
also these octahedrons are kinda sus
are you detecting divergence with a square (|x| > 2, |y| > 2) ?
the standard way is to use a circle
compact tartan
odd-looking
vivid halo
Yea they are the artifact of a nonstandard coloring procedure
If you do the standard thing you shouldn’t see these
compact tartan
yup
here's my take https://glslsandbox.com/e#76482.0
using a square indeed gives it a more angular look
nimble shuttle
looks cool still
compact tartan
a website where you can just type GLSL fragment shader code and it renders in your browser
neat lintel
I have a social inquiry. With math mostly whenever I want to ask a question I feel hesitant because ive been told before that I should just read something or google it. This is true for most of my online experience and my experience with some professors. Ive also been told that math is supposed to be a social activity. Is this not what is meant by that? Is it only a social activity under the premise that you have complete mastery over assumed mastery of prerequisites? Sometimes I wonder why do I even bother asking others for help, profs/students included. Am I taking the correct approach or is it too pessimistic?
limber thunder
well, most people are busy and thus don't have the time to walk people through a concept or a given problem. As much as people love maths that's why tutoring and teaching positions are paid -- they take an important portion of your time and energy
math is a social activity in the sense of studying with your peers and discussing stuff from time to time
as well as the whole academic ecosystem e.g. seminars, talks
neat lintel
Ok thanks for clarifying the statement I understand better now
pale orchid
the issue might be mixing up open-ended discussion with asking for simple definitions and explanations
not everyone likes teaching
light needle
Well I like teaching but i prefer questions like "why are we defining X like Y" , and not questions that someone could easily just have googled "what is the definition of X". but it depends on context ig, like I want to see some effort on part ofthe student
leaden torrent
ops wrong channel
uncut totem
hey guys should i learn high school algebra before college algebra
high school algebra = algebra 1 and 2
pls anyone??
but see this
guy
he dosent even say to learn high school algebra before college algebra
wild lantern
These things don't always have the same name from institution to institution.
uncut totem
?
wild lantern
Like, some places college algebra is the same thing as thr kind of algebra you'd learn in a hs other places it might cover a little more.
wild lantern
Where I am from you would probably do something like algebra 1, geometry, algebra 2, trig, precalc then start calculus.
wild lantern
algebra 3 is known as college algebra
Not everywhere.
uncut totem
i am currently learning algebra 1
uncut totem
Where I am from you would probably do something like algebra 1, geometry, algebr...
literally the correct order i am doing
wild lantern
So then keep doing it?
wild lantern
What about it?
uncut totem
like when do we learn it
wild lantern
It depends on what it covers.
uncut totem
oh ok
neat lintel
Differs from place to place
uncut totem
Where I am from you would probably do something like algebra 1, geometry, algebr...
for now i will just be doin this
wild lantern
That is a good path to start with.
wild lantern
Where I am from you do something like calc 1-3, differential equations, linear algebra.
uncut totem
ok
wild lantern
Then you might do an intro to proofs course then abstract algebra courses, real analysis courses, a complex analysis and a more rigorous linear algebra course. Also some electives along the way
uncut totem
so after calulus 1 i should do calc 2?
wild lantern
But all that is far off from where you are at.
uncut totem
bruh i am in 8th
wild lantern
Yeah, calc 2 is after calc 1.
uncut totem
ok so anything is between calc 1 and 2?
wild lantern
Not usually.
uncut totem
i am desperate to learn calc
uncut totem
Some people will do some physics and stuff too.
ye i am also doin that
do u know the order to learn physics
?
wild lantern
Where I am from it's just called physics 1-3
wild lantern
The first is mainly about like mechanics, dynamics newtons law stuff like that.
The second was electricity and magnetism.
uncut totem
ok so what are all the contents in phy 1
wild lantern
Its been a couple yrs and I'm not physics expert.
uncut totem
man its ok
wild lantern
If you google the openstax physics textbook that was what we used
uncut totem
but i know newtons laws are part of it
wild lantern
It comes in three volumes iirc
wild lantern
Our physics 1 used volume 1 and so on.
wild lantern
Yep.
uncut totem
lol ok
wild lantern
Also, sometimes people do versions of physics that don't require calc.
wild lantern
The calc versi9n here required calc 2 iirc
uncut totem
and i am doin algebra 1 in khan academy
wait so what is physics called in openstax
like which shall i start with
there are a ton
wild lantern
It's the "university physics" one
wild lantern
No, I'm saying it requires some calculus
uncut totem
ohh ok
wild lantern
Like, people take it along with calc 2.
wild lantern
I think even the "college physics" type books and courses require trig
Idk about hs physics tho
teal condor
need help guys
wild lantern
I do not know I've never done a hs physics course
Those openstax textbooks aren't the greatest, but they are free
wild lantern
So you can poke around in them and see if they're too advanced for you or not
wild lantern
AP courses are like college prep courses some kids do in hs I think.
wild lantern
No problem.
neat lintel
advanced placement
errant junco
are you detecting divergence with a square (|x| > 2, |y| > 2) ?
Yeah, I was squares (as you said |x| < 2 && |y| < 2). Now I swapped to circles and it does look different
errant junco
I was messing around with it and came to this fractal. Is it just a julia set, or what? And I can't find the reason why this formed instead of the normal mandelbrot set. Can someone give me the formula to generate this one, so I can see where I went wrong?
so apparently, this is z^4 + c
compact tartan
julia sets are something else
distant vortex
What are you using to make these? 
i want to try too
compact tartan
in a mandelbrot set you're iterating $f_c(z) = z^2 + c$, and considering the behavior of $\lim_n f_c^n(0)$ as $c \in \bC$
fathom swallowBOT
mniip
distant vortex
I meant like what software 
pale orchid
for the worst experience, use several nested loops in python
compact tartan
in the julia set of this process you're fixing a $c \in \bC$ and considering behavior of $\lim_n f_c^n(z_0)$ as $z_0 \in \bC$
fathom swallowBOT
mniip
compact tartan
I meant like what software <:kek:817820192579125248>
wasn't talking to you
distant vortex
Oh, lol, I didn't see they had asked a q 👍
fathom swallowBOT
You have switched to the white colourscheme.
compact tartan
$$\begin{tikzpicture}
\pgfdeclarefunctionalshading{test}{\pgfpoint{-25bp}{-25bp}}{\pgfpoint{25bp}{25bp}}{
}{
12 div exch 12 div exch
2 copy
4 copy mul 2 mul add exch 4 2 roll dup mul exch dup mul exch sub add exch
4 copy mul 2 mul add exch 4 2 roll dup mul exch dup mul exch sub add exch
4 copy mul 2 mul add exch 4 2 roll dup mul exch dup mul exch sub add exch
4 copy mul 2 mul add exch 4 2 roll dup mul exch dup mul exch sub add exch
4 copy mul 2 mul add exch 4 2 roll dup mul exch dup mul exch sub add exch
4 copy mul 2 mul add exch 4 2 roll dup mul exch dup mul exch sub add exch
4 copy mul 2 mul add exch 4 2 roll dup mul exch dup mul exch sub add exch
4 copy mul 2 mul add exch 4 2 roll dup mul exch dup mul exch sub add exch
4 copy mul 2 mul add exch 4 2 roll dup mul exch dup mul exch sub add exch
4 copy mul 2 mul add exch 4 2 roll dup mul exch dup mul exch sub add exch
dup mul exch dup mul add 4 lt {
100 mul sin 1 add 2 div exch 100 mul sin 1 add 2 div 1 3 2 roll
} {
100 mul sin 1 add 2 div exch 100 mul sin 1 add 2 div 0
} ifelse
}
\shade[shading=test] (0,0) rectangle (3,3);
\end{tikzpicture}$$
fathom swallowBOT
mniip
compact tartan
I wonder what happens if you compute the derivative of f^n when you stop
compact tartan
not much interesting
vivid halo
Oh wow nice 
compact tartan
I wonder if there's a way to rewrite mandelbrot in a way where it doesn't completely die due to floating point precision when I keep iterating outside the circle
toxic schooner
pale orchid
why horrendous?
compact tartan
ok so like the problem is that in $f(z) = z^2 + c$, if it becomes that $|f^n(0)| > 2$, then the function starts growing as $|f^n(0)| \sim 2^{2^n}$
fathom swallowBOT
mniip
compact tartan
even if we do the computation in some sort of floating point, the width of the exponent runs out really quickly
we can do the computation in polar, but that doesn't really change anything except we're now actually storing $|f^n(0)|$ as one of the values
fathom swallowBOT
mniip
compact tartan
so what if instead of storing $|f^n(0)|$ we stored $\log |f^n(0)|$
fathom swallowBOT
mniip
compact tartan
so what we can do is basically represent a complex number by its complex Log
pale orchid
sounds reasonable