The h Bar

9:40 AM

I am trying to understand the Fourier transform, at least in a condensed matter context. Consider $f(x) = 1/\sqrt{N} \sum_k e^{ikx} f(k)$ where $f(x)$ has the lattice periodicity. Then I usually say that $k$ is summed over the first Brillouin zone. Is this an intrinsic definition of the Fourier transform, or only a result from the fact that summing over more only yields double counting in the expression that I consider?

ACuriousMind

11:35 AM

@B.Brekke it's part of Bloch's theorem that summing over the Brillouin zone suffices

a priori, the Fourier transform alone is not restricted to the zone

B. Brekke

Thanks, that is very helpful. I am dealing with multiple periodicities and have to be careful in what to sum over and what not to.

JingleBells

4:54 PM

A wild JingleBells has joined the server.

I'm learning some Machine Learning math (linear algebra) and I have some difficulty understanding this thing:

I understand that when we do the matrix multiplication it cleverly turns out that x1 multiplies with the first column, x2 with the second column, etc. but so what, I don't understand why the text says that the solution to the system of linear equations is this simple b = ... thing

What I'd regard as a solution would be 4 values for the 4 xs so that when you plug them into the thing, the left side equals the right side

I don't understand how the text concluded that [42, 8, 0, 0]T is the solution

I'm very advanced in maths I know 😎

I'm giving 10 BTC to whoever helps me

@ACuriousMind ^

ACuriousMind

@JingleBells You have $c_1 = (1,0)^T, c_2 = (0,1)^T$, so eq. (2.39) is a special case of the equation $\sum_i x_ic_i = b$ in the previous paragraph with $x_1 = 42, x_2 = 8, x_3 = 0, x_4 = 0$.

JingleBells

5:13 PM

gotchya, so x1*c1 = b, x2*c2 = b, etc. but I don't understand how we go from that, to knowing what the xs must be so they're considered a solution

so I tried plugging stuff, and I ended up with:

x1*c1 = b, x1*[1, 0]T = b, [x1, 0]T = [42, 8]T

^ I don't know what to do with this

oh yeah

actually, I don't understand why ∑ixici=b would spit out the correct x values

ACuriousMind

I don't think you understood what I meant

Any correct solution (where a solution is a tuple $x_1, x_2, x_3, x_4$) fulfills $\sum_i x_i c_i = b$

you know what the $c_i$ are, and you know what $b$ is

now, eq. (2.39) suggestively writes $b$ in terms of the $c_i$

the rightmost part of eq. (2.39) in particular is just $\sum_i x_i c_i$ for $x_1 = 42, x_2 = 8, x_3 = 0, x_4 = 0$

therefore, that tuple of $x_i$ is a solution

JingleBells

Gotchya, I don't understand though how doing x1*c1 = b for example will get us to a correct value for x1 so that when plugged in both linear equations together with the rest of (still unknown) xs will give us b

ACuriousMind

who's "doing x1*c1 = b"?

JingleBells

@ACuriousMind this?

ohh

hmm

ACuriousMind

you know that $\sum_i$ is a sum, right?

$\sum_i x_i c_i$ is just "written down in formulae" what the result of the matrix multiplication is/how appllying a matrix to a vector works

JingleBells

5:25 PM

@ACuriousMind I do, it's just that my mind decided to ignore it for whatever reason

lemme think about it a bit, I think i'll get it

JingleBells

5:40 PM

I decided to expand on the sum thing and I ended up with:
|x1 + 8x3 - 4x4 | = |42|
|x2 + 2x3 + 12x4| = |8 |
Which is what you end up when you do matrix multiplication on the eq 2.39, but I'm not sure how convert that into the system of 2 linear equations which I'm used to working with

is x1 + 8x3 - 4x4 = 42 the first linear equation, and there fore x2 + 2x3 + 12x4 = 8 the second linear equation?

ACuriousMind

nothing in your screenshot above is about solving any equations

they're just observing that $x_1 = 42, x_2 = 8, x_3 = 0, x_4 = 0$ is one possible solution for these two equations.

JingleBells

|x1 + 8x3 - 4x4 | = |42|
|x2 + 2x3 + 12x4| = |8 |
So on the left side there's a matrix 2x1 and on the right side again a matrix 2x1 (2 rows and 1 column) so I don't know how to take these 2 matrices and convert them into the 2 linear equations to which the x1 = 42, x2 = 8, x3 = 0, x4 = 0 is one solution

so if I do the above as $A = B, AB^(-1)$

omg im so bad at latex

$A = B$ -> $AB^{-1} = 0$

I'm sorry if I'm being confusing, it's probably because I'm confused, I'm trying my best to explain my misunderstanding

Right now I'm trying to figure out how to take the $\sum x_i\bf c_i = \bf b$ and convert it to the 2 linear equations

and I've reached the point that I'm explaining above

$\begin{bmatrix}
x_1 + 8x_3 - 4x_4 \\
x_2 + 2x_3 + 12x_4
\end{bmatrix}$ = $\begin{bmatrix}
42 \\
8
\end{bmatrix}$

and I don't know how to convert that ^ to:

$x_1 + 8x_3 - 4x_4 = 42$
$x_2 + 2x_3 + 12x_4 = 8$
Which are supposedly the 2 linear equations the eq. (2.39) represents

So once I know how the $\sum x_i\bf c_i = \bf b$ converts to the 2 linear equations, I'll finally understand why solving the sum thing spits out correct x values

Correction: I meant eq. (2.38) instead of eq. (2.39)

OH!

I got it, everything

@ACuriousMind Thanks for help, how's life?

JingleBells

6:28 PM

Though I still don't understand how to convert that:
$\begin{bmatrix} x_1 + 8x_3 - 4x_4 \\ x_2 + 2x_3 + 12x_4 \end{bmatrix} = \begin{bmatrix} 42 \\ 8 \end{bmatrix}$
to:
$x_1 + 8x_3 - 4x_4 = 42$
$x_2 + 2x_3 + 12x_4 = 8$

ACuriousMind

@JingleBells I don't really know what you mean there - the first way of writing it is the same as the individual equations in your second way pretty much by definition

there aren't any "steps" to the conversion or anything, that's just what it means

@JingleBells overall nice but currently rather annoying because I've got covid :P

JingleBells

@ACuriousMind gotchya

@ACuriousMind ah that's bad, I got a vaccine and stuff but yeah, covid is bad

@ACuriousMind u still working at the company as a software engineer?

ACuriousMind

I'm fully vaccinated and the symptoms are "mild", but "mild" still means I've been feeling sick for the last three days (my flatmate is on day 6, I think) :P

@JingleBells yes, still essentially the same job except they call me a senior engineer now

JingleBells

@ACuriousMind yeah that's happening all over the world, sad that it has become so common

@ACuriousMind senior engineer = big money bank

ACuriousMind

it does look nice on the pay check, yes ;)

JingleBells

6:38 PM

I'm making Roblox games to make money for now and learning machine learning on the side

ACuriousMind

have you actually made any money with that? all I've heard is that getting that to be lucrative is pretty rare

JingleBells

I'm getting there, slowly, but I know I'm super close to making it. I have an innovative solution that will allow me to beat the competition by creating games that kids wanna play much faster than the competition

I'm hustling, every day all day

@ACuriousMind Some people make money from Roblox, others don't, it depends on what game you're making and how you're marketing it, and after spending a year and half hustling and trying to figure it out, I believe I'm pretty close

but I'm not doing it only for the money, if I'm gonna be doing something, I wanna bring something new to the table, something much better than what already exists and push the limits, so I have a way of doing that and currently working on it

@ACuriousMind Do you know about DALL-E?

dope stuff

just describe what you want and it draws it, amazing

I can get royalty-free assets for my games for free (if DALL-E goes public for free)

ACuriousMind

I've seen some of its stuff

I think like the text-based GPT-3 it's a really interesting thing that still falls short of being actually useful because you can very often still tell that its output is uncanny/artificial

JingleBells

yeah that's true but I assume for trivial stuff like a 2d coin asset, it'll do it pretty well

ACuriousMind

AI Dungeon is a really nice demonstration of that - it's fun to play around with, and the AI can be surprisingly coherent, but it's not coherent consistently enough to actually generate anything someone would play for any reason other than the novelty

JingleBells

6:53 PM

how come we haven't seen deep fakes of famous people saying bad stuff?

maybe in a few years

if deep fakes become undistinguishable from the real videos, if a famous person says something bad, they can just say "nah that's deep fake" lol

Will Smith slapping Chris Rock was a deep fake btw

Sir Cumference

8:36 PM

@JingleBells Ey you're back

uhoh

9:59 PM

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uhoh

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I've added a +100 bounty

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