The h Bar

@Obliv $\frac1{1-x}=1+x+x^2+x^3+\cdots$ and then substitute $x=e^{-\beta\varepsilon}$

Oh thank u, whats the name of this type of series btw?

nvm i got it, it's a geometric series

1:17 AM

I hope this is the worst we'll get for partition functions

geometric series that is

1:47 AM

for the high temperature limit $k_BT\gg \varepsilon$, how come $\beta \varepsilon \ll 1$ means $e^{\beta \varepsilon}\approx 1+\beta\varepsilon$?

3:12 AM

@Obliv For small $|x|$, we get $\ln(1+x)=x$. There are various ways to show that limit. Eg, integrate that geometric series $\frac1{1-x}=1+x+x^2+x^3+\cdots$

Ohh right

It's a very handy limit. Similar to $\lim_{x\to 0} \sin(x) = x$

Also see en.wikipedia.org/wiki/Bernoulli%27s_inequality

thank you. I see it all the time I just forget because I don't know the name of it. I guess just first order taylor approximation?

like $e^x = 1 + \frac{x}{1!}+\frac{x^2}{2!}+...$

Yep

there's also another one I see when we pull out terms like $\sqrt{x^2+a^2}=x\sqrt{1+\frac{a^2}{x^2}}$ where if $x\gg a$ or something then this approximates to something..

perhaps I wrote it wrong, i'll look it up lol

it's $(1+x)^a \approx 1+ax$

i guess for sufficiently small $x$ in that situation

3:21 AM

Right. $(1+x/2)^2 = (1 + x + x^2/4)$ So for small $x$ we can ignore the $x^2$ term.

ooh

i'm guessing this only works for exponents $\geq 2$?

Hence $\sqrt{1+x} \approx 1 + x/2$

nvm with higher exponents means more terms to ignore as well..

@Obliv no, this is a general binomial theorem and will work for much more general cases

Veritasium had a video about how Newton changed the game about finding the value of $\pi$, and in it there is a lot of generalisation of the binomial theorem right there.

Also, $(1+x)(1-x)=1-x^2$, so for small $x$, we get $1/(1+x) \approx 1-x$. Which is related to that geometric series.

There are a ton of arctan identities that are handy for computing $\pi$

See machination.eclipse.co.uk

3:31 AM

@PM2Ring my brain is trying to remember what those arctan-identity ways of computing pi is called and failing

oh. machin-ation

i get it

what i personally got interested a few days ago was how people computed tables of logarithms

which mostly seems to come down to "get really good at computing square roots"

But there are all sorts of fun ways to compute pi. Depending on how many digits you want, and what arithmetic functions you're prepared to use. If you can do square roots, you can use AGM (arithmetic-geometric mean) algorithms that converge quadratically.

AGM is crazy cool

And you can use AGM for log, too.

i do like how closely related AGM is to the pendulum

A: Solution to pendulum differential equation

Traditionally, people computed log tables with fairly simple arithmetic though. Creating 7 figure log tables by hand was a lot of work, and many early books of log & trig tables had a few typos / arithmetic errors. People spent years doing this stuff.

@Semiclassical Me too. :)

5

This answer continues on from the answer by wong tom and uses the same notation. As Tom said, the equation of motion of the simple undamped pendulum with maximum swing angle $\theta_m$ leads to this integral: $$t=\sqrt{\frac{l}{g}}{\Large\int_{0}^{\phi}}\frac{ds}{\sqrt{1-\sin^{2}(\theta_{m}/2) \s...

3:40 AM

@PM2Ring i'm thinking of Briggs method as described here: matheducators.stackexchange.com/a/7607/1940

Some golfed pi code: codegolf.stackexchange.com/a/265731/46655 & codegolf.stackexchange.com/a/265749/46655

@Semiclassical Ah, right. That sqrt method also been used on Puzzling a few times.

"Briggs used 54 square roots, accurate to 32 dp.!"

that's kinda amazing tbh

There's a more efficient way, using Jacobi theta functions & the AGM. But it also uses a bit of range reduction, so you don't have to compute lots of terms in the theta series.

Those recursive sqrt things (and AGMs) can often recycle previous values for initial approximations in Newton's method, so they aren't as bad as doing each sqrt from scratch.

i forgot were log tables initially used for astronomical data?

or maybe im thinking of the reaaally small angles in trig functions for said data idk

Log tables were used for everything that required multiplication, division, powers or roots.

3:51 AM

napier bones ;0

it really was revolutionary

Before log tables, there were tricks involving trig identities that were used to do multiplication via addition.

@PM2Ring i never remember the spelling of that method

something phaeresis

Prosthaphaeresis

Prosthaphaeresis (from the Greek προσθαφαίρεσις) was an algorithm used in the late 16th century and early 17th century for approximate multiplication and division using formulas from trigonometry. For the 25 years preceding the invention of the logarithm in 1614, it was the only known generally applicable way of approximating products quickly. Its name comes from the Greek prosthen (πρόσθεν) meaning before and aphaeresis (ἀφαίρεσις), meaning taking away or subtraction. In ancient times the term was used to mean a reduction to bring the apparent place of a moving point or planet to the mean place...

And after logs were invented, they made log trig tables, so you could just look up the log of a sin or tan directly.

somewhat related chat in the math room about a similar inquiry @Semiclassical

I found an old pdf I linked later on and it still works thankfully :)

3:55 AM

Here's a demo of that AGM log algorithm. Which I got from an article by Brent.

sagecell.sagemath.org/…

i occasionally get bored and implement the AGM pi algorithm in Excel

tho the precision runs out fast

@Semiclassical I will hopefully find some time to read more about this. It looks very interesting

I have a AGM pi version that uses fixed point arithmetic, with no divisions, only bit shifts. And of course I had to write fixed point sqrt code with no division to go with it. ;)

en.m.wikipedia.org/wiki/Pentagramma_mirificum

why do we love shapes so much ;P

sagecell.sagemath.org/…

4:04 AM

whats "quake style"

Fast inverse square root

Fast inverse square root, sometimes referred to as Fast InvSqrt() or by the hexadecimal constant 0x5F3759DF, is an algorithm that estimates 1 x {\textstyle {\frac {1}{\sqrt {x}}}} , the reciprocal (or multiplicative inverse) of the square root of a 32-bit floating-point number x {\displaystyle x} in IEEE 754 floating-point format. The algorithm is best known for its implementation in 1999 in Quake III Arena, a first-person shooter...

^ That

i had a feeling it was related to one of my favorite games of all time ;)

to this day, no game/engine has felt as responsive and amazing as those oldies

same with the half life 2 engine i think its the same one just modified

doom vs quake is an interesting comparison in terms of historical legacy

Here's something visual. It draws various dodecahedra, depending on a parameter, 0 < h < 1. Actually, h can go outside that range, and you get strange non-convex shapes.

sagecell.sagemath.org/…

4:09 AM

in a wider cultural sense i think it's doom, but in terms of actual technology it's definitely quake

90,525,801,730 Cannon Balls - Numberphile

@PM2Ring that reminds me of that numberphile video about the cannonball problem

not really that relevant though lol

►

Cannonball problem

In the mathematics of figurate numbers, the cannonball problem asks which numbers are both square and square pyramidal. The problem can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be arranged into a square pyramid. Equivalently, which squares can be represented as the sum of consecutive squares, starting from 1. == Formulation as a Diophantine equation == When cannonballs are stacked within a square frame, the number of balls is a square pyramidal number; Thomas Harriot gave a formula for this number around 1587, answering a question...

@Semiclassical a bit before my time unfortunately. I was born in '97 so I only found out about quake via the steam launch of quake live but I still think that and quake pro mod/cpma are some of the most fun shooters. I heard doom was great though

@PM2Ring yes so the man in the video extends this problem to search for what other shape bases does this work for and he found some large number it worked for

31,265-agon lol

Diophantine equations can be fun. Or very frustrating. Even quadratic ones can be hard, and once you go beyond the 2nd degree, they can be insanely hard. Sometimes you can brute-force it & look for patterns. But that's not much use if there are no smallish solutions. Or no solutions at all.

Here's a "fun" quadratic one. puzzling.stackexchange.com/q/99377/36040 I eventually found an efficient algorithm to generate all solutions. But I was too burned-out by then, and didn't have the energy to convert my results to LaTeX. And now I only vaguely remember the details... but I do still have my notes, on paper, somewhere...

6:18 AM

does anyone know how to make such expressions look nicer in latex :P

a long cases expression...

Becoming a funded independent researcher seems like the dream

Sanjana

6:36 AM

@SillyGoose After making this note, are you going to put it up on ArXiv or some webpage?

@Sanjana for this note i don't think so--maybe in a few years it will be part of a paper or something :P

is there nicer formatting on there?

Sanjana

no idea

@SillyGoose Is this related to the Fubini-Sturdy stuff you were doing?

yes this is a computation that i needed to operationally know the fubini-study measure for

i was computing $S = 1 - \text{Tr}\left(\int_{\mathbb{C}P^n} d \Omega_n \rho^2\right)$ where $\rho$ is a time-evolved reduced density matrix with some $\mathbb{C}P^n$ dependence

the answer turns out to be nice and simple >:D which makes me wonder if there is a simpler way to do the computation

but i tried to ask on stack about coordinate free/other coordintizations of $\mathbb{C}P^n$ expectation values to no avail

fqq

7:32 AM

@SillyGoose i think it'll look nicer if you just don't use cases and have separate equations

@SillyGoose your cases are about $i=1$ and $i\neq1$, but some of the expressions themselves don't depend upon $i$ in the correct way. Must be some typoes. Also, these seem to be complete angular integrals. If they are, you can just use standard Gaußian integrals to extract the results directly.

8:12 AM

@naturallyInconsistent the $i$ dependency comes from the fact that $n_i$ when $i = 2$ is different from $n_i$ for all other $i$. (i did mistype $i = 1$ instead of $i = 2$ though)

and these $n_i$ are what is in the integrand

@fqq hmm okay thanks i might try this out

10:22 AM

hi

which multiverse argument do u find more realistic out of the quantum mechanics one and the general relativity one

John Rennie

10:46 AM

Is there a GR multiverse?

11:27 AM

@JohnRennie yes it's in this video youtu.be/6akmv1bsz1M?si=ouZuq282JH6jlkkg

it comes from continuing Penrose diagrams

John Rennie

What Veritasium isn't telling you is that the Kerr geometry is (probably) unstable to perturbations and cannot exist outside of a mathematician's head. So the Penrose diagrams showing multiple universes linked through the Kerr geometries don't actually exist.

The same applies to the Reissner-Nordström geometry.

Veritasium said (and I think the guest physicists too) that it's theoretical but not impossible

@JohnRennie oh u r saying the same thing

this is why i asked "which multiverse argument do u find relatively more realistic between the QM one and the GR one?" @JohnRennie

John Rennie

I'm not sure if it has been proven that the Kerr and RN geometries are unstable, but everyone believes that they are.

So unless everyone is wrong the GR multiverses are exceedingly unlikely to exist.

On the other hand eternal and chaotic inflation seem plausible.

oh

chaotic inflation could produce chaotically different realities in the future i guess

Veritasium also said that black holes were thought of as purely mathematical solutions a 100 years ago

so these multiverses could be a prediction like black holes were

@JohnRennie do you mean in the sense that you would have different spacetimes embedded into something else ("the multiverse")?

I find calling maximally extended spacetimes "multiverses" a bit strange - it's still one spacetime, just parts of it are causally disconnected

11:43 AM

i find it to be within the bounds of what we mean by multiverse in daily life

if that's enough to be a "multiverse", then if you accept that inflation may have causally separated patches of the early universe, it's actually not that far fetched that we live in a multiverse

you don't need white holes or weird Penrose diagrams to get causal disconnectedness

oh...

i just love the portal aspect of those multiverses

all of string theory, QM, and GR give a multiverse in their own different way

this means that there is probably a multiverse out there

in QM, we have decohered universes. in GR, causally disconnected universes. and in string theory, a landscape

the point of the string landscape is not that all the points in the landscape exist simultaneously

what do u mean

the string landscape is not a "multiverse" of simultaneously existing distinct universes

that's not the point of it at all

it's just the "landscape of choices" you have for possible universes resulting from different compactification choices

and ideally someone would find a dynamic for the "real universe" that explains why the universe would end up as the one we're in

11:52 AM

oh

i interpreted it naively that all vacua were embedded in a bigger space and that explained fine tuning

when people say "they all exist", do they mean they exist at different points of time?

and which multiverse reasoning do u find the most realistic out of these three?

Relativisticcucumber

12:57 PM

HOOOOOOOOONK

user 70432

1:35 PM

h o n k

2:18 PM

H O N K ~

@SillyGoose Oh, that is an important mistake to catch. What is the standard result? What are $\mathrm d\Omega_n$? Usually that has trigonometric functions in there, but it seems to be missing. Are you sure $\theta_1$ does not have a wider integration region? In 3D polar we have an $\int_0^\pi$ and a $\int_0^{2\pi}$ to keep track of.

2:59 PM

Like, in the usual 3D polar $\int\mathrm d\Omega=\int_0^{2\pi}\int_0^\pi\sin\vartheta\,\mathrm d\vartheta\mathrm d\varphi$

3:26 PM

Maxwells equations are highly underrated compared to Einsteins Field Equations

Sir Cumference

3:52 PM

@DIRAC1930 I've never heard of anyone saying Maxwell's equations are underrated lol

Plenty of people use them all the time :P

4:19 PM

It seems though that alot of people who are interested in QFT seem to want to get past QED as quickly as possible to move onto QCD or something so Maxwells equations get relagated

@DIRAC1930 what do you think all the quantum optics people are doing?

I'm willing to bet there's more people "doing QED" in a broad sense than doing QCD

Yes, but amongst those interested in fundamental theory

4:36 PM

In another life, I wish I could have done quantum optics ngl

A couple years ago I almost had the chance to do a PhD in that field but funding fell through

5:15 PM

Is there anything broader than QFT? 💀

Slereah

quantum theory :p

My man is using set theory at its deepest level

For what I've seen, QFT books seems the messiest of all

ST books are so tidy in comparison :P

the QFT books are messy because they're trying to get to QED/the Standard Model/cross section computations in too short a time

string theory books have no such constraint to have to reach actually useful results :P

@ACuriousMind they sure have other constraints :P

ahahaha

5:24 PM

Oh my god I got ACM laughing

you have to imagine this as a very pained laugh :P

I wanted to ask but I decided not to give away my lack of understanding of basic human emotions on chat

Like that time Qmechanic replied with "Ha-Ha" and I'm still haunted by the thought he was being sarcastic

oh I'm human now?

You were programmed to appear human and therefore display human-like emotions

>:|

5:29 PM

My biggest gripe with QFT books is that they are too heavily reliant on literature. I mean, that's how it should be in science but QFT books push it to the limit

Every two lines there is a paper that proved something in a very specific setting

That's probably because as you say they want to get you into the business of Feynman diagrams quickly

> For a one-dimensional compact manifold, two topologies are possible - corresponding to the closed and open strings

Aren't all 1D compact manifolds homeomorphic to $S^1$?

they forgot "with boundary"

I don't think I've read a QFT book that I would recommend

Hopefully going through this pictures.abebooks.com/inventory/30905879986.jpg will change my mind

@DIRAC1930 probably only Weinberg

Weinberg needs to learn how to write equations properly

That was God's nerf to that guy

5:42 PM

His historical introduction is fantastic in volume 1 however

He made sure to express his resentment towards Dirac though :P

7:35 PM

@DIRAC1930 XD

Boo scattering cross sections

@naturallyInconsistent The integral is not a spherical integration. $dOmega$ is the Fubini-Study measure in local octant coordinates. It is proportional to $\prod_k \cos\theta_k (\sin \theta_k)^{2k-1}d\theta_k d\nu_k$. The $d\theta$ is my shorthand for integrating over all theta variables. It indeed only goes from $0$ to $\pi/2$. I think this is part of why this coordination is called octant. I immediately integrated over the other variables $\nu_k$ because the integrand is independent of them

Say I know 10% of some atom is in the excited state $s_1$ and I know the partition function is $9(e^{0})+5(e^{-\beta \varepsilon})$

9-fold degenerate ground state and 5-fold degenerate first excited state

I want to find the temperature, given the energy of the first excited state so I know it's $0.1 = \frac{1}{9+5\exp(-\beta\varepsilon)}(...)$ idk what to multiply

like whether I should use only 1 of the 5 excited state boltzmann factors

7:53 PM

I wonder if any of the qft books here physics.stackexchange.com/questions/673224/… would be good

nvm found the solution supposed to do $5\exp(-\beta\varepsilon)$ which makes sense cuz there are 5 available states to occupy and that's what the probability is meant to respresent