vacuumbubbles

vacuumbubbles, 14 days ago to random

It's interesting how expressive Feynman diagram notation can get if you just think of them as tensor contractions instead of particles bouncing all around the place.

Or in other words, I'd like to enter the "unhinged notation" contest please

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vacuumbubbles, 14 days ago to random

Right now I'm having the problem that I vaguely remember a (seemingly) useful equation, but can't find the reference. Now I'm questioning whether I actually saw this equation in reality or if it was just a nonsensical dream. Does anyone else here have this problem?

shadeow, 1 month ago to mathematics French

Alors la les matheux j'ai besoin de vous. Je suis tombé la dessus et je suis bouche bée 👀
#maths #mathematics #mathematiques #lycee #exercice

vacuumbubbles, 1 month ago to random

Geometric algebra can be represented by matrix representations. For instance, the geometric algebra of space Cl(3) can be represented by the Pauli matrix algebra, and the complex spacetime algebra Cl(1, 3, C) can be represented by the Dirac gamma matrix algebra. This way, every multivector is represented by a matrix.

Spinors are then defined as the "vectors" belonging to these matrices. They transform under the action of the matrices representing rotors.

Viewed this way, it makes more sense to think of matrix representations of geometric algebras as "spinor bases". If we choose a basis for spinors, we choose a matrix representation too, similar to how choosing basis states in QM also determines the "basis matrices" for matrix operators.

johncarlosbaez, 2 months ago (edited 2 months ago) to random

So you wake up one day wanting to invent a 2-dimensional number system. This requires a new number 𝑖 that's at right angles to 1. So you figure multiplying by 𝑖 must rotate numbers by 90°. So multiplying by 𝑖² rotates by 180°, so

𝑖² = -1

Cool!

Then you notice something else. The derivative of a function in the 𝑦 direction must be 𝑖 times its derivative in the 𝑥 direction, because the derivative is linear and you get the 𝑦 direction by rotating the 𝑥 direction by 90°: that is, multiplying it by 𝑖. So you get this equation:

[ \frac{\partial f}{\partial y} = i \frac{\partial f}{\partial x} ]

Cool!

Then you notice something else. If you use this equation twice you get

[ \frac{\partial^2 f}{\partial y^2} = i \frac{\partial f}{\partial x\partial y} = i^2 \frac{\partial^2 f}{\partial x^2} = - \frac{\partial^2 f}{\partial x^2} ]

so

[ \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0 ]

Wow! Every function with a second derivative obeys the Laplace equation!

You decide this one is a keeper.

https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations

vacuumbubbles, 2 months ago to random

I am explicitly not celebrating pi day. Here's why: https://hexnet.org/files/documents/tau-manifesto.pdf

ColinTheMathmo, 3 months ago (edited 3 months ago) to random

[I'm not asking for advice ... please don't try to help me]

OK, so ...

Trying to assist #ElderlyRelative to to things on the computer. These are things that they are absolutely capable of, and will (I believe) greatly enhance their quality of life.

Then Oh. My. God.

The misconceptions are f'n unbelievably. The way they think things work, the things they believe happen when they perform certain operations, it's just ... inconceivable.

I've started to unpick some of what's going on, and it's horrendous.

To start ...

johncarlosbaez, 3 months ago (edited 3 months ago) to random

The CEO of Exxon just said "we've waited too long" to tackle climate change. He blames "society" and "activists".

Yes, Exxon. Yes, Exxon. Yes, Exxon: the corporation that for decades has been spending millions to slow progress on climate change, despite its own research showing the problem was urgent.

He said:

"We've waited too long to open the aperture on the solution sets in terms of what we need, as a society, to start reducing emissions.... Frankly, society, and the activist — the dominant voice in this discussion — has tried to exclude the industry that has the most capacity and the highest potential for helping with some of the technologies."

What is this — some sort of dark, twisted joke?

The interview is here:

https://web.archive.org/web/20240228015207/https://fortune.com/2024/02/27/exxon-ceo-darren-woods-interview-pay-the-price-for-net-zero/

https://www.salon.com/2024/02/29/exxon-ceo-to-world-climate-isnt-our-fault-now-pay-the-price/

vacuumbubbles, 4 months ago to random

Here's a calculation that surprised me: People talk about Heisenberg's uncertainty relation as if it was something that only shows up on QM scales. But let' say we have an electron Gaussian wave packet with a width of σₓ=1𝑚𝑚. The uncertainty in velocity will be:
σᵥ=σₚ/𝑚=ℏ/(𝑚σₓ)≈11𝑐𝑚/𝑠
That's something my primate brain can imagine quite easily

parismarx, 4 months ago to tech

I used to look at these kinds of statements as deceptive PR, but increasingly I see them more through the lens of faith.

The tech billionaires are true believers and don’t accept they’re misunderstanding things like intelligence because they believe themselves to be geniuses.

https://www.wired.co.uk/article/deepmind

#tech #ai #agi #deepmind

vacuumbubbles, 4 months ago to random

In the hard sci-fi Xeelee Sequence by Stephen Baxter (a theoretical physicist), the first intelligent alien species humanity encounters are the Squeem (/stʃiːm/). They are a hivemind species of symbiotic aquatic organisms utterly unlike anything humanity had imagined - for a start, they had no concept of integer numbers except as a very outlandish mathematical concept, a bit like spinors are a very outlandish concept to humans.

This got me thinking - is something like this plausible mathematically? I'd argue that it is. Mathematics is an axiomatic science - you postulate some axioms and derive stuff from them. In principle, you can postulate any axioms you want, but most of the time, you want to do something with the results from mathematics, so they should have some relation to your lived reality. For instance, humanity first developed geometry in Ancient Egypt to record land claims, and integer algebra to facilitate trade. More advanced concepts of mathematics emerged relatively slowly after that - negative numbers during the Han dynasty, calculus in Medieval Europe, etc, whenever they were needed. The concept of doing abstract mathematics with no practical use is relatively new.

Analogously, a symbiotic aquatic hivemind species probably would only have little use for 2D geometry and trading in our sense, and would instead be concerned with wave mechanics, real numbers, etc.. Only after Squeem and human physicists reach the Standard Model of particle physics, their mathematics would be expected to converge in some sense.

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christianp, 4 months ago to random

Heard on a food podcast today: "they might not be infinite, but they're definitely uncountable!"

Should come with a content warning for mathematicians.

gregeganSF, 4 months ago to random

Me: Of course I’m not stupid enough to deliberately pour bleach onto metal!

Also me: Leaves a bottle of bleach on the floor, and fails to notice that it’s sprung a minuscule leak until a nearby drain is coated in cupric chloride.

johncarlosbaez, 4 months ago (edited 4 months ago) to random

Camembert is an endangered species! It relies on a mold that had lost the ability to reproduce sexually... and now, thanks to inbreeding, this mold has also lost the ability to reproduce using spores. Roquefort is also in trouble, but for Camembert it's worse:

"The world over, this other symbol of French gastronomy is inoculated exclusively with one single strain of Penicillium camemberti, a white mutant that was selected for Brie cheeses in 1898 and Camemberts in 1902.

The problem is that ever since then the strain has been replicated by vegetative propagation only. Until the 1950s, Camemberts still had grey, green or in some cases orange-tinged moulds on their surface. But the industry was not fond of these colours, considering them unappealing, and staked everything on the albino strain of P. camemberti, which is completely white and moreover has a silky texture. This is how Camembert acquired its now-characteristic pure white rind.

Year after year, generation after generation, the albino strain of P. camemberti, which was already incapable of sexual reproduction, lost its ability to produce asexual spores. As a result it is now very difficult for the entire industry to obtain enough P. camemberti spores to inoculate their production of the famous Norman cheese.

Worse still, while the Roquefort PDO (Protected Designation of Origin) standard retains a degree of microbial biodiversity, the PDO specifications for Camembert require farmers and other producers to use P. camemberti exclusively."

What to do about it? Switch to eating American cheese, or Velveeta? Read on....

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gregeganSF, 5 months ago to random

TIL that “spinor” is pronounced “spin”+“or”, which makes perfect sense ... but either none of my physics lecturers used the word, or I forgot how they said it. When reading it in text for the last 4 decades, I’ve been mentally pronouncing it “spine”+“or”.

https://www.bbc.co.uk/programmes/m000fw0p

ARBB, 1 year ago to random Portuguese

The Lorentz Group ( O(3,1) ), can, as usual, be neatly decomposed into four subgroups ( \mathfrak{L}(\cdot,\cdot) ) . The often considered decomposition is the one that conserves both parity and time direction, which the proper orthochronous subgroup ( \mathfrak{L}(\uparrow, +) ).

However, one can be interested in the subgroup of transforms that reverse the time direction, rarely called the "antichronous" group. (1/n)