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#equations

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Soh Kam Yung

"In the paper, the researchers suggest they have figured out how to unify three physical theories that explain the motion of fluids. [...] This breakthrough won’t change the theories themselves, but it mathematically justifies them and strengthens our confidence that the equations work in the way we think they do."

scientificamerican.com/article

Digital illustration of lines representing a river or sea flowing over and around pebbles
Scientific American · Lofty Math Problem Called Hilbert’s Sixth Closer to Being SolvedBy Jack Murtagh
#Physics#Mathematics#FluidDynamics
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Pustam | पुस्तम | পুস্তম🇳🇵

A cycloidal pendulum - one suspended from the cusp of an inverted cycloid - is isochronous, meaning its period is constant regardless of the amplitude of the swing. Please find the proof using energy methods: Lagrange's equations (in the images attached to the reply).

Background:
The standard pendulum period of 2πL/g or frequency g/L holds only for small oscillations. The frequency becomes smaller as the amplitude grows. If you want to build a pendulum whose frequency is independent of the amplitude, you should hang it from the cusp of a cycloid of a certain size, as shown in the gif. As the string wraps partially around the cycloid, the effect decreases the length of the string in the air, increasing the frequency back up to a constant value.

In more detail:
A cycloid is the path taken by a point on the rim of a rolling wheel. The upside-down cycloid in the gif can be parameterized by (x,y)=R(θ−sin⁡θ,−1+cos⁡θ), where θ=0 corresponds to the cusp. Consider a pendulum of length L=4R hanging from the cusp, and let α be the angle the string makes with the vertical, as shown (in the proof).

GIF
#Pendulum#Cycloid#Period
Public
Soh Kam Yung

"[A]bout a decade ago, I was helping to refine a system for fabricating chips using extreme ultraviolet (EUV) light.
[...]
I realized that many aspects of this process have intriguing similarities to what happens during a supernova: a sudden explosion, an expanding cloud of plasma debris, and a shock wave that slams into a thin hydrogen environment. (Interstellar material consists mostly of hydrogen.)"

spectrum.ieee.org/euv-light-so

IEEE Spectrum · The Tiny Star Explosions Powering Moore’s LawBy Jayson Stewart
#Equations#Supernovas#Fabrication
Public
jwls

So, I learned something this past weekend about sequential exponential numbers, and how they can be verified through addition and subtraction no matter how high the exponents go.

Is this a known thing?

For example:

Every sequential ^2 has a once nested difference that increases by 2 and it's every odd number.

Every sequential ^3 has a twice nested difference of 6, which the first set of differences create a number which must then be subtracted from the next difference to see this pattern.

Every sequential ^4 has a thrice nested difference of 24, which the first set of differences create a number which must then be subtracted from the next set, followed by the third set to see this pattern.

Every sequential ^5 has a quaternary nested difference of 120, which means requires subtracting the sets four times to see the pattern.

So this validates ^6 at the fifth set of differences will provide a pattern of 720, ^7 at the sixth set will provide a pattern of 5040 and so on.

What is this called? Is there a simplified equation that shows this? Any and all help is much appreciated. Thanks!

#math#sums#difference
Public
Kirk

I just muted a guy trying to post a 'universal equation' and who used no citations to good peer reviewed papers for it. Beware and don't trust everything you see on the internet, as Gandalf famously said.

The closest thing we have now (early 2025 if you are reading this from the future) to a 'universal equation' is the standard model Lagrangian and it is SUPER LONG (a full page), and would be really awkward to try to put into a post.
#science #physics #equations
Take a look at it yourself:
symmetrymagazine.org/article/t

symmetry magazineThe deconstructed Standard Model equationThe Standard Model is far more than elementary particles arranged in a table.
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cli345

Les #équations peuvent produire des dessins. 🥳
Par exemple, voici une équation d'un col de vêtement : |y| + |x²+sin(y)-1| = 1

Si vous voulez l'essayer dans G'MIC -> Rendering -> Equation Plot [Implicit], il faut la traduire : abs(y) + abs(x^2+sin(y)-1) - 1

Vous voudrez peut-être retourner l'image : abs(-y) + abs(x^2+sin(-y)-1) - 1

Vous voudrez peut-être faire un zoom "×2", il faut faire "/2" pour x et pour y : abs(-y/2) + abs(x^2/4+sin(-y/2)-1) - 1

Amusez-vous bien 🙂

Dessin d'un col de vêtement dans G'MIC -> Rendering -> Equation Plot [Implicit]
#math#geometry#gmic
Public
doctorambient

So I am using an equation editor for the first time in years, because a lot of word processors *still* don't properly support equations (i.e. they don't support LaTeX for real).

Equation editors still suck. Equation editors still connect parts of the equations (that shouldn't be connected) so that you can't edit them and minor mistakes require a full redo.

How is this the case in 2024!?!

#math#latex#equations
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