Ciro Santilli @cirosantilli 37

Approximates an original function by sines. If the function is "well behaved enough", the approximation is to arbitrary precision.

Fourier's original motivation, and a key application, is solving partial differential equations with the Fourier series.

Can only be used to approximate for periodic functions (obviously from its definition!). The Fourier transform however overcomes that restriction:

The Fourier series behaves really nicely in $L^2$, where it always exists and converges pointwise to the function: Carleson's theorem.

The list: en.wikipedia.org/wiki/List_of_Nvidia_graphics_processing_unit

These are basically technically minded people that Ciro Santilli feels have similar interests/psychology to him, and who write too much for their own good:

Maybe one day these will also be legendary, who knows:

Another category Ciro admires are the "computational physics visualization" people, these people will go to Heaven:

Related:

Institution led:

Other mentions:

The BBC 1979-1982 adaptations of John Le Carré's novels are the best miniseries ever made:They are the most realistic depiction of spycraft ever made.

Some honorable mentions:

Bibliography:

In this section we classify some functions by the type of inputs and outputs they take and produce.

Some linker related answers by Ciro Santilli:

Ciro Santilli used to read books when he was younger (Harry Potter up to the 4th, Lord of the Rings), but once you are reading code, technical articles and news the whole day, you really just want to watch videos of people doing useless things on YouTube to rest, enough text.

Books are slow. No patience. Need faster immediate satisfaction.

Paradoxically Ciro feels like he's becoming a writer of sorts though, one semi independent section/answer/piece of knowledge at a time.

Writing is not just giving out information. It is re-feeling it.

TODO. I think this is the key point. Notably, $U(1)$ symmetry implies charge conservation.

More precisely, each generator of the corresponding Lie algebra leads to one separate conserved current, such that a single symmetry can lead to multiple conserved currents.

This is basically the local symmetry version of Noether's theorem.

Then to maintain charge conservation, we have to maintain local symmetry, which in turn means we have to add a gauge field as shown at Video "Deriving the qED Lagrangian by Dietterich Labs (2018)".

Forces can then be seen as kind of a side effect of this.

Bibliography:

Subgroup of the Poincaré group without translations. Therefore, in those, the spacetime origin is always fixed.

Or in other words, it is as if two observers had their space and time origins at the exact same place. However, their space axes may be rotated, and one may be at a relative speed to the other to create a Lorentz boost. Note however that if they are at relative speeds to one another, then their axes will immediately stop being at the same location in the next moment of time, so things are only valid infinitesimally in that case.

This group is made up of matrix multiplication alone, no need to add the offset vector: space rotations and Lorentz boost only spin around and bend things around the origin.

One definition: set of all 4x4 matrices that keep the Minkowski inner product, mentioned at Physics from Symmetry by Jakob Schwichtenberg (2015) page 63. This then implies:

The equation that allows us to calculate stuff in special relativity!

Take two observers with identical rules and stopwatch, and aligned axes, but one is on a car moving at towards the $+x$ direction at speed $v$.

TODO image.

When both observe an event, if we denote:It is of course arbitrary who is standing and who is moving, we will just use the term "standing" for the one without primes.

Then the coordinates of the event observed by the observer on the car are:where:

Note that if $\frac{v}{c}$ tends towards zero, then this reduces to the usual Galilean transformations which our intuition expects:

This explains why we don't observe special relativity in our daily lives: macroscopic objects move too slowly compared to light, and $\frac{v}{c}$ is almost zero.

The name makes absolutely no sense in modern terms, as nor colors nor light are used directly in the measurements. It is purely historical.