Boooooring. Except for Navier-Stokes existence and smoothness. That's OK.

Local symmetries appear to be a synonym to internal symmetry, see description at: Section "Internal and spacetime symmetries".

As mentioned at Quote , local symmetries map to forces in the Standard Model.

Appears to be a synonym for: gauge symmetry.

A local symmetry is a transformation that you apply a different transformation for each point, instead of a single transformation for every point.

TODO what's the point of a local symmetry?

Bibliography:

Affiliation: University of São Paulo.

The dominating model of a computer.

The model is extremely simple, but has been proven to be able to solve all the problems that any reasonable computer model can solve, thus its adoption as the "default model".

The smallest known Turing machine that cannot be proven to halt or not as of 2019 is 7,918-states: www.scottaaronson.com/blog/?p=2725. Shtetl-Optimized by Scott Aaronson is just the best website.

A bunch of non-reasonable-looking computers have also been proven to be Turing complete for fun, e.g. Magic: The Gathering.

The list: complexityzoo.uwaterloo.ca/Complexity_Zoo

Applications:

Born: 1965

Died: 2010+-ish

This is the lowest level of abstraction computer, at which the basic gates and power are described.

At this level, you are basically thinking about the 3D layered structure of a chip, and how to make machines that will allow you to create better, usually smaller, gates.

It is quite amazing to read through books such as The Supermen: The Story of Seymour Cray by Charles J. Murray (1997), as it makes you notice that earlier CPUs (all before the 70's) were not made with integrated circuits, but rather smaller pieces glued up on PCBs! E.g. the arithmetic logic unit was actually a discrete component at one point.

The reason for this can also be understood quite clearly by reading books such as Robert Noyce: The Man Behind the Microchip by Leslie Berlin (2006). The first integrated circuits were just too small for this. It was initially unimaginable that a CPU would fit in a single chip! Even just having a very small number of components on a chip was already revolutionary and enough to kick-start the industry. Just imagine how much money any level of integration saved in those early days for production, e.g. as opposed to manually soldering point-to-point constructions. Also the reliability, size an weight gains were amazing. In particular for military and spacial applications originally.

Register transfer level is the abstraction level at which computer chips are mostly designed.

The only two truly relevant RTL languages as of 2020 are: Verilog and VHDL. Everything else compiles to those, because that's all that EDA vendors support.

Much like a C compiler abstracts away the CPU assembly to:

Compilers for RTL languages such as Verilog and VHDL abstract away the details of the specific semiconductor technology used for those exact same reasons.

The compilers essentially compile the RTL languages into a standard cell library.

Examples of companies that work at this level include: