It basically replaces a bunch of discrete digital components with a single chip. So you don't have to wire things manually.
Particularly fundamental if you would be putting those chips up a thousand cell towers for signal processing, and ever felt the need to reprogram them! Resoldering would be fun, would it? So you just do a over the wire update of everything.
Vs a microcontroller: same reason why you would want to use discrete components: speed. Especially when you want to do a bunch of things in parallel fast.
One limitation is that it only handles digital electronics: electronics.stackexchange.com/questions/25525/are-there-any-analog-fpgas There are some analog analogs, but they are much more restricted due to signal loss, which is exactly what digital electronics is very good at mitigating.
The History of the FPGA by Asianometry (2022)
Source. The cool thing about parallel evolution is that it shows how complex phenotype can evolve from very different initial genetic conditions, highlighting the great power of evolution.
The Nerdy Kid in the White Shirt scene from Charlie Wilson's War
. Source. The old man lost his horse scene from Charlie Wilson's War
. Source. Effect of a change of basis on the matrix of a bilinear form by 
Ciro Santilli 35 Updated 2025-04-24 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-04-24 +Created 1970-01-01If is the change of basis matrix, then the matrix representation of a bilinear form that looked like:then the matrix in the new basis is:Sylvester's law of inertia then tells us that the number of positive, negative and 0 eigenvalues of both of those matrices is the same.
Proof: the value of a given bilinear form cannot change due to a change of basis, since the bilinear form is just a function, and does not depend on the choice of basis. The only thing that change is the matrix representation of the form. Therefore, we must have:and in the new basis:and so since:
List of magnetic confinement fusion reactors by Ciro Santilli 35 Updated 2025-04-24 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-04-24 +Created 1970-01-01Same motivation as Galilean invariance, but relativistic version of that: we want the laws of physics to have the same form on all inertial frames, so we really want to write them in a way that is Lorentz covariant.
This is just the relativistic version of that which takes the Lorentz transformation into account instead of just the old Galilean transformation.
The key question is: why is this not symmetrical?
One answer is: because one of the twin accelerates, and therefore changes inertial frames.
But the better answer is: understand what happens when the stationary twin sends light signals at constant time intervals to each other. When does the travelling twin receives them?
Another way of understanding it is: you have to make all calculations on a single inertial frame for the entire trip.
Supposing the sibling quickly accelerates out (or magically starts moving at constant speed), travels at constant speed, and quickly accelerates back, and travels at constant speed setup, there are three frames that seem reasonable:
- the frame of the non-accelerating sibling
- the outgoing trip of the accelerating sibling
- the return trip of the accelerating sibling
If you do that, all three calculations give the exact same result, which is reassuring.
Another way to understand it is to do explicit integrations of the acceleration: physics.stackexchange.com/questions/242043/what-is-the-proper-way-to-explain-the-twin-paradox/242044#242044 This is the least insightful however :-)
Bibliography:
It is said, that once upon a time, programmers used CSV and collaborated on SourceForge, and that everyone was happy.
These days, are however, long gone in the mists of time as of 2020, and beyond Ciro Santilli's programming birth.
Infinitely many SQL answers.
As mentioned at Ciro Santilli's Stack Overflow contributions, he just answers every semi-duplicate immediatly as it is asked, and is therefore able to overcome the Stack Overflow maximum 200 daily reputation limit by far. E.g. in 2018, Gordon reached 135k (archive), thus almost double the 73k yearly limit due to the 200 daily limit, all of that with accepts.
This is in contrast to Ciro Santilli's contribution style which is to only answer questions as he needs the subject, or generally important questions that aroused his interest.
2014 Blog post describing his activity: blog.data-miners.com/2014/08/an-achievement-on-stack-overflow.html, key quote:so that suggests his contributions also take a meditative value.
For a few months, I sporadically answered questions. Then, in the first week of May, my Mom's younger brother passed away. That meant lots of time hanging around family, planning the funeral, and the like. Answering questions on Stack Overflow turned out to be a good way to get away from things. So, I became more intent.
www.data-miners.com/linoff.htm mentions he's an SQL consultant that consulted for several big companies.
There are unlisted articles, also show them or only show them.