Center (group theory) by Ciro Santilli 35 Updated +Created
Group operation by Ciro Santilli 35 Updated +Created
Ring (mathematics) by Ciro Santilli 35 Updated +Created
A Ring can be seen as a generalization of a field where:
Addition however has to be commutative and have inverses, i.e. it is an Abelian group.
The simplest example of a ring which is not a full fledged field and with commutative multiplication are the integers. Notably, no inverses exist except for the identity itself and -1. E.g. the inverse of 2 would be 1/2 which is not in the set. More specifically, the integers are a commutative ring.
A polynomial ring is another example with the same properties as the integers.
The simplest non-commutative, non-division is is the set of all 2x2 matrices of real numbers:
  • we know that 2x2 matrix multiplication is non-commutative in general
  • some 2x2 matrices have a multiplicative inverse, but others don't
Note that is not a ring because you can by addition reach the zero matrix.
Nabla symbol by Ciro Santilli 35 Updated +Created
As if Greek letters weren't enough, physicists and mathematicians also like to make up tons of symbols, some of which look like the could actually be Greek letters!
Nabla is one of those: it was completely made up in modern times, and just happens to look like an inverted upper case delta to make things even more confusing!
Nabla means "harp" in Greek, which looks like the symbol.
Applications of the Fourier series by Ciro Santilli 35 Updated +Created
Formal system by Ciro Santilli 35 Updated +Created
NeoCarbonFood by Ciro Santilli 35 Updated +Created
Previously known as "Food From Electricity", "NeoCarbonFood" sounds like a more commercializable version of it.
Uses electricity to electrolyse water into hydrogen and oxygen molecules, and then use bacteria that do hydrogen chemosynthesis to convert it into food.
Endurance athlete by Ciro Santilli 35 Updated +Created
Electronic lab equipment by Ciro Santilli 35 Updated +Created
Video 1.
A Perfect Electronics Bench? by Keysight (2021)
Source.
Electronics bibliography by Ciro Santilli 35 Updated +Created
List of e-learning websites by Ciro Santilli 35 Updated +Created
Dirac delta function by Ciro Santilli 35 Updated +Created
The "0-width" pulse distribution that integrates to a step.
There's not way to describe it as a classical function, making it the most important example of a distribution.
Applications:
Ordinary differential equation by Ciro Santilli 35 Updated +Created
Chain rule by Ciro Santilli 35 Updated +Created
Here's an example of the chain rule. Suppose we want to calculate:
So we have:
and so:
Therefore the final result is:
Total derivative by Ciro Santilli 35 Updated +Created
The total derivative of a function assigns for every point of the domain a linear map with same domain, which is the best linear approximation to the function value around this point, i.e. the tangent plane.
E.g. in 1D:
and in 2D:
Simulation by Ciro Santilli 35 Updated +Created
Computer music bibliography by Ciro Santilli 35 Updated +Created
Holomorphic function by Ciro Santilli 35 Updated +Created
Being a complex holomorphic function is an extremely strong condition.
The existence of the first derivative implies the existence of all derivatives.
Another extremely strong consequence is the identity theorem.
"Holos" means "entire" in Greek, so maybe this is a reference to the fact that due to the identity theorem, knowing the function on a small open ball implies knowing the function everywhere.
Skunk Works by Ciro Santilli 35 Updated +Created

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