Ciro Santilli @cirosantilli  40

Yet, all breakthroughs, comes from them, because the people who are crazy enough to believe they can change the world are the ones who actually do ;-)

How to deal articles:

Basically the Royal Society's scientific vulgarization cousin.

Ciro Santilli hates it when an expert does this!!!

If you estimate that the audience won't know the name of the concept, that's fine, do explain it as well.

But you must also give the name!!!

This also manifests itself when news outlets omit foreign names from headlines, notably Chinese, but likely happens to all non-european languages too.

Polynomial (possibly a multivariate polynomial) with integer coefficients.

Sometimes systems of Diophantine equations are considered.

Problems generally involve finding integer solutions to the equations, notably determining if any solution exists, and if infinitely solutions exist.

The general problem is known to be undecidable: Hilbert's tenth problem.

The Pythagorean triples, and its generalization Fermat's last theorem, are the quintessential examples.

A polynomial with multiple input arguments, e.g. with two inputs $x$ and $y$:as opposed to a polynomial with a single argument e.g. one with just $x$:

By default, we think of polynomials over the real numbers or complex numbers.

However, a polynomial can be defined over any other field just as well, the most notable example being that of a polynomial over a finite field.

For example, given the finite field of order 9, $GP(3)$ and with elements $\{0,1,2\}$, we can denote polynomials over that ring aswhere $x$ is the variable name.

For example, one such polynomial could be:and another one:Note how all the coefficients are members of the finite field we chose.

Given this, we could evaluate the polynomial for any element of the field, e.g.:and so on.

We can also add polynomials as usual over the field:and multiplication works analogously.

The polynomials together with polynomial addition and multiplication form a commutative ring.

All those dedicated applied mathematicians languages are a waste of society's time, Ciro Santilli sure applied mathematicians are capable of writing a few extra braces in exchange for a sane general purpose language, we should instead just invest in good libraries with fast C bindings for those languages like NumPy where needed, and powerful mainlined integrated development environments.

And when Ciro Santilli see the closed source ones like MATLAB being used, it makes him lose all hope on humanity. Why. As of 2020. Why? In the 1980s, maybe. But in the 2020s?

Used a lot in quantum mechanics, where the equations are really hard to solve. There's even a dedicated wiki page for it: en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics). Notably, Feynman diagrams are a way to represent perturbation calculations in quantum field theory.

Let's gather some of the best results we come across here:

Definition: "easy" number theory learnt in primary school, notably the operations of addition, subtraction, multiplication and division.