Ciro Santilli @cirosantilli 35

Let's start with the one dimensional case. Let the $q:\mathbb{R}\to \mathbb{R}$ and a Functional $I$ defined by a function of three variables $F:{\mathbb{R}}^3\to \mathbb{R}$:

Then, the Euler-Lagrange equation gives the maxima and minima of the that type of functional. Note that this type of functional is just one very specific type of functional amongst all possible functionals that one might come up with. However, it turns out to be enough to do most of physics, so we are happy with with it.

Given $F$, the Euler-Lagrange equations are a system of ordinary differential equations constructed from that $F$ such that the solutions to that system are the maxima/minima.

In the one dimensional case, the system has a single ordinary differential equation:

By $\frac{\partial F}{\partial q}$ and $\frac{\partial F}{\partial \dot{q}}$ we simply mean "the partial derivative of $F$ with respect to its second and third arguments". The notation is a bit confusing at first, but that's all it means.

Therefore, that expression ends up being at most a second order ordinary differential equation where $q$ is the unknown, since:

Now let's think about the multi-dimensional case. Instead of having $q:\mathbb{R}\to \mathbb{R}$, we now have $q:\mathbb{R}\to {\mathbb{R}}^n$. Think about the Lagrangian mechanics motivation of a double pendulum where for a given time we have two angles.

Let's do the 2-dimensional case then. In that case, $F$ is going to be a function of 5 variables $F:{\mathbb{R}}^5\to \mathbb{R}$ rather than 3 as in the one dimensional case, and the functional looks like:

This time, the Euler-Lagrange equations are going to be a system of two ordinary differential equations on two unknown functions $q_1$ and $q_2$ of order up to 2 in both variables:At this point, notation is getting a bit clunky, so people will often condense the $q_i$ vectoror just omit the arguments of $F$ entirely:

Module bound above, possibly multiplied by a constant:is defined as:

E.g.:

It just has RNA that can be transcribed directly by the host ribosome.

It is a bit hard to decide if those people are serious or not. Sometimes it feels scammy, but sometimes it feels fun and right!

Particularly concerning is the fact that they are not a not-for-profit entity, and it is hard to understand how they might make money.

Charles Simon, the founder, is pretty focused in how natural neurons work vs artificial neural network models. He has some good explanations of that, and one major focus of the project is their semi open source spiking neuron simulator BrainSimII. While Ciro Santilli believes that there might be insight in that, he also has doubts if certain modules of the brain wouldn't be more suitable coded directly in regular programming languages with greater ease and performance.

FutureAI appears to be Charles' retirement for fun project, he is likely independently wealthy. Well done.

Photon hits excited electron, makes that electron go down, and generates a new identical photon in the process, with the exact same:This is the basis of lasers.

Bibliography:

Vimscript unit testing!!!

Low budget.

Good theory of Jesus.

List of similar feeling films: www.youtube.com/watch?v=zwYwFoanrNg 11 Underrated Hard Sci-fi Movies by Marvelous Videos (2021)