Ciro Santilli @cirosantilli 35

Ciro Santilli's favorite religion. He does not believe fully in it, nor has he studied it besides through brief Wikipedia and Googling.

Ciro likes Buddhism because it feels like the least "metaphysical explanations to things you can't see" of the religions he knows.

Rather, it feels more like "a plausible theory of the mind" and highly compatible with physics.

Ciro also believes that there is a positive correlation between being a software engineer and liking Buddhist-like things, see also: the correlation between software engineers and Buddhism.

This does not seem to go deep into the Standard Model as Physics from Symmetry by Jakob Schwichtenberg (2015), appears to focus more on more basic applications.

But because it is more basic, it does explain some things quite well.

Every Lie algebra corresponds to a single simply connected Lie group.

The Baker-Campbell-Hausdorff formula basically defines how to map an algebra to the group.

Bibliography:

Cardinality $\le$ dimension of the vector space.

An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee (2011) shows that this is a tensor that represents the volume of a parallelepiped.

It takes as input three vectors, and outputs one real number, the volume. And it is linear on each vector. This perfectly satisfied the definition of a tensor of order (3,0).

Given a basis $(e_i,e_j,e_k)$ and a function that return the volume of a parallelepiped given by three vectors $V(v_1,v_2,v_3)$, ${\epsilon }_{ikj}=V(e_i,e_j,e_k)$.

A measurable function defined on a closed interval is square integrable (and therefore in $L^2$) if and only if Fourier series converges in $L^2$ norm the function:

Both are harmonic oscillators.

In the LC circuit:

You can kickstart motion in either of those systems in two ways:

The function that fully describes a physical system in Lagrangian mechanics.

A waste of time like the rest of the knowledge olympiads.

Wikipedia mentions quoting his Nobel Prize biography:

Like a heat equation but for functions without time dependence, space-only.

TODO confirm: does the solution of the heat equation always converge to the solution of the Laplace equation as time tends to infinity?

In one dimension, the Laplace equation is boring as it is just a straight line since the second derivative must be 0. That also matches our intuition of the limit solution of the heat equation.

Uniqueness: Uniqueness theorem for Poisson's equation.

Invertible matrices. Or if you think a bit more generally, an invertible linear map.

When the field $F$ is not given, it defaults to the real numbers.

Non-invertible are excluded "because" otherwise it would not form a group (every element must have an inverse). This is therefore the largest possible group under matrix multiplication, other matrix multiplication groups being subgroups of it.

TODO motivation. Motivation. Motivation. Motivation. The definitin with quotient group is easy to understand.

Our notation: $D_n$, called "dihedral group of degree n", means the dihedral group of the regular polygon with $n$ sides, and therefore has order $2n$ (all rotations + flips), called the "dihedral group of order 2n".